Topological aspects of low-dimensional systems

UJF NATO ASI LES HOUCHES Session LXIX 1998 ASPECTS TOPOLOGIQUES DE LA PHYSIQUE EN BASSE DIMENSION TOPOLOGICAL ASPECTS ...

Author: A. Comtet | T. Jolicoeur | S. Ouvry | F. David

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UJF NATO ASI LES HOUCHES Session LXIX 1998

ASPECTS TOPOLOGIQUES DE LA PHYSIQUE EN BASSE DIMENSION

TOPOLOGICAL ASPECTS OF LOW DIMENSIONAL SYSTEMS

CONFERENCIERS G. DUNNE B. DUPLANTIER M.P.A. FISHER S. GIRVIN J. MYRHEIM S. NECHAEV A.P. POLYCHRONAKOS H. SALEUR M. SHAYEGAN D. THOULESS

A. AKKERMANS J.T. CHALKER V. CROQUETTE J. DESBOIS D.C. GLATTLI

ÉCOLE DE PHYSIQUE DES HOUCHES - UJF & INPG - GRENOBLE

a NATO Advanced Study Institute

LES HOUCHES SESSION LXIX 7-31 July 1998 Aspects topologiques de la physique en basse dimension Topological aspects of low dimensional systems Edited by A. COMTET, T. JOLICŒUR, S. OUVRY and F. DAVID

SCIENCES

Springer 7 avenue du Hoggar, PA de Courtabœuf, B.P. 112, 91944 Les Ulis cedex A, France 875-81 Massachusetts Avenue, Cambridge,

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Published in cooperation with the NATO Scientific Affair Division

ISBN 3-540-66909-4 ISBN 2-86883-424-8

Springer-Verlag Berlin Heidelberg New York EDP Sciences Les Ulis

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the French and German Copyright laws of March 11, 1957 and September 9, 1965, respectively. Violations fall under the prosecution act of the French and German Copyright Laws. © EDP Sciences; Springer-Verlag 1999 Printed in France

LES HOUCHES - ECOLE DE PHYSIQUE ÉCOLE D'ÉTÉ DE PHYSIQUE THÉORIQUE SERVICE INTER-UNIVERSITAIRE COMMUN À L'UNIVERSITÉ JOSEPH FOURIER DE GRENOBLE ET À L'INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE, SUBVENTIONNÉ PAR LE MINISTÈRE DE L'ÉDUCATION NATIONALE, DE LA RECHERCHE ET DE LA TECHNOLOGIE, LE CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE ET LE COMMISSARIAT À L'ÉNERGIE ATOMIQUE Membres du Conseil d'Administration : Claude Feuerstein (président), Yves Brunet (vice-président), Cécile De Witt, Daniel Decamps, Thierry Dhombre, Hubert Flocard, Jean-François Joanny, Michèle Leduc, James Lequeux, Marcel Lesieur, Giorgio Parisi, Michel Peyrard, Jean-Paul Poirier, Claude Weisbuch, Joseph Zaccai, Jean Zinn-Justin Directeur : François David

ECOLE D'ETE DE PHYSIQUE THEORIQUE SESSION LXIX INSTITUT D'ÉTUDES AVANCÉES DE L'OTAN NATO ADVANCED STUDY INSTITUTE 7 juillet — 31 juillet 1998 Directeurs Scientifiques de la session : Alain COMTET, LPTMS, bâtiment 100, 91406 Orsay, France, Thierry JOLICŒUR, SPhT, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France et Stéphane OUVRY, LPTMS, bâtiment 100, 91406 Orsay Cedex, France

SESSIONS PRECEDENTES

1951 -1997 I 1951 II 1952 III 1953 IV 1954 V 1955

VI 1956

VII 1957

VIII IX X XI XII XIII XIV XV XVI XVII XVIII

1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968

Quantum mechanics. Quantum field theory Quantum mechanics. Statistical mechanics. Nuclear physics Quantum mechanics. Solid state physics. Statistical mechanics. Elementary particle physics Quantum mechanics. Collision theory. Nucleon-nucleon interaction. Quantum electrodynamics Quantum mechanics. Non-equilibrium phenomena. Nuclear reactions. Interaction of a nucleus with atomic and molecular fields Quantum perturbation theory. Low temperature physics. Quantum theory of solids; dislocations and plastic properties. Magnesium; ferromagnetism Scattering theory. Recent developments in field theory. Nuclear interaction; strong interactions. High energy electrons. Experiments in high energy nuclear physics The many body problem The theory of neutral and ionized gases* Elementary particles and dispersion relations* Low temperature physics* Geophysics; the earth's environment* Relativity groups and topology* Quantum optics and electronics High energy physics High energy astrophysics* Many body physics* Nuclear physics*

Sessions ayant reçu l'appui du Comité Scientifique de l'OTAN.

VIII

XIX XX XXI XXII XXIII XXIV XXV XXVI June Inst. XXVII XXVIII XXIX XXX XXXI XXXII XXXIII XXXIV XXXV XXXVI XXXVII XXXVIII XXXIX XL XLI XLII XLIII XLIV

1969 1970 1971 1972 1972 1973 1973 1974 1975 1975 1975 1976 1977 1978 1979 1979 1980 1980 1981 1981 1982 1982 1983 1983 1984 1984 1985

XLV XLVI XLVII XLVIII XLIX L LI LII LIII LIV

1985 1986 1986 1988 1988 1988 1989 1989 1990 1990

Physical problems in biological systems Statistical mechanics and quantum field theory Particle physics Plasma physics Black holes Fluids dynamics Molecular fluids* Atomic and molecular physics and the interstellar matter* Structural analysis of collision amplitudes Frontiers in laser spectroscopy* Methods in field theory* Weak and electromagnetic interactions at high energy* Nuclear physics with heavy ions and mesons* Ill-condensed matter Membranes and intercellular communication Physical cosmology Laser plasma interaction Physics of defects Chaotic behaviour of deterministic systems* Gauge theories in high energy physics* New trends in atomic physics* Recent advances in field theory and statistical mechanics* Relativity, groups and topology* Birth and infancy of stars* Cellular and molecular aspects of developmental biology* Critical phenomena, random systems, gauge theories* Architecture of fundamental interactions at short distances* Signal processing* Chance and matter Astrophysical fluid dynamics Liquids at interfaces Fields, strings and critical phenomena Oceanographic and geophysical tomography Liquids, freezing and glass transition Chaos and quantum physics* Fundamental systems in quantum optics* Supernovae*

Sessions ayant reçu l'appui du Comité Scientifique de l'OTAN.

IX

LV 1991 LVI 1991 LVII LVIII LIX LX LXI LXII

1992 1992 1993 1993 1994 1994

LXIII LXIV LXV LXVI LXVII LXVIII

1995 1995 1996 1996 1997 1997

Particles in the nineties* Strongly interacting fermions and high Tc superconductivity Gravitation and quantizations Progress in picture processing* Computational fluid dynamics Cosmology and large scale structure Mesoscopic quantum physics Fluctuating geometries in statistical mechanics and quantum field theory Quantum fluctuations* Quantum symmetries* From cell to brain* Trends in nuclear physics, 100 years later Modélisation du climat de la terre et de sa variabilité Particules et interactions : le modèle standard mis à l'épreuve*

Sessions ayant reçu l'appui du Comité Scientifique de l'OTAN. Publishers: Session VIII: Dunod, Wiley, Methuen; Sessions IX & X: Herman, Wiley - Session XI: Gordon and Breach, Presses Universitaires - Sessions XIIXXV: Gordon and Breach - Sessions XXVI-LXVIII: North-Holland.

ORGANISERS A. COMTET, LPTMS, bâtiment 100, 91406 Orsay, France T. JOLICŒUR, SPhT, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France S. OUVRY, LPTMS, bâtiment 100, 91406 Orsay Cedex, France F. DAVID, École de Physique des Houches & SPhT, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France

LECTURERS G.V. DUNNE, Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A. B. DUPLANTIER, SPhT, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France M.P.A. FISHER, Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, U.S.A. S.M. GIRVIN, Indiana University, Department of Physics, Bloomington, IN 47405, U.S.A. J. MYRHEIM, Department of Physics, The Norwegian University of Science and Technology (NTNU), N-7034 Trondheim, Norway S.NECHAEV, UMR 8626, CNRS-Université Paris XI, LPTMS, bâtiment 100, Université Paris Sud, 91405 Orsay Cedex, France, and L.D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia A.P. POLYCHRONAKOS, Institutionen för Teoretisk Fysik, Box 803, 751 08 Uppsala, Sweden, and Physics Department, University of Ionnina, 45110 Ionnina, Greece H. SALEUR, Department of Physics, University of Southern California, Los-Angeles, CA 90089-0484, U.S.A. M. SHAYEGAN, Department of Electrical Engineering, Princeton University, Princeton, NJ, U.S.A. D.J. THOULESS, Department of Physics, Box Washington, Seattle, WA 98195, U.S.A.

351560,

University

of

XII

SEMINAR SPEAKERS E. AKKERMANS, Technion, Israel Institute of Technology, Department of Physics, 32000 Haifa, Israel J. CHALKER, Theoretical Physics, Oxford University, Oxford, 0X1 3NP, U.K.

1

Keble road,

V. CROQUETTE, E.N.S., 24 rue Lhomond, 75231 Paris Cedex, France J. DESBOIS, I.P.N., Service de Physique Théorique, 91406 Orsay Cedex, France C. GLATTLI, Service de Physique de l'État Condensé, L'Orme des Merisiers, CEA Saclay, 91191 Gif-sur-Yvette, France

STUDENTS

M. AGUADO MARTINEZ DE CONTRASTA, Departamento de Fisica Teorica, Facultad de Ciencias, Cindad Universitaria s/n, 50009 Zaragoza, Spain L. AMICO, Dpto. Fisica Teorica de la Materia Condensada, Facultad de Ciencias, c-v, Universidad Autonoma de Madrid, 28049 Madrid, Spain D. BAZZALI, Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, 2 avenue Adolph Chauvin, 95302 Cergy-Pontoise, France J. BETOURAS, University of Oxford, Department of Physics, Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, U.K. M. BOCQUET, Service de Physique Théorique, L'Orme des Merisiers, CEA Saclay, 91191 G if-sur-Yvette Cedex, France V. BRUNEL, SPhT, L'Orme des Merisiers, CEA Saclay, 91191 G if-sur-Yvette Cedex, France J. BÜRKI, Institut Romand de Recherche Numérique en Physique des Matériaux, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Suisse D. CARPENTIER, Laboratoire de Physique Théorique de l'École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France C.R. CASSANELLO, Institüt für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany H. CASTILLO, University of Illinois at Urbana-Champain, Dept. of Physics, 1110W. Green St., Urbana, IL 61801, U.S.A. J.-S. CAUX, Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, U.K. C. CHAUBET, Université Montpellier 2, Groupe d'Étude des Semiconducteurs, Place E. Bataillon, 34095 Montpellier Cedex 5, France V. CHEIANOV, Institut for Theoretical Physics, Uppsala University, Lägerhyddsv. 19, Uppsala, Sweden N.R. COOPER, T.C.M. Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 OHE, U.K. P.R. EASTHAM, Cavendish Laboratory, Madingley Road, Cambridge, CB3 OHE, U.K. T. FUKUI, Institut für Theoretische Physik, Universität zu Köln, Zülpicher str.77, 50937 Köln, Germany C. FURTLEHNER, Max-Planck-Institut für Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany J. GORYO, Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan A. GREEN, Physics Department, Princeton University, Jadwin Hall, Princeton NJ 08544, U.S.A.; Trinity College, Cambridge, CB2 ITQ, U.K.

XIV

T. HALL, University of Connecticut U-46, Physics Department, 2152 Hillside Road, Storrs, CT 06269, U.S.A. J.H. HAN, APCTP, 207-43 Cheongryangri-Dong Dongdaemun-Gu, Seoul 130012, Korea A. HO, Dept. Physics, Rutgers University, P.O. Box 849, Piscatawy, NJ 088550849, U.S.A. K. IMURA, Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan J.L. JACOBSEN, Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, U.K. D. KHVESHCHENKO, NORDITA, Blegdamsvej 17, Copenhagen 2100, Denmark S. KIRCHNER, Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany J. KONDEV, Institute for Advanced study, Olden Lane, Princeton, NJ, U.S.A. K. LE HUR, Laboratoire de Physique des Solides, bâtiment 510, 91405 Orsay Cedex, France D. LILLIEHÖÖK, Department of Physics, Stockholm University, Box 6730, 113 85, Stockholm, Sweden A. MALTSEV, L.D. Landau, Institute for Theoretical Physics, ul. Kosygina 2, 117940 Moscow, Russia R. MELIN, CRTBT-CRNS, 25 avenue de Martyrs, BP. 166 X, 38042 Grenoble Cedex 09, France S. MELINTE, Unité de Physico-Chimie et de Physique des Matériaux, Université Catholique de Louvain, Place Croix du Sud 1, 1348 Louvainla-Neuve, Belgium M. MILOVANOVIC, Technion, Physics Department, 32000 Haifa, Israel G. MISGUICH, LPTL, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex, France J. MOORE, Massachusetts Inst. of Technology, 77 Masachusetts Ave., Cambridge, MA 02139, U.S.A. E. ORIGNAC, Laboratoire de Physique des Solides, Univ. de Paris-Sud, bâtiment 510, Centre Universitaire d'Orsay, 91405 Orsay Cedex, France S. PEYSSON, Laboratoire de Physique, URA 13-25 du CNRS associée à l'ENS Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France K.-V. PHAM, Laboratoire de Physique des Solides, bâtiment 510, Centre Universitaire Paris XI, 91405 Orsay Cedex, France B. PONSOT, Laboratoire de Physique Mathématique, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex, France R. RAMAZASHVILI, Physics Department, Rutgers University, Piscataway, NJ 08855-0849, U.S.A., Loomis Laboratory, University of Illinois at UrbanaChampaign, Urbana, IL 61801-3080, U.S.A.

XV

N. SANDLER, Univ. of Illinois at Urbana-Champaign, Dept. of Physics, 1110 West Green St., Urbana, IL 61801, U.S.A. F. SIANO, Univ. of Southern California, Dept. of Physics, Los Angeles, CA 90089-0484, U.S.A. J. SINOVA, Indiana University, Physics Department, Swain Hall West 117, Bloomington, IN 47405-4201, U.S.A. B. SKORIC, Inst. for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands E. SUKHORUKOV, Institut für Physik, Universität Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland M. TITOV, Theoretical Department, Petersburg Nuclear Physics Institute, Gatchina 188350, Petersburg District, Russia A. TSCHERSICH, Ruhr-Universität Bochum, Lehrstuhl Theoretische Physik III, Gebäude NB 6/127, Universitätstrasse 150, 44780 Bochum, Germany R. VAN ELBURG, Institute for Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands S. VILLAIN-GUILLOT, Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany S. VISHVESHWARA, Dept. of Physics, University of California, Santa Barbara, CA 93108, U.S.A. A. VISHWANATH, Dept. of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, U.S.A. X. WAINTAL, CEA, Service de Physique de l'État Condensé, Centre d'Étude de Saclay, 91191 Gif-sur-Yvette, France

FREE AUDITORS/AUDITEURS LIBRES A.-Z. EZZINE DE BLAS, Laboratoire de Physique des Solides, bâtiment 510, Université Paris Sud, 91405 Orsay, France S. ISAKOV, Division de Physique Théorique, IPN, 91406 Orsay, France, and Medical Radiological Research Centre, Obninsk, Kaluga Region 249020, Russia E. TUTUC, Department of Elec. Engineering, Princeton University, Princeton, NJ 08544, U.S.A.

PREFACE

L'utilisation en physique théorique de concepts empruntes a la topologie a conduit depuis plusieurs décennies à des développements intéressants dans des directions variées. En théorie quantique des champs, les travaux précurseurs de Skyrme suivis de ceux sur les solutions classiques des équations de Yang-MillsHiggs ont fait largement appel à ces notions et ont ainsi permis d'explorer certains secteurs non perturbatifs des théories de jauge. Les concepts empruntés à la topologie ont trouvé d'autres champs d'application, en particulier en physique de la matière condensée. Citons par exemple les travaux sur la classification des défauts dans les milieux ordonnés (Kleman, Toulouse, École des Houches XXXV, 1980). Plus récemment, un domaine où topologie et physique de la matière condensée ont connu une synergie remarquable est l'effet Hall quantique. Les rapides et impressionnants progrès expérimentaux dans la fabrication d'hétérojonctions (par épitaxie moléculaire) où un gaz bidimensionnel d'électrons peut être piégé ont été accompagnés de progrès théoriques dans la compréhension des systèmes tridimensionnels. Ces développements majeurs, spécifiques des années 80-90, ont été couronnés en automne 1998 par le prix Nobel de Physique, attribué à deux expérimentateurs, Stornier et Tsui, pour la découverte expérimentale indirecte de porteurs de charges fractionnaires dans les systèmes Hall quantique, et à un théoricien, Laughlin, pour leur prédiction théorique. Les notions de charge et de statistique fractionnaires ont précisément une interprétation théorique en terme d'interaction topologique de portée infinie. Il n'est donc pas fortuit qu'une École des Houches "Aspects topologiques de la physique en basse dimension" ait été organisée pendant l'été 1998. Les thèmes principaux de l'École ont porté sur la physique de l'effet Hall, et les concepts théoriques spécifiques à la physique bidimensionnelle, tels les statistiques intermédiaires (modèle des anyons) ou les théories de Chern-Simons. Des incursions ont été effectuées dans les systèmes unidimensionnels, tels les liquides de Luttinger et les modèles de Calogero-Sutherland.

Un autre domaine dans lequel des considérations topologiques ont apporté un éclairage intéressant est celui de la physique des polymères. Les contraintes topologiques peuvent en effet être décrites par des concepts empruntés à la théorie des noeuds et à la physique statistique. C'est dans ce contexte qu'ont été abordés à l'École l'étude du mouvement Brownien et ses relations avec la théorie des nœuds. Le déroulement de l'Ecole a été le suivant : Steve Girvin a ouvert l'Ecole par un cours théorique sur l'effet Hall quantique et certains développements récents comme les Skyrmions. En parallèle, Mansour Shayegan couvrait les aspects expérimentaux de l'effet Hall. Les théories de Chern-Simons ont été abordées par Gerald Dunne. Le modèle des anyons et le problème de la quantification d'un système de particules identiques en dimension 2 ont été discutés en détail par Jan Myrheim. Les aspects purement unidimensionnels des statistiques intermédiaires ont été couverts par Alexios Polychronakos. Hubert Saleur a donné un cours introductif aux théories conformes et à leurs applications au problème de la transmission tunnel à travers une impureté dans un système Hall fractionnaire. Ce sujet a connu un regain d'intérêt certain depuis la mise en évidence expérimentale récente de charges fractionnaires dans les systèmes Hall par la mesure du bruit de grenaille du courant tunnel à travers l'échantillon Hall. Il s'agit là de la confirmation directe de l'existence de charges fractionnaires transportant le courant Hall, entrevues dans les expériences de Störmer et Tsui du début des années 80. Certains développements expérimentaux de ce domaine particulièrement chaud ont été couverts dans un séminaire donné par Christian Glattli. Serguei Nechaev et Bertrand Duplantier ont clôt l'École par deux revues sur le mouvement Brownien, le groupe des tresses et leurs relations avec la théorie des nœuds. Plusieurs de ces concepts interviennent dans l'étude des propriétés d'élasticité et de torsion des molécules d'ADN, sujet qui a fait l'objet d'un séminaire de Vincent Croquette. Il nous a paru opportun de replacer les considérations topologiques évoquées dans les différents cours et séminaires dans un contexte plus général : c'était là l'objectif du cours de David Thouless. La notion de nombre topologique a été illustrée par de nombreux exemples allant de la physique de l'effet Hall à celle des superfluides. Les vortex qui sont naturellement au cœur de ce dernier sujet sont réapparus dans le séminaire d'Éric Ackermans consacré à la supraconductivité dans les systèmes mésoscopiques. Les questions de l'effet du désordre sur un gaz d'électrons bidimensionnel en présence d'un champ magnétique jouent certainement un rôle central, encore mal compris, dans la compréhension de l'effet Hall. Faute de pouvoir y consacrer un cours entier, ces problèmes ont été traités dans un séminaire sur l'état de l'art par John Chalker, et dans un séminaire par Jean Desbois

XIX

sur un modèle dans lequel la source du désordre se trouve dans le champ magnétique. Matthew Fisher a été malheureusement empêché à la dernière minute de venir aux Houches donner son cours sur de nouvelles phases dans des systèmes de spins unidimensionnels. Il a néanmoins mis ses notes de cours à la disposition des étudiants et nous a autorisé à publier son cours dans ce volume. Nous lui en sommes très reconnaissant. Malheureusement le cours de Bertrand Duplantier n'a pu donner lieu ni à des notes pour les étudiants, ni à un cours écrit, comme la tradition l'exige. Nous avons enfin tenu à ce que les étudiants présents à l'Ecole puissent présenter des séminaires sur leur travail. Deux sessions ont ainsi été consacrées à ces exposés, dont la liste se trouve à la fin de ce volume. Faute de place, de nombreux étudiants brillants et motivés n'ont pu participer à cette session. Nous espérons que la publication rapide de ce volume leur permettra de profiter du programme de cette École. Cette LXIXe session de l'École d'Été des Houches a été rendue possible grâce : - au soutien de l'Université Joseph Fourier de Grenoble et aux soutiens financiers du Ministère de l'Éducation Nationale, de la Recherche et de la Technologie (MENRT), du Centre National de la Recherche Scientifique (CNRS) et du Commissariat à l'Énergie Atomique (CEA) ; - tout spécialement au soutien de la Division des Affaires Scientifiques de l'OTAN, dont le programme des Advanced Study Institutes (ASI) incluait cette session, et enfin au soutien complémentaire de la National Science Foundation (NSF) des U.S.A. ; - aux orientations données par le Conseil d'Administration de l'École de Physique ; - au travail de Ghislaine d'Henry, Isabel Lelièvre et Brigitte Rousset tout à long de la préparation, du déroulement et de l'administration de la session ; - et bien sûr à la contribution dévouée de l'ensemble du personnel de l'École de Physique.

A. Comtet T. Jolicœur S. Ouvry F. David

PREFACE The use of concepts borrowed from topology has led to major advances in theoretical physics in recent years. In quantum field theory, the pioneering work by Skyrme and follow-ups on classical solutions of Yang-Mills-Higgs theories has lead to the discovery of the non-peturbative sectors of gauge theory. Topology has also found its way into condensed matter physics. Classification of defects in ordered media by homotopy theory is a well-known example (see e.g. Kleman and Toulouse, Les Houches XXXV, 1980). More recently, topology and condensed matter physics have again met in the realm of the fractional quantum Hall effect. Experimental progress in molecular beam epitaxy techniques leading to high-mobility samples allowed the discovery of this remarkable and novel phenomenon. These developments lead also to the attribution of the 1998 Nobel Prize in physics to Laughlin, Störmer and Tsui. The notions of fractional charge as well as fractional statistics can be interpreted by a topological interaction of infinite range. So it is natural to find in the Les Houches series a school devoted to quantum Hall physics, intermediate statistics and Chern-Simons theory. This session also included some one-dimensional physics topics like the Calogero-Sutherland model and some Luttinger-liquid physics. Polymer physics is also related to topology. In this field topological constraints may be described by concepts from knot theory and statistical physics. Hence this session also included Brownian motion theory related to knot theory. The school started with a theoretical survey by Steve M. Girvin on the quantum Hall effect, including recent developments on skyrmions. An experimental review was given at the same time by Mansour Shayegan. Chern-Simons theories were discussed by Gerald Dunne. The physics of anyons and quantization in two dimensions was presented by Jan Myrheim. One-dimensional statistics was reviewed by Alexios Polychronakos. Hubert Saleur discussed conformai field theory and recent applications to impurity problems. The evidence for fractional charge in shot noise measurements was presented by D. Christian Glattli. Serguei Nechaev and Bertrand Duplantier presented Brownian motion, braid group theory and the link with knot theory. A seminar by Vincent Croquette was devoted to recent applications to DNA physics.

XX11

A general overview of the role of topology in physics was given by David Thouless. The very notion of topological quantum numbers was illustrated by various examples from quantum Hall physics to superfluids. Vortices were also a common theme in a seminar given by Eric Akkermans. The all-important role of disorder in the quantum Hall effect was discussed in a review seminar by John Chalker and a more specialized talk by Jean Desbois, who concentrated on a model with a random magnetic field. Matthew P.A. Fisher was unfortunately unable attend the session as originally scheduled. However, he kindly produced the lecture notes that are included in this volume. We are very grateful to him for this. The lectures by Bertrand Duplantier led to no written version at all, contrary to the school tradition. There were two sessions devoted to participant's seminars and the list of these is given at the end of the book. We were able to admit only a limited number of participants among all the many highly qualified people who applied. We hope that the quick publication of this volume will give everyone access to some of the benefits of this school. This session LXIX was possible thanks to support from: - Université Joseph Fourier, Grenoble, the Ministère de l'Education Nationale, de la Recherche et de la Technologie (MENRT), the Centre National de la Recherche Scientifique (CNRS) and the Commissariat à l'Énergie Atomique (CEA); - the Division for Scientific Affairs of NATO whose ASI program included this session; - thanks are also due to the NSF of U.S.A. Orientations and choices were approved by the Scientific Board of the École de Physique des Houches. Last, but not least, very special thanks are due to Ghislaine d'Henry, Isabel Lelièvre and Brigitte Rousset for their valuable assistance during the preparation of this session as well as during the session. Thanks are also due to "Le Chef as well as to all the people in Les Houches who made this wonderful session possible. A. Comtet T. Jolicceur S. Ouvry F. David

CONTENTS

Lecturers

xi

Participants

xiii

Pre´face

xvii

Preface

xxi

Contents

xxiii

Course 1. Electrons in a Flatland by M. Shayegan

1

1 Introduction

3

2 Samples and measurements 2.1 2D electrons at the GaAs/AlGaAs interface . . . . . . . . . . . . . . 2.2 Magnetotransport measurement techniques . . . . . . . . . . . . . .

6 6 10

3 Ground states of the 2D System in a strong magnetic ﬁeld 10 3.1 Shubnikov-de Haas oscillations and the IQHE . . . . . . . . . . . . . 10 3.2 FQHE and Wigner crystal . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Composite fermions

16

5 Ferromagnetic state at ν = 1 and Skyrmions

19

6 Correlated bilayer electron states 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Electron System in a wide, single, quantum well . . . 6.3 Evolution of the QHE states in a wide quantum well 6.4 Evolution of insulating phases . . . . . . . . . . . . . 6.5 Many-body, bilayer QHE at ν = 1 . . . . . . . . . . . 6.6 Spontaneous interlayer Charge transfer . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

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21 21 26 29 34 41 44 48

xxiv

Course 2. The Quantum Hall Effect: Novel Excitations and Broken Symmetries by S.M.Girvin 1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26

quantum Hall eﬀect Introduction . . . . . . . . . . . . . . . . . . . Why 2D is important . . . . . . . . . . . . . Constructing the 2DEG . . . . . . . . . . . . Why is disorder and localization important? Classical dynamics . . . . . . . . . . . . . . . Semi-classical approximation . . . . . . . . . Quantum dynamics in strong B Fields . . . IQHE edge states . . . . . . . . . . . . . . . . Semiclassical percolation picture . . . . . . . Fractional QHE . . . . . . . . . . . . . . . . . The ν = 1 many-body state . . . . . . . . . . Neutral collective excitations . . . . . . . . . Charged excitations . . . . . . . . . . . . . . FQHE edge states . . . . . . . . . . . . . . . Quantum hall ferromagnets . . . . . . . . . . Coulomb exchange . . . . . . . . . . . . . . . Spin wave excitations . . . . . . . . . . . . . Eﬀective action . . . . . . . . . . . . . . . . . Topological excitations . . . . . . . . . . . . Skyrmion dynamics . . . . . . . . . . . . . . Skyrme lattices . . . . . . . . . . . . . . . . . Double-layer quantum Hall ferromagnets . . Pseudospin analogy . . . . . . . . . . . . . . Experimental background . . . . . . . . . . . Interlayer phase coherence . . . . . . . . . . Interlayer tunneling and tilted ﬁeld eﬀects .

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53 . . . . . . . . . . . . . . . . . . . . . . . . . .

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55 55 57 57 58 61 64 65 72 76 80 85 94 104 113 116 118 119 124 129 141 147 152 154 156 160 162

Appendix A Lowest Landau level projection

165

Appendix B Berry’s phase and adiabatic transport

168

Course 3. Aspects of Chern-Simons Theory by G.V. Dunne

177

1 Introduction

179

2 Basics of planar ﬁeld theory 2.1 Chern-Simons coupled to matter ﬁelds - “anyons” . . . . . . . . . . 2.2 Maxwell-Chern-Simons: Topologically massive gauge theory . . . 2.3 Fermions in 2 + 1-dimensions . . . . . . . . . . . . . . . . . . . . . . 2.4 Discrete symmetries: P, C and T . . . . . . . . . . . . . . . . . . .

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182 182 186 189 190

xxv 2.5 2.6

Poincar´e algebra in 2 + 1-dimensions . . . . . . . . . . . . . . . . . . 192 Nonabelian Chern-Simons theories . . . . . . . . . . . . . . . . . . . . 193

3 Canonical quantization of Chern-Simons theories 3.1 Canonical structure of Chern-Simons theories . . . . . . . . . 3.2 Chern-Simons quantum mechanics . . . . . . . . . . . . . . . . 3.3 Canonical quantization of abelian Chern-Simons theories . . 3.4 Quantization on the torus and magnetic translations . . . . . 3.5 Canonical quantization of nonabelian Chern-Simons theories 3.6 Chern-Simons theories with boundary . . . . . . . . . . . . . .

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195 195 198 203 205 208 212

4 Chern-Simons vortices 4.1 Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices 4.2 Relativistic Chern-Simons vortices . . . . . . . . . . . . . . . 4.3 Nonabelian relativistic Chern-Simons vortices . . . . . . . . 4.4 Nonrelativistic Chern-Simons vortices: Jackiw-Pi model . . 4.5 Nonabelian nonrelativistic Chern-Simons vortices . . . . . . 4.6 Vortices in the Zhang-Hansson-Kivelson model for FQHE . 4.7 Vortex dynamics . . . . . . . . . . . . . . . . . . . . . . . . .

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214 214 219 224 225 228 231 234

5 Induced Chern-Simons terms 5.1 Perturbatively induced Chern-Simons terms: Fermion loop . . . 5.2 Induced currents and Chern-Simons terms . . . . . . . . . . . . . 5.3 Induced Chern-Simons terms without fermions . . . . . . . . . . 5.4 A ﬁnite temperature puzzle . . . . . . . . . . . . . . . . . . . . . . 5.5 Quantum mechanical ﬁnite temperature model . . . . . . . . . . 5.6 Exact ﬁnite temperature 2 + 1 eﬀective actions . . . . . . . . . . 5.7 Finite temperature perturbation theory and Chern-Simons terms

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237 238 242 243 246 248 253 256

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Course 4. Anyons by J. Myrheim

265

1 Introduction 1.1 The concept of particle statistics . . . . . . . . . . 1.2 Statistical mechanics and the many-body problem 1.3 Experimental physics in two dimensions . . . . . 1.4 The algebraic approach: Heisenberg quantization 1.5 More general quantizations . . . . . . . . . . . . .

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269 270 273 275 277 279

2 The 2.1 2.2 2.3 2.4

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280 281 283 283 285

conﬁguration space The Euclidean relative space for two Dimensions d = 1, 2, 3 . . . . . . . . Homotopy . . . . . . . . . . . . . . . The braid group . . . . . . . . . . .

particles . . . . . . . . . . . . . . . . . .

3 Schr¨ odinger quantization in one dimension

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286

xxvi 4 Heisenberg quantization in one dimension 290 4.1 The coordinate representation . . . . . . . . . . . . . . . . . . . . . . 291 5 Schr¨ odinger quantization in dimension d 5.1 Scalar wave functions . . . . . . . . . . . 5.2 Homotopy . . . . . . . . . . . . . . . . . . 5.3 Interchange phases . . . . . . . . . . . . . 5.4 The statistics vector potential . . . . . . 5.5 The N-particle case . . . . . . . . . . . . 5.6 Chern-Simons theory . . . . . . . . . . . .

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295 296 298 299 301 303 304

6 The 6.1 6.2 6.3 6.4 6.5

Feynman path integral for anyons Eigenstates for Position and momentum . . . . . The path integral . . . . . . . . . . . . . . . . . . . Conjugation classes in SN . . . . . . . . . . . . . . The non-interacting case . . . . . . . . . . . . . . Duality of Feynman and Schr¨ odinger quantization

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306 307 308 312 314 315

7 The 7.1 7.2 7.3 7.4

harmonic oscillator The two-dimensional harmonic oscillator . . . Two anyons in a harmonic oscillator potential More than two anyons . . . . . . . . . . . . . . The three-anyon problem . . . . . . . . . . . .

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317 317 320 323 332

8 The 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12

anyon gas The cluster and virial expansions . . . . . . . . . . . . First and second order perturbative results . . . . . . . Regularization by periodic boundary conditions . . . . Regularization by a harmonic oscillator potential . . . Bosons and fermions . . . . . . . . . . . . . . . . . . . . Two anyons . . . . . . . . . . . . . . . . . . . . . . . . . Three anyons . . . . . . . . . . . . . . . . . . . . . . . . The Monte Carlo method . . . . . . . . . . . . . . . . . The path integral representation of the coeﬃcients GP Exact and approximate polynomials . . . . . . . . . . . The fourth virial coeﬃcient of anyons . . . . . . . . . . Two polynomial theorems . . . . . . . . . . . . . . . . .

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338 339 340 344 348 350 352 354 356 358 362 364 368

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373 374 377 380

9 Charged particles in a constant magnetic 9.1 One particle in a magnetic ﬁeld . . . . . 9.2 Two anyons in a magnetic ﬁeld . . . . . . 9.3 The anyon gas in a magnetic ﬁeld . . . .

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xxvii 10 Interchange phases and geometric phases 10.1 Introduction to geometric phases . . . . . . . . . . . . . 10.2 One particle in a magnetic ﬁeld . . . . . . . . . . . . . 10.3 Two particles in a magnetic ﬁeld . . . . . . . . . . . . . 10.4 Interchange of two anyons in potential wells . . . . . . 10.5 Laughlin’s theory of the fractional quantum Hall eﬀect

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383 383 385 387 390 392

Course 5. Generalized Statistics in One Dimension by A.P. Polychronakos

415

1 Introduction

417

2 Permutation group approach 418 2.1 Realization of the reduced Hilbert space . . . . . . . . . . . . . . . . 418 2.2 Path integral and generalized statistics . . . . . . . . . . . . . . . . . 422 2.3 Cluster decomposition and factorizability . . . . . . . . . . . . . . . . 424 3 One-dimensional systems: Calogero model 427 3.1 The Calogero-Sutherland-Moser model . . . . . . . . . . . . . . . . . 428 3.2 Large-N properties of the CSM model and duality . . . . . . . . . . 431 4 One-dimensional systems: Matrix model 4.1 Hermitian matrix model . . . . . . . . . . 4.2 The unitary matrix model . . . . . . . . . 4.3 Quantization and spectrum . . . . . . . . 4.4 Reduction to spin-particle systems . . . .

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433 433 437 438 443

5 Operator approaches 5.1 Exchange operator formalism . . . . . . . 5.2 Systems with internal degrees of freedom 5.3 Asymptotic Bethe ansatz approach . . . 5.4 The freezing trick and spin models . . . .

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448 448 453 455 457

6 Exclusion statistics 6.1 Motivation from the CSM model . . . 6.2 Semiclassics – Heuristics . . . . . . . . 6.3 Exclusion statistical mechanics . . . . 6.4 Exclusion statistics path integral . . . 6.5 Is this the only “exclusion” statistics?

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459 459 460 462 465 467

7 Epilogue

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469

xxviii

Course 6. Lectures on Non-perturbative Field Theory and Quantum Impurity Problems by H. Saleur 1 Some notions of conformal ﬁeld theory 1.1 The free boson via path integrals . . . . . . 1.2 Normal ordering and OPE . . . . . . . . . . 1.3 The stress energy tensor . . . . . . . . . . . . 1.4 Conformal in(co)variance . . . . . . . . . . . 1.5 Some remarks on Ward identities in QFT . 1.6 The Virasoro algebra: Intuitive introduction 1.7 Cylinders . . . . . . . . . . . . . . . . . . . . 1.8 The free boson via Hamiltonians . . . . . . . 1.9 Modular invariance . . . . . . . . . . . . . . .

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483 483 485 488 490 493 494 497 500 502

2 Conformal invariance analysis of quantum impurity ﬁxed points 503 2.1 Boundary conformal ﬁeld theory . . . . . . . . . . . . . . . . . . . . . 503 2.2 Partition functions and boundary states . . . . . . . . . . . . . . . . 506 2.3 Boundary entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 3 The 3.1 3.2 3.3 3.4

boundary sine-Gordon model: General results The model and the ﬂow . . . . . . . . . . . . . . . . . . . . Perturbation near the UV ﬁxed point . . . . . . . . . . . . Perturbation near the IR ﬁxed point . . . . . . . . . . . . An alternative to the instanton expansion: The conformal invariance analysis . . . . . . . . . . . . . . . . . . . . . . .

512 . . . . . . 512 . . . . . . 513 . . . . . . 515 . . . . . . 518

4 Search for integrability: Classical analysis

520

5 Quantum integrability 524 5.1 Conformal perturbation theory . . . . . . . . . . . . . . . . . . . . . . 524 5.2 S-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 5.3 Back to the boundary sine-Gordon model . . . . . . . . . . . . . . . 531 6 The thermodynamic Bethe-ansatz: The gas of particles with “Yang-Baxter statistics” 6.1 Zamolodchikov Fateev algebra . . . . . . . . . . . . . . . . . . . . 6.2 The TBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Standard computation: The central Charge . . . . . . . . . . . 6.4 Thermodynamics of the ﬂow between N and D ﬁxed points . . . 7 Using the TBA to compute static 7.1 Tunneling in the FQHE . . . . . 7.2 Conductance without impurity . 7.3 Conductance with impurity . . .

transport . . . . . . . . . . . . . . . . . . . . .

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532 532 534 536 538

properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

541 541 542 543

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xxix

Seminar 1. Quantum Partition Noise and the Detection of Fractionally Charged Laughlin Quasiparticles by D.C. Glattli

551

1 Introduction

553

2 Partition noise in quantum conductors 2.1 Quantum partition noise . . . . . . . . . . . . . . . . . . . . . . 2.2 Partition noise and quantum statistics . . . . . . . . . . . . . . 2.3 Quantum conductors reach the partition noise limit . . . . . . 2.4 Experimental evidences of quantum partition noise in quantum conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

554 . . . 554 . . . 555 . . . 557 . . . 558

3 Partition noise in the quantum Hall regime and determination of the fractional Charge 3.1 Edge states in the integer quantum Hall eﬀect regime . . . . . . . 3.2 Tunneling between IQHE edge channels and partition noise . . . . 3.3 Edge channels in the fractional regime . . . . . . . . . . . . . . . . 3.4 Noise predictions in the fractional regime . . . . . . . . . . . . . . . 3.5 Measurement of the fractional Charge using noise . . . . . . . . . . 3.6 Beyond the Poissonian noise of fractional charges . . . . . . . . . .

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562 562 563 564 567 569 570

Course 7. Mott Insulators, Spin Liquids and Quantum Disordered Superconductivity by Matthew P.A. Fisher

575

1 Introduction

577

2 Models and metals 579 2.1 Noninteracting electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 579 2.2 Interaction eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 3 Mott insulators and quantum magnetism 583 3.1 Spin models and quantum magnetism . . . . . . . . . . . . . . . . . . 584 3.2 Spin liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 4 Bosonization primer 5 2 Leg Hubbard ladder 5.1 Bonding and antibonding bands 5.2 Interactions . . . . . . . . . . . . 5.3 Bosonization . . . . . . . . . . . 5.4 d-Mott phase . . . . . . . . . . . 5.5 Symmetry and doping . . . . . .

588

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592 592 596 598 601 603

xxx 6 d-Wave superconductivity 6.1 BGS theory re-visited . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 d-wave symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Continuum description of gapless quasiparticles . . . . . . . . . . . .

604 604 609 610

7 Eﬀective ﬁeld theory 612 7.1 Quasiparticles and phase ﬂucutations . . . . . . . . . . . . . . . . . . 612 7.2 Nodons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 8 Vortices 623 8.1 ic/2e versus hc/e vortices . . . . . . . . . . . . . . . . . . . . . . . . . 623 8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 9 Nodal liquid phase 628 9.1 Half-ﬁlling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 9.2 Doping the nodal liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 632 9.3 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Appendix A Lattice duality

635

A.1 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 A.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Course 8. Statistics of Knots and Entangled Random Walks by S. Nechaev

643

1 Introduction

645

2 Knot diagrams as disordered Spin Systems 2.1 Brief review of statistical problems in topology . . . . . . . . . 2.2 Abelian problems in statistics of entangled random walks and incompleteness of Gauss invariant . . . . . . . . . . . . . . . . . 2.3 Nonabelian algebraic knot invariants . . . . . . . . . . . . . . . 2.4 Lattice knot diagrams as disordered Potts model . . . . . . . . 2.5 Notion about annealed and quenched realizations of topological disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

647 . . . 647 . . . 651 . . . 656 . . . 663 . . . 669

3 Random walks on locally non-commutative groups 3.1 Brownian bridges on simplest non-commutative groups and knot statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Random walks on locally free groups . . . . . . . . . . . . . . . . 3.3 Analytic results for random walks on locally free groups . . . . . 3.4 Brownian bridges on Lobachevskii plane and products of non-commutative random matrices . . . . . . . . . . . . . . . .

675 . . 676 . . 689 . . 692 . . 697

xxxi 4 Conformal methods in statistics of random walks with topological constraints 701 4.1 Construction of nonabelian connections for Γ2 and P SL(2, ) from conformal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 4.2 Random walk on double punctured plane and conformal ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 4.3 Statistics of random walks with topological constraints in the two–dimensional lattices of obstacles . . . . . . . . . . . . . . . . . . 709 5 Physical applications. Polymer language in statistics of entangled chain-like objects 715 5.1 Polymer chain in 3D-array of obstacles . . . . . . . . . . . . . . . . . 716 5.2 Collapsed phase of unknotted polymer . . . . . . . . . . . . . . . . . 719 6 Some “tight” problems of physics 6.1 Remarks and comments 6.2 Remarks and comments 6.3 Remarks and comments

the probability theory and statistical 727 to Section 2 . . . . . . . . . . . . . . . . . . 728 to Sections 3 and 4 . . . . . . . . . . . . . . 728 to Section 5 . . . . . . . . . . . . . . . . . . 729

Seminar 2. Twisting a Single DNA Molecule: Experiments and Models by T. Strick, J.-F. Allemand, D. Bensimon, V. Croquette, C. Bouchiat, M. Me´zard and R. Lavery

735

1 Introduction

737

2 Single molecule micromanipulation 739 2.1 Forces at the molecular scale . . . . . . . . . . . . . . . . . . . . . . . 739 2.2 Brownian motion: A sensitive tool for measuring forces . . . . . . . 740 3 Stretching B-DNA is well described by the worm-like chain model 740 3.1 The Freely jointed chain elasticity model . . . . . . . . . . . . . . . . 740 3.2 The overstretching transition . . . . . . . . . . . . . . . . . . . . . . . 743 4 The 4.1 4.2 4.3 4.4 4.5

torsional buckling instability The buckling instability at T = 0 . . . . . . . . . . . . . . . The buckling instability in the rod-like chain (RLC) model Elastic rod model of supercoiled DNA . . . . . . . . . . . . Theoretical analysis of the extension versus supercoiling experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical torques are associated to phase changes . . . . . . .

744 . . . . . 744 . . . . . 746 . . . . . 746 . . . . . 751 . . . . . 754

xxxii 5 Unwinding DNA leads to denaturation 754 5.1 Twisting rigidity measured through the critical torque of denaturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 5.2 Phase coexistence in the large torsional stress regime . . . . . . . . . 758 6 Overtwisting DNA leads to P-DNA 760 6.1 Phase coexistence of B-DNA and P-DNA in the large torsional stress regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 6.2 Chemical evidence of exposed bases . . . . . . . . . . . . . . . . . . . 762 7 Conclusions

762

Course 9. Introduction to Topological Quantum Numbers by D.J. Thouless

767

Preface

769

1 Winding numbers and topological classiﬁcation 769 1.1 Precision and topological invariants . . . . . . . . . . . . . . . . . . . 769 1.2 Winding numbers and line defects . . . . . . . . . . . . . . . . . . . . 770 1.3 Homotopy groups and defect classiﬁcation . . . . . . . . . . . . . . . 772 2 Superﬂuids and superconductors 775 2.1 Quantized vortices and ﬂux lines . . . . . . . . . . . . . . . . . . . . . 775 2.2 Detection of quantized circulation and ﬂux . . . . . . . . . . . . . . . 781 2.3 Precision of circulation and ﬂux quantization measurements . . . . . 784 3 The 3.1 3.2 3.3

Magnus force 786 Magnus force and two-ﬂuid model . . . . . . . . . . . . . . . . . . . . 786 Vortex moving in a neutral superﬂuid . . . . . . . . . . . . . . . . . . 788 Transverse force in superconductors . . . . . . . . . . . . . . . . . . . 792

4 Quantum Hall eﬀect 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proportionality of current density and electric ﬁeld 4.3 Bloch’s theorem and the Laughlin argument . . . . 4.4 Chern numbers . . . . . . . . . . . . . . . . . . . . . 4.5 Fractional quantum Hall eﬀect . . . . . . . . . . . . 4.6 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . .

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794 794 795 796 799 803 806

5 Topological phase transitions 5.1 The vortex induced transition in superﬂuid helium ﬁlms 5.2 Two-dimensional magnetic Systems . . . . . . . . . . . . 5.3 Topological order in solids . . . . . . . . . . . . . . . . . 5.4 Superconducting ﬁlms and layered materials . . . . . . .

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807 807 813 814 817

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xxxiii 6 The A phase of superﬂuid 3 He 819 6.1 Vortices in the A phase . . . . . . . . . . . . . . . . . . . . . . . . . . 819 6.2 Other defects and textures . . . . . . . . . . . . . . . . . . . . . . . . 823 7 Liquid crystals 826 7.1 Order in liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 826 7.2 Defects and textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

Seminar 3. Geometrical Description of Vortices in Ginzburg-Landau Billiards by E. Akkermans and K. Mallick

843

1 Introduction

845

2 Diﬀerentiable manifolds 2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Diﬀerential forms and their integration . . . . . . . . . . 2.3 Topological invariants of a manifold . . . . . . . . . . . . 2.4 Riemannian manifolds and absolute diﬀerential calculus 2.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

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846 846 847 853 855 858 860

3 Fiber bundles and their topology 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Local symmetries. Connexion and curvature . . . 3.3 Chern classes . . . . . . . . . . . . . . . . . . . . . 3.4 Manifolds with a boundary: Chern-Simons classes 3.5 The Weitzenb¨ ock formula . . . . . . . . . . . . . .

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860 860 861 862 865 869

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4 The dual point of Ginzburg-Landau equations for an inﬁnite System 870 4.1 The Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . . . 870 4.2 The Bogomol’nyi identities . . . . . . . . . . . . . . . . . . . . . . . . 871 5 The 5.1 5.2 5.3

superconducting billiard 872 The zero current line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 A selection mechanism and topological phase transitions . . . . . . . 874 A geometrical expression of the Gibbs potential for ﬁnite Systems . 874

Seminar 4. The Integer Quantum Hall Effect and Anderson Localisation by J.T. Chalker

879

1 Introduction

881

2 Scaling theory and localisation transitions

882

xxxiv 3 The plateau transitions as quantum critical points

885

4 Single particle models

887

5 Numerical studies

890

6 Discussion and outlook

892

Seminar 5. Random Magnetic Impurities and Quantum Hall Effect by J. Desbois

895

1 Average density of states (D.O.S.) [1]

897

2 Hall conductivity [2]

901

3 Magnetization and persistent currents [3]

904

Seminars by participants

911

COURSE 1

ELECTRONS IN A FLATLAND

M. SHAYEGAN Department of Electrical Engineering, Princeton University, Princeton, New Jersey, U.S.A.

Contents 1 Introduction

3

2 Samples and measurements 2.1 2D electrons at the GaAs/AlGaAs interface . . . . . . . . . . . . . 2.2 Magnetotransport measurement techniques . . . . . . . . . . . . .

6 6 10

3 Ground states of the 2D system in a strong magnetic field 3.1 Shubnikov-de Haas oscillations and the IQHE . . . . . . . . . . . . 3.2 FQHE and Wigner crystal . . . . . . . . . . . . . . . . . . . . . . .

10 10 12

4 Composite Fermions

16

5 Ferromagnetic state at = 1 and Skyrmions

19

6 Correlated bilayer electron states 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Electron system in a wide, single, quantum well . . . 6.3 Evolution of the QHE states in a wide quantum well 6.4 Evolution of insulating phases . . . . . . . . . . . . . 6.5 Many-body, bilayer QHE at ν = 1 . . . . . . . . . . 6.6 Spontaneous interlayer charge transfer . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . .

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21 21 26 29 34 41 44 48

ELECTRONS IN A FLATLAND

M. Shayegan

1

Introduction

Electrons in a “ﬂatland” are amazing! A simple low-temperature measurement of the resistance of a two-dimensional electron system (2DES) as a function of perpendicular magnetic ﬁeld (B) reveals why (Fig. 1). In this ﬁgure the resistivities along (ρxx ) and perpendicular (ρxy ) to the direction of current are shown, and the vertical markings denote the Landau-level ﬁlling factor (ν). Look how the behavior of ρxx with temperature (T ), shown schematically in the inset, changes as a function of the magnetic ﬁeld. At certain ﬁelds, marked A, ρxx drops exponentially with decreasing temperature and approaches zero as T → 0. This is the quantum Hall eﬀect (QHE) and, as you can see in the other trace of Figure 1, the Hall resistance (ρxy ) becomes quantized near these ﬁelds. The QHE is best described as an incompressible quantum liquid which often possesses a high degree of shortrange electron correlation. Next, look at the T -dependence of ρxx at the ﬁelds marked B (near 13 and 14 T for this sample). Here ρxx exponentially increases with decreasing T , signaling an insulating behavior. The nature of this insulating state is not entirely clear, but it is generally believed that it is a pinned Wigner solid, a “crystal” of electrons with long-range positional order. Now look at what happens at the magnetic ﬁeld marked C. At this ﬁeld, ρxx shows a nearly temperature-independent behavior, reminiscent of a metal. It turns out that at this particular ﬁeld there are two ﬂux quanta per each electron. The electron magically combines with the two ﬂux quanta and forms the celebrated “composite Fermion”, a quasiparticle which now moves around in the 2D plane as if no external magnetic ﬁeld was applied. So in one sweep, just changing the magnetic ﬁeld, the 2DES shows a variety of ground states ranging from insulating to metallic to “superconductinglike”. And, as it turns out, these ground states are stabilized primarily by strong electron-electron correlations. The data of Figure 1 reveals the extreme richness of this system, one which has rendered the ﬁeld of 2D carrier systems in a high magnetic ﬁeld among the most active and exciting in solid state physics. It has already led to two physics Nobel prizes, one in 1985 c EDP Sciences, Springer-Verlag 1999

4

Topological Aspects of Low Dimensional Systems

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to von Klitzing for the integral QHE (IQHE) [1,2], and another in 1998 to Laughlin, Stormer and Tsui for the fractional QHE (FQHE) [3,4], but surprises don’t seem to stop. Although both IQHE and FQHE have been studied extensively since their discoveries (see e.g. [5-8]), there have been a number of signiﬁcant developments in recent years. These developments, on the one hand, have unveiled new subtleties of the basic QHE and on the other hand, have led to a more global and unifying picture of the physics of the 2DES at high

M. Shayegan: Electrons in a Flatland

5

magnetic ﬁelds. Among these are the descriptions of the 2DESs at high B in terms of quasi-particles which consist of electrons and magnetic ﬂux. The ﬂux attachment treatment, which is based on Chern-Simons gauge transformation, maps the 2DES at high B onto a Fermionic or Bosonic system at a diﬀerent, eﬀective, magnetic ﬁeld Beﬀ . Such mappings provide elegant explanations, as well as predictions, for some of the most striking, observable QHE phenomena. Examples include the existence of a Fermi surface for the composite Fermions at ν = 12 ﬁlling where Beﬀ = 0, the similarity of the IQHE and FQHE, the transitions between QHE states and the transitions between QHE and insulating states at low ﬁllings. The purpose of these notes is to provide a glimpse of some of the exciting recent experimental results in this ﬁeld. I will focus on the following ﬁve areas; I will be very brief when covering these topics except in the part dealing with the bilayer systems, where I will go a bit more in depth: 1. a quick summary of some of the sample parameters and experimental aspects; 2. some basic and general remarks on the ground states of a 2DES in a strong magnetic ﬁeld; 3. a simple magnetic focusing experiment near ν = 12 which provides a clear demonstration of the presence of a composite Fermion Fermi surface and the semiclassical, ballistic motion of the composite Fermions; 4. recent experimental results near the ν = 1 QHE providing evidence for yet another set of quasi-particles, namely electron spin textures known as Skyrmions; and 5. bilayer electron systems in which the additional (layer) degree of freedom leads to unique QHE and insulating states which are stabilized by strong intralayer and interlayer correlations. I’d like to emphasize that these notes cannot and do not deal with all the important and exciting aspects of the QHE and related phenomena. They provide only a limited and selective sample of recent experimental developments. Readers interested in more details are referred to the original papers as well as extensive review articles and books [1-8]. Also, there will be a minimal treatment of theory here; for more details and insight, I suggest reading the comprehensive and illuminating notes by Steve Girvin in this volume and those by Allan MacDonald in proceedings of the 1994 Les Houches Summer School [9].

6

Topological Aspects of Low Dimensional Systems (

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Fig. 2. Schematic description of a modulation-doped GaAs/AlGaAs interface. Since the conduction-band edge (ECB ) of GaAs lies lower in energy than that of AlGaAs, electrons transfer from the doped AlGaAs region to the undoped GaAs to form a quasi-2D electron system (2DES) at the interface. The 2DES is separated from the doped AlGaAs by an undoped AlGaAs (spacer) layer to minimize electron scattering by the ionized impurities. Note that the electron wavefunction, ψ(z), has a ﬁnite extent in the direction perpendicular to the plane in which the electrons move freely. In (b) and (c) two common doping techniques are shown: bulk doping where the AlGaAs is uniformly doped and δ-doping where the dopants are themselves conﬁned to a plane (to two planes in the structure shown in (c)).

2

Samples and measurements

2.1 2D electrons at the GaAs/AlGaAs interface One of the simplest ways to place electrons in a ﬂatland is to conﬁne them to the interface between two semiconductors which have diﬀerent bandgaps. An example is shown in Figure 2 where a 2DES is formed at the interface between undoped GaAs and AlGaAs. The larger bandgap of AlGaAs leads to its conduction-band energy (ECB ) being higher than GaAs. The system is “modulation-doped” [10] meaning that the dopant atoms (in this case,

M. Shayegan: Electrons in a Flatland

7

Si donors) are placed in AlGaAs at some distance away from the interface. The electrons from the donors ﬁnd it energetically favorable to transfer to the lower energy conduction-band of GaAs. But as they transfer, an electric ﬁeld sets up between the positively-charged (ionized) donors in AlGaAs and the transferred electrons in GaAs. This electric ﬁeld limits the amount of charge transfer. Figures 2b and 2c schematically show ECB as a function of position, at equilibrium, after the charge transfer has taken place [11,12]. A key point in the structure of Figure 2 is that the 2DES is separated from the ionized dopants. As a result, the scattering of electrons by the ionized impurity potential is signiﬁcantly reduced, meaning that the 2D electrons are essentially “free” to move in the plane. It turns out this is crucial for much of the phenomena that is observed in these systems: by reducing the disorder and the electron-impurity interaction, electrons are allowed to interact with each other, and the result is a host of new manybody ground and excited states. Another important message here is that although we call the system “two-dimensional”, the electron wavefunction ψ (z) spreads in the z direction by a ﬁnite amount, typically ∼ 100 ˚ A. This ﬁnite layer-thickness plays an important role and should be taken into account when comparing theoretical calculations and experimental results: it distinguishes between “ideal” 2D system assumed in many calculations and the “real” quasi-2D, experimental system. How does one fabricate a structure like in Figure 2 and what are the details of a typical sample structure? Figures 3 and 4 provide schematic illustrations. The best quality GaAs/AlGaAs samples are presently grown by molecular beam epitaxy (MBE) [13]. The MBE system (Fig. 3) is essentially a very “clean” high-vacuum evaporation chamber. A GaAs substrate, heated to about 600 ◦ C, is positioned in front of eﬀusion cells (ovens) each of which contains one of the required elements (Ga, Al, As, and Si). The ovens are heated to appropriate temperatures to produce ﬂuxes of these elements which can impinge on the GaAs substrate. Each oven also has a shutter which is controlled, often via a computer, to produce a desired structure such as the one shown in Figure 4. Under these circumstances, and with a growth rate of about one monolayer of GaAs per second (which is roughly 1 µm/hour), one can grow very high quality, single-crystal structures with nearly any design. What determines the “quality” of the 2DES? For the electroninteraction-dominated phenomena in which we are interested here, the best sample is typically one with the least amount of imperfections such as interface irregularities, ionized impurities, etc. It is this consideration that leads to a complicated-looking structure such as the one shown in Figure 4. For example, the 2DES is separated from the Si dopants by a very thick spacer layer of undoped AlGaAs. The double-δ-doping is used to reduce the autocompensation of Si and to maximize the distance between

8

Topological Aspects of Low Dimensional Systems

Fig. 3. Cross-sectional view of a molecular beam epitaxy (MBE) growth chamber (after Ref. [13]), essentially a very high-vacuum evaporation chamber with a base pressure of 10−14 atmosphere. The chamber is equipped with various vacuum pumps, such as ion-pumps and cryopumps, and also can have analytical equipment such as a reﬂection high-energy electron diﬀractometer (RHEED) to monitor in-situ the substrate surface morphology as well as growth rate.

the ionized dopants and the 2DES [14,15]. Details and rationale for other fabrication procedures such as growth interruptions, the use of spacer with graded Al composition, etc., can be found in References 15 and 16. But a very important factor determining the quality of the 2DES, one which is not explicitly apparent in the structure of Figure 4, is the amount of residual (or unintentional) impurities that are incorporated throughout the structure during the MBE growth. These impurities are always present because the vacuum in the MBE chamber is not perfect, and also because the source materials (Ga, Al, etc.) used in the ovens are not 100% pure. It turns out in fact that in a structure like in Figure 4, with a large (> 2000 ˚ A) spacer layer thickness, the most important factor in obtaining very low-disorder 2DES is the purity of the grown material and not the speciﬁc details of the structural parameters. The vacuum integrity of the MBE growth chamber and the cleanliness and purity of the source materials and the GaAs substrate are therefore of paramount importance for the fabrication of state-of-the-art 2DES.

M. Shayegan: Electrons in a Flatland

9

Fig. 4. Layer structure of a modulation-doped GaAs/AlGaAs heterojunction grown by molecular beam epitaxy (after Ref. [15]). The measured magnetotransport data for this sample are shown in Figure 1.

A measure of the electronic “quality” of a 2DES is its low-temperature mobility, µ. Over the years, the mobility of modulation-doped GaAs/AlGaAs heterostructures has improved tremendously and the record stands at about 107 cm2 /Vs for a 2DES density (n) of ∼ 2 × 1011 cm−2 , implying a mean-free-path of tens of microns [17]. This mobility is more than ∼ 104 times higher than µ for a uniformly-doped piece of GaAs, demonstrating the striking power of modulation-doping. As mentioned in the last paragraph, the mobility in such thick-spacer structures is in fact limited by the concentration of the non-intentional (residual) impurities. This is evidenced by the observation [16-18] that µ ∼ nγ with γ 0.6; this is the

10

Topological Aspects of Low Dimensional Systems

dependence expected if the dominant source of scattering is the residual impurities in the close proximity of the 2DES [19]. The residual impurity concentration, deduced from the mobility values for state-of-the-art 2DES with µ 106 cm2 /Vs for n 5 × 1010 cm−2 is ni 1 × 1014 cm−3 , consistent with the residual GaAs doping expected in very clean MBE systems. An ni ∼ 1014 cm−3 means that the average distance between the residual impurities (∼ 2000 ˚ A) is smaller than the spacer layer thickness and, more importantly, is much larger than the typical inter-electron distance in the 2DES (∼ 450 ˚ A for n = 5 × 1010 cm−2 ). Clearly in such low-disorder 2D systems it is reasonable to expect that the physics can be dominated by electron-electron interaction. 2.2 Magnetotransport measurement techniques A variety of experimental techniques have been used to probe the electrical, optical, thermal, and other properties of the 2DES in a high magnetic ﬁeld. The bulk of the measurements, however, have been on the magnetotransport properties. Magnetotransport measurements are also by far the main topic of this paper. I therefore brieﬂy discuss such measurements here. In typical dc (or low-frequency, 100 Hz) transport experiments, the diagonal and Hall resistivities are measured in a Hall bridge or van der Pauw geometry with ∼1 mm distance between the contacts. Contacts to the 2DES are made by alloying In or InSn in a reducing atmosphere at ∼ 450 ◦ C for about 10 minutes. High-frequency measurements often involve more specialized geometries and contacting schemes. The low-T 2D carrier concentration can be varied by either illuminating the sample with a light-emitting diode or applying voltage (with respect to the 2DES) to a back- and/or front-gate electrode. Low temperatures are achieved using a 3 He/4 He dilution refrigerator, while the magnetic ﬁeld is provided either by a superconducting solenoid or a Bitter magnet, or a combination of both. The low-frequency magnetotransport measurements are typically performed with a current excitation of 10−9 A, corresponding to an electric ﬁeld of 10−4 V cm−1 and using the lock-in technique. 3

Ground states of the 2D system in a strong magnetic field

3.1 Shubnikov-de Haas oscillations and the IQHE A large magnetic ﬁeld applied perpendicular to the plane of a 2DES acts like a harmonic oscillator potential and leads to the quantization of the orbital motion. The allowed energies are quantized and are given by the “Landau Levels” (LLs), (N + 12 )ωc , where N = 0, 1, 2, ... and ωc = eB/m∗ is the cyclotron energy. For a system with a ﬁnite eﬀective Lande g-factor (g ∗ ), the energy spectrum is further quantized as each LL is spin-split to

M. Shayegan: Electrons in a Flatland

11

G( (

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Fig. 5. Density-of-states as a function of energy for a 2D carrier system: (a) in the absence of a magnetic ﬁeld, (b) with a magnetic ﬁeld (B) applied perpendicular to the 2D plane but neglecting the spin-splitting of the resulting Landau levels, and (c) with spin-splitting included. As is typical for a 2D electron system in a standard, single GaAs/AlGaAs heterojunction, here it is assumed that only one (size-quantized) electric subband, whose edge energy is marked by Eo , is occupied.

two levels separated by the Zeeman energy |g ∗ µB B| where µB is the Bohr magneton. This evolution of the density-of-states, D(E), for a 2D system in a magnetic ﬁeld is schematically shown in Figure 5. Note that for 2D electrons in GaAs, m∗ = 0.067mo and g ∗ −0.44, so that the cyclotron energy is about 70 times larger than the (bare) Zeeman energy. The degeneracy of each spin-split quantized energy level is eB/h. Since this degeneracy increases with B, to keep the total 2D density (n) of

12

Topological Aspects of Low Dimensional Systems

the system constant, the Fermi energy (EF ) has to move so that fewer and fewer LLs are occupied with increasing B. The number of spin-split LLs occupied at a given B is deﬁned as the ﬁlling factor and is given by ν = n/ (eB/h) = nh/eB. Equivalently, ν is the number of electrons per ﬂux quantum Φo = h/e. As B is increased and EF passes through the oscillating D(E), nearly all properties of the system, such as electrical resistivity, magnetic susceptibility, heat capacity, etc., oscillate. (The magnetoresistance oscillations are often called Shubnikov-de Haas oscillations.) The oscillations are periodic in 1/B with frequency nh/e or nh/2e, depending on whether or not the spin-splitting is resolved. This means that from a measurement of the frequency of the oscillations one can deduce the density. We will return to these oscillations in Section 6 where we analyze them to characterize the bilayer electron systems. The delta-function-like energy levels shown in Figure 5 are for an ideally pure 2DES. In the presence of disorder, the levels are broadened with their width, Γ, being of the order of /τq where τq is the quantum lifetime of the carriers. The states in the LLs’ tails are localized and only the centers of the LLs contain current-carrying extended states. Now suppose the ﬁlling factor is i, or nearly i, so that EF lies in the localized states between the i and i + 1 LL. If the disorder and temperature are suﬃciently small so that Γ and kB T are both smaller than the LL separation, then as T → 0 the longitudinal conductivity (σxx ) vanishes and σxy becomes quantized at a value that is equal to ie2 /h. This is the integral QHE. That σxx → 0 is simply a consequence of there being no extended states in the bulk of the 2D system to carry current. There are, however, i current-carrying “edge states” near the edge of the sample and this leads to σxy being quantized although demonstrating this quantization is more subtle (see, e.g., Steve Girvin’s notes). Note also that, according to the simple relations which convert the elements of the conductivity tensor to those of the resistivity 2 2 2 2 and ρxy = σxy / σxx . Therefore, + σxy + σxy tensor, ρxx = σxx / σxx σxx = 0 and σxy = ie2 /h means that ρxx = 0 and ρxy = h/ie2 . This explains the experimental result in Figure 1 for the Hall bar sample shown in the inset. To summarize, the IQHE is a consequence of: (1) the quantization of the 2D system’s energy levels into a set of well-deﬁned (but broadened) LLs with separation greater than kB T , and (2) the presence of localized states in between these LLs. Note that no electron-electron interaction is needed to bring about or to explain the IQHE. 3.2 FQHE and Wigner crystal Suppose B is suﬃciently raised so that ν < 1. At T = 0 the kinetic energy of the 2DES is quenched and the system enters a regime where, in the absence of disorder, its ground state is determined entirely by the electron-electron

M. Shayegan: Electrons in a Flatland

13

interaction. In the inﬁnite B limit, the system approaches a classical 2D system which is known to be an electron crystal (Wigner Crystal) with the electrons localized at the sites of a triangular lattice [20]. At ﬁnite B, the electrons cannot be localized to a length smaller than the cyclotron orbit radius of the lowest LL, or the magnetic length lB = (/eB)1/2 = 1/2 (ν/2πn) , and the ground state is typically a gas or liquid. However, when lB is much smaller than the average distance between electrons, i.e. when ν << 1, a crystalline state is possible [21]. We will return to this crystalline state shortly. A competing ground state of the 2D system at high B is the FQH liquid [3,4]. Ironically, the work that led to the discovery of the new and totally unexpected FQH phenomenon [3] was itself one of the early experimental searches for the magnetic-ﬁeld-induced Wigner crystal! The FQHE, observed at the principal ﬁllings ν = 1/q and other rational fractional ﬁllings ν = p/q (q=odd integer) is characterized by the vanishing of ρxx and the quantization of ρxy at (q/p) h/e2 as T → 0. The eﬀect is phenomenologically similar to the IQHE but its origin is very diﬀerent. The FQHE state is an intrinsically many-body, incompressible quantum liquid, described by the Laughlin wavefunction: Ψνm

|zi |2 m ∼ (zi − zj ) × exp − · 4lB i,j i

Here zi and zj are (complex) coordinates of pairs of electrons in the plane, and m = 1/ν is an odd integer so that the wavefunction is antisymmetric when two electrons are interchanged (Pauli exclusion principle). Note also that the Coulomb repulsion between electrons is built into this wavefunction as it becomes small when two electrons come close to each other. The wavefunction has strong short-range correlation but it does not describe a crystalline phase as it has no long-range order (see the “snap shots” shown in Fig. 12 of Steve Girvin’s notes). The FQHE has many fundamental and interesting characteristics among which we brieﬂy mention three here. First, the incompressibility implies that the ground state is separated from its excitations by a ﬁnite energy gap, ∆. Experimentally ∆ can be measured from the activated T -dependence of ρxx according to ρxx ∼ exp (−∆/2kB T ). This is superﬁcially similar to the energy gap between the LLs which leads to the IQHE, but the origin of the FQHE gap is entirely many-body. The theoretical ∆ for the ν = 13 FQHE in an ideal 2DES, with no disorder, zero layer-thickness, and inﬁnitely separated LLs, is ∼ 0.1e2 /εlB , where ε is the dielectric constant of the host material (GaAs). In real samples, however, the ubiquitous disorder, ﬁnitelayer thickness, and LL mixing lead to a gap which is much smaller than 0.1e2 /εlB (see, e.g., Ref. [22]). Finite layer thickness, for example, leads to

14

Topological Aspects of Low Dimensional Systems

a softening of the short-range Coulomb interaction, and results in the weakening of the FQHE. In fact, experiments [23] and calculations [24] have revealed that once the layer thickness exceeds ∼ 3lB , the FQHE quickly collapses. A second, quite intriguing yet fundamental feature of the FQHE is that its elementary excitations carry fractional charge e∗ = e/m (see Steve Girvin’s notes for details and insight). There have been several reports of measuring this fractional charge, the latest being measurements of the quantum shot noise which is proportional to the charge of the conducting carriers [25]. In these measurements, the current noise was monitored as a function of the backscattered current which results from the tunneling between the FQHE edge states in a point-contact (constriction). The results near ν = 13 indeed reveal that the current carrying particles have charge e/3. The third noteworthy feature is the existence of FQHE states not only at the primary ﬁllings ν = 13 and 15 , but also at a host of other odd-denominator ﬁllings. Examples are the states at ν = 25 , 37 , ... and ν = 35 , 47 , ... which can be seen in Figure 1. The strength of these states, namely their measured energy gaps, typically decreases as the denominator of their ﬁlling gets larger. Also, they appear to form a sequence of decreasing strength as one goes from the primary state, such as ν = 13 , towards the even-denominator ﬁlling ν = 12 at which there is no FQHE state. These observations led to an explanation for the higher order FQHE states based on a “hierarchical” scheme where each state is considered the “parent” state for the adjacent (in ﬁlling factor) weaker state. The idea is that as one deviates from the exact ﬁlling for a given FQHE state, quasiparticles are created above the energy gap; these quasiparticles then interact and form an incompressible liquid once their density to magnetic ﬂux ratio reaches certain values. Such a hierarchical construction can generate all the odd-denominator fractions, and explains certain features of the observed FQHE sequences. However, it fails to account for the observed strength/weakness of all FQHE states. Also, in this scheme, the wavefunctions of the higher order states turn out to be much more complex than 1 (m an odd integer). Moreover, such those for the primary states at ν = m description of the FQHE diﬀers entirely from that of the IQHE while it is hard to overlook the striking similarity between the FQHE sequence, e.g., at ν = 13 , 25 , 37 ... and the IQHE sequence at ν = 1, 2, 3, ... In fact, if we “slide” the 5.5 < B < 9T portion of the magnetoresistance trace in Figure 1 to the left so that the position of ν = 12 is now the “zero” of (eﬀective) magnetic ﬁeld, we can see a one-to-one correspondence between the above IQHE and FQHE sequences, both in terms of the ﬁeld positions of ρxx minima and their relative strength. (This is true if we assume that the 2DES is fully spin-polarized at all ﬁelds.)

M. Shayegan: Electrons in a Flatland

15

Such observations prompted the search for a description of the QHE which somehow links the integral and fractional eﬀects. This has culminated in a remarkable description in which an even number of ﬂuxes combine with an electron to form a new, “composite” Fermion (CF) [26]. The electron-electron interaction and the large magnetic ﬁeld are embedded in this ﬂux-electron quasiparticle so that the system now behaves as if it contains (essentially) non-interacting particles moving in an “eﬀective” magnetic ﬁeld which is the balance of the external ﬁeld once the attached ﬂuxes are deducted. For example, focusing on the range 13 ≤ ν ≤ 12 and attaching two ﬂuxes to each electron, the CF description maps the FQHE at ν = 13 , 25 , 37 , ... to the IQHE at ν = 1, 2, 3, ... Perhaps even more intriguing is the notion that the eﬀective ﬁeld at ν = 12 is zero for the CFs so that, at ν = 12 , they ignore the large external magnetic ﬁeld and move about as if there is no magnetic ﬁeld. In fact, the CFs should have a Fermi surface at ν = 12 , just like electrons do at zero magnetic ﬁeld! The CF description and its implications have recently been among the most active areas in the physics of 2D systems at high B, both experimentally and theoretically [26]. In Section 4, I will brieﬂy describe a simple experiment probing the Fermi surface of CFs. A few words regarding the competition between the FQHE and Wigner crystal (WC) are in order. It turns out that the Laughlin FQH liquid states at ν = p/q are particularly robust and have ground state energies which are lower than the WC state energy, at least for ν > 15 . This is illustrated in Figure 6 where the estimated energies are plotted as a function of ν (for details of estimations see Refs. [27] and [28]). The downward “cusps” in energy reﬂect the incompressibility of the FQHE states and the presence of energy gaps which are proportional to the discontinuties in the derivative of energy vs. ν. Also shown schematically in Figure 6 (dashed curve) is the expected dependence of the WC ground state energy on ν. Theoretical calculations predict that, in an ideal 2D system, the WC should be the ground state for ν smaller than about 16 . It is evident from Figure 6 that while at ν = 15 the FQHE can be the ground state, the WC state may win as the ﬁlling deviates slightly from 15 . It is possible therefore to have a WC which is reentrant around a FQH liquid state. The above picture has been used to rationalize the general current belief that the insulating behavior observed around the ν = 15 FQHE in the best quality GaAs/AlGaAs 2DESs is the signature of a pinned WC state. The solid is presumably “pinned” by the disorder potential, and can be made to slide if a suﬃciently large electric ﬁeld is applied. Such depinning would result in a nonlinear current-voltage characteristic, consistent with numerous measurements. The magnetic-ﬁeld-induced WC crystal problem in 2D systems has been studied extensively during the past ten years; for recent reviews see reference [21].

16

Topological Aspects of Low Dimensional Systems

*URXQGVWDWHHQHUJ\0>.@ >.@

)4+(

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)LOOLQJIDFWRU3 Fig. 6. Energies of two competing ground states of a 2D electron system at high magnetic ﬁeld. The FQHE incompressible liquid states occur at special odddenominator ﬁllings as the downward energy “cusps” indicate (solid curve). The Wigner crystal (WC) state has monotically decreasing energy as a function of inverse ﬁlling (dashed curve) and is expected to win for ﬁllings less than about 16 . (After Ref. [28]).

4

Composite Fermions

Exploiting the transmutability of the statistics in 2D, a gauge transformation that binds an even number of magnetic ﬂux quanta (2mΦ0 where m is an integer and Φ0 ≡ h/e is the ﬂux quantum) to each electron maps the 2DES at even-denominator ﬁllings to a system of CFs at a vanishing Beﬀ [26]. Such transformation elegantly maps a FQHE observed at the 2DES ﬁlling ν to an IQHE for the CF system at ﬁlling ν where ν = ν/ (1 − 2mν). Moreover, since Beﬀ = 0 at ν = 1/2m, the CF system should possess certain Fermi-liquid-like properties. Most notably, a CF Fermi surface should exist at and near ν = 12 , supporting phenomena such as geometrical resonances and CF ballistic transport.

M. Shayegan: Electrons in a Flatland

17

Fig. 7. Magnetic focusing spectra are shown for 2D electrons near zero external magnetic ﬁeld (bottom trace) and for composite Fermions near ν = 12 (top trace) where the external ﬁeld is about 9 T. In the top trace, the position of ν = 12 marks the zero of the eﬀective magnetic ﬁeld (Beﬀ ) for CFs. Both traces exhibit peaks at ﬁelds where the distance between the injector and collector point-contacts (L 5.3 µm in this case) matches a multiple integer of the classical cyclotron orbit diameter. The inset schematically shows the top view of the sample. (After Ref. [29]).

Here I ﬁrst present, as an example, the results of an experiment which provide a clear demonstration of the surprisingly simple behavior of CFs near ν = 12 , namely their semiclassical, ballistic motion under the inﬂuence of Beﬀ . I then attempt to give a perspective of the ﬁeld by listing some earlier, key experimental results and identifying current puzzles. For more details, I suggest reading more comprehensive review articles [26]. Figure 7 shows data from a magnetic focusing experiment [29] near B = 0 (bottom trace) and ν = 12 (top trace). The geometry of the experiment is sketched in the inset, which shows the top view of the sample. Parts of the sample are etched (thick lines in Fig. 7 inset) so that the 2DES is separated into three regions which are connected by two narrow constrictions (point-contacts). The distance between the two constrictions L, is chosen to be smaller than or of the order of the mean-free-path of the electrons. Ballistic electrons are then injected from the lower-left section to the upper section through the “injector” constriction by passing a current between the ohmic contacts 1 and 2. Now a small B-ﬁeld is applied

18

Topological Aspects of Low Dimensional Systems

perpendicular to the plane to “bend” the semiclassical, ballistic trajectory of the injected electrons as they travel in the upper section. As B is increased, whenever L matches a multiple integer of the electron’s semiclassical cyclotron orbit diameter, dc = 2m∗ v F /eB = 2kF /eB, the ballistic electrons impinge on the “collector” constriction, either directly or after one or more bounces oﬀ the focusing barrier separating the two constrictions. At these B, one observes a maximum in the voltage measured between the lowerright and the upper sections (contacts 3 and 4). The traces shown in Figure 7 are the voltages measured between contacts 3 and 4, normalized to the current injected between contacts 1 and 2. Maxima can be clearly seen in the lower trace of Figure 7 for B > 0 and their positions are indeed consistent with the values of L and kF for this 2DES. Note that for B < 0, the electrons are deﬂected to the left and no magnetic focusing is expected, consistent with the absence of any observed maxima. The experiments of Goldman et al. [29] reveal oscillations of the resistance not only near B = 0 for electrons, but also near ν = 12 (upper trace of Fig. 7). The data provide a remarkable demonstration of the “classical”, ballistic motion of the CFs under the inﬂuence of Beﬀ . Note that Beﬀ is only a few tenths of a Tesla while the real external magnetic ﬁeld is about 12 Tesla! The large external magnetic ﬂux felt by the interacting electrons is replaced by the much smaller ﬂux inﬂuencing the apparently simple ﬂux-electron composites. The data of Figure 7 also provide a direct determination of the Fermi wavevector as well as an estimate for the ballistic mean-free-path of the CFs ( 1 µm). To bring the current status of the CFs into focus, I list some key experiments done so far (original references can be found in [26] or in [30]. These include measurements of the surface acoustic wave propagation, FQHE activation energies, CF eﬀective mass, resistance oscillations in antidot arrays, magnetic focusing, low-T thermopower, magnetooptics, CF spin, temperature dependence of the CF conductivity at ν = 12 and 32 , and ballistic CF transport in nanostructures. The results of most of these experiments are in general agreement with each other and with the CF picture although some inconsistencies exist. Among the most controversial and hot current topics are the CF eﬀective mass (m∗CF ) and the degree of CF spin polarization. Theoretically, m∗CF is expected to be strongly inﬂuenced by electronelectron interaction; it should be much larger than the bare (B = 0) electron eﬀective mass, should scale with the Coulomb energy and therefore increase with B as ∼ B 1/2 and may diverge as ν → 12 [26]. While most experimental results agree with a rather large m∗CF , there is no quantitative agreement. Neither is there an experimental consensus on the functional dependence of m∗CF on B although most measurements agree with ∼ B 1/2 dependence and some experiments have even reported a diverging m∗CF as ν → 12 . The spinpolarization of the CFs near ν = 32 is also unclear. The surface acoustic

M. Shayegan: Electrons in a Flatland

19

wave and antidot transport measurements suggest a spin-polarized state near ν = 32 . The tilted-B magnetotransport measurements, however, are consistent with an unpolarized spin. In summary, while there is now compelling experimental evidence for the validity of the CF picture of the FQHE, important uncertainties regarding the properties of these fascinating composite particles still remain.

5

Ferromagnetic state at ν = 1 and Skyrmions

For 2DESs in GaAs, while the IQHE at even ν arises from the single-particle energy gaps separating the LLs, the spin splitting of these levels leads to IQHE at odd ν. The electron-electron interaction and in particular the exchange energy, however, play a dominant role for odd-ν IQHE and often lead to a substantially larger QHE energy gap than expected from the bare eﬀective g-factor (g ∗ −0.44) for GaAs [31]. In fact, according to theory [32], the odd-ν IQHE states should exist even in the limit of zero Zeeman energy (g ∗ → 0); there should be a spontaneous ferromagnetic order with a spin polarized 2DES ground state. Perhaps even more interesting are the predicted excitations of these ferromagnetic states: provided that g ∗ is suﬃciently small, the charged excitations of the system are ﬁnite-size “Skyrmions”, termed so after the work of Skyrme in 1958 [33], rather than single spin ﬂips. Skyrmions are spin textures, smooth distortions of the spin ﬁeld involving several spin ﬂips [32,34]. The spin and size of the Skyrmions are determined by the competition between the Zeeman and the exchange energies: a large ratio of the exchange energy over the Zeeman energy would favor large-size Skyrmions over single spin ﬂips as the (exchange) energy gained by the near parallelism of the spins would outweigh the (Zeeman) energy cost of the extra spin ﬂips. Skyrmions are relevant at ν = 1 (at ﬁnite T ) and near ν = 1 where the 2DES is not fully spin polarized Clear experimental evidence for ﬁnite-size Skyrmions was recently provided by the pioneering nuclear magnetic resonance measurements of Barrett et al. [35]. On either side of ν = 1, they observed a rapid drop of the Knight-shift of the 71 Ga nuclei which are in contact with or are near the 2DES. Associating this Knight-shift with the spin polarization of the 2DES, they deduced that the charged excitations of the ν = 1 QHE carry large (4) eﬀective spins [35]. Subsequent theoretical calculations have shown excellent quantitative agreement with the Knight-shift data and the spin polarization of the 2DES, thereby providing additional credence to the Skyrmionic picture near ν = 1 [36]. Finally, magnetotransport [37] and magnetooptical [38] data have provided further evidence for Skyrmions and their size.

20

Topological Aspects of Low Dimensional Systems

Implied by the Knight-shift data [35,39] is a strong coupling of the nuclear and 2DES spin systems near ν = 1 where Skyrmions are present. Here I would like to discuss some recent 2DES heat capacity (C) data near ν = 1 at very low T [40] which dramatically manifest the consequences of this Skyrmion-induced coupling. Moreover, a remarkably sharp peak observed in C vs. T is suggestive of a phase transition in the electronic system, possibly signaling a crystallization of the Skyrmions at very low T . Bayot et al. [40] have succeeded in measuring C vs. B and T in a multiple-quantum-well sample in the QHE regime and at very low T (down to 25 mK). Their C vs. B data, shown in Figure 8, is striking in that at high B (near ν = 1) C becomes many orders of magnitude larger than its low B value. Figure 9 reveals yet another intriguing feature of their data; in a small range of ν near 0.8 (and also near 1.2), C vs. T exhibits a very sharp peak at a temperature Tc which sensitively depends on ν (Tc quickly drops as ν deviates from 0.8 or 1.2) [40]. The low B data of Figure 8a can be understood based on the 2DES electronic heat capacity and its oscillating density of states at the Fermi energy [40,41]. The high B data (Figs. 8b and 9) near ν = 1, on the other hand, are unexpected and cannot be accounted for based on the thermodynamic properties of the 2DES alone. Both the very large magnitude of C and the T −2 dependence of C at high T (dashed line in Fig. 9) hint at the nuclear Schottky eﬀect. Utilizing this clue, Bayot et al. were able to semi-quantitatively explain the magnitude and the dependence of C on B and T (for T > 0.1 K) based on a simple Schottky model for the nuclear spins of the Ga and As atoms in the quantum wells. Implicit in this interpretation of course is a coupling between the nuclear spins and the lattice; this coupling is assumed to be provided by the Skyrmions. The Schottky model, however, predicts a smooth maximum in C at T ∼ δ/2kB 2 mK for B 7 T and cannot explain the sharp peak observed at Tc ∼ 35 mK (δ is the nuclear spin splitting). It is possible that this peak may be a signature of the expected Skyrmion crystallization and the associated magnetic ordering near ν = 1 [40,42,43]. Such crystallization has indeed been proposed theoretically [36,43] although the details of the Skyrmion liquid-solid transition and, in particular, how it would aﬀect the coupling to the nuclear spin system are unknown. One feature of the data that qualitatively agrees with the Skyrmion crystallization is worth emphasizing. As shown in the upper inset in Figure 9, the observed Tc decreases rapidly as ν deviates from 0.8 or 1.2 [40]; this is consistent with the expectation that as the Skyrmion density decreases, the Skyrme crystal melting T should decrease.

M. Shayegan: Electrons in a Flatland

21

Fig. 8. Heat capacity C of a multiple-quantum-well sample, showing orders of magnitude enhancement of the high-B data (b) over the low-B data (a). The line through the data points is a guide to the eye. (After Bayot et al. [40].)

6

Correlated bilayer electron states

6.1 Overview The introduction of an additional degree of freedom can have a profound eﬀect on the many-body ground states of the 2DES at high B. For example, the addition of a spin degree of freedom stabilizes particular spinunpolarized FQHE observed at lower B [44,45], while substantially

22

Topological Aspects of Low Dimensional Systems

Fig. 9. The temperature dependence of C at B = 7 T (ν = 0.81) is shown in the main ﬁgure in a log-log plot. The dashed line shows the T −2 dependence expected for the Schottky model. The lower inset shows a linear plot of C vs. T at B = 6.7 T (ν = 0.85). The temperature Tc , at which the heat capacity exhibits the sharp peak depends on the ﬁlling factor as shown in the upper inset. (After Bayot et al. [40].)

increasing the layer thickness (thus introducing an additional spatial degree of freedom) leads to a weakening and eventual collapse of the FQHE [23,24]. In this section, I’d like to review magnetotransport results, obtained primarily in my laboratory at Princeton University, on a novel bilayer electron system. The data show how the additional (layer) degree of freedom results in new correlated states. We have been studying an electron system, conﬁned in a wide GaAs quantum well, which can be tuned from a single-layer-like (albeit thick) system to a bilayer system by increasing the electron density n in the well [46]. This evolution with n and, in particular, the transition to a bilayer system where interlayer as well as intralayer interactions are dominant, has a dramatic eﬀect on the correlated states of the electron system, as manifested in the magnetotransport data [47-51]. Figures 10 to 12 provide examples for an electron system in a 750 ˚ A-wide GaAs quantum well. In certain ranges of n, there are well-developed FQH states at the even-denominator ﬁllings ν = 12 [47-49,52] and 32 [49] which have no counterparts in standard 2DESs in single-heterostructures. (Note that ν = 12 is the total ﬁlling for the system; it corresponds to 1/4 ﬁlling

M. Shayegan: Electrons in a Flatland

23

7*P.

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for each layer). Figure 11 shows that at n 1.26 × 1011 cm−2 , insulating phases (IPs), which are reentrant around the ν = 12 FQH state, develop. The data of Figure 11 have a remarkable resemblance to the IPs observed in very high-quality, standard, single-layer, GaAs 2DESs (e.g., see Fig. 1) except that here the IP is reentrant around the much higher ﬁlling ν = 12 rather than ν = 15 ! As we will discuss, the IPs of Figure 11 are suggestive of a pinned, bilayer Wigner crystal which is stabilized at high ν thanks to the interlayer correlation among electrons. Figure 12 demonstrates yet another surprising aspect of the QHE in this system. Here the Arrhenius plots of

24

Topological Aspects of Low Dimensional Systems

ν

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resistance at ν = 1 as the density is varied are very unusual: the activated behavior of Rxx vs. 1/T starts rather abruptly below a temperature T ∗ which is much smaller than the deduced QHE gaps ( 20 K) and, even more surprisingly, is n-dependent. The evolution of the magnetotransport data in this system as a function of n, and the intriguing interplay between the incompressible liquid states (including the ν = 12 FQHE) and the IP, which displays behavior profoundly diﬀerent from any observed in a standard 2DES, are the subject of this

M. Shayegan: Electrons in a Flatland

25

Fig. 12. Arrhenius plots of Rxx at ν = 1 vs. 1/T for the sample of Figures 10 and 11 for diﬀerent n. For n > 10 × 1010 cm−2 , the activated behavior of Rxx ends abruptly above a temperature T ∗ which strongly depends on n. (After Lay et al. [50].)

section. We will ﬁrst give a brief overview of the sample structure and measurements in Section 6.2. Sections 6.3 and 6.4 summarize the evolution of the QHE and insulating states as the electron system is tuned from single-layer to bilayer. We will see that the interlayer as well as intralayer correlations play a key role in stabilizing the unique ground states of the system. In Section 6.5 we highlight our results for the ν = 1 QHE in this system [50] and suggest that its very unusual dependence on T and n may be indicative of an unusual ﬁnite-temperature transition, from a QHE state to a compressible state, which is unique to interacting bilayer systems. Finally in Section 6.6 we present recent experimental results which reveal that a bilayer system with two equally-populated layers at zero magnetic ﬁeld can spontaneously break its charge distribution symmetry through an interlayer charge transfer near the magnetic quantum 1 limit [27]. New FQH states at 2 + = unusual total ﬁllings such as ν = 11 15 3 5 stabilize as signatures that the system deforms itself, at substantial electrostatic energy cost, in order to gain correlation energy by “locking in” separate incompressible liquid phases at unequal ﬁllings in the two layers (e.g., layered 13 and 25 states in the case of ν = 11 15 ).

26

Topological Aspects of Low Dimensional Systems

6.2 Electron system in a wide, single, quantum well ˚ is a The electron system in a wide, GaAs quantum well of width ∼ 1000 A particularly interesting one (Figs. 13 and 14). At low n the electrons occupy the lowest electric subband and have a single-layer-like (but rather “thick” in the z -direction) charge distribution (Fig. 13). As more electrons are added to the well, their electrostatic repulsion forces them to pile up near the well’s sides and the resulting electron charge distribution appears increasingly bilayer-like. A relevant parameter that quantiﬁes this evolution is the energy diﬀerence between the two lowest subbands which, for a symmetric charge distribution, corresponds to symmetric-to-antisymmetric energy splitting ∆SAS ; this is a measure of the coupling between the two layers. Also relevant is the interlayer distance, deﬁned by the parameter d as shown in Figure 13. A crucial property of the electron system in a wide quantum well is that, for a given well width, both ∆SAS and d depend on n: increasing n makes d larger and ∆SAS smaller so that the system can be tuned from bilayer to (thick) single-layer by decreasing n (Fig. 13). This evolution with density plays a decisive role in the properties of the correlated electron states in this system. Experimentally, we control both n and the charge distribution symmetry in the samples via front- and back-side gates (Fig. 15), and by measuring the occupied subband electron densities from Fourier transforms of the lowB magnetoresistance (Shubnikov-de Haas) oscillations (Fig. 16). One of the simplest ways to ﬁnd the symmetric charge distribution at a given density n is to measure and minimize the subband separation (∆01 ) as a function of pairs of applied front- and back-gate biases while n is kept constant. The basic idea is that in our wide quantum well, as in a double-quantum-well, at a ﬁxed n, ∆01 is smallest when the charge distribution is symmetric. In practice, we start with a given n and measure the subband densities from the Fourier transforms of the Shubnikov-de Haas oscillations (Fig. 16). Note that the diﬀerence between these densities can be simply converted to subband separation ∆01 by dividing by the 2D density of states m∗ /π2 . Next we lower n by an amount ∆n by applying a negative bias to the frontgate VFG (with respect to an Ohmic contact made to the electron system), and then raise the density by the same amount ∆n via the application of a positive bias VBG to the back-gate. We then measure new subband densities. By repeating this procedure we can ﬁnd the pair of VFG and VBG that results in the minimum measured ∆01 . This pair of VFG and VBG gives the symmetric (“balanced”) charge distribution, and the minimum ∆01 is ∆SAS at this n. Balanced states at a new density (n + n´) can now be achieved by changing one of the gate biases to reach (n + n´/2) and the other gate to reach (n + n´) (n´ can be positive or negative). The ∆SAS data of Figure 13 were in fact obtained from Shubnikov-de Haas measurements on such “balanced” states, i.e., the gates were tuned to preserve symmetric

M. Shayegan: Electrons in a Flatland

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Fig. 13. The evolution of an electron system in a 750 ˚ A-wide single quantum well as a function of total electron density n. On the left the results of Hartree-Fock simulations for the self-consistent conduction band potentials (solid curves) and charge distributions (dotted curves) are shown. On the right the calculated ∆SAS (solid curve) is compared to the measured ∆SAS (closed circles); also shown is the calculated layer separation d. (After Manoharan et al. [28].)

charge distributions in the well as shown in Figure 15a. The remarkable agreement of the data with the self-consistent calculations in Figure 13, and the controlled variation of ∆SAS and d with n, attest to the excellent tunability of the electron system in a wide quantum well. Besides this tunability, the bilayer electron system in a wide GaAs well has another great advantage over its counterpart in a double-quantumwell. Here the eﬀective “barrier” separating the two electron layers is GaAs (Fig. 14) while in a double-quantum-well the barrier is AlGaAs or AlAs. The purity of GaAs grown in a molecular beam epitaxy chamber is typically higher than that of AlGaAs or AlAs (because of the high reactivity of Al and its sensitivity to impurities). Moreover, the AlGaAs or AlAs barrier in a double-quantum-well introduces additional interfaces which often adversely aﬀect the quality of the electron system, e.g., because of the additional interface roughness scattering.

28

Topological Aspects of Low Dimensional Systems

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M. Shayegan: Electrons in a Flatland

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6.3 Evolution of the QHE states in a wide quantum well Light is shed on the origin of the FQH states at ν = 12 and other ﬁllings in a wide quantum well by examining the competition between (1) ∆SAS , (2) the in-plane correlation energy Ce/εlB [where C is a constant ∼ 0.1 1/2 and lB ≡ (/eB) is the magnetic length], and (3) the interlayer Coulomb 2 interaction ∼ e /εd. To quantify behavior it is useful to construct the ra tios γ ≡ e2 /εlB /∆SAS and e2 /εlB / e2 /εd = d/lB . As n is increased,

30

Topological Aspects of Low Dimensional Systems

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Q

I48.>7@ Fig. 16. Determination of subband gap from low-ﬁeld Shubnikov-de Haas oscillations. Data are shown for a balanced charge distribution with n = 8.8 × 1010 cm2 . The oscillations, shown in (a), are periodic in 1/B. Their Fourier power spectrum, shown in (b), exhibits two frequencies which can be converted to subband densities by multiplying by the (spin-degenerate) Landau level degeneracy (2e/h). The subband separation (∆SAS ) can in turn be found by dividing the diﬀerence between these subband densities by the 2D density of states, m∗ /π 2 . (After Manoharan [28].)

γ increases since both ∆SAS and lB (for a FQH state at a given ν) decrease, and d/lB increases. When γ is small, the system should exhibit only “one-component” (1C) FQH states (standard single-layer odd-denominator states) constructed from only the symmetric subband, while for large γ

M. Shayegan: Electrons in a Flatland

31

the in-plane Coulomb energy becomes suﬃciently strong to allow the antisymmetric subband to mix into the correlated ground state to lower its energy and a “two-component” (2C) state ensues. These 2C states, constructed out of the now nearly degenerate symmetric and antisymmetric basis states, have a generalized Laughlin wavefunction of the form [53-56]: Ψνmmn ∼

m

(ui − uj )

i,j

× exp −

(wi − wj )

i,j

|ui |2 i

2 4lB

−

|wi |2 i

2 4lB

m

n

(ui − wj )

i,j

,

(6.1)

where in a more intuitive, pseudo-spin or two-layer language, ui and wi denote the pseudo-spin or the complex 2D coordinates of an electron in the two layers. The integer exponents m and n determine the intralayer and interlayer correlations, respectively, and the total ﬁlling factor for the Ψνmmn state is ν = 2/ (m + n). Now the 2C states described by Ψνmmn come in two classes. For large d/lB , the system behaves as two independent layers in parallel, each with half the total density. FQH states in this regime therefore have even nu2/3 merator and odd denominator. An example is the Ψ330 state which has a 2 1 total ﬁlling of 3 ( 3 ﬁlling in each layer). Note that the exponent n = 0 means that there is no interlayer correlation. For small enough d/lB , on the other hand, the interlayer interaction can become comparable to the in-plane (intralayer) interaction and fundamentally new FQH states become possible. Such states have strong interlayer correlation and can be 1/2 at even-denominator ν; a special example is the Ψ331 state with ν = 12 . 1 Another example is the Ψ111 state at ν = 1 which we will come back to in Section 6.5. Figure 17 captures some of the possible 1C and 2C QHE states. A careful study of the evolution of the FQH states in a wide quantum well as a function of n reveals that this evolution is compatible with the above picture [49]. Shown in Figure 18, for example, are three traces taken at low, high and intermediate n. The trace at low n exhibits, besides the usual integer QHE, the standard (odd-denominator) FQH states observed in high-quality single-layer 2D systems. The FQH states observed in the high n trace, on the other hand, while also having odd-denominators, have predominantly even-numerators (exceptions are the QHE states observed 2 4 at ν = 1 and at 11 15 , between 3 and 5 states, to which we will return in Sects. 6.5 and 6.6 respectively). The trace taken at intermediate n is most unusual as it appears to exhibit both single-layer FQH states (such as ν = 35 ) and the unique, even-denominator ν = 12 FQH state. Figure 19 provides a summary of the data taken at diﬀerent n on this sample. The quasiparticle excitation gaps of several FQH states, determined

32

Topological Aspects of Low Dimensional Systems

Fig. 17. Examples of one-component (1C) and two-component (2C) FQH states in a wide quantum well. Numbers refer to the total ﬁlling factor ν of the electron system in the well. The even-numerator FQH states in (a) can exist in both single-layer and bilayer systems; in the bilayer case, they are essentially two independent FQH states in two parallel layers without interlayer correlation. The odd-numerator FQH state in (b) exists only in a single-layer system, while the even-denominator 12 state is unique to bilayer systems and possesses interlayer correlation.

via thermal activation measurements, depend on γ as shown in this ﬁgure. As expected, increasing γ suppresses 1C states (such as 35 ) and enhances 2C states (such as 45 ). Two states, ν = 23 and ν = 43 , undergo a 1C to 2C phase transition as γ is increased. The critical point for this transition, γ 13.5, is consistent with the ratio of the in-plane correlation energy and ∆SAS ∼ 0.1 e2 /εlB /∆SAS = 0.1γ being of the order of unity, and matches the point where the energy gaps of other 1C and 2C states emerge from zero. Surrounding this point is a region where the ν = 12 FQH liquid stabilizes. Note that since this is a 2C state which also possesses interlayer correlation (the 2C ν = 23 and 43 states are simply 13 and 23 states in parallel layers), it exists only within a ﬁnite range of γ.

M. Shayegan: Electrons in a Flatland

/RZGHQVLW\

Q=ÓFP− D

+LJKGHQVLW\

Q=ÓFP−

E

7[\>KH @

7[[>N:V@

33

,QWHUPHGLDWHGHQVLW\

Q=ÓFP− F

%>7@ ˚ Fig. 18. Examples of QHE data in a 750 A-wide quantum well, with balanced charge distribution, at three densities. The low-density data in (a) are essentially the same as in a high-quality, monolayer 2D electron system. The high density trace in (b) resembles what is observed in two parallel and independent layers as it exhibits primarily FQH states at ﬁllings with even numerators and odd denominators (exceptions to these will be discussed in the text). At intermediate density (b), we have a bilayer system with strong interlayer interaction which leads to the stability of the new FQHE at ν = 12 . (After Manoharan [28]).

A particularly interesting experiment is to study the eﬀect of the charge distribution asymmetry on the FQH states observed in this system [28,49,57]. The results for the ν = 12 state are shown in Figure 20 where n is kept ﬁxed at 1.03 × 1011 cm−2 but the charge distribution is made asymmetric by applying VFG and VBG in opposite polarities (Fig. 15b); here nt is the electron density transferred from the back layer to the front layer. It can be seen that the ν = 12 FQH state quickly collapses as nt is increased by a small amount. Note that increasing nt means pushing the system to a single-layer

34

Topological Aspects of Low Dimensional Systems

Fig. 19. Measured energy gaps ∆ν of several FQH states vs. γ. The number of components (C) in each state is marked in parentheses. As γ increases, the 1C to 2C transition is observed at γ 13.5. The ν = 12 state is correlated both in the plane and between layers, and thus exists only within a ﬁnite range of γ. (After Suen [49] and Manoharan [28].)

(1C) situation; indeed, for nt = n/2 the system would become single-layer with all the charge residing in the front layer. (Note also that, as expected, the subband separation, ∆01 increases with nt .) These observations are consistent with the ν = 12 FQH state being a 2C state. For additional experimental results on the variation of FQHE energy gaps in this system with nt , see reference [49]. 6.4 Evolution of insulating phases Concurrent with the evolution of the FQH states in this wide quantum well, we observe an insulating phase (IP) which moves to higher ν as n is increased. The data are summarized in Figure 21 where ρxx at base T is plotted vs. ν −1 ∝ B for several representative n. Experimentally, the IP is identiﬁed by a resistivity that is both large (ρxx > h/e2 26 kΩ/2, the quantum unit of resistance) [58], and strongly increasing as T → 0 (see, e.g., Fig. 11). For very low n, the IP appears near ν = 15 , while at the 1 highest n there The IP observed in the intermediate is an10IP for ν 2 . 10 density range 10×10 < n < 14×10 cm−2 is most remarkable as it very quickly moves to larger ν with small increases in n (see, e.g., traces B, C,

M. Shayegan: Electrons in a Flatland

35

−

Q=Ó FP

D Q =

W

∆

6$6

=.

−

E Q =Ó FP

W

∆ =.

5[[>NΩ@

−

F Q =Ó FP

W

∆ =.

−

G Q =Ó FP

W

∆ =.

%>7@ Fig. 20. Collapse of the ν = 12 FQHE state with asymmetry. In this experiment, total density is kept ﬁxed at n = 1.03 × 1011 cm−2 while the charge distribution is made increasingly more asymmetric by applying front- and back- gate biases of opposite polarity [see Fig. 15b]. The ν = 12 FQHE is strongest when the charge distribution is symmetric but quickly weakens and eventually collapses as the layer charge imbalance nt is increased. (After Suen [49,57] and Manoharan [28].)

and D in Fig. 21); along the way, it also shows reentrant behavior around well-developed FQH states at ν = 27 (trace B), ν = 13 (traces C and D), and ν = 12 (trace E). Then, as n increases past this point, the IP begins to move

36

Topological Aspects of Low Dimensional Systems

)

ν

(

'

&

ρ[[>NΩV@

$× FP % & ' ( )

−

&

% $ %

)LOOLQJIDFWRUν

Fig. 21. Evolution of the insulating phase (IP) at T 25 mK. As n is increased, the IP moves quickly in to higher ν, becoming reentrant around several FQH states along the way, until it stabilizes around the ν = 12 bilayer state (bold trace E). As n is further increased from this point, the IP reverses direction and begins to move out toward lower ν. (After Manoharan et al. [51].)

in the opposite direction to lower ν (trace F). The data (trace E) shown in Figure 21 (see also Fig. 11) for n = 12.6 × 1010 cm−2 bear a striking resemblance to the IP observed reentrant around ν = 15 in low-disorder, single-layer 2DESs (Fig. 1), generally interpreted as a pinned Wigner solid [21]; here, however, the IP is reentrant around the bilayer ν = 12 FQH state, with the reentrant peak reaching the prominently high ﬁlling of ν = 0.54. The IPs presented in Figure 21 cannot be explained by single-particle localization. First, in the case of standard, single-layer 2DESs it is well known that as n is lowered, the quality of the 2DES deteriorates and the sample shows a disorder-induced IP at progressively larger ν [18]. This is opposite the behavior observed here: as n decreases from 10.9 × 1010 to 3.7 × 1010 cm−2 , the quality worsens as expected (e.g. mobility decreases monotonically from 1.4 × 106 to 5.3 × 105 cm2 /Vs) but the IP moves to smaller ν. Second, the observation of IPs which are reentrant around correlated FQH states, and particularly around the very fragile ν = 12 state [49], strongly suggests that electron interactions are also important in stabilizing the IP. We associate these IPs with pinned, bilayer Wigner crystal (WC) states which are stabilized at high ν thanks to the interlayer as well as intralayer electron correlation.

M. Shayegan: Electrons in a Flatland

37

To illustrate that the behavior of this IP is indeed consistent with the WC picture, it is instructive to ﬁrst examine the three main reentrant ρxx peaks in Figure 21 (from traces B, D, and E), which appear at ν = 0.30, 0.39, and 0.54 for the IPs surrounding the ν = 27 , 13 , and 12 FQH states, respectively. The values of γ at these peaks are respectively 16.9, 16.3, and 16.5. The peak positions span a large region of ν, and yet the associated γ are remarkably similar. Moreover, at a γ of 16.5, interlayer interactions are clearly important as this point is straddled by the 2C 12 state in Figure 19. The construction of a phase diagram [51] for the observed IPs facilitates a clear correlation between the IP evolution, the 1C to 2C transition, and the development of the ν = 12 liquid. To this end, we ﬁrst collected a ρxx data set for a fairly dense grid of points in the n−B plane by incrementally changing to a color interpolating n and sweeping B at base T . Next, ρxx was mapped between blue (ρxx = 0) and red ρxx ≥ h/e2 . Finally, using the B, n, and ∆SAS values at each point, the color-mapped ρxx data set was plotted vs. ν and γ (Fig. 22b; for the original color plot, see Ref. [51]). By utilizing h/e2 as a natural resistivity scale for demarcating the IP and noninsulating states [58], the result is a comprehensive phase diagram depicting incompressible phases (dark blue) together with compressible phases, both insulating (dark red) and metallic (all other colors). Immediately obvious in the phase diagram are the various FQH transitions, manifested by the appearance or disappearance of dark blue FQH phases at several ν (see, e.g., 35 , 45 , and the 12 “gulf”), or by a change in vertical width of the FQH phase (see, e.g., 23 ); these transitions correlate directly with the measured energy gaps (Fig. 22a). Another striking feature is the wrinkling in the IP boundary: this is caused by the aforementioned IP reentrance around several FQH states, perhaps most picturesque near ν 0.55 due to the formation of an IP “peninsula” above the 12 gulf. The limiting critical ν at low n (low γ) is close to 15 , consistent with a lowdisorder monolayer 2DES (Fig. 1). For the highest n when the electron system is eﬀectively two weakly coupled layers in parallel, one would expect (and measurements on wider quantum wells directly indicate) that the IP boundary moves to ν 25 , consistent with two high-quality independent layers becoming insulating near 15 ﬁlling in each layer. As our system is tuned through coupled layers, however, the IP boundary moves vividly above both of these limits to ν 0.55, and then only at higher n does it begin to fall back toward the 25 weak-coupling limit (lying outside the density limits of this sample). We can examine in more detail the evolution of the IP as depicted in the phase diagram of Figure 22b by making comparisons to Figure 22a. For intermediate n, as γ increases, the IP ﬁrst remains close to ν= 15 but then begins to move to higher ν in the range of 12 < γ < 15. This range

38

Topological Aspects of Low Dimensional Systems

Fig. 22. (a) Measured energy gaps ∆ν of several FQH states vs. γ. The number of components (C) in each state is shown in parentheses. (b) Phase diagram showing ρxx , chromatically mapped according to the color bar (right), vs. ν and γ. (After Manoharan et al. [51].)

is precisely bisected by γ 13.5 (Fig. 22a), where 1C to 2C transition occurs. Then the IP moves very quickly to ν 12 as evidenced by the nearly vertical phase boundary at γ 16. As discussed earlier, and as evident from Figure 22, this γ is centrally located in the parameter range in which the 1 2 state stabilizes. A quick glance at the phase diagram underscores this

M. Shayegan: Electrons in a Flatland

39

Fig. 23. phase (IP). Traces (a) and phase diagram (b) are at ﬁxed total n, with varying amounts of charge nt transferred between layers. [In (b), ρxx is mapped using the same color scale shown in Fig. 22.] Slight imbalance |nt | destabilizes the IP. (After Manoharan et al. [51].)

central point: the γ extent of the ν = 12 gulf coincides directly with the rapid ν shift in the phase boundary of the insulator. The most convincing evidence for the formation of a pinned bilayer crystal comes from perturbing the symmetric (“balanced”) charge distributions. Intuitively, the strength of a bilayer WC should be diminished under unbalanced conditions due to incommensurability eﬀects. This is indeed observed quite prominently in our system, and can be highlighted by examining the high-ν reentrant IPs. Figure 23 shows the eﬀect of asymmetry on the IP reentrant around ν = 13 at ﬁxed n = 11.0 × 1010 cm−2 and for varying nt , where nt is the electron density transferred from the back layer to the front by proper gate biasing from the balanced condition. We construct an “imbalance” phase diagram by plotting ρxx , color mapped to the same scale shown in Figure 22 vs. nt and B (Fig. 23b), and include in Figure 23a three representative ρxx traces (horizontal slices through the phase diagram of Fig. 23b). It is very clear that, while the 1C ν = 13 state is strengthened

40

Topological Aspects of Low Dimensional Systems

Fig. 24. Plots of the measured ν = 1 QHE energy gap (∆1 ) and T ∗ vs. γ. The boundary for the collapse of the ν = 1 QHE to a compressible state for this sample is shown by the vertical dashed line. As indicated by the dotted line, the measured T ∗ extrapolates to zero at this boundary. For comparison, the measured ∆SAS and the ν = 12 QHE gap (∆1/2 ) are also shown.

as expected, the IP is weakened by increasing imbalance |nt |: The IP is most stable in a perfectly balanced state (nt = 0) with a phase boundary at B 10 T, while the IP peak at ν = 0.38 (B = 12 T) drops dramatically even for small imbalance nt = 4.6×109 cm−2 (Fig. 23a). As |nt | isincreased past 7 × 109 cm−2 , the reentrant IP is destroyed ρxx < h/e2 and the 1 2 FQH state disappears; the phase boundary (which has now jumped to

M. Shayegan: Electrons in a Flatland

41

B 14 T) continues to be pushed back as nt increase further (Fig. 23b). For the IP reentrant around ν = 12 at n = 12.6×1010 cm−2 (see Fig. 11), the corresponding destabilization of the insulator (not shown here) occurs at an imbalance of less than 3% (|nt | /n 0.027). In all cases, note that both the ν = 12 FQH state and the reentrant IP are strongest in the balanced condition; asymmetry simultaneously destroys both the bilayer quantum Hall liquid and the insulator − for example, the vertical boundaries of the red reentrant IP “island” closely match those of the dark blue 12 liquid phase (Fig. 23b). Recently, several theoretical papers have examined Wigner crystallization in 2D systems with an additional spatial degree of freedom by considering multiple [59], wide [60], and double [61] quantum wells. While extracting details of the bilayer lattice (see, e.g. Ref. [61]) is beyond the scope of the present work, our observation of a 2C insulator at the large ﬁllings we identify sharply resonates with the fundamental principle underlying these theoretical investigations [59-61]: there is an additional potential energy gain due to the interlayer Coulomb interaction so that for equivalent layer densities a 2C WC can form at higher ν (e.g., ν 15 per layer) than a 1C WC. In addition, interlayer coupling may concomitantly weaken the FQH eﬀect, making a crossing of the liquid and solid ground-state energies even more favorable [23,60]. To summarize, our bilayer electron system provides a unique means of tuning the eﬀective electron-electron interactions underpinning the formation of various many-particle ground states. The crux of this reasoning is that this system possesses two vital “yardsticks” for gauging the relative importance of interlayer and intralayer interactions: the 1C to 2C transition and the novel bilayer ν = 12 condensate. Utilizing these measuring sticks, we can connect the fascinating evolution of the IP with the signiﬁcance and critical counterbalance of electron-electron interactions. In this light, the data conclusively indicate that the IP we observe for γ 13 is a collective 2C state with comparable interlayer and intralayer correlations. The characteristics of this bilayer electron insulator are remarkably consistent [59,60] with the formation of a novel pinned bilayer-correlated Wigner solid, a unique 2D electron crystal stabilized through the introduction of an additional quantum degree of freedom. 6.5 Many-body, bilayer QHE at ν = 1 In bilayer systems with appropriate parameters, the interlayer interactions can also lead to correlated QHE at integral ﬁllings [62]. A particularly interesting example is the 2C Ψ1111 state at ν = 1. In contrast to the 1C ν = 1 QHE associated with ∆SAS , the many-body, bilayer ν = 1 incompressible state associated with Ψ1111 has been predicted to exhibit exotic properties such as neutral superﬂuid modes and a Kosterlitz-Thouless

42

Topological Aspects of Low Dimensional Systems

transition [63,64], and has already revealed an unexpected in-plane-B-driven transition (to another incompressible state) [65]. Data on electron systems in wide wells [50] reveal that in these systems the ground state at ν = 1 evolves continuously from a QHE state stabilized by large ∆SAS at low n to a many-body QHE state stabilized by strong interlayer interaction at intermediate n. As n is further increased, we observe an incompressible-to-compressible transition. The unusual T and n dependence of the data as the transition boundary is approached is suggestive of an additional ﬁnite-temperature transition from a QHE to a compressible state, which is unique to bilayer systems. Here we brieﬂy summarize these data. Figure 12 presents Arrhenius plots of Rxx at ν = 1 for several n in our 750 ˚ A-wide GaAs quantum well. The quasiparticle QHE excitation gaps ∆1 determined from the slopes of the (low-T ) activated regions of these plots, together with the measured and calculated ∆SAS are shown in Figure 24a as a function of γ. For comparison, the gaps for the ν = 12 QH state in the same sample are also shown in Figure 24a. Several features of the data of Figures 12 and 24 are noteworthy: (1) while ∆SAS decreases with increasing γ, ∆1 increases and exceeds ∆SAS by more than a factor of 3 at the highest measured γ. (2) For n 10 × 1010 cm−2 (γ > 10), the activated behavior of Rxx vs. 1/T starts rather abruptly below an n-dependent temperature T ∗ . Above T ∗ , the Rxx minimum at ν = 1 vanishes, i.e., Rxx becomes nearly independent of B and T . For n 10 × 1010 cm−2 (γ 10), however, the Arrhenius plots show a smoother behavior and ∆1 gaps start to decrease with decreasing n. (3) The measured ∆1 for n > 10 × 1010 cm−2 (γ > 12) are approximately constant (≈20 K) and exceed T ∗ by more than an order of magnitude. (4) A plot of the measured T ∗ vs. γ presented in Figure 24b, shows that T ∗ decreases with increasing γ and extrapolates to zero at γ 29. This γ corresponds to an incompressible-to-compressible phase boundary for ν = 1 which we have observed in electron systems in a number of wide quantum wells with varying width [50]: in these systems, for suﬃciently large γ, we observe a collapse of the ν = 1 QHE to a compressible state. In the present sample, we cannot reach this boundary because the needed high n is not experimentally achievable but, based on our data on other samples, we expect the boundary to be at γ = 29 for this sample. The above data demonstrate that the ground state of the electron system in this wide quantum well at ν = 1 evolves continuously from a 1C QHE state at low n (γ 10) stabilized by a large ∆SAS to a 2C QHE state at intermediate n (10 γ 20), and then makes a transition to a metallic (compressible) state at large n (γ > 29). We believe that for intermediate n we are observing a bilayer QHE state stabilized by comparable interlayer and intralayer correlations, possibly a 2C, Ψ1111 -like state [50]. Note in Figure 24a that in this density range, γ for the ν = 1 QHE state overlaps

M. Shayegan: Electrons in a Flatland

43

Fig. 25. Arrhenius plots of Rxx at ν = 3 vs. 1/T for the sample of Figure 12. Note the very diﬀerent and much smoother behavior of this data compared to the ν = 1 data in Figure 12.

with γ of the ν = 12 QH state, a 2C liquid state which certainly requires interlayer correlations. Finally, the data of Figures 12 and 24 are collectively very unusual and qualitatively diﬀerent from what is observed for the integral or fractional QHE in standard, single-layer 2DESs, or for the QHE at higher ﬁllings such as ν = 3 in the same wide quantum well sample (Fig. 25). The single-layer ν = 1 QHE data typically exhibit a smoother saturation of the activated behavior at high T and, as n is lowered, they show a larger Rxx (at any given T ) and a smaller excitation gap [66]. This behavior is very similar to what we observe for the ν = 3 QHE in this sample (Fig. 25) and for the ν = 1 QHE at low n < 10 × 1010 cm−2 , far away from the compressible boundary (Fig. 12). It is insharp contrast to the ν = 1 data at high n near the compressible boundary n > 10 × 1010 cm−2 , where Rxx vs. T −1 data appear to simply shift horizontally to lower T as n is raised. While we do not have a clear understanding of this peculiar data, it is possible that a ﬁnite-temperature transition from a QHE to a compressible state is taking place, with T ∗ marking this transition. Although unprecedented in a single-layer 2DES, ﬁnite-T transitions may occur in bilayer systems with appropriate parameters: examples include the

44

Topological Aspects of Low Dimensional Systems

Kosterlitz-Thouless transition theoretically proposed for the Ψ1111 state ([63,64], also see Steve Girvin’s notes in this volume) or a transition from a correlated (Ψ1111 -like) incompressible state to an uncorrelated, compressible state with ν = 1/2 in each layer [67]. Details of such transitions and how they will quantitatively aﬀect the transport properties are not known, however. 6.6 Spontaneous interlayer charge transfer Another noteworthy feature of the data taken on this electron system is the 19 appearance of FQH states at unusual ﬁllings such as 11 15 and 15 which again have no counterparts in single-layer systems. The strongest of these occurs 2 4 at ν = 11 15 , between ν = 3 and 5 , and can be seen in Figure 22b as a dark blue strip for γ 15. We observe such states near the ν = 23 and 43 FQH states as these become of 2C origin. Here we ﬁrst present additional data on these states and then argue that they signal a spontaneous interlayer charge transfer at high magnetic ﬁelds [27]. Data of Figure 26, taken as the angle θ between the magnetic ﬁeld and the normal to the 2D plane is varied, provide additional examples (see Fig. 15c for experimental geometry). Note that with increasing θ, for a given ﬁlling factor, the in-plane component of the magnetic ﬁeld B increases and drives the system from 1C to 2C. This is because the in-plane magnetic ﬁeld suppresses the tunneling and reduces ∆SAS [68]. Data of Figure 26 reﬂect this expectation. The top trace of Figure 26a, taken at θ = 0, shows essentially 1C FQH states while the lower trace which was taken at large θ exhibits predominantly 2C features: even-numerator FQH states at ν = 23 , 45 , 43 , 65 , etc. The lower trace, however, reveals the presence of several other Rxx minima at unusual ﬁllings between these even-numerator ﬁllings, 19 29 e.g., at ν = 11 15 , 15 , and 35 . Figure 26b summarizes the evolution of the FQH states in this electron system as a function of increasing θ. We have condensed a large set of traces onto the (B⊥ , θ) plane by mapping Rxx (normalized to its maximum value within the plotted parameter range) to a grayscale color between black and white. In such a plot, the IQHE and FQHE phases show up as dark black regions, whose width along the B⊥ axis is a reﬂection of the strength of the associated state, i.e., the magnitude of its energy gap. The traces in Figure 26a can be interpreted as constant-θ slices through the image of Figure 26b. As θ is increased, the system is swept from the 1C through the 2C regime; a visible measure of this general evolution is the weakening and eventual collapse of the ν = 1 QH state. The ν = 35 FQH state, another 1C state, is also destroyed by the increasing B . For the states that can exist as both 1C and 2C phases, transitions between the two ground states are evident. For example, the ν = 23 and 45 states undergo a 1C to 2C transition at θ 18◦ and 27◦ , respectively. Nestled between these two states and in

M. Shayegan: Electrons in a Flatland

45

Fig. 26. Tuning the bilayer electron system (at ﬁxed n = 11.2 × 1010 cm−2 ) from the 1C to the 2C regime by increasing θ (hence B ) (a) Rxx vs. B⊥ “slices” through the image of (b), where the normalized Rxx is mapped to a gray scale and also plotted vs. θ (total ﬁllings ν are labeled). Spontaneous interlayer charge transfer engenders new FQH liquids (see features marked by vertical arrows and bold fractions). (After Manoharan et al. [27].)

close proximity to their 1C↔2C transitions, an 11 15 FQH state develops and becomes quite strong. At the same time, ρ exhibits a quantized plateau xy 2 h/e (see lower right of Fig. 26a). Very similar behavior is observed at 15 11

46

Topological Aspects of Low Dimensional Systems

Fig. 27. Competition between capacitive charging energy εCAP (insets) and liquid correlation energy εCOR (arrows) governing the susceptibility toward spontaneous interlayer charge transfer. Displayed fractions are layer ﬁllings ν . (After Manoharan et al. [27].)

on the other side of ν = 1 in Figure 26b. A 19 15 state develops in the vicinity of the ν = 43 1C↔2C transition (at θ 35◦ ) along with the appearance of the 2C 65 state (at θ 38◦ ). What is the origin of these “special” states? We argue that at these ﬁllings, interlayer charge transfer takes place so that two strong FQH states (at diﬀerent layer ﬁllings) stabilize in separate layers. The idea is best 11 illustrated in Figure 27 (insets): the system at ν = 11 15 (layer ﬁlling 30 ) 1 deforms itself so that one layer locks in the strong 3 FQH state and the other 1 2 the 25 state (note that 11 15 = 3 + 5 ). What drives this “phase-separation” into two compressible FQH liquids is the presence of downward “cusps” in the ground state energy of the system at the magic FQHE fractions (Fig. 27). An estimate of the energy savings from such transfer is made in Figure 27 where the calculated ground states are plotted as a function of layer ﬁlling ν . For details of how this calculation was made, see references [27] and [28]. The estimated correlation energy gained by forming the incompressible 13 and 25 FQH liquid states in the two separate layers, 1.3 K, is indeed quite comparable to the electrostatic (capacitive) cost, Q2 /2nC 0.9 K, 1energy for transferring the appropriate fraction 30 of electrons from one layer to the other (Q is the transferred charge and C is the interlayer capacitance) [27]. The interlayer charge transfer interpretation therefore seems plausible. The data presented so far show the presence of FQH states such as ν = 11 15 when the electron system is made 2C by either going to high density (Fig. 15a) or at large θ (Fig. 15c). In both these cases the electron system

M. Shayegan: Electrons in a Flatland

47

Fig. 28. Intentionally imposed interlayer charge transfer nt will stabilize (b) an incompressible layered 13 + 25 FQH state at ν = 11 when the charge distribution 15 is imbalanced close to the expected (5 : 6) layer density ratio [dotted line in (d)]. (After Manoharan et al. [27].)

has a symmetric (balanced) charge distribution at zero magnetic ﬁeld, and we are conjecturing that at high B⊥ the interlayer charge transfer takes place. To verify this conjecture, we did the following experimental test at θ = 0. We start with the electron system at an n where the ν = 23 FQH

48

Topological Aspects of Low Dimensional Systems

state has just become 2C so that the incompressible state at 11 15 has barely developed (e.g., n = 12.6 × 1010 cm−2 ; see Fig. 28a). Now suppose we keep n ﬁxed but intentionally impose an interlayer charge transfer nt by applying a perpendicular electric ﬁeld (physically generated via front- and back-gate biases of opposite sign, as schematically shown in Fig. 15b). As we transfer 11 should get stronger as 2nt /n approaches the ratio charge, the 215 FQH state 2 1 1 1 − + , and then should become weaker once 2nt /n exceeds / = 5 3 5 3 11 1 . The data shown in Figures 28a-c demonstrate that this behavior is in11 deed observed in our experiment. In particular, the quasiparticle excitation when the 2nt /n exceeds gap ∆11/15 measured for the 11 15 FQH state is largest 1 2 1 , i.e. layer densities imbalanced in the ratio : 11 3 5 = (5 : 6) (Fig. 28d). Two additional features of the data in Figure 26 are noteworthy. First, ◦ the ν = 11 15 state appears to become weaker with increasing θ 40 . This is reasonable and stems from the fact that spontaneous charge transfer will only occur if the correlation energy savings overcome the capacitive energy cost. At very large B (or n), the two layers become increasingly more isolated and the capacitive energy opposing charge transfer begins to dominate any correlation energy savings that would come from a 13 + 25 state. Thus, the system remains compressible, as expected for two distant 11 and weakly-coupled parallel 2DESs at ν = 11 15 ( 30 ﬁlling in each layer). 29 2 3 Second, the Rxx minimum near ν = 35 = 5 + 7 suggests a developing FQH state at this ﬁlling (Fig. 26). Such a state can be stabilized if, at ν = 29 35 , there is an interlayer charge transfer so that one layer supports a FQH state at 25 ﬁlling and the other at 37 . Similarly, the weak Rxx minimum observed near ν = 13 21 (Fig. 26a) may hint at a developing FQH state stabilized by the formation of 13 and 27 FQH states in the separate layers. 6.7 Summary Magnetotransport data taken on an electron system in a wide quantum well with variable density reveal a striking evolution of its correlated states. While the data at low and high densities are consistent with single-layer and weakly-coupled bilayer states respectively, at intermediate densities the data exhibit new QHE and insulating phases which are stabilized by both intralayer and interlayer electron correlations. Much of this presentation (Sect. 6) is based on the work of my former students H.C. Manoharan, Y.W. Suen, M.B. Santos, and T.S. Lay at Princeton University. I thank them and my other colleagues, especially V. Bayot, for their hard work and for many illuminating discussions. I am indebted to H.C. Manoharan for providing me with most of the ﬁgures, including some unpublished ones from his Ph.D. thesis. I also thank Ms. Connie Brown for her patience and care, and for her excellent typing of the mauscript. The work at Princeton University has been supported primarily by the National Science Foundation.

M. Shayegan: Electrons in a Flatland

49

References [1] [2] [3] [4] [5]

[6] [7] [8] [9]

[10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23]

von Klitzing K. , Dorda G. and Pepper M., Phys. Rev. Lett. 45 (1980) 494. Laughlin R.B., Phys. Rev. B 23 (1981) 5632. Tsui D.C., Stormer H.L. and Gossard A.C., Phys. Rev. Lett. 48 (1982) 1559. Laughlin R.B., Phys. Rev. Lett. 50 (1983) 1395. Many of the reports on the physics of 2D systems can be found in the proceedings of the international conferences on the Electronic Properties of Two-dimensional Systems (EP2DS), which are held every two years. For recent proceedings, see: Surf. Sci. 305 (1994); Surf. Sci. 361/362 (1996); Physica B 249-251 (1998). Prange R.E. and Girvin S.M., The Quantum Hall Eﬀect, 2nd ed. (Springer-Verlag, New York, 1990). Chakraborty T. and Pietilainen P., The Quantum Hall Eﬀects: Fractional and Integral (Springer-Verlag, Berlin 1995). Das S. Sarma and Pinczuk A., Perspectives in Quantum Hall Eﬀects (Wiley, New York, 1997). MacDonald’s A.H., Les Houches 1994 Notes, edited by E. Akkermans, G. Montambaux, J.-L. Pichard and J. Zinn-Justin, Mesoscopic Quantum Physics (Elsevier, Amsterdam, 1995) p. 659. Stormer H.L., Dingle R., Gossard A.C., Wiegmann W. and Sturge M.D., Solid State Commun. 29 (1979) 705. For a simple and useful text on semiconductor heterostructures, see J.H. Davies, The Physics of Low-Dimensional Semiconductors (Cambridge University Press Cambridge, 1998). Other useful books and review articles are: Bastard G., Wave Mechanics Applied to Semiconductor Heterostructures (Halsted Press, New York, 1988); Weisbuch C. and Vinter B., Quantum Semiconductor Structures, Academic Press, New York, 1991; M.J. Kelly, Low-Dimensional Semiconductors, Materials, Physics, Technology, Devices (Clarendon Press, Oxford, 1995). For a comprehensive review of the 2D physics in the pre-QHE era see Ando T., Fowler A.B. and Stern F., Rev. Mod. Phys. 54 (1982) 437. Cho A.Y., Mater. Res. Soc. (MRS) Bull. 20 (1995) 21. Etienne B. and Paris E., J. Phys. France 48 (1987) 2049. Shayegan M., Goldman V.J., Santos M., Sajoto T., Engel L. and Tsui D.C., Appl. Phys. Lett. 53 (1988) 2080. Shayegan M., Goldman V.G., Jiang C., Sajoto T. and Santos M., Appl. Phys. Lett. 52 (1988) 1086. Pfeiﬀer L.N., West K.W., Stormer H.L. and Baldwin K.W., Appl. Phys. Lett. 55 (1989) 1888. Sajoto T., Suen Y.W., Engel L.W., Santos M.B. and Shayegan M., Phys. Rev. B 41 (1990) 8449. Stern F., Appl. Phys. Lett. 43 (1983) 974; Gold A., Phys. Rev. B 44 (1991) 8818. Grimes C.C. and Adams G., Phys. Rev. Lett. 42 (1979) 795; for a review, see reference [12]. For a review of recent developments in the magnetic-ﬁeld-induced Wigner crystal states of 2D systems see articles by M. Shayegan (experiments) and H.A. Fertig (theory) in reference [8]. Willett R.L., Stormer H.L., Tsui D.C., Gossard A.C. and English J.H., Phys. Rev. B 37 (1988) 8476. Shayegan M., Jo J., Suen Y.W., Santos M. and Goldman V.J., Phys. Rev. Lett. 65 (1990) 2916.

50

Topological Aspects of Low Dimensional Systems

[24] He S., Zhang F.C., Xie X.C. and Das Sarma S., Phys Rev. B 42 (1990) 11376. [25] de-Picciotto R., Reznikov M., Heiblum M., Umansky V., Bunin G. and Mahalu D., Nature 389 (1997) 162; Saminadayar L., Glattli D.C., Jin Y. and Etienne B., Phys. Rev. Lett. 79 (1997) 2526. [26] For a review of composite Fermions and the FQHE, see theory articles by Jain J.K. and by Halperin B.I. in reference [8]. Also included in reference [8] is a comprehensive review, by Stormer H.L. and Tsui D.C., of the experimental results supporting the CF picture. [27] Manoharan H.C., Suen Y.W., Lay T.S., Santos M.B. and Shayegan M., Phys. Rev. Lett. 79 (1997) 2722. [28] Manoharan H.C., Ph.D. Thesis (Princeton University, 1998). [29] Goldman V.J., Su B. and Jain J.K., Phys. Rev. Lett. 72 (1994) 2065. [30] Shayegan M., Solid State Commun. 102 (1997) 155. [31] Nicholas R.J., Haug R.J., K.v. Klitzing and Weimann G., Phys. Rev. B 37 (1988) 1294. [32] Sondhi S.L., Karlhede A., Kivelson S.A. and Rezayi E.H., Phys. Rev. B 47 (1993) 16419. [33] Skyrme T.H.R., Proc. Soc R.. London, Ser. A 247 (1958) 260. [34] Moon K., Mori H., Kun Yang, Girvin S.M., MacDonald A.H., Zheng L., Yoshioka D. and Shou-Cheng Zhang, Phys. Rev. B 51 (1995) 5138. [35] Barrett S.E., Dabbagh G., Pfeiﬀer L.N., West K. and Tycko R., Phys. Rev. Lett. 74 (1995) 5112. [36] Brey L., Fertig H.A., Cote R. and MacDonald A.H., Phys. Rev. Lett. 75 (1995) 2562. [37] Schmeller A., Eisenstein J.P., Pfeiﬀer L.N. and West K.W., Phys. Rev. Lett. 75 (1995) 4290. [38] Aifer E.H., Goldberg B.B. and Broido D.A., Phys. Rev. Lett. 76 (1996) 680. [39] Tycko R., Barrett S.E., Dabbagh G. and Pfeiﬀer L.N., Science 268 (1995) 1460. [40] Bayot V., Grivei E., Santos M.B. and Shayegan M., Phys. Rev. Lett. 76 (1996) 4584. [41] See, e.g. Wang J.K., Tsui D.C., Santos M.B. and Shayegan M., Phys. Rev. B 45 (1992) 4384. [42] Bayot V., Grivei E., Beuken J.-M., Melinte S. and Shayegan M., Phys. Rev. Lett. 79 (1997) 1718. [43] Cˆ ot´ e R., MacDonald A.H., Brey L., Fertig H.A., Girvin S.M. and Stoof H.T.C., Phys. Rev. Lett. 78 (1997) 4825. [44] Willett R., Eisenstein J.P., Stormer H.L., Tsui D.C., Gossard A.C. and English J.H., Phys. Rev. Lett. 59 (1987) 1776; Haldane F.D.M. and Rezayi E.H., Phys. Rev. Lett. 60 (1988) 956. [45] Clark R.G., Haynes S.R., Suckling A.M., Mallett J.R., Wright P.A., Harris J.J. and Foxon C.T., Phys. Rev. Lett. 62 (1989) 1536; Eisenstein J.P., Stormer H.L., Pfeiﬀer L. and West K.W., ibid (1989) 1540. [46] Suen Y.W., Jo J., Santos M., Engel L.W., Hwang S.W. and Shayegan M., Phys. Rev. B 44 (1991) 5947. [47] Suen Y.W., Engel L.W., Santos M.B., Shayegan M. and Tsui D.C., Phys. Rev. Lett. 68 (1992) 1379. [48] Suen Y.W., Santos M.B. and Shayegan M., ibid 69 (1992) 3551. [49] Suen Y.W., Manoharan H.C., Ying X., Santos M.B. and Shayegan M., Phys. Rev. Lett. 72 (1994) 3405. [50] Lay T.S., Suen Y.W., Manoharan H.C., Ying X., Santos M.B. and Shayegan M., Phys. Rev. B 50 (1994) 17725.

M. Shayegan: Electrons in a Flatland

51

[51] Manoharan H.C., Suen Y.W., Santos M.B. and Shayegan M., Phys. Rev. Lett. 77 (1996) 1813. [52] The ν = 1/2 FQH state is also observed in bilayer electron systems in double quantum wells. [Eisenstein J.P., Boebinger G.S., Pfeiﬀer L.N., West K.W. and He S., Phys. Rev. Lett. 68 (1992) 1383]. [53] Halperin B.I., Helv. Phys. Acta. 56 (1983) 75. [54] Yoshioka D., MacDonald A.H. and Girvin S.M., Phys. Rev. B 39 (1989) 1932. [55] He S., Das Sarma S. and Xie X.C., Phys. Rev. B 47 (1993) 4394. [56] MacDonald A.H., Surf. Sci. 229 (1990) 1. [57] Suen Y.W., Ph.D. Thesis (Princeton University, 1994). [58] Shahar D., Tsui D.C., Shayegan M., Bhatt R.N. and Cunningham J.E., Phys. Rev. Lett. 74 (1995) 4511. [59] Oji H.C., MacDonald A.H. and Girvin S.M., Phys. Rev. Lett. 58 (1987) 824; L. ´ Swierkowski, D. Neilson and J. Szyma´ nski, Phys. Rev. Lett. 67 (1991) 240. [60] Price R., Zhu X., DasSarma S. and Platzman P.M., Phys. Rev. B 51 (1995) 2017. [61] Zheng L. and Fertig H.A., Phys. Rev. B 52 (1995) 12282; Navasimhan S. and Ho T.-L., Phys. Rev. B 52 (1995) 12291. [62] Chakraborty T. and Pietilainen P., Phys. Rev. Lett. B 59 (1987) 2784; Fertig H.A., Phys. Rev. B 40 (1989) 1087. [63] Wen X.G. and Zee A., Phys. Rev. Lett. 69 (1992) 1811; Ezawa F. and Iwazaki A., Int. J. Mod. Phys. B 6 (1992) 3205. [64] Moon K., Mori H., Yang K., Girvin S.M., MacDonald A.H., Zheng A.H., Yoshioka D. and Zhang S.C., Phys. Rev. B 51 (1995) 5138. [65] Murphy S.Q., Eisenstein J.P., Boebinger G.S., Pfeiﬀer L.N. and West K.W., Phys. Rev. Lett. 72 (1994) 728. [66] See, e.g., Usher A., Nicholas R.J., Harris J.J. and Foxon C.T., Phys. Rev. B 41 (1990) 1129. [67] Wen X.G. (private communication). [68] Hu J. and MacDonald A.H., Phys. Rev. B 46 (1992) 12554.

COURSE 2

THE QUANTUM HALL EFFECT: NOVEL EXCITATIONS AND BROKEN SYMMETRIES

S.M. GIRVIN Indiana University, Department of Physics, Bloomington, IN 47405, U.S.A.

Contents 1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26

quantum Hall eﬀect Introduction . . . . . . . . . . . . . . . . . . Why 2D is important . . . . . . . . . . . . . Constructing the 2DEG . . . . . . . . . . . Why is disorder and localization important? Classical dynamics . . . . . . . . . . . . . . Semi-classical approximation . . . . . . . . Quantum Dynamics in Strong B Fields . . . IQHE edge states . . . . . . . . . . . . . . . Semiclassical percolation picture . . . . . . Fractional QHE . . . . . . . . . . . . . . . . The ν = 1 many-body state . . . . . . . . . Neutral collective excitations . . . . . . . . Charged excitations . . . . . . . . . . . . . FQHE edge states . . . . . . . . . . . . . . Quantum hall ferromagnets . . . . . . . . . Coulomb exchange . . . . . . . . . . . . . . Spin wave excitations . . . . . . . . . . . . Eﬀective action . . . . . . . . . . . . . . . . Topological excitations . . . . . . . . . . . . Skyrmion dynamics . . . . . . . . . . . . . . Skyrme lattices . . . . . . . . . . . . . . . . Double-layer quantum hall ferromagnets . . Pseudospin analogy . . . . . . . . . . . . . . Experimental background . . . . . . . . . . Interlayer phase coherence . . . . . . . . . . Interlayer tunneling and tilted ﬁeld eﬀects .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 57 57 58 61 64 65 72 76 80 85 94 104 113 116 118 119 124 129 141 147 152 154 156 160 162

Appendix

165

A Lowest Landau level projection

165

B Berry’s phase and adiabatic transport

168

THE QUANTUM HALL EFFECT: NOVEL EXCITATIONS AND BROKEN SYMMETRIES

S.M. Girvin

1

The quantum Hall eﬀect

1.1 Introduction The Quantum Hall Eﬀect (QHE) is one of the most remarkable condensedmatter phenomena discovered in the second half of the 20th century. It rivals superconductivity in its fundamental signiﬁcance as a manifestation of quantum mechanics on macroscopic scales. The basic experimental observation is the nearly vanishing dissipation σxx → 0

(1.1)

and the quantization of the Hall conductance σxy = ν

e2 h

(1.2)

of a real (as opposed to some theorist’s fantasy) transistor-like device (similar in some cases to the transistors in computer chips) containing a two-dimensional electron gas subjected to a strong magnetic ﬁeld. This quantization is universal and independent of all microscopic details such as the type of semiconductor material, the purity of the sample, the precise value of the magneticﬁeld, and so forth. As a result, the eﬀect is now used

These lectures are dedicated to the memory of Heinz Schulz, a great friend and a wonderful physicist. c EDP Sciences, Springer-Verlag 1999

56

Topological Aspects of Low Dimensional Systems

to maintain1 the standard of electrical resistance by metrology laboratories around the world. In addition, since the speed of light is now deﬁned, a measurement of e2 /h is equivalent to a measurement of the ﬁne structure constant of fundamental importance in quantum electrodynamics. In the so-called Integer Quantum Hall Eﬀect (IQHE) discovered by von Klitzing in 1980, the quantum number ν is a simple integer with a precision of about 10−10 and an absolute accuracy of about 10−8 (both being limited by our ability to do resistance metrology). In 1982, Tsui et al. discovered that in certain devices with reduced (but still non-zero) disorder, the quantum number ν could take on rational fractional values. This so-called Fractional Quantum Hall Eﬀect (FQHE) is the result of quite diﬀerent underlying physics involving strong Coulomb interactions and correlations among the electrons. The particles condense into special quantum states whose excitations have the bizarre property of being described by fractional quantum numbers, including fractional charge and fractional statistics that are intermediate between ordinary Bose and Fermi statistics. The FQHE has proven to be a rich and surprising arena for the testing of our understanding of strongly correlated quantum systems. With a simple twist of a dial on her apparatus, the quantum Hall experimentalist can cause the electrons to condense into a bewildering array of new “vacua”, each of which is described by a diﬀerent quantum ﬁeld theory. The novel order parameters describing each of these phases are completely unprecedented. We begin with a brief description of why two-dimensionality is important to the universality of the result and how modern semiconductor processing techniques can be used to generate a nearly ideal two-dimensional electron gas (2DEG). We then give a review of the classical and semi-classical theories of the motion of charged particles in a magnetic ﬁeld. Next we consider the limit of low temperatures and strong ﬁelds where a full quantum treatment of the dynamics is required. After that we will be in a position to understand the localization phase transition in the IQHE. We will then study the origins of the FQHE and the physics described by the novel wave function invented by Robert Laughlin to describe the special condensed state of the electrons. Finally we will discuss topological excitations and broken symmetries in quantum Hall ferromagnets. 1 Maintain does not mean define. The SI ohm is deﬁned in terms of the kilogram, the second and the speed of light (formerly the meter). It is best realized using the reactive impedance of a capacitor whose capacitance is computed from ﬁrst principles. This is an extremely tedious procedure and the QHE is a very convenient method for realizing a ﬁxed, reproducible impedance to check for drifts of resistance standards. It does not however define the ohm. Equation (1.2) is given in cgs units. When converted to SI Z units the quantum of resistance is h/e2 (cgs) → 2α ≈ 25, 812.80 Ω (SI) where α is the

ﬁne structure constant and Z ≡

µ0 /0 is the impedance of free space.

S.M. Girvin: The Quantum Hall Eﬀect

57

The review presented here is by no means complete. It is primarily an introduction to the basics followed by a more advanced discussion of recent developments in quantum Hall ferromagnetism. Among the many topics which receive little or no discussion are the FQHE hierarchical states, interlayer drag eﬀects, FQHE edge state tunneling and the composite boson [1] and fermion [2] pictures of the FQHE. A number of general reviews exist which the reader may be interested in consulting [3–11] 1.2 Why 2D is important As one learns in the study of scaling in the localization transition, resistivity (which is what theorists calculate) and resistance (which is what experimentalists measure) for classical systems (in the shape of a hypercube) of size L are related by [12, 13] (1.3) R = ρL(2−d) . Two dimensions is therefore special since in this case the resistance of the sample is scale invariant and (e2 /h)R is dimensionless. This turns out to be crucial to the universality of the result. In particular it means that one does not have to measure the physical dimensions of the sample to one part in 1010 in order to obtain the resistivity to that precision. Since the locations of the edges of the sample are not well-deﬁned enough to even contemplate such a measurement, this is a very fortunate feature of having available a 2DEG. It further turns out that, since the dissipation is nearly zero in the QHE states, even the shape of the sample and the precise location of the Hall voltage probes are almost completely irrelevant. 1.3 Constructing the 2DEG There are a variety of techniques to construct two-dimensional electron gases. Figure 1.1 shows one example in which the energy bands in a GaAs/AlAs heterostructure are used to create a “quantum well”. Electrons from a Si donor layer fall into the quantum well to create the 2DEG. The energy level (“electric subband”) spacing for the “particle in a box” states of the well can be of order 103 K which is much larger than the cryogenic temperatures at which QHE experiments are performed. Hence all the electrons are frozen into the lowest electric subband (if this is consistent with the Pauli principle) but remain free to move in the plane of the GaAs layer forming the well. The dynamics of the electrons is therefore eﬀectively two-dimensional even though the quantum well is not literally two-dimensional. Heterostructures that are grown one atomic layer at a time by Molecular Beam Epitaxy (MBE) are nearly perfectly ordered on the atomic scale. In addition the Si donor layer can be set back a considerable distance

58

Topological Aspects of Low Dimensional Systems

conduction band Si donor

AlAs

GaAs

AlAs

valence band Fig. 1.1. Schematic illustration of a GaAs/AlAs heterostructure quantum well. The vertical axis is band energy and the horizontal axis is position in the MBE growth direction. The dark circles indicate the Si+ ions which have donated electrons into the quantum well. The lowest electric subband wave function of the quantum well is illustrated by the dashed line. It is common to use an alloy of GaAs and AlAs rather than pure AlAs for the barrier region as illustrated here.

(∼ 0.5 µm) to minimize the random scattering from the ionized Si donors. Using these techniques, electron mobilities of 107 cm2 /Vs can be achieved at low temperatures corresponding to incredibly long mean free paths of ∼ 0.1 mm. As a result of the extremely low disorder in these systems, subtle electronic correlation energies come to the fore and yield a remarkable variety of quantum ground states, some of which we shall explore here. The same MBE and remote doping technology is used to make GaAs quantum well High Electron Mobility Transistors (HEMTs) which are used in all cellular telephones and in radio telescope receivers where they are prized for their low noise and ability to amplify extremely weak signals. The same technology is widely utilized to produce the quantum well lasers used in compact disk players. 1.4 Why is disorder and localization important? Paradoxically, the extreme universality of the transport properties in the quantum Hall regime occurs because of, rather than in spite of, the random disorder and uncontrolled imperfections which the devices contain. Anderson localization in the presence of disorder plays an essential role in the quantization, but this localization is strongly modiﬁed by the strong magnetic ﬁeld. In two dimensions (for zero magnetic ﬁeld and non-interacting electrons) all states are localized even for arbitrarily weak disorder. The essence of this weak localization eﬀect is the current “echo” associated with the quantum interference corrections to classical transport [14]. These quantum

S.M. Girvin: The Quantum Hall Eﬀect

59

interference eﬀects rely crucially on the existence of time-reversal symmetry. In the presence of a strong quantizing magnetic ﬁeld, time-reversal symmetry is destroyed and the localization properties of the disordered 2D electron gas are radically altered. We will shortly see that there exists a novel phase transition, not between a metal and insulator, but rather between two distinctly diﬀerent insulating states. In the absence of any impurities the 2DEG is translationally invariant and there is no preferred frame of reference2 . As a result we can transform to a frame of reference moving with velocity −v relative to the lab frame. In this frame the electrons appear to be moving at velocity +v and carrying current density J = −nev, (1.4) where n is the areal density and we use the convention that the electron charge is −e. In the lab frame, the electromagnetic ﬁelds are E B

= 0

(1.5)

=

(1.6)

B zˆ.

In the moving frame they are (to lowest order in v/c) E B

1 = − v × B c = B zˆ.

(1.7) (1.8)

This Lorentz transformation picture is precisely equivalent to the usual statement that an electric ﬁeld must exist which just cancels the Lorentz in order for the device to carry the current straight through force −e v×B c without deﬂection. Thus we have = B J × B. ˆ E nec

(1.9)

The resistivity tensor is deﬁned by E µ = ρµν J ν . Hence we can make the identiﬁcation B 0 +1 ρ= . −1 0 nec

(1.10)

(1.11)

2 This assumes that we can ignore the periodic potential of the crystal which is of course ﬁxed in the lab frame. Within the eﬀective mass approximation this potential modiﬁes the mass but does not destroy the Galilean invariance since the energy is still quadratic in the momentum.

60

Topological Aspects of Low Dimensional Systems

The conductivity tensor is the matrix inverse of this so that J µ = σµν E ν , and nec σ= B

0 −1 +1 0

(1.12) .

(1.13)

Notice that, paradoxically, the system looks insulating since σxx = 0 and yet it looks like a perfect conductor since ρxx = 0. In an ordinary insulator σxy = 0 and so ρxx = ∞. Here σxy = nec B = 0 and so the inverse exists. The argument given above relies only on Lorentz covariance. The only property of the 2DEG that entered was the density. The argument works equally well whether the system is classical or quantum, whether the electron state is liquid, vapor, or solid. It simply does not matter. Thus, in the absence of disorder, the Hall eﬀect teaches us nothing about the system other than its density. The Hall resistivity is simply a linear function of magnetic ﬁeld whose slope tells us about the density. In the quantum Hall regime we would therefore see none of the novel physics in the absence of disorder since disorder is needed to destroy translation invariance. Once the translation invariance is destroyed there is a preferred frame of reference and the Lorentz covariance argument given above fails. Figure 1.2 shows the remarkable transport data for a real device in the quantum Hall regime. Instead of a Hall resistivity which is simply a linear function of magnetic ﬁeld, we see a series of so-called Hall plateaus in which ρxy is a universal constant 1 h (1.14) ρxy = − 2 νe independent of all microscopic details (including the precise value of the magnetic ﬁeld). Associated with each of these plateaus is a dramatic decrease in the dissipative resistivity ρxx −→ 0 which drops as much as 13 orders of magnitude in the plateau regions. Clearly the system is undergoing some sort of sequence of phase transitions into highly idealized dissipationless states. Just as in a superconductor, the dissipationless state supports persistent currents. These can be produced in devices having the Corbino ring geometry shown in Figure 1.3. Applying additional ﬂux through the ring produces a temporary azimuthal electric ﬁeld by Faraday induction. A current pulse is induced at right angles to the E ﬁeld and produces a radial charge polarization as shown. This polarization induces a (quasi-) permanent radial electric ﬁeld which in turn causes persistent azimuthal currents. Torque magnetometer measurements [16] have shown that the currents can persist ∼ 103 s at very low temperatures. After this time the tiny σxx gradually allows the radial charge polarization to dissipate. We can think of the azimuthal currents as gradually spiraling outwards due to the Hall

S.M. Girvin: The Quantum Hall Eﬀect

61

Fig. 1.2. Integer and fractional quantum Hall transport data showing the plateau regions in the Hall resistance RH and associated dips in the dissipative resistance R. The numbers indicate the Landau level ﬁlling factors at which various features occur. After reference [15].

angle (between current and electric ﬁeld) being very slightly less than 90◦ (by ∼ 10−13 ). We have shown that the random impurity potential (and by implication Anderson localization) is a necessary condition for Hall plateaus to occur, but we have not yet understood precisely how this novel behavior comes about. That is our next task.

1.5 Classical dynamics The classical equations of motion for an electron of charge −e moving in caused by two dimensions under the inﬂuence of the Lorentz force −e v×B c = B zˆ are a magnetic ﬁeld B eB y˙ c eB m¨ y = + x. ˙ c

m¨ x

= −

(1.15) (1.16)

62

Topological Aspects of Low Dimensional Systems

-

-

-

++

-

+ + + +

-

++

Φ (t) + ++

+ + + +

-

-

-

+

-

-

Fig. 1.3. Persistent current circulating in a quantum Hall device having the Corbino geometry. The radial electric ﬁeld is maintained by the charges which can not ﬂow back together because σxx is nearly zero. These charges result from the radial current pulse associated with the azimuthal electric ﬁeld pulse produced by the applied ﬂux Φ(t).

The general solution of these equations corresponds to motion in a circle of arbitrary radius R r = R (cos(ωc t + δ), sin(ωc t + δ)) .

(1.17)

Here δ is an arbitrary phase for the motion and ωc ≡

eB mc

(1.18)

is known as the classical cyclotron frequency. Notice that the period of the orbit is independent of the radius and that the tangential speed v = Rωc

(1.19)

controls the radius. A fast particle travels in a large circle but returns to the starting point in the same length of time as a slow particle which (necessarily) travels in a small circle. The motion is thus isochronous much like that of a harmonic oscillator whose period is independent of the amplitude of the motion. This apparent analogy is not an accident as we shall see when we study the Hamiltonian (which we will need for the full quantum solution). Because of some subtleties involving distinctions between canonical and mechanical momentum in the presence of a magnetic ﬁeld, it is worth reviewing the formal Lagrangian and Hamiltonian approaches to this problem.

S.M. Girvin: The Quantum Hall Eﬀect

63

The above classical equations of motion follow from the Lagrangian L=

e 1 mx˙ µ x˙ µ − x˙ µ Aµ , 2 c

(1.20)

is the vector potential where µ = 1, 2 refers to x and y respectively and A evaluated at the position of the particle. (We use the Einstein summation convention throughout this discussion.) Using e δL = − x˙ µ ∂ν Aµ δxν c

(1.21)

and

δL e = mx˙ ν − Aν δ x˙ ν c the Euler-Lagrange equation of motion becomes e m¨ xν = − [∂ν Aµ − ∂µ Aν ] x˙ µ . c

(1.22)

(1.23)

Using B Bα

= =

×A ∇ αβγ ∂β Aγ

(1.24) (1.25)

shows that this is equivalent to equations (1.15–1.16). Once we have the Lagrangian we can deduce the canonical momentum pµ

≡

δL δ x˙ µ

e = mx˙ µ − Aµ , c

(1.26)

and the Hamiltonian H[ p, x] ≡ =

x˙ µ pµ − L(x˙ , x) 1 µ e µ µ e µ p + A . p + A 2m c c

(1.27)

(Recall that the Lagrangian is canonically a function of the positions and velocities while the Hamiltonian is canonically a function of the positions and momenta.) The quantity e pµmech ≡ pµ + Aµ c

(1.28)

is known as the mechanical momentum. Hamilton’s equations of motion x˙ µ

=

p˙ µ

=

∂H 1 = pµmech µ ∂p m ∂H e ν e ν p + A ∂µ Aν − µ =− ∂x mc c

(1.29) (1.30)

64

Topological Aspects of Low Dimensional Systems

show that it is the mechanical momentum, not the canonical momentum, which is equal to the usual expression related to the velocity pµmech = mx˙ µ .

(1.31)

Using Hamilton’s equations of motion we can recover Newton’s law for the Lorentz force given in equation (1.23) by simply taking a time derivative of x˙ µ in equation (1.29) and then using equation (1.30). The distinction between canonical and mechanical momentum can lead to confusion. For example it is possible for the particle to have a ﬁnite velocity while having zero (canonical) momentum! Furthermore the canonical momentum is dependent (as we will see later) on the choice of gauge for the vector potential and hence is not a physical observable. The mechanical momentum, being simply related to the velocity (and hence the current) is physically observable and gauge invariant. The classical equations of motion only involve the curl of the vector potential and so the particular gauge choice is not very important at the classical level. We will therefore delay discussion of gauge choices until we study the full quantum solution, where the issue is unavoidable. 1.6 Semi-classical approximation Recall that in the semi-classical approximation used in transport theory (r, t) made up of a linear superposition we consider wave packets ΨR(t), K(t) of Bloch waves. These packets are large on the scale of the de Broglie wavelength so that they have a well-deﬁned central wave vector K(t), but they are small on the scale of everything else (external potentials, etc.) so that they simultaneously can be considered to have well-deﬁned mean and R are parameters labeling the wave packet position R(t). (Note that K not arguments.) We then argue (and will discuss further below) that the solution of the Schr¨ odinger equation in this semiclassical limit gives a wave packet whose parameters K(t) and R(t) obey the appropriate analog of the classical Hamilton equations of motion R˙ µ

=

∂ΨR, K |H|ΨR, K

hK˙ µ ¯

=

−

∂¯ hK µ ∂ΨR, K |H|ΨR, K ∂Rµ

(1.32) ·

(1.33)

Naturally this leads to the same circular motion of the wave packet at the classical cyclotron frequency discussed above. For weak ﬁelds and fast electrons the radius of these circular orbits will be large compared to the size of the wave packets and the semi-classical approximation will be valid. However at strong ﬁelds, the approximation begins to break down because the orbits are too small and because h ¯ ωc becomes a signiﬁcant (large) energy.

S.M. Girvin: The Quantum Hall Eﬀect

65

Thus we anticipate that the semi-classical regime requires ¯hωc F , where F is the Fermi energy. We have already seen hints that the problem we are studying is really a harmonic oscillator problem. For the harmonic oscillator there is a characteristic energy scale h ¯ ω (in this case ¯hωc ) and a characteristic length scale for the zero-point ﬂuctuations of the position in the ground state. The analog quantity in this problem is the so-called magnetic length 257 ˚ A hc ¯ ≡ = · (1.34) eB B 1tesla

The physical interpretation of this length is that the area 2π 2 contains one quantum of magnetic ﬂux Φ0 where3 hc · e That is to say, the density of magnetic ﬂux is Φ0 =

(1.35)

Φ0 · (1.36) 2π 2 To be in the semiclassical limit then requires that the Fermi wavelength be small on the scale of the magnetic length so that kF 1. This condition turns out to be equivalent to h ¯ ωc F so they are not separate constraints. B=

Exercise 1.1. Use the Bohr-Sommerfeld quantization condition that the orbit have a circumference containing an integral number of de Broglie wavelengths to ﬁnd the allowed orbits of a 2D electron moving in a uniform magnetic ﬁeld. Show that each successive orbit encloses precisely one additional quantum of ﬂux in its interior. Hint: It is important to make the distinction between the canonical momentum (which controls the de Broglie wavelength) and the mechanical momentum (which controls the = velocity). The calculation is simpliﬁed if one uses the symmetric gauge A 1 − 2 r × B in which the vector potential is purely azimuthal and independent of the azimuthal angle. 1.7 Quantum Dynamics in Strong B Fields Since we will be dealing with the Hamiltonian and the Schr¨ odinger equation, our ﬁrst order of business is to choose a gauge for the vector potential. One convenient choice is the so-called Landau gauge: r ) = xB yˆ A(

(1.37)

3 Note that in the study of superconductors the ﬂux quantum is deﬁned with a factor of 2e rather than e to account for the pairing of the electrons in the condensate.

66

Topological Aspects of Low Dimensional Systems y

x

= xB yˆ. The magFig. 1.4. Illustration of the Landau gauge vector potential A netic ﬁeld is perfectly uniform, but the vector potential has a preferred origin and orientation corresponding to the particular gauge choice.

×A = B zˆ. In this gauge the vector potential points in the which obeys ∇ y direction but varies only with the x position, as illustrated in Figure 1.4. Hence the system still has translation invariance in the y direction. Notice that the magnetic ﬁeld (and hence all the physics) is translationally invariant, but the Hamiltonian is not! (See exercise 1.2). This is one of many peculiarities of dealing with vector potentials. Exercise 1.2. Show for the Landau gauge that even though the Hamiltonian is not invariant for translations in the x direction, the physics is still invariant since the change in the Hamiltonian that occurs under translation is simply equivalent to a gauge change. Prove this for any arbitrary gauge, assuming only that the magnetic ﬁeld is uniform. The Hamiltonian can be written in the Landau gauge as 2 1 eB 2 H= px + py + x . 2m c

(1.38)

Taking advantage of the translation symmetry in the y direction, let us attempt a separation of variables by writing the wave function in the form ψk (x, y) = eiky fk (x).

(1.39)

This has the advantage that it is an eigenstate of py and hence we can make hk in the Hamiltonian. After separating variables the replacement py −→ ¯

S.M. Girvin: The Quantum Hall Eﬀect

67

we have the eﬀective one-dimensional Schr¨odinger equation hk fk (x) = k fk (x), where 1 2 1 p + hk ≡ 2m x 2m

2 eB x . hk + ¯ c

(1.40)

(1.41)

This is simply a one-dimensional displaced harmonic oscillator4 hk =

2 1 2 1 px + mωc2 x + k 2 2m 2

(1.42)

whose frequency is the classical cyclotron frequency and whose central position Xk = −k 2 is (somewhat paradoxically) determined by the y momentum quantum number. Thus for each plane wave chosen for the y direction there will be an entire family of energy eigenvalues 1 (1.43) kn = n + hω c ¯ 2 which depend only on n are completely independent of the y momentum ¯ k. The corresponding (unnormalized) eigenfunctions are h 2 2 1 1 ψnk (r ) = √ eiky Hn (x + k 2 )e− 22 (x+k ) , L

(1.44)

where Hn is (as usual for harmonic oscillators) the nth Hermite polynomial (in this case displaced to the new central position Xk ). Exercise 1.3. Verify that equation (1.44) is in fact a solution of the Schr¨ odinger equation as claimed. These harmonic oscillator levels are called Landau levels. Due to the lack of dependence of the energy on k, the degeneracy of each level is enormous, as we will now show. We assume periodic boundary conditions in the y direction. Because of the vector potential, it is impossible to simultaneously have periodic boundary conditions in the x direction. However since the basis wave functions are harmonic oscillator polynomials multiplied by strongly converging Gaussians, they rapidly vanish for positions away from the center position X0 = −k 2 . Let us suppose that the sample is rectangular with dimensions Lx , Ly and that the left hand edge is at x = −Lx and the right hand edge is at x = 0. Then the values of the wavevector k 4 Thus we have arrived at the harmonic oscillator hinted at semiclassically, but paradoxically it is only one-dimensional, not two. The other degree of freedom appears (in this gauge) in the y momentum.

68

Topological Aspects of Low Dimensional Systems

for which the basis state is substantially inside the sample run from k = 0 to k = Lx / 2 . It is clear that the states at the left edge and the right edge diﬀer strongly in their k values and hence periodic boundary conditions are impossible5 . The total number of states in each Landau level is then N=

Ly 2π

Lx /2

dk = 0

Lx Ly = NΦ 2π 2

(1.45)

where NΦ ≡

BLx Ly Φ0

(1.46)

is the number of ﬂux quanta penetrating the sample. Thus there is one state per Landau level per ﬂux quantum which is consistent with the semiclassical result from Exercise (1.1). Notice that even though the family of allowed wavevectors is only one-dimensional, we ﬁnd that the degeneracy of each Landau level is extensive in the two-dimensional area. The reason for this is that the spacing between wave vectors allowed by the periodic 2π decreases while the range of allowed wave boundary conditions ∆k = L y 2 vectors [0, Lx / ] increases with increasing L. The reader may also worry that for very large samples, the range of allowed values of k will be so large that it will fall outside the ﬁrst Brillouin zone forcing us to include band mixing and the periodic lattice potential beyond the eﬀective mass approximation. This is not true however, since the canonical momentum is a gauge dependent quantity. The value of k in any particular region of the sample can be made small by shifting the origin of the coordinate system to that region (thereby making a gauge transformation). The width of√the harmonic oscillator wave functions in the nth Landau level is of order n . This is microscopic compared to the system size, but note that the spacing between the centers ∆ = ∆k 2 =

2π 2 Ly

(1.47)

is vastly smaller (assuming Ly ). Thus the supports of the diﬀerent basis states are strongly overlapping (but they are still orthogonal).

5 The best one can achieve is so-called quasi-periodic boundary conditions in which the phase diﬀerence between the left and right edges is zero at the bottom and rises linearly with height, reaching 2πNΦ ≡ Lx Ly /2 at the top. The eigenfunctions with these boundary conditions are elliptic theta functions which are linear combinations of the gaussians discussed here. See the discussion by Haldane in reference [3].

S.M. Girvin: The Quantum Hall Eﬀect

69

Exercise 1.4. Using the fact that the energy for the nth harmonic oscilhωc , present a semi-classical argument explaining the lator state is (n + 12 )¯ result claimed above that the width of the support of the wave function √ scales as n . Exercise 1.5. Using the Landau gauge, construct a gaussian wave packet in the lowest Landau level of the form +∞ 2 2 1 ak eiky e− 22 (x+k ) , Ψ(x, y) = −∞

choosing ak in such a way that the wave packet is localized as closely as What is the smallest size wave packet that possible around some point R. can be constructed without mixing in higher Landau levels?

Having now found the eigenfunctions for an electron in a strong magnetic ﬁeld we can relate them back to the semi-classical picture of wave packets undergoing circular cyclotron motion. Consider an initial semiclassical wave packet located at some position and having some speciﬁed momentum. In the semiclassical limit the mean energy of this packet will greatly exceed 2 K2

¯ hωc and hence it will be made up of a linear the cyclotron energy h¯ 2m combination of a large number of diﬀerent Landau level states centered h ¯2K2 around n ¯ = 2m¯ hωc Ψ(r, t) =

n

Ly

1 dk an (k)ψnk (r )e−i(n+ 2 )ωc t . 2π

(1.48)

Notice that in an ordinary 2D problem at zero ﬁeld, the complete set of plane wave states would be labeled by a 2D continuous momentum label. Here we have one discrete label (the Landau level index) and a 1D continuous labels (the y wave vector). Thus the “sum” over the complete set of states is actually a combination of a summation and an integration. The details of the initial position and momentum are controlled by the amplitudes an (k). We can immediately see however, that since the energy levels are exactly evenly spaced that the motion is exactly periodic: 2π = Ψ(r, t). Ψ r, t + ωc

(1.49)

If one works through the details, one ﬁnds that the motion is indeed circular and corresponds to the expected semi-classical cyclotron orbit. For simplicity we will restrict the remainder of our discussion to the lowest Landau level where the (correctly normalized) eigenfunctions in the

70

Topological Aspects of Low Dimensional Systems

Landau gauge are (dropping the index n = 0 from now on): 2 2 1 1 eiky e− 22 (x+k ) ψk (r ) = √ 1/2 π L

(1.50)

and every state has the same energy eigenvalue k = 12 ¯hωc . We imagine that the magnetic ﬁeld (and hence the Landau level splitting) is very large so that we can ignore higher Landau levels. (There are some subtleties here to which we will return.) Because the states are all degenerate, any wave packet made up of any combination of the basis states will be a stationary state. The total current will therefore be zero. We anticipate however from semiclassical considerations that there should be some remnant of the classical circular motion visible in the local current density. To see this note that the expectation value of the current in the kth basis state is e 1 (1.51) Ψk p + A J = −e Ψk . m c The y component of the current is 2 2 1 e eB − 212 (x+k2 )2 Jy = − x e− 22 (x+k ) h ¯ k + dx e 1/2 c mπ 2 2

1 eωc (1.52) = − 1/2 dx e− 2 (x+k ) x + k 2 . π We see from the integrand that the current density is antisymmetric about the peak of the gaussian and hence the total current vanishes. This antisymmetry (positive vertical current on the left, negative vertical current on the right) is the remnant of the semiclassical circular motion. Let us now consider the case of a uniform electric ﬁeld pointing in the x direction and giving rise to the potential energy V (r ) = +eEx.

(1.53)

This still has translation symmetry in the y direction and so our Landau gauge choice is still the most convenient. Again separating variables we see that the solution is nearly the same as before, except that the displacement of the harmonic oscillator is slightly diﬀerent. The Hamiltonian in equation (1.54) becomes hk =

2 1 2 1 px + mωc2 x + k 2 + eEx. 2m 2

(1.54)

Completing the square we see that the oscillator is now centered at the new position eE Xk = −k 2 − (1.55) mωc2

S.M. Girvin: The Quantum Hall Eﬀect ε

71

ε n=2

n=2

n=1

n=1

n=0

n=0 x

x

(a)

(b)

Fig. 1.5. Illustration of electron Landau energy levels n + 12 ¯ hωc vs. position xk = −k2 . (a) Zero electric ﬁeld case. (b) Case with ﬁnite electric ﬁeld pointing in the +ˆ x direction.

and the energy eigenvalue is now linearly dependent on the particle’s peak position Xk (and therefore linear in the y momentum) k =

1 1 hωc + eEXk + m¯ ¯ v2 , 2 2

(1.56)

where

E · (1.57) B Because of the shift in the peak position of the wavefunction, the perfect antisymmetry of the current distribution is destroyed and there is a net current Jy = −e¯ v (1.58) v¯ ≡ −c

× B/B 2 drift velocity. This result showing that v¯yˆ is simply the usual cE can be derived either by explicitly doing the integral for the current or by noting that the wave packet group velocity is eE ∂Xk 1 ∂k = = v¯ h ∂k ¯ ¯ ∂k h

(1.59)

independent of the value of k (since the electric ﬁeld is a constant in this case, giving rise to a strictly linear potential). Thus we have recovered the correct kinematics from our quantum solution. It should be noted that the applied electric ﬁeld “tilts” the Landau levels in the sense that their energy is now linear in position as illustrated in Figure 1.5. This means that there are degeneracies between diﬀerent Landau level states because diﬀerent kinetic energy can compensate different potential energy in the electric ﬁeld. Nevertheless, we have found

72

Topological Aspects of Low Dimensional Systems

the exact eigenstates (i.e., the stationary states). It is not possible for an electron to decay into one of the other degenerate states because they have diﬀerent canonical momenta. If however disorder or phonons are available to break translation symmetry, then these decays become allowed and dissipation can appear. The matrix elements for such processes are small if the electric ﬁeld is weak because the degenerate states are widely separated spatially due to the small tilt of the Landau levels. Exercise 1.6. It is interesting to note that the exact eigenstates in the presence of the electric ﬁeld can be viewed as displaced oscillator states in the original (zero E ﬁeld) basis. In this basis the displaced states are linear combinations of all the Landau level excited states of the same k. Use ﬁrstorder perturbation theory to ﬁnd the amount by which the n = 1 Landau level is mixed into the n = 0 state. Compare this with the exact amount of mixing computed using the exact displaced oscillator state. Show that the two results agree to ﬁrst order in E. Because the displaced state is a linear combination of more than one Landau level, it can carry a ﬁnite current. Give an argument, based on perturbation theory why the amount of this current is inversely proportional to the B ﬁeld, but is independent of the mass of the particle. Hint: how does the mass aﬀect the Landau level energy spacing and the current operator? 1.8 IQHE edge states Now that we understand drift in a uniform electric ﬁeld, we can consider the problem of electrons conﬁned in a Hall bar of ﬁnite width by a nonuniform electric ﬁeld. For simplicity, we will consider the situation where the potential V (x) is smooth on the scale of the magnetic length, but this is not central to the discussion. If we assume that the system still has translation symmetry in the y direction, the solution to the Schr¨ odinger equation must still be of the form 1 ψ(x, y) = eiky fk (x). Ly

(1.60)

The function fk will no longer be a simple harmonic wave function as we found in the case of the uniform electric ﬁeld. However we can anticipate that fk will still be peaked near (but in general not precisely at) the point Xk ≡ −k 2 . The eigenvalues k will no longer be precisely linear in k but will still reﬂect the kinetic energy of the cyclotron motion plus the local potential energy V (Xk ) (plus small corrections analogous to the one in Eq. (1.56)). This is illustrated in Figure 1.6. We see that the group velocity vk =

1 ∂k yˆ ¯ ∂k h

(1.61)

S.M. Girvin: The Quantum Hall Eﬀect

x

73

x

k

Fig. 1.6. Illustration of a smooth conﬁning potential which varies only in the x direction. The horizontal dashed line indicates the equilibrium fermi level. The dashed curve indicates the wave packet envelope fk which is displaced from its nominal position xk ≡ −k2 by the slope of the potential.

y x

Fig. 1.7. Semi-classical view of skipping orbits at the fermi level at the two edges ×B drift. The circular of the sample where the conﬁning electric ﬁeld causes E orbit illustrated in the center of the sample carries no net drift current if the local electric ﬁeld is zero.

has the opposite sign on the two edges of the sample. This means that in the ground state there are edge currents of opposite sign ﬂowing in the sample. The semi-classical interpretation of these currents is that they represent “skipping orbits” in which the circular cyclotron motion is interrupted by collisions with the walls at the edges as illustrated in Figure 1.7. One way to analyze the Hall eﬀect in this system is quite analogous to the Landauer picture of transport in narrow wires [17,18]. The edge states play the role of the left and right moving states at the two fermi points. Because (as we saw earlier) momentum in a magnetic ﬁeld corresponds to position,

74

Topological Aspects of Low Dimensional Systems

the edge states are essentially real space realizations of the fermi surface. A Hall voltage drop across the sample in the x direction corresponds to a diﬀerence in electrochemical potential between the two edges. Borrowing from the Landauer formulation of transport, we will choose to apply this in the form of a chemical potential diﬀerence and ignore any changes in electrostatic potential6 . What this does is increase the number of electrons in skipping orbits on one edge of the sample and/or decrease the number on the other edge. Previously the net current due to the two edges was zero, but now there is a net Hall current. To calculate this current we have to add up the group velocities of all the occupied states +∞ Ly 1 ∂k e nk , dk (1.62) I =− Ly −∞ 2π ¯ h ∂k where for the moment we assume that in the bulk, only a single Landau level is occupied and nk is the probability that state k in that Landau level is occupied. Assuming zero temperature and noting that the integrand is a perfect derivative, we have e µL e I=− d = − [µL − µR ] . (1.63) h µR h (To understand the order of limits of integration, recall that as k increases, Xk decreases.) The deﬁnition of the Hall voltage drop is7 (+e)VH ≡ (+e) [VR − VL ] = [µR − µL ] .

(1.64)

Hence

e2 VH , (1.65) h where we have now allowed for the possibility that ν diﬀerent Landau levels are occupied in the bulk and hence there are ν separate edge channels contributing to the current. This is the analog of having ν “open” channels in the Landauer transport picture. In the Landauer picture for an ordinary wire, we are considering the longitudinal voltage drop (and computing σxx ), while here we have the Hall voltage drop (and are computing σxy ). I = −ν

6 This has led to various confusions in the literature. If there is an electrostatic potential gradient then some of the net Hall current may be carried in the bulk rather than at the edges, but the ﬁnal answer is the same. In any case, the essential part of the physics is that the only place where there are low lying excitations is at the edges. 7 To get the signs straight here, note that an increase in chemical potential brings in more electrons. This is equivalent to a more positive voltage and hence a more negative potential energy −eV. Since H − µN enters the thermodynamics, electrostatic potential energy and chemical potential move the electron density oppositely. V and µ thus have the same sign of eﬀect because electrons are negatively charged.

S.M. Girvin: The Quantum Hall Eﬀect

75

The analogy is quite precise however because we view the right and left movers as having distributions controlled by separate chemical potentials. It just happens in the QHE case that the right and left movers are physically separated in such a way that the voltage drop is transverse to the current. Using the above result and the fact that the current ﬂows at right angles to the voltage drop we have the desired results σxx σxy

=

0

(1.66)

=

e2 −ν , h

(1.67)

with the quantum number ν being an integer. So far we have been ignoring the possible eﬀects of disorder. Recall that for a single-channel one-dimensional wire in the Landauer picture, a disordered region in the middle of the wire will reduce the conductivity to I=

e2 2 |T | , h

(1.68)

where |T |2 is the probability for an electron to be transmitted through the disordered region. The reduction in transmitted current is due to back scattering. Remarkably, in the QHE case, the back scattering is essentially zero in very wide samples. To see this note that in the case of the Hall bar, scattering into a backward moving state would require transfer of the electron from one edge of the sample to the other since the edge states are spatially separated. For samples which are very wide compared to the magnetic length (more precisely, to the Anderson localization length) the matrix element for this is exponentially small. In short, there can be nothing but forward scattering. An incoming wave given by equation (1.60) can only be transmitted in the forward direction, at most suﬀering a simple phase shift δk 1 ψout (x, y) = eiδk eiky fk (x). (1.69) Ly This is because no other states of the same energy are available. If the disorder causes Landau level mixing at the edges to occur (because the conﬁning potential is relatively steep) then it is possible for an electron in one edge channel to scatter into another, but the current is still going in the same direction so that there is no reduction in overall transmission probability. It is this chiral (unidirectional) nature of the edge states which is responsible for the fact that the Hall conductance is correctly quantized independent of the disorder. Disorder will broaden the Landau levels in the bulk and provide a reservoir of (localized) states which will allow the chemical potential to vary smoothly with density. These localized states will not contribute to the

76

Topological Aspects of Low Dimensional Systems

transport and so the Hall conductance will be quantized over a plateau of ﬁnite width in B (or density) as seen in the data. Thus obtaining the universal value of quantized Hall conductance to a precision of 10−10 does not require ﬁne tuning the applied B ﬁeld to a similar precision. The localization of states in the bulk by disorder is an essential part of the physics of the quantum Hall eﬀect as we saw when we studied the role of translation invariance. We learned previously that in zero magnetic ﬁeld all states are (weakly) localized in two dimensions. In the presence of a quantizing magnetic ﬁeld, most states are strongly localized as discussed above. However if all states were localized then it would be impossible to have a quantum phase transition from one QHE plateau to the next. To understand how this works it is convenient to work in a semiclassical percolation picture to be described below. Exercise 1.7. Show that the number of edge channels whose energies lie in the gap between two Landau levels scales with the length L of the sample, while the number of bulk states scales with the area. Use these facts to show that the range of magnetic ﬁeld in which the chemical potential lies in between two Landau levels scales to zero in the thermodynamic limit. Hence ﬁnite width quantized Hall plateaus can not occur in the absence of disorder that produces a reservoir of localized states in the bulk whose number is proportional to the area.

1.9 Semiclassical percolation picture Let us consider a smooth random potential caused, say, by ionized silicon donors remotely located away from the 2DEG in the GaAs semiconductor host. We take the magnetic ﬁeld to be very large so that the magnetic length is small on the scale over which the potential varies. In addition, we ignore the Coulomb interactions among the electrons. What is the nature of the eigenfunctions in this random potential? We have learned how to solve the problem exactly for the case of a constant electric ﬁeld and know the general form of the solution when there is translation invariance in one direction. We found that the wave functions were plane waves running along lines of constant potential energy and having a width perpendicular to this which is very small and on the order of the magnetic length. The reason for this is the discreteness of the kinetic energy in a strong magnetic ﬁeld. It is impossible for an electron stuck in a given Landau level to continuously vary its kinetic energy. Hence energy conservation restricts its motion to regions of constant potential energy. In the limit of inﬁnite magnetic ﬁeld where Landau level mixing is completely negligible, this conﬁnement to lines of constant potential becomes exact (as the magnetic length goes to zero).

S.M. Girvin: The Quantum Hall Eﬀect

77

We are led to the following somewhat paradoxical picture. The strong magnetic ﬁeld should be viewed as putting the system in the quantum limit in the sense that h ¯ ωc is a very large energy (comparable to F ). At the same time (if one assumes the potential is smooth) one can argue that since the magnetic length is small compared to the scale over which the random potential varies, the system is in a semi-classical limit where small wave ×B drift trajectories. packets (on the scale of ) follow classical E From this discussion it then seems very reasonable that in the presence of a smooth random potential, with no particular translation symmetry, the eigenfunctions will live on contour lines of constant energy on the random energy surface. Thus low energy states will be found lying along contours in deep valleys in the potential landscape while high energy states will be found encircling “mountain tops” in the landscape. Naturally these extreme states will be strongly localized about these extrema in the potential. Exercise 1.8. Using the Lagrangian for a charged particle in a magnetic ﬁeld with a scalar potential V (r ), consider the high ﬁeld limit by setting the mass to zero (thereby sending the quantum cyclotron energy to inﬁnity). 1. Derive the classical equations of motion from the Lagrangian and ×B drift along isopotential contours. show that they yield simple E 2. Find the momentum conjugate to the coordinate x and show that (with an appropriate gauge choice) it is the coordinate y: px = −

h ¯ y 2

(1.70)

so that we have the strange commutation relation [x, y] = −i 2 .

(1.71)

In the inﬁnite ﬁeld limit where → 0 the coordinates commute and we recover the semi-classical result in which eﬀectively point particles drift along isopotentials. To understand the nature of states at intermediate energies, it is useful to imagine gradually ﬁlling a random landscape with water as illustrated in Figure 1.8. In this analogy, sea level represents the chemical potential for the electrons. When only a small amount of water has been added, the water will ﬁll the deepest valleys and form small lakes. As the sea level is increased the lakes will grow larger and their shorelines will begin to take on more complex shapes. At a certain critical value of sea level a phase transition will occur in which the shoreline percolates from one side of the system to the other. As the sea level is raised still further, the ocean will

78

Topological Aspects of Low Dimensional Systems

Fig. 1.8. Contour map of a smooth random landscape. Closed dashed lines indicate local mountain peaks. Closed solid lines indicate valleys. From top to bottom, the gray ﬁlled areas indicate the increasing “sea level” whose shoreline ﬁnally percolates from one edge of the sample to the other (bottom panel). The particle-hole excitations live along the shoreline and become gapless when the shoreline becomes inﬁnite in extent.

cover the majority of the land and only a few mountain tops will stick out above the water. The shore line will no longer percolate but only surround the mountain tops. As the sea level is raised still higher additional percolation transitions will occur successively as each successive Landau level passes under water. If Landau level mixing is small and the disorder potential is symmetrically distributed about zero, then the critical value of the chemical potential for the nth percolation transition will occur near the center of the nth Landau level 1 ∗ (1.72) hω c . ¯ µn = n + 2 This percolation transition corresponds to the transition between quantized Hall plateaus. To see why, note that when the sea level is below the percolation point, most of the sample is dry land. The electron gas is therefore insulating. When sea level is above the percolation point, most of the sample is covered with water. The electron gas is therefore connected throughout the majority of the sample and a quantized Hall current can be carried. Another way to see this is to note that when the sea level is above the percolation point, the conﬁning potential will make a shoreline along the full length of each edge of the sample. The edge states will then carry current from one end of the sample to the other.

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Fig. 1.9. Illustration of edge states that wander deep into the bulk as the quantum Hall localization transition is approached from the conducting side. Solid arrows indicate the direction of drift along the isopotential lines. Dashed arrows indicate quantum tunneling from one semi-classical orbit (edge state) to the other. This backscattering localizes the eigenstates and prevents transmission through the sample using the “edge” states (which become part of the bulk localized states).

We can also understand from this picture why the dissipative conductivity σxx has a sharp peak just as the plateau transition occurs. (Recall the data in Fig. 1.2). Away from the critical point the circumference of any particular patch of shoreline is ﬁnite. The period of the semiclassical orbit around this is ﬁnite and hence so is the quantum level spacing. Thus there are small energy gaps for excitation of states across these real-space fermi levels. Adding an inﬁnitesimal electric ﬁeld will only weakly perturb these states due to the gap and the ﬁniteness of the perturbing matrix element which will be limited to values on the order of ∼ eED where D is the diameter of the orbit. If however the shoreline percolates from one end of the sample to the other then the orbital period diverges and the gap vanishes. An inﬁnitesimal electric ﬁeld can then cause dissipation of energy. Another way to see this is that as the percolation level is approached from above, the edge states on the two sides will begin taking detours deeper and deeper into the bulk and begin communicating with each other as the localization length diverges and the shoreline zig zags throughout the bulk of the sample. Thus electrons in one edge state can be back scattered into the other edge states and ultimately reﬂected from the sample as illustrated in Figure 1.9.

80

Topological Aspects of Low Dimensional Systems

Because the random potential broadens out the Landau level density of states, the quantized Hall plateaus will have ﬁnite width. As the chemical potential is varied in the regime of localized states in between the Landau level peaks, only the occupancy of localized states is changing. Hence the transport properties remain constant until the next percolation transition occurs. It is important to have the disorder present to produce this ﬁnite density of states and to localize those states. It is known that as the (classical) percolation point is approached in two dimensions, the characteristic size (diameter) of the shoreline orbits diverges like (1.73) ξ ∼ |δ|−4/3 , where δ measures the deviation of the sea level from its critical value. The shoreline structure is not smooth and in fact its circumference diverges with a larger exponent 7/3 showing that these are highly ramiﬁed fractal objects whose circumference scales as the 7/4th power of the diameter. So far we have assumed that the magnetic length is essentially zero. That is, we have ignored the fact that the wave function support extends a small distance transverse to the isopotential lines. If two diﬀerent orbits with the same energy pass near each other but are classically disconnected, the particle can still tunnel between them if the magnetic length is ﬁnite. This quantum tunneling causes the localization length to diverge faster than the classical percolation model predicts. Numerical simulations ﬁnd that the localization length diverges like [19–22] ξ ∼ |δ|−ν

(1.74)

where the exponent ν (not to be confused with the Landau level ﬁlling factor!) has a value close (but probably not exactly equal to) 7/3 rather than the 4/3 found in classical percolation. It is believed that this exponent is universal and independent of Landau level index. Experiments on the quantum critical behavior are quite diﬃcult but there is evidence [23], at least in selected samples which show good scaling, that ν is indeed close to 7/3 (although there is some recent controversy on this point [24]) and that the conductivity tensor is universal at the critical point. [21, 25] Why Coulomb interactions that are present in real samples do not spoil agreement with the numerical simulations is something of a mystery at the time of this writing. For a discussion of some of these issues see [13]. 1.10 Fractional QHE Under some circumstances of weak (but non-zero) disorder, quantized Hall plateaus appear which are characterized by simple rational fractional quantum numbers. For example, at magnetic ﬁelds three times larger than

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those at which the ν = 1 integer ﬁlling factor plateau occurs, the lowest Landau level is only 1/3 occupied. The system ought to be below the percolation threshold and hence be insulating. Instead a robust quantized Hall plateau is observed indicating that electrons can travel through the sample and that (since σxx −→ 0) there is an excitation gap. This novel and quite unexpected physics is controlled by Coulomb repulsion between the electrons. It is best understood by ﬁrst ignoring the disorder and trying to discover the nature of the special correlated many-body ground state into which the electrons condense when the ﬁlling factor is a rational fraction. For reasons that will become clear later, it is convenient to analyze the problem in a new gauge = − 1 r × B (1.75) A 2 known as the symmetric gauge. Unlike the Landau gauge which preserves translation symmetry in one direction, the symmetric gauge preserves rotational symmetry about the origin. Hence we anticipate that angular momentum (rather than y linear momentum) will be a good quantum number in this gauge. For simplicity we will restrict our attention to the lowest Landau level only and (simply to avoid some awkward minus signs) change the sign of the = −B zˆ. With these restrictions, it is not hard to show that the B ﬁeld: B solutions of the free-particle Schr¨ odinger equation having deﬁnite angular momentum are 2 1 1 z m e− 4 |z| (1.76) ϕm = √ 2π 2 2m m! where z = (x + iy)/ is a dimensionless complex number representing the position vector r ≡ (x, y) and m ≥ 0 is an integer. Exercise 1.9. Verify that the basis functions in equation (1.76) do solve the Schr¨ odinger equation in the absence of a potential and do lie in the lowest Landau level. Hint: Rewrite the kinetic energy in such a way that becomes B · L. p · A The angular momentum of these basis states is of course h ¯ m. If we restrict our attention to the lowest Landau level, then there exists only one state with any given angular momentum and only non-negative values of m are allowed. This “handedness” is a result of the chirality built into the problem by the magnetic ﬁeld. It seems rather peculiar that in the Landau gauge we had a continuous one-dimensional family of basis states for this two-dimensional problem. Now we ﬁnd that in a diﬀerent gauge, we have a discrete one dimensional label for the basis states! Nevertheless, we still end up with the correct density of states per unit area. To√see this note that the peak value of |ϕm |2 occurs at a radius of Rpeak = 2m 2 . The area 2π 2 m of a circle of

82

Topological Aspects of Low Dimensional Systems

this radius contains m ﬂux quanta. Hence we obtain the standard result of one state per Landau level per quantum of ﬂux penetrating the sample. Because all the basis states are degenerate, any linear combination of them is also an allowed solution of the Schr¨ odinger equation. Hence any function of the form [26] 1

Ψ(x, y) = f (z)e− 4 |z|

2

(1.77)

is allowed so long as f is analytic in its argument. In particular, arbitrary polynomials of any degree N f (z) =

N

(z − Zj )

(1.78)

j=1

are allowed (at least in the thermodynamic limit) and are conveniently deﬁned by the locations of their N zeros {Zj ; j = 1, 2, . . . , N }. Another useful solution is the so-called coherent state which is a particular inﬁnite order polynomial 1 1 ∗ 1 ∗ e 2 λ z e− 4 λ λ . (1.79) fλ (z) ≡ √ 2 2π The wave function using this polynomial has the property that it is a narrow gaussian wave packet centered at the position deﬁned by the complex number λ. Completing the square shows that the probability density is given by 1 − 1 |z−λ|2 e 2 . (1.80) 2π 2 This is the smallest wave packet that can be constructed from states within the lowest Landau level. The reader will ﬁnd it instructive to compare this gaussian packet to the one constructed in the Landau gauge in Exercise (1.5). Because the kinetic energy is completely degenerate, the eﬀect of Coulomb interactions among the particles is nontrivial. To develop a feel for the problem, let us begin by solving the two-body problem. Recall that the standard procedure is to take advantage of the rotational symmetry to write down a solution with the relative angular momentum of the particles being a good quantum number and then solve the Schr¨ odinger equation for the radial part of the wave function. Here we ﬁnd that the analyticity properties of the wave functions in the lowest Landau level greatly simpliﬁes the situation. If we know the angular behavior of a wave function, analyticity uniquely deﬁnes the radial behavior. Thus for example for a single particle, knowing that the angular part of the wave function is eimθ , we know that the 2 2 1 1 full wave function is guaranteed to uniquely be rm eimθ e− 4 |z| = z m e− 4 |z| . 1

2

|Ψλ |2 = |fλ |2 e− 2 |z| =

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83

Haldane Pseudopotential Vm

1.0

− 0.8

0.6

−

0.4

− 0.2

0.0

0

2

− − − − − − − − 4

6

8

10

relative angular momentum Fig. 1.10. The Haldane pseudopotential Vm vs. relative angular momentum m for two particles interacting via the Coulomb interaction. Units are e2 /, where is the dielectric constant of the host semiconductor and the ﬁnite thickness of the quantum well has been neglected.

Consider now the two body problem for particles with relative angular momentum m and center of mass angular momentum M . The unique analytic wave function is (ignoring normalization factors) 1

ΨmM (z1 , z2 ) = (z1 − z2 )m (z1 + z2 )M e− 4 (|z1 |

2

+|z2 |2 )

.

(1.81)

If m and M are non-negative integers, then the prefactor of the exponential is simply a polynomial in the two arguments and so is a state made up of linear combinations of the degenerate one-body basis states ϕm given in equation (1.76) and therefore lies in the lowest Landau level. Note that if the particles are spinless fermions then m must be odd to give the correct exchange symmetry. Remarkably, this is the exact (neglecting Landau level mixing) solution for the Schr¨ odinger equation for any central potential V (|z1 − z2 |) acting between the two particles8 . We do not need to 8 Note

that neglecting Landau level mixing is a poor approximation for strong

84

Topological Aspects of Low Dimensional Systems

solve any radial equation because of the powerful restrictions due to analyticity. There is only one state in the (lowest Landau level) Hilbert space with relative angular momentum m and center of mass angular momentum M . Hence (neglecting Landau level mixing) it is an exact eigenstate of any central potential. ΨmM is the exact answer independent of the Hamiltonian! The corresponding energy eigenvalue vm is independent of M and is referred to as the mth Haldane pseudopotential

vm =

mM |V |mM · mM |mM

(1.82)

The Haldane pseudopotentials for the repulsive Coulomb potential are shown in Figure 1.10. These discrete energy eigenstates represent bound states of the repulsive potential. If there were no magnetic ﬁeld present, a repulsive potential would of course have only a continuous spectrum with no discrete bound states. However in the presence of the magnetic ﬁeld, there are eﬀectively bound states because the kinetic energy has been quenched. Ordinarily two particles that have a lot of potential energy because of their repulsive interaction can ﬂy apart converting that potential energy into kinetic energy. Here however (neglecting Landau level mixing) the particles all have ﬁxed kinetic energy. Hence particles that are repelling each other are stuck and can not escape from each other. One can view this semiclassically as the two particles orbiting each other under the inﬂuence of ×B drift with the Lorentz force preventing them from ﬂying apart. In E the presence of an attractive potential the eigenvalues change sign, but of course the eigenfunctions remain exactly the same (since they are unique)! The fact that a repulsive potential has a discrete spectrum for a pair of particles is (as we will shortly see) the central feature of the physics underlying the existence of an excitation gap in the fractional quantum Hall eﬀect. One might hope that since we have found analyticity to uniquely determine the two-body eigenstates, we might be able to determine manyparticle eigenstates exactly. The situation is complicated however by the fact that for three or more particles, the various relative angular momenta L12 , L13 , L23 , etc. do not all commute. Thus we can not write down general exact eigenstates. We will however be able to use the analyticity to great advantage and make exact statements for certain special cases.

potentials V ¯ hωc unless they are very smooth on the scale of the magnetic length.

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85

Exercise 1.10. Express the exact lowest Landau level two-body eigenstate 2 2 1 Ψ(z1 , z2 ) = (z1 − z2 )3 e− 4 {|z1 | +|z2 | } in terms of the basis of all possible two-body Slater determinants. Exercise 1.11. Verify the claim that the Haldane pseudopotential vm is independent of the center of mass angular momentum M . Exercise 1.12. Evaluate the Haldane pseudopotentials for the Coulomb 2 2 potential e r . Express your answer in units of e . For the speciﬁc case of = 10 and B = 10 T, express your answer in Kelvin. Exercise 1.13. Take into account the ﬁnite thickness of the quantum well by assuming that the one-particle basis states have the form ψm (z, s) = ϕm (z)Φ(s), where s is the coordinate in the direction normal to the quantum well. Write down (but do not evaluate) the formal expression for the Haldane pseudo-potentials in this case. Qualitatively describe the eﬀect of ﬁnite thickness on the values of the diﬀerent pseudopotentials for the case where the well thickness is approximately equal to the magnetic length. 1.11 The ν = 1 many-body state So far we have found the one- and two-body states. Our next task is to write down the wave function for a fully ﬁlled Landau level. We need to ﬁnd −1 |z |2 (1.83) ψ[z] = f [z] e 4 j j where [z] stands for (z1 , z2 , . . . , zN ) and f is a polynomial representing the Slater determinant with all states occupied. Consider the simple example of two particles. We want one particle in the orbital ϕ0 and one in ϕ1 , as illustrated schematically in Figure 1.11a. Thus (again ignoring normalization) (z1 )0 (z2 )0 = (z1 )0 (z2 )1 − (z2 )0 (z1 )1 f [z] = (z1 )1 (z2 )1 =

(z2 − z1 ).

(1.84)

This is the lowest possible order polynomial that is antisymmetric. For the case of three particles we have (see Fig. 1.11b) (z1 )0 (z2 )0 (z3 )0 f [z] = (z1 )1 (z2 )1 (z3 )1 = z2 z32 − z3 z22 − z11 z32 + z31 z12 + z1 z22 − z21 z12 (z1 )2 (z2 )2 (z3 )2

86

Topological Aspects of Low Dimensional Systems

(a) m=0

m=1

m=2

m=3

m=4

(b) Fig. 1.11. Orbital occupancies for the maximal density ﬁlled Landau level state with (a) two particles and (b) three particles. There are no particle labels here. In the Slater determinant wave function, the particles are labeled but a sum is taken over all possible permutations of the labels in order to antisymmetrize the wave function.

=

−(z1 − z2 )(z1 − z3 )(z2 − z3 )

=

−

3 (zi − zj ).

(1.85)

i<j

This form for the Slater determinant is known as the Vandermonde polynomial. The overall minus sign is unimportant and we will drop it. The single Slater determinant to ﬁll the ﬁrst N angular momentum states is a simple generalization of equation (1.85) fN [z] =

N (zi − zj ).

(1.86)

i<j

To prove that this is true for general N , note that the polynomial is fully antisymmetric and the highest power of any z that appears is z N −1 . Thus the highest angular momentum state that is occupied is m = N − 1. But since the antisymmetry guarantees that no two particles can be in the same state, all N states from m = 0 to m = N − 1 must be occupied. This proves that we have the correct Slater determinant. Exercise 1.14. Show carefully that the Vandermonde polynomial for N particles is in fact totally antisymmetric. One can also use induction to show that the Vandermonde polynomial is the correct Slater determinant by writing fN +1 (z) = fN (z)

N

(zi − zN +1 )

(1.87)

i=1

which can be shown to agree with the result of expanding the determinant of the (N + 1) × (N + 1) matrix in terms of the minors associated with the (N + 1)st row or column.

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Note that since the Vandermonde polynomial corresponds to the ﬁlled Landau level it is the unique state having the maximum density and hence is an exact eigenstate for any form of interaction among the particles (neglecting Landau level mixing and ignoring the degeneracy in the center of mass angular momentum). The (unnormalized) probability distribution for particles in the ﬁlled Landau level state is 2

|Ψ[z]| =

N

|zi − zj |2 e

− 12

N j=1

|zj |2

.

(1.88)

i<j

This seems like a rather complicated object about which it is hard to make any useful statements. It is clear that the polynomial term tries to keep the particles away from each other and gets larger as the particles spread out. It is also clear that the exponential term is small if the particles spread out too much. Such simple questions as, “Is the density uniform?”, seem hard to answer however. It turns out that there is a beautiful analogy to plasma physics developed by Laughlin which sheds a great deal of light on the nature of this many particle probability distribution. To see how this works, let us pretend that the norm of the wave function 2 (1.89) Z ≡ d z1 . . . d2 zN |ψ[z] |2 is the partition function of a classical statistical mechanics problem with Boltzmann weight 2 (1.90) |Ψ[z]| = e−βUclass where β ≡

2 m

and Uclass ≡ m2

i<j

(− ln |zi − zj |) +

m |zk |2 . 4

(1.91)

k

(The parameter m = 1 in the present case but we introduce it for later convenience.) It is perhaps not obvious at ﬁrst glance that we have made tremendous progress, but we have. This is because Uclass turns out to be the potential energy of a fake classical one-component plasma of particles of charge m in a uniform (“jellium”) neutralizing background. Hence we can bring to bear well-developed intuition about classical plasma physics to study the properties of |Ψ|2 . Please remember however that all the statements we make here are about a particular wave function. There are no actual long-range logarithmic interactions in the quantum Hamiltonian for which this wave function is the approximate groundstate.

88

Topological Aspects of Low Dimensional Systems

To understand this, let us ﬁrst review the electrostatics of charges in three dimensions. For a charge Q particle in 3D, the surface integral of the electric ﬁeld on a sphere of radius R surrounding the charge obeys ·E = 4πQ. dA (1.92) Since the area of the sphere is 4πR2 we deduce

and

r) E(

= Q

ϕ(r )

=

rˆ r2

(1.93)

Q r

(1.94)

·E = −∇2 ϕ = 4πQ δ 3 (r ) ∇

(1.95)

where ϕ is the electrostatic potential. Now consider a two-dimensional world where all the ﬁeld lines are conﬁned to a plane (or equivalently consider the electrostatics of inﬁnitely long charged rods in 3D). The analogous equation for the line integral of the normal electric ﬁeld on a circle of radius R is = 2πQ ds · E (1.96) where the 2π (instead of 4π) appears because the circumference of a circle is 2πR (and is analogous to 4πR2 ). Thus we ﬁnd r) = E(

Qˆ r r

ϕ(r ) =

Q − ln

r r0

(1.97) (1.98)

and the 2D version of Poisson’s equation is ·E = −∇2 ϕ = 2πQ δ 2 (r ). ∇

(1.99)

Here r0 is an arbitrary scale factor whose value is immaterial since it only shifts ϕ by a constant. We now see why the potential energy of interaction among a group of objects with charge m is

(− ln |zi − zj |) . (1.100) U 0 = m2 i<j

(Since z = (x + iy)/ we are using r0 = .) This explains the ﬁrst term in equation (1.91).

S.M. Girvin: The Quantum Hall Eﬀect

89

To understand the second term notice that −∇2

1 2 1 |z| = − 2 = 2πρB 4

(1.101)

where ρB ≡ −

1 · 2π 2

(1.102)

Equation (1.101) can be interpreted as Poisson’s equation and tells us that 1 2 4 |z| represents the electrostatic potential of a constant charge density ρB . Thus the second term in equation (1.91) is the energy of charge m objects interacting with this negative background. Notice that 2π 2 is precisely the area containing one quantum of ﬂux. Thus the background charge density is precisely B/Φ0 , the density of ﬂux in units of the ﬂux quantum. The very long range forces in this fake plasma cost huge (fake) “energy” unless the plasma is everywhere locally neutral (on length scales larger than the Debye screening length which in this case is comparable to the particle spacing). In order to be neutral, the density n of particles must obey nm + ρB

=

⇒

=

n

0 1 1 m 2π 2

(1.103) (1.104)

since each particle carries (fake) charge m. For our ﬁlled Landau level with m = 1, this is of course the correct answer for the density since every single-particle state is occupied and there is one state per quantum of ﬂux. We again emphasize that the energy of the fake plasma has nothing to do with the quantum Hamiltonian and the true energy. The plasma analogy is merely a statement about this particular choice of wave function. It says that the square of the wave function is very small (because Uclass is large) for conﬁgurations in which the density deviates even a small amount from 1/(2π 2 ). The electrons can in principle be found anywhere, but the overwhelming probability is that they are found in a conﬁguration which is locally random (liquid-like) but with negligible density ﬂuctuations on long length scales. We will discuss the nature of the typical conﬁgurations again further below in connection with Figure 1.12. When the fractional quantum Hall eﬀect was discovered, Robert Laughlin realized that one could write down a many-body variational wave function at ﬁlling factor ν = 1/m by simply taking the mth power of the polynomial that describes the ﬁlled Landau level m [z] = fN

N (zi − zj )m . i<j

(1.105)

90

Topological Aspects of Low Dimensional Systems

In order for this to remain analytic, m must be an integer. To preserve the antisymmetry m must be restricted to the odd integers. In the plasma analogy the particles now have fake charge m (rather than unity) and the 1 1 1 density of electrons is n = m 2π2 so the Landau level ﬁlling factor ν = m = 1 1 1 3 , 5 , 7 , etc. (Later on, other wave functions were developed to describe more general states in the hierarchy of rational fractional ﬁlling factors at which quantized Hall plateaus were observed [3, 4, 6, 8, 9].) The Laughlin wave function naturally builds in good correlations among the electrons because each particle sees an m-fold zero at the positions of all the other particles. The wave function vanishes extremely rapidly if any two particles approach each other, and this helps minimize the expectation value of the Coulomb energy. Since the kinetic energy is ﬁxed we need only concern ourselves with the expectation value of the potential energy for this variational wave function. Despite the fact that there are no adjustable variational parameters (other than m which controls the density) the Laughlin wave functions have proven to be very nearly exact for almost any realistic form of repulsive interaction. To understand how this can be so, it is instructive to consider a model for which this wave function actually is the exact ground state. Notice that the form of the wave function guarantees that every pair of particles has relative angular momentum greater than or equal to m. One should not make the mistake of thinking that every pair has relative angular momentum precisely equal to m. This would require the spatial separation between particles to be very nearly the same for every pair, which is of course impossible. Suppose that we write the Hamiltonian in terms of the Haldane pseudopotentials ∞

vm Pm (ij) (1.106) V = m =0 i<j

where Pm (ij) is the projection operator which selects out states in which particles i and j have relative angular momentum m. If Pm (ij) and Pm (jk) commuted with each other things would be simple to solve, but this is not the case. However if we consider the case of a “hard-core potential” deﬁned by vm = 0 for m ≥ m, then clearly the mth Laughlin state is an exact, zero energy eigenstate V ψm [z] = 0.

(1.107)

Pm (ij)ψm = 0

(1.108)

This follows from the fact that

for any m < m since every pair has relative angular momentum of at least m.

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91

Fig. 1.12. Comparison of typical conﬁgurations for a completely uncorrelated (Poisson) distribution of 1000 particles (left panel) to the distribution given by the Laughlin wave function for m = 3 (right panel). The latter is a snapshot taken during a Monte Carlo simulation of the distribution. The Monte Carlo procedure consists of proposing a random trial move of one of the particles to a new position. If this move increases the value of |Ψ|2 it is always accepted. If the move decreases the value of |Ψ|2 by a factor p, then the move is accepted with probability p. After equilibration of the plasma by a large number of such moves one ﬁnds that the conﬁgurations generated are distributed according to |Ψ|2 . (After Laughlin, Chap. 7 in [3].)

Because the relative angular momentum of a pair can change only in discrete (even integer) units, it turns out that this hard core model has an excitation gap. For example for m = 3, any excitation out of the Laughlin ground state necessarily weakens the nearly ideal correlations by forcing at least one pair of particles to have relative angular momentum 1 instead of 3 (or larger). This costs an excitation energy of order v1 . This excitation gap is essential to the existence of dissipationless (σxx = ρxx = 0) current ﬂow. In addition this gap means that the Laughlin state is stable against perturbations. Thus the diﬀerence between the Haldane pseudopotentials vm for the Coulomb interaction and the pseudopotentials for the hard core model can be treated as a small perturbation (relative to the excitation gap). Numerical studies show that for realistic pseudopotentials the overlap between the true ground state and the Laughlin state is extremely good. To get a better understanding of the correlations built into the Laughlin wave function it is useful to consider the snapshot in Figure 1.12 which shows a typical conﬁguration of particles in the Laughlin ground state (obtained from a Monte Carlo sampling of |ψ|2 ) compared to a random (Poisson) distribution. Focussing ﬁrst on the large scale features we see that density

92

Topological Aspects of Low Dimensional Systems

ﬂuctuations at long wavelengths are severely suppressed in the Laughlin state. This is easily understood in terms of the plasma analogy and the desire for local neutrality. A simple estimate for the density ﬂuctuations ρq at wave vector q can be obtained by noting that the fake plasma potential energy can be written (ignoring a constant associated with self-interactions being included) 1 2πm2 ρq ρ−q (1.109) Uclass = 2L2 q2 q =0

where L is the area of the system and 2π q2 is the Fourier transform of the logarithmic potential (easily derived from ∇2 (− ln (r)) = −2π δ 2 (r ) ). At long wavelengths (q 2 n) it is legitimate to treat ρq as a collective coordinate of an elastic continuum. The distribution e−βUclass of these coordinates is a gaussian and so obeys (taking into account the fact that ρ−q = (ρq )∗ ) 2

ρq ρ−q = L2

q2 · 4πm

(1.110)

We clearly see that the long-range (fake) forces in the (fake) plasma strongly suppress long wavelength density ﬂuctuations. We will return more to this point later when we study collective density wave excitations above the Laughlin ground state. The density ﬂuctuations on short length scales are best studied in real space. The radial correlation g(r) function is a convenient object to consider. g(r) tells us the density at r given that there is a particle at the origin N (N − 1) 2 2 z . . . d2 zN |ψ(0, r, z3 , . . . , zN )| (1.111) g(r) = d 3 n2 Z where Z ≡ ψ|ψ, n is the density (assumed uniform) and the remaining factors account for all the diﬀerent pairs of particles that could contribute. The factors of density are included in the denominator so that limr→∞ g(r) = 1. Because the m = 1 state is a single Slater determinant g(z) can be computed exactly 2 1 (1.112) g(z) = 1 − e− 2 |z| . Figure 1.13 shows numerical estimates of h(r) ≡ 1 − g(r) for the cases m = 3 and 5. Notice that for the ν = 1/m state g(z) ∼ |z|2m for small distances. Because of the strong suppression of density ﬂuctuations at long wavelengths, g(z) converges exponentially rapidly to unity at large distances. For m > 1, g develops oscillations indicative of solid-like correlations

S.M. Girvin: The Quantum Hall Eﬀect

93

Fig. 1.13. Plot of the two-point correlation function h(r) ≡ 1 − g(r) for the Laughlin plasma with ν −1 = m = 3 (left panel) and m = 5 (right panel). Notice that, unlike the result for m = 1 given in equation (1.112), g(r) exhibits the oscillatory behavior characteristic of a strongly coupled plasma with short-range solid-like local order.

and, the plasma actually freezes9 at m ≈ 65. The Coulomb interaction energy can be expressed in terms of g(z) as10 nN ψ|V |ψ = ψ|ψ 2

d2 z

e2 [g(z) − 1] |z|

(1.113)

where the (−1) term accounts for the neutralizing background and is the dielectric constant of the host semiconductor. We can interpret g(z) − 1 as the density of the “exchange-correlation hole” surrounding each particle. The correlation energies per particle for m = 3 and 5 are [27]

and

1 ψ3 |V |ψ3 = −0.4100 ± 0.0001 N ψ3 |ψ3

(1.114)

1 ψ5 |V |ψ5 = −0.3277 ± 0.0002 N ψ5 |ψ5

(1.115)

9 That

is, Monte Carlo simulation of |Ψ|2 shows that the particles are most likely to be found in a crystalline conﬁguration which breaks translation symmetry. Again we emphasize that this is a statement about the Laughlin variational wave function, not necessarily a statement about what the electrons actually do. It turns out that for m ≥ ∼ 7 the Laughlin wave function is no longer the best variational wave function. One can write down wave functions describing Wigner crystal states which have lower variational energy than the Laughlin liquid. 10 This expression assumes a strictly zero thickness electron gas. Otherwise one must 2 +∞ |F (s)|2 e2 by e ds √ where F is the wavefunction factor describing the replace |z| −∞

quantum well bound state.

|z|2 +s2

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Topological Aspects of Low Dimensional Systems

in units of e2 / which is ≈ 161 K for = 12.8 (the value in GaAs), B = 10 T. For the ﬁlled Landau level (m = 1) the exchange energy is − π8 as can be seen from equations (1.112) and (1.113). Exercise 1.15. Find the radial distribution function for a onedimensional spinless free electron gas of density n by writing the ground state wave function as a single Slater determinant and then integrating out all but two of the coordinates. Use this ﬁrst quantization method even if you already know how to do this calculation using second quantization. Hint: Take advantage of the following representation of the determinant of a N × N matrix M in terms of permutations P of N objects. Det M =

P

(−1)P

N

MjPj .

j=1

Exercise 1.16. Using the same method derive equation (1.112). 1.12 Neutral collective excitations So far we have studied one particular variational wave function and found that it has good correlations built into it as graphically illustrated in Figure 1.12. To further bolster the case that this wave function captures the physics of the fractional Hall eﬀect we must now demonstrate that there is ﬁnite energy cost to produce excitations above this ground state. In this section we will study the neutral collective excitations. We will examine the charged excitations in the next section. It turns out that the neutral excitations are phonon-like excitations similar to those in solids and in superﬂuid helium. We can therefore use a simple modiﬁcation of Feynman’s theory of the excitations in superﬂuid helium [28, 29]. By way of introduction let us start with the simple harmonic oscillator. The ground state is of the form 2

ψ0 (x) ∼ e−αx .

(1.116)

Suppose we did not know the excited state and tried to make a variational ansatz for it. Normally we think of the variational method as applying only to ground states. However it is not hard to see that the ﬁrst excited state energy is given by ψ|H|ψ 1 = min (1.117) ψ|ψ provided that we do the minimization over the set of states ψ which are constrained to be orthogonal to the ground state ψ0 . One simple way to

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produce a variational state which is automatically orthogonal to the ground state is to change the parity by multiplying by the ﬁrst power of the coordinate 2 (1.118) ψ1 (x) ∼ x e−αx . Variation with respect to α of course leads (in this special case) to the exact ﬁrst excited state. With this background let us now consider the case of phonons in superﬂuid 4 He. Feynman argued that because of the Bose statistics of the particles, there are no low-lying single-particle excitations. This is in stark contrast to a fermi gas which has a high density of low-lying excitations around the fermi surface. Feynman argued that the only low-lying excitations in 4 He are collective density oscillations that are well-described by the following family of variational wave functions (that has no adjustable parameters) labeled by the wave vector 1 ψk = √ ρk Φ0 N

(1.119)

where Φ0 is the exact ground state and ρk ≡

N

e−ik·rj

(1.120)

j=1

is the Fourier transform of the density. The physical picture behind this is that at long wavelengths the ﬂuid acts like an elastic continuum and ρk can be treated as a generalized oscillator normal-mode coordinate. In this sense equation (1.119) is then analogous to equation (1.118). To see that ψk is orthogonal to the ground state we simply note that Φ0 |ψk

= =

1 √ Φ0 |ρk |Φ0 N 1 √ d3 R e−ik·R Φ0 |ρ(r )|Φ0 N

where ρ(r ) ≡

N

δ 3 (rj − R)

(1.121)

(1.122)

j=1

is the density operator. If Φ0 describes a translationally invariant liquid ground state then the Fourier transform of the mean density vanishes for k = 0. There are several reasons why ψk is a good variational wave function, especially for small k. First, it contains the ground state as a factor. Hence it contains all the special correlations built into the ground state to make

96

Topological Aspects of Low Dimensional Systems

(a)

λ

λ

2π λ = __ k (b)

Fig. 1.14. (a) Conﬁguration of particles in which the Fourier transform of the density at wave vector k is non-zero. (b) The Fourier amplitude will have a similar magnitude for this conﬁguration but a diﬀerent phase.

sure that the particles avoid close approaches to each other without paying a high price in kinetic energy. Second, ψk builds in the features we expect on physical grounds for a density wave. To see this, consider evaluating ψk for a conﬁguration of the particles like that shown in Figure 1.14a which has a density modulation at wave vector k. This is not a conﬁguration that maximizes |Φ0 |2 , but as long as the density modulation is not too large and the particles avoid close approaches, |Φ0 |2 will not fall too far below its maximum value. More importantly, |ρk |2 will be much larger than it would for a more nearly uniform distribution of positions. As a result |ψk |2 will be large and this will be a likely conﬁguration of the particles in the excited state. For a conﬁguration like that in Figure 1.14b, the phase of ρk will shift but |ψk |2 will have the same magnitude. This is analogous to the parity change in the harmonic oscillator example. Because all diﬀerent phases of the density wave are equally likely, ρk has a mean density which is uniform (translationally invariant). To proceed with the calculation of the variational estimate for the excitation energy ∆(k) of the density wave state we write ∆(k) = where

f (k) s(k)

f (k) ≡ ψk |(H − E0 )|ψk ,

(1.123)

(1.124)

with E0 being the exact ground state energy and s(k) ≡ ψk |ψk =

1 Φ0 |ρ† ρ |Φ0 · k k N

(1.125)

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97

We see that the norm of the variational state s(k) turns out to be the static structure factor of the ground state. It is a measure of the mean square density ﬂuctuations at wave vector k. Continuing the harmonic oscillator analogy, we can view this as a measure of the zero-point ﬂuctuations of the normal-mode oscillator coordinate ρk . For superﬂuid 4 He s(k) can be directly measured by neutron scattering and can also be computed theoretically using quantum Monte Carlo methods [30]. We will return to this point shortly. Exercise 1.17. Show that for a uniform liquid state of density n, the static structure factor is related to the Fourier transform of the radial distribution function by s(k) = N δk,0 + 1 + n d3 r eik·r [g(r) − 1]

The numerator in equation (1.124) is called the oscillator strength and can be written 1 Φ0 |ρ†k [H, ρk ]|Φ0 · f (k) = (1.126) N For uniform systems with parity symmetry we can write this as a double commutator 1 † (1.127) Φ0 ρk , [H, ρk ] Φ0 f (k) = 2N from which we can derive the justiﬁably famous oscillator strength sum rule f (k) =

¯ 2 k2 h 2M

(1.128)

where M is the (band) mass of the particles11 . Remarkably (and conveniently) this is a universal result independent of the form of the interaction potential between the particles. This follows from the fact that only the kinetic energy part of the Hamiltonian fails to commute with the density. Exercise 1.18. Derive equation (1.127) and then equation (1.128) from equation (1.126) for a system of interacting particles. We thus arrive at the Feynman-Bijl formula for the collective mode excitation energy h2 k 2 1 ¯ · (1.129) ∆(k) = 2M s(k) 11 Later on in equation (1.137) we will express the oscillator strength in terms of a frequency integral. Strictly speaking if this is integrated up to very high frequencies including interband transitions, then M is replaced by the bare electron mass.

98

Topological Aspects of Low Dimensional Systems

We can interpret the ﬁrst term as the energy cost if a single particle (initially at rest) were to absorb all the momentum and the second term is a renormalization factor describing momentum (and position) correlations among the particles. One of the remarkable features of the Feynman-Bijl formula is that it manages to express a dynamical quantity ∆(k), which is a property of the excited state spectrum, solely in terms of a static property of the ground state, namely s(k). This is a very powerful and useful approximation. Returning to equation (1.119) we see that ψk describes a linear superposition of states in which one single particle has had its momentum boosted by h ¯k. We do not know which one however. The summation in equation (1.120) tells us that it is equally likely to be particle 1 or particle 2 or . . . , etc. This state should not be confused with the state in which boost is applied to particle 1 and particle 2 and . . . , etc. This state is described by a product   N eik·rj  Φ0 (1.130) Φk ≡  j=1

which can be rewritten

 N 

1 Φk = exp iN k ·  rj  Φ0   N j=1  



(1.131) 2 2

k showing that this is an exact energy eigenstate (with energy N h¯2M ) in which the center of mass momentum has been boosted by N ¯hk. In superﬂuid 4 He the structure factor vanishes linearly at small wave vectors s(k) ∼ ξk (1.132)

so that ∆(k) is linear as expected for a sound mode 2 h 1 ¯ ∆(k) = k 2M ξ

(1.133)

from which we see that the sound velocity is given by cs =

h 1 ¯ · 2M ξ

(1.134)

This phonon mode should not be confused with the ordinary hydrodynamic sound mode in classical ﬂuids. The latter occurs in a collision dominated regime ωτ 1 in which collision-induced pressure provides the restoring force. The phonon mode described here by ψk is a low-lying eigenstate of the quantum Hamiltonian.

S.M. Girvin: The Quantum Hall Eﬀect

ε

99

k

10K k 20 Fig. 1.15. Schematic illustration of the phonon dispersion in superﬂuid liquid 4 He. For small wave vectors the dispersion is linear, as is expected for a gapless Goldstone mode. The roton minimum due to the peak in the static structure factor occurs at a wave vector k of approximately 20 in units of inverse ˚ A. The roton energy is approximately 10 in units of Kelvins.

At larger wave vectors there is a peak in the static structure factor caused by the solid-like oscillations in the radial distribution function g(r) similar to those shown in Figure 1.13 for the Laughlin liquid. This peak in s(k) leads to the so-called roton minimum in ∆(k) as illustrated in Figure 1.15. To better understand the Feynman picture of the collective excited states recall that the dynamical structure factor is deﬁned (at zero temperature) by $ % † H − E0 2π S(q, ω) ≡ (1.135) Φ0 ρq δ ω − ρq Φ0 · N h ¯ The static structure factor is the zeroth frequency moment ∞ ∞ dω dω S(q, ω) = S(q, ω) s(q) = 2π 2π −∞ 0

(1.136)

(with the second equality valid only at zero temperature). Similarly the oscillator strength in equation (1.124) becomes (at zero temperature) ∞ ∞ dω dω f (q) = hω S(q, ω) = ¯ hω S(q, ω). ¯ (1.137) 2π −∞ 2π 0 Thus we arrive at the result that the Feynman-Bijl formula can be rewritten ∞ dω hω S(q, ω) ¯ ∆(q) = 0 ∞2πdω · (1.138) 0 2π S(q, ω) That is, ∆(q) is the mean excitation energy (weighted by the square of the density operator matrix element). Clearly the mean exceeds the minimum

100

Topological Aspects of Low Dimensional Systems

and so the estimate is variational as claimed. Feynman’s approximation is equivalent to the assumption that only a single mode contributes any oscillator strength so that the zero-temperature dynamical structure factor contains only a single delta function peak 1 (1.139) S(q, ω) = 2π s(q) δ ω − ∆(q) . h ¯ Notice that this approximate form satisﬁes both equation (1.136) and equation (1.137) provided that the collective mode energy ∆(q) obeys the Feynman-Bijl formula in equation (1.129). Exercise 1.19. For a system with a homogeneous liquid ground state, the (linear response) static susceptibility of the density to a perturbation U = Vq ρ−q is deﬁned by ρq = χ(q)Vq . (1.140) Using ﬁrst order perturbation theory show that the static susceptibility is given in terms of the dynamical structure factor by ∞ dω 1 S(q, ω). (1.141) χ(q) = −2 2π ¯ hω 0 Using the single mode approximation and the oscillator strength sum rule, derive an expression for the collective mode dispersion in terms of χ(q). (Your answer should not involve the static structure factor. Note also that equation (1.140) is not needed to produce the answer to this part. Just work with equation (1.141).) As we mentioned previously Feynman argued that in 4 He the Bose symmetry of the wave functions guarantees that unlike in Fermi systems, there is only a single low-lying mode, namely the phonon density mode. The paucity of low-energy single particle excitations in boson systems is what helps make them superﬂuid–there are no dissipative channels for the current to decay into. Despite the fact that the quantum Hall system is made up of fermions, the behavior is also reminiscent of superﬂuidity since the current ﬂow is dissipationless. Indeed, within the “composite boson” picture, one views the FQHE ground state as a bose condensate [1,9,10]. Let us therefore blindly make the single-mode approximation and see what happens. From equation (1.110) we see that the static structure factor for the mth Laughlin state is (for small wave vectors only) s(q) =

1 L2 q 2 = q 2 2 , N 4πm 2

where we have used L2 /N = 2π 2 m.

(1.142)

The Feynman-Bijl formula then

S.M. Girvin: The Quantum Hall Eﬀect yields12 ∆(q) =

¯ 2 q2 2 h =h ¯ ωc . 2M q 2 2

101

(1.143)

This predicts that there is an excitation gap that is independent of wave vector (for small q) and equal to the cyclotron energy. It is in fact correct that at long wavelengths the oscillator strength is dominated by transitions in which a single particle is excited from the n = 0 to the n = 1 Landau level. Furthermore, Kohn’s theorem guarantees that the mode energy is precisely hωc . Equation (1.143) was derived speciﬁcally for the Laughlin state, but ¯ it is actually quite general, applying to any translationally invariant liquid ground state. One might expect that the single mode approximation (SMA) will not work well in an ordinary Fermi gas due to the high density of excitations around the Fermi surface13 . Here however the Fermi surface has been destroyed by the magnetic ﬁeld and the continuum of excitations with diﬀerent kinetic energies has been turned into a set of discrete inter-Landau-level excitations, the lowest of which dominates the oscillator strength. For ﬁlling factor ν = 1 the Pauli principle prevents any intra-level excitations and the excitation gap is in fact h ¯ ωc as predicted by the SMA. However for ν < 1 there should exist intra-Landau-level excitations whose energy scale is set by the interaction scale e2 / rather than the kinetic energy scale h ¯ ωc . Indeed we can formally think of taking the band mass to zero (M → 0) which would send h ¯ ωc → ∞ while keeping e2 / ﬁxed. Unfortunately the SMA as it stands now is not very useful in this limit. What we need is a variational wave function that represents a density wave but is restricted to lie in the Hilbert space of the lowest Landau level. This can be formally accomplished by replacing equation (1.119) by ψk = ρ¯k ψm

(1.144)

where the overbar indicates that the density operator has been projected into the lowest Landau level. The details of how this is accomplished are presented in Appendix A. The analog of equation (1.123) is ∆(k) =

f¯(k) s¯(k)

(1.145)

where f¯ and s¯ are the projected oscillator strength and structure factor, 12 We will continue to use the symbol M here for the band mass of the electrons to avoid confusion with the inverse ﬁlling factor m. 13 This expectation is only partly correct however as one discovers when studying collective plasma oscillations in systems with long-range Coulomb forces.

102

Topological Aspects of Low Dimensional Systems

respectively. As shown in Appendix A s¯(k)

2 2 1 1 ψm |¯ ρ† ρ¯k |ψm = s(k) − 1 − e− 2 |k| k N = s(k) − sν=1 (k).

≡

(1.146)

This vanishes for the ﬁlled Landau level because the Pauli principle forbids all intra-Landau-level excitations. For the mth Laughlin state equation (1.142) shows us that the leading term in s(k) for small k is 12 k 2 2 . Putting this into equation (1.146) we see that the leading behavior for s¯(k) is therefore quartic (1.147) s¯(k) ∼ a(k )4 + . . . . We can not compute the coeﬃcient a without ﬁnding the k 4 correction to equation (1.142). It turns out that there exists a compressibility sum rule for the fake plasma from which we can obtain the exact result [29] a=

m−1 · 8

(1.148)

The projected oscillator strength is given by equation (1.127) with the density operators replaced by their projections. In the case of 4 He only the kinetic energy part of the Hamiltonian failed to commute with the density. It was for this reason that the oscillator strength came out to be a universal number related to the mass of the particles. Within the lowest Landau level however the kinetic energy is an irrelevant constant. Instead, after projection the density operators no longer commute with each other (see Appendix A). It follows from these commutation relations that the projected oscillator strength is proportional to the strength of the interaction term. The leading small k behavior is [29] e2 f¯(k) = b (k )4 + . . .

(1.149)

where b is a dimensionless constant that depends on the details of the interaction potential. The intra-Landau level excitation energy therefore has a ﬁnite gap at small k ∆(k) =

b e2 f¯(k) ∼ + O(k 2 ) + . . . s¯(k) a

(1.150)

This is quite diﬀerent from the case of superﬂuid 4 He in which the mode is gapless. However like the case of the superﬂuid, this “magnetophonon” mode has a “magnetoroton” minimum at ﬁnite k as illustrated in Figure 1.16. The ﬁgure also shows results from numerical exact diagonalization studies which demonstrate that the single mode approximation

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103

Fig. 1.16. Comparison of the single mode approximation (SMA) prediction of the collective mode energy for ﬁlling factors ν = 1/3, 1/5, 1/7 (solid lines) with small-system numerical results for N particles. Crosses indicate the N = 7, ν = 1/3 spherical system, triangles indicate the N = 6, ν = 1/3 hexagonal unit cell system results of Haldane and Rezayi [31]. Solid dots are for N = 9, ν = 1/3 and N = 7, ν = 1/5 spherical system calculations of Fano et al. [32] Arrows at the top indicate the magnitude of the reciprocal lattice vector of the Wigner crystal at the corresponding ﬁlling factor. Notice that unlike the phonon collective mode in superﬂuid helium shown in Figure 1.15, the mode here is gapped.

is extremely accurate. Note that the magnetoroton minimum occurs close to the position of the smallest reciprocal lattice vector in the Wigner crystal of the same density. In the crystal the phonon frequency would go exactly to zero at this point. (Recall that in a crystal the phonon dispersion curves have the periodicity of the reciprocal lattice.) Because the oscillator strength is almost entirely in the cyclotron mode, the dipole matrix element for coupling the collective excitations to light is very small. They have however been observed in Raman scattering [33] and found to have an energy gap in excellent quantitative agreement with the single mode approximation. Finally we remark that these collective excitations are characterized by a well-deﬁned wave vector k despite the presence of the strong magnetic ﬁeld. This is only possible because they are charge neutral which allows one to deﬁne a gauge invariant conserved momentum [34].

104

Topological Aspects of Low Dimensional Systems

1.13 Charged excitations Except for the fact that they are gapped, the neutral magnetophonon excitations are closely analogous to the phonon excitations in superﬂuid 4 He. We further pursue this analogy with a search for the analog of vortices in superﬂuid ﬁlms. A vortex is a topological defect which is the quantum version of the familiar whirlpool. A reasonably good variational wave function for a vortex in a two-dimensional ﬁlm of 4 He is   N   ± e±iθ(rj −R) Φ0 . ψR f |rj − R| (1.151) =  j=1

Here θ is the azimuthal angle that the particle’s position makes relative to the location of the vortex center. The function f vanishes as r approaches R, and goes to unity far away. The choice of sign in the phase determines R whether the vortex is right or left handed. The interpretation of this wave function is the following. The vortex is a topological defect because if any particle is dragged around a closed loop the phase of the wave function winds by ±2π. This phase surrounding R, gradient means that current is circulating around the core. Consider a large The phase change of 2π around the circle circle of radius ξ centered on R. ˆ occurs in a distance 2πξ so the local gradient seen by every particle is θ/ξ. Recalling equation (1.131) we see that locally the center of mass momentum has been boosted by ± h¯ξ θˆ so that the current density of the whirlpool falls oﬀ inversely with distance from the core14 . Near the core f falls to zero because of the “centrifugal barrier” associated with this circulation. In a more accurate variational wave function the core would be treated slightly diﬀerently but the asymptotic large distance behavior would be unchanged. What is the analog of all this for the lowest Landau level? For ψ + we see that every particle has its angular momentum boosted by one unit. In the lowest Landau level analyticity (in the symmetric gauge) requires us to replace eiθ by z = x + iy. Thus we are led to the Laughlin “quasi-hole” wave function N + [z] = (zj − Z) ψm [z] (1.152) ψZ j=1

where Z is a complex number denoting the position of the vortex and ψm is the Laughlin wave function at ﬁlling factor ν = 1/m. The corresponding 14 This slow algebraic decay of the current density means that the total kinetic energy of a single vortex diverges logarithmically with the size of the system. This in turn leads to the Kosterlitz Thouless phase transition in which pairs of vortices bind together below a critical temperature. As we will see below there is no corresponding ﬁnite temperature transition in a quantum Hall system.

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105

antivortex (“quasi-electron” state) involves zj∗ suitably projected (as discussed in Appendix A): − [z] = ψZ

N ∂ − Z ∗ ψm [z] 2 ∂zj j=1

(1.153)

where as usual the derivatives act only on the polynomial part of ψm . All these derivatives make ψ − somewhat diﬃcult to work with. We will therefore concentrate on the quasi-hole state ψ + . The origin of the names quasihole and quasi-electron will become clear shortly. Unlike the case of a superﬂuid ﬁlm, the presence of the vector potential allows these vortices to cost only a ﬁnite energy to produce and hence the electrical dissipation is always ﬁnite at any non-zero temperature. There is no ﬁnite temperature transition into a superﬂuid state as in the Kosterlitz Thouless transition. From a ﬁeld theoretic point of view, this is closely analogous to the Higg’s mechanism [1]. Just as in our study of the Laughlin wave function, it is very useful to see how the plasma analogy works for the quasi-hole state + 2 | = e−βUclass e−βV |ψZ

(1.154)

where Uclass is given by equation (1.91), β = 2/m as before and V ≡m

N

(− ln |zj − Z|) .

(1.155)

j=1

Thus we have the classical statistical mechanics of a one-component plasma of (fake) charge m objects seeing a neutralizing jellium background plus a new potential energy V representing the interaction of these objects with an “impurity” located at Z and having unit charge. Recall that the chief desire of the plasma is to maintain charge neutrality. Hence the plasma particles will be repelled from Z. Because the plasma particles have fake charge m, the screening cloud will have to have a net reduction of 1/m particles to screen the impurity. But this means that the quasi-hole has fractional fermion number! The (true) physical charge of the object is a fraction of the elementary charge q∗ =

e · m

(1.156)

This is very strange! How can we possibly have an elementary excitation carrying fractional charge in a system made up entirely of electrons? To understand this let us consider an example of another quantum system that seems to have fractional charge, but in reality doesn’t. Imagine three protons arranged in an equilateral triangle as shown in Figure 1.17. Let

106

Topological Aspects of Low Dimensional Systems

1S

1S

1S Fig. 1.17. Illustration of an electron tunneling among the 1S orbitals of three protons. The tunneling is exponentially slow for large separations which leads to only exponentially small lifting of what would otherwise be a three-fold degenerate ground state.

there be one electron in the system. In the spirit of the tight-binding model we consider only the 1S orbital on each of the three “lattice sites”. The Bloch states are 3 1 ikj e |j (1.157) ψk = √ 3 j=1 where |j is the 1S orbital for the jth atom. The equilateral triangle is like a linear system of length 3 with periodic boundary conditions. 'Hence & the allowed values of the wavevector are kα = 2π 3 α; α = −1, 0, +1 . The energy eigenvalues are kα = −E1S − 2J cos kα

(1.158)

where E1S is the isolated atom energy and −J is the hopping matrix element related to the orbital overlap and is exponentially small for large separations of the atoms. The projection operator that measures whether or not the particle is on site n is (1.159) Pn ≡ |n n|. Its expectation value in any of the three eigenstates is ψkα |Pn |ψkα =

1 · 3

(1.160)

This equation simply reﬂects the fact that as the particle tunnels from site to site it is equally likely to be found on any site. Hence it will, on average, be found on a particular site n only 1/3 of the time. The average electron number per site is thus 1/3. This however is a trivial example because the

S.M. Girvin: The Quantum Hall Eﬀect

107

value of the measured charge is always an integer. Two-thirds of the time we measure zero and one third of the time we measure unity. This means that the charge ﬂuctuates. One measure of the ﬂuctuations is √ 1 1 2 − = , (1.161) Pn2 − Pn 2 = 3 9 3 which shows that the ﬂuctuations are larger than the mean value. This result is most easily obtained by noting Pn2 = Pn . e A characteristic feature of this “imposter” fractional charge m that guarantees that it ﬂuctuates is the existence in the spectrum of the Hamiltonian of a set of m nearly degenerate states. (In our toy example here, m = 3.) The characteristic time scale for the charge ﬂuctuations is τ ∼ ¯h/∆ where ∆ is the energy splitting of the quasi-degenerate manifold of states. In our tight-binding example τ ∼ ¯ h/J is the characteristic time it takes an electron to tunnel from the 1S orbital on one site to the next. As the separation between the sites increases this tunneling time grows exponentially large and the charge ﬂuctuations become exponentially slow and thus easy to detect. In a certain precise sense, the fractional charge of the Laughlin quasiparticles behaves very diﬀerently from this. An electron added at low energies to a ν = 1/3 quantum Hall ﬂuid breaks up into three charge 1/3 Laughlin quasiparticles. These quasiparticles can move arbitrarily far apart from each other15 and yet no quasi-degenerate manifold of states appears. The excitation gap to the ﬁrst excited state remains ﬁnite. The only degeneracy is that associated with the positions of the quasiparticles. If we imagine that there are three impurity potentials that pin down the positions of the three quasiparticles, then the state of the system is uniquely speciﬁed. Because there is no quasidegeneracy, we do not have to specify any more information other than the positions of the quasiparticles. Hence in a deep sense, they are true elementary particles whose fractional charge is a sharp quantum observable. Of course, since the system is made up only of electrons, if we capture the charges in some region in a box, we will always get an integer number of electrons inside the box. However in order to close the box we have to locally destroy the Laughlin state. This will cost (at a minimum) the excitation gap. This may not seem important since the gap is small — only a few Kelvin or so. But imagine that the gap were an MeV or a GeV. Then we would have to build a particle accelerator to “close the box” and probe the ﬂuctuations in the charge. These ﬂuctuations would be analogous to the ones seen in quantum electrodynamics at energies above 2me c2 where electron-positron pairs are produced during the measurement of charge form factors by means of a scattering experiment. 15 Recall

that unlike the case of vortices in superﬂuids, these objects are unconﬁned.

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Topological Aspects of Low Dimensional Systems

Put another way, the charge of the Laughlin quasiparticle ﬂuctuates but only at high frequencies ∼ ∆/¯ h. If this frequency (which is ∼ 50 GHz) is higher than the frequency response limit of our voltage probes, we will see no charge ﬂuctuations. We can formalize this by writing a modiﬁed projection operator [35] for the charge on some site n by Pn(Ω) ≡ P Ω Pn P Ω

(1.162)

where Pn = |n n| as before and P (Ω) ≡ θ(Ω − H + E0 )

(1.163)

is the operator that projects onto the subset of eigenstates with excitation (Ω) energies less than Ω. Pn thus represents a measurement with a highfrequency cutoﬀ built in to represent the ﬁnite bandwidth of the detector. Returning to our tight-binding example, consider the situation where J is

large enough that the excitation gap ∆ = 1 − cos 2π 3 J exceeds the cutoﬀ Ω. Then P (Ω)

=

+1

|ψkα θ(Ω − kα + k0 ) ψkα |

α=−1

= |ψk0 ψk0 |

(1.164)

is simply a projector on the ground state. In this case Pn(Ω) = |ψk0 and

1 ψk0 | 3

2 ψk0 [Pn(Ω) ]2 ψk0 − ψk0 |Pn(Ω) |ψk0 = 0.

(1.165)

(1.166)

The charge ﬂuctuations in the ground state are then zero (as measured by the ﬁnite bandwidth detector). The argument for the Laughlin quasiparticles is similar. We again emphasize that one can not think of a single charge tunneling among three sites because the excitation gap remains ﬁnite no matter how far apart the quasiparticle sites are located. This is possible only because it is a correlated many-particle system. To gain a better understanding of fractional charge it is useful to compare this situation to that in high energy physics. In that ﬁeld of study one knows the physics at low energies – this is just the phenomena of our everyday world. The goal is to study the high energy (short length scale) limit to see where this low energy physics comes from. What force laws lead to our world? Probing the proton with high energy electrons we can temporarily break it up into three fractionally charged quarks, for example.

S.M. Girvin: The Quantum Hall Eﬀect

109

Condensed matter physics in a sense does the reverse. We know the phenomena at “high” energies (i.e. room temperature) and we would like to see how the known dynamics (Coulomb’s law and non-relativistic quantum mechanics) leads to unknown and surprising collective eﬀects at low temperatures and long length scales. The analog of the particle accelerator is the dilution refrigerator. To further understand Laughlin quasiparticles consider the point of view of “ﬂatland” physicists living in the cold, two-dimensional world of a ν = 1/3 quantum Hall sample. As far as the ﬂatlanders are concerned the “vacuum” (the Laughlin liquid) is completely inert and featureless. They discover however that the universe is not completely empty. There are a few elementary particles around, all having the same charge q. The ﬂatland equivalent of Benjamin Franklin chooses a unit of charge which not only makes q negative but gives it the fractional value −1/3. For some reason the Flatlanders go along with this. Flatland cosmologists theorize that these objects are “cosmic strings”, topological defects left over from the “big cool down” that followed the creation of the universe. Flatland experimentalists call for the creation of a national accelerator facility which will reach the unprecedented energy scale of 10 Kelvin. With great eﬀort and expense this energy scale is reached and the accelerator is used to smash together three charged particles. To the astonishment of the entire world a new short-lived particle is temporarily created with the bizarre property of having integer charge! There is another way to see that the Laughlin quasiparticles carry fractional charge which is useful to understand because it shows the deep connection between the sharp fractional charge and the sharp quantization of the Hall conductivity. Imagine piercing the sample with an inﬁnitely thin magnetic solenoid as shown in Figure 1.18 and slowly increasing the magnetic ﬂux Φ from 0 to Φ0 = hc e the quantum of ﬂux. Because of the existence of a ﬁnite excitation gap ∆ the process is adiabatic and reversible if performed slowly on a time scale long compared to h ¯ /∆. Faraday’s law tells us that the changing ﬂux induces an electric ﬁeld obeying ( = − 1 ∂Φ (1.167) dr · E c ∂t Γ where Γ is any contour surrounding the ﬂux tube. Because the electric ﬁeld contains only Fourier components at frequencies ω obeying h ¯ ω < ∆, there is no dissipation and σxx = σyy = ρxx = ρyy = 0. The electric ﬁeld induces a current density obeying = ρxy J × zˆ (1.168) E (

so that ρxy

Γ

1 dΦ J · (ˆ z × dr) = − . c dt

(1.169)

110

Topological Aspects of Low Dimensional Systems

Φ (t)

J(t) E(t)

Fig. 1.18. Construction of a Laughlin quasiparticle by adiabatically threading ﬂux Φ(t) through a point in the sample. Faraday induction gives an azimuthal electric ﬁeld E(t) which in turn produces a radial current J(t). For each quantum of ﬂux added, charge νe ﬂows into (or out of) the region due to the quantized Hall conductivity νe2 /h. A ﬂux tube containing an integer number of ﬂux quanta is invisible to the particles (since the Aharanov phase shift is an integer multiple of 2π) and so can be removed by a singular gauge transformation.

The integral on the LHS represents the total current ﬂowing into the region enclosed by the contour. Thus the charge inside this region obeys ρxy

1 dΦ dQ =− · dt c dt

(1.170)

After one quantum of ﬂux has been added the ﬁnal charge is Q=

h 1 σxy Φ0 = σxy · c e

(1.171) 2

Thus on the quantized Hall plateau at ﬁlling factor ν where σxy = ν eh we have the result Q = νe. (1.172) Reversing the sign of the added ﬂux would reverse the sign of the charge. The ﬁnal step in the argument is to note that an inﬁnitesimal tube containing a quantum of ﬂux is invisible to the particles. This is because

S.M. Girvin: The Quantum Hall Eﬀect

111

the Aharonov-Bohm phase factor for traveling around the ﬂux tube is unity. ( e exp i δ A · dr = e±2πi = 1. (1.173) hc Γ ¯ is the additional vector potential due to the solenoid. Assuming Here δ A the ﬂux tube is located at the origin and making the gauge choice = Φ0 δA

θˆ , 2πr

(1.174)

one can see by direct substitution into the Schr¨ odinger equation that the only eﬀect of the quantized ﬂux tube is to change the phase of the wave function by zj ψ→ψ eiθj . (1.175) =ψ |zj | j

j

The removal of a quantized ﬂux tube is thus a “singular gauge change” which has no physical eﬀect. Let us reiterate. Adiabatic insertion of a ﬂux quantum changes the state of the system by pulling in (or pushing out) a (fractionally) quantized amount of charge. Once the ﬂux tube contains a quantum of ﬂux it eﬀectively becomes invisible to the electrons and can be removed by means of a singular gauge transformation. Because the excitation gap is preserved during the adiabatic addition of the ﬂux, the state of the system is fully speciﬁed by the position of the resulting quasiparticle. As discussed before there are no low-lying quasidegenerate states. This version of the argument highlights the essential importance of the fact that σxx = 0 and σxy is quantized. The existence of the fractionally quantized Hall transport coeﬃcients guarantees the existence of fractionally charged elementary excitations These fractionally charged objects have been observed directly by using an ultrasensitive electrometer made from a quantum dot [36] and by the reduced shot noise which they produce when they carry current [37]. Because the Laughlin quasiparticles are discrete objects they cost a nonzero (but ﬁnite) energy to produce. Since they are charged they can be thermally excited only in neutral pairs. The charge excitation gap is therefore (1.176) ∆c = ∆+ + ∆− where ∆± is the vortex/antivortex (quasielectron/quasihole) excitation energy. In the presence of a transport current these thermally excited charges can move under the inﬂuence of the Hall electric ﬁeld and dissipate energy. The resulting resistivity has the Arrhenius form ρxx ∼ γ

h −β∆c /2 e e2

(1.177)

112

Topological Aspects of Low Dimensional Systems

where γ is a dimensionless constant of order unity. Note that the law of mass action tells us that the activation energy is ∆c /2 not ∆c since the charges are excited in pairs. There is a close analogy between the dissipation described here and the ﬂux ﬂow resistance caused by vortices in a superconducting ﬁlm. Theoretical estimates of ∆c are in good agreement with experimental values determined from transport measurements [38]. Typical values of ∆c are only a few percent of e2 / and hence no larger than a few Kelvin. In a superﬂuid time-reversal symmetry guarantees that vortices and antivortices have equal energies. The lack of time reversal symmetry here means that ∆+ and ∆− can be quite diﬀerent. Consider for example the hard-core model for which the Laughlin wave function ψm is an exact zero energy ground state as shown in equation (1.107). Equation (1.152) shows that the quasihole state contains ψm as a factor and hence is also an exact zero energy eigenstate for the hard-core interaction. Thus the quasihole costs zero energy. On the other hand equation (1.153) tells us that the derivatives reduce the degree of homogeneity of the Laughlin polynomial and therefore the energy of the quasielectron must be non-zero in the hard-core model. At ﬁlling factor ν = 1/m this asymmetry has no particular signiﬁcance since the quasiparticles must be excited in pairs. Consider now what happens when the magnetic ﬁeld is increased slightly or the particle number is decreased slightly so that the ﬁlling factor is slightly smaller than 1/m. The lowest energy way to accommodate this is to inject m quasiholes into the Laughlin state for each electron that is removed (or for each mΦ0 of ﬂux that is added). The system energy (ignoring disorder and interactions in the dilute gas of quasiparticles) is E+ = Em − δN m∆+

(1.178)

where Em is the Laughlin ground state energy and −δN is the number of added holes. Conversely for ﬁlling factors slightly greater than 1/m the energy is (with +δN being the number of added electrons) E− = Em + δN m∆− .

(1.179)

This is illustrated in Figure 1.19. The slope of the lines in the ﬁgure determines the chemical potential µ± =

∂E± = ∓m∆± . ∂δN

(1.180)

The chemical potential suﬀers a jump discontinuity of m(∆+ + ∆− ) = m∆c just at ﬁlling factor µ = 1/m. This jump in the chemical potential is the signature of the charge excitation gap just as it is in a semiconductor or insulator. Notice that this form of the energy is very reminiscent of the

S.M. Girvin: The Quantum Hall Eﬀect

113

Ε(δΝ) m ∆ − δΝ m ∆ + δΝ

ν=1/m

δΝ

Fig. 1.19. Energy cost for inserting δN electrons into the Laughlin state near ﬁlling factor ν = 1/m. The slope of the line is the chemical potential. Its discontinuity at ν = 1/m measures the charge excitation gap.

energy of a type-II superconductor as a function of the applied magnetic ﬁeld (which induces vortices and therefore has an energy cost ∆E ∼ |B|). Recall that in order to have a quantized Hall plateau of ﬁnite width it is necessary to have disorder present. For the integer case we found that disorder localizes the excess electrons allowing the transport coeﬃcients to not change with the ﬁlling factor. Here it is the fractionally-charged quasiparticles that are localized by the disorder16 . Just as in the integer case the disorder may ﬁll in the gap in the density of states but the DC value of σxx can remain zero because of the localization. Thus the fractional plateaus can have ﬁnite width. If the density of quasiparticles becomes too high they may delocalize and condense into a correlated Laughlin state of their own. This gives rise to a hierarchical family of Hall plateaus at rational fractional ﬁlling factors ν = p/q (generically with q odd due to the Pauli principle). There are several diﬀerent but entirely equivalent ways of constructing and viewing this hierarchy which we will not delve into here [3, 4, 6]. 1.14 FQHE edge states We learned in our study of the integer QHE that gapless edge excitations exist even when the bulk has a large excitation gap. Because the bulk is incompressible the only gapless neutral excitations must be area-preserving shape distortions such as those illustrated for a disk geometry in Figure 1.20a. 16 Note again the essential importance of the fact that the objects are “elementary particles”. That is, there are no residual degeneracies once the positions are pinned down.

114

Topological Aspects of Low Dimensional Systems

Because of the conﬁning potential at the edges these shape distortions have ×B drift. It is possible to show a characteristic velocity produced by the E that this view of the gapless neutral excitations is precisely equivalent to the usual Fermi gas particle-hole pair excitations that we considered previously in our discussion of edge states. Recall that we argued that the contour line of the electrostatic potential separating the occupied from the empty states could be viewed as a real-space analog of the Fermi surface (since position and momentum are equivalent in the Landau gauge). The charged excitations at the edge are simply ordinary electrons added or removed from the vicinity of the edge. In the case of a fractional QHE state at ν = 1/m the bulk gap is caused by Coulomb correlations and is smaller but still ﬁnite. Again the only gapless excitations are area-preserving shape distortions. Now however the charge of each edge can be varied in units of e/m. Consider the annulus of Hall ﬂuid shown in Figure 1.20b. The extension of the Laughlin wave function ψm to this situation is   N ψmn [z] =  zjn  ψm . (1.181) j=1

This simply places a large number n 1 of quasiholes at the origin. Following the plasma analogy we see that this looks like a highly charged impurity at the origin which repels the plasma, producing the annulus shown in Figure 1.20b. Each time we increase n by one unit, the annulus expands. We can view this expansion as increasing the electron number at the outer edge by 1/m and reducing it by 1/m at the inner edge. (Thereby keeping the total electron number integral as it must be.) It is appropriate to view the Laughlin quasiparticles, which are gapped in the bulk, as being liberated at the edge. The gapless shape distortions in the Hall liquid are thus excitations in a “gas” of fractionally charged quasiparticles. This fact produces a profound alteration in the tunneling density of states to inject an electron into the system. An electron which is suddenly added to an edge (by tunneling through a barrier from an external electrode) will have very high energy unless it breaks up into m Laughlin quasiparticles. This leads to an “orthogonality catastrophe” which simply means that the probability for this process is smaller and smaller for ﬁnal states of lower and lower energy. As a result the current-voltage characteristic for the tunnel junction becomes non-linear [17, 39, 40] I ∼ V m.

(1.182)

For the ﬁlled Landau level m = 1 the quasiparticles have charge q = em = e and are ordinary electrons. Hence there is no orthogonality catastrophe and the I-V characteristic is linear as expected for an ordinary metallic

S.M. Girvin: The Quantum Hall Eﬀect

115

(a)

Φ

(b)

Fig. 1.20. Area-preserving shape distortions of the incompressible quantum Hall state. (a) IQHE Laughlin liquid “droplet” at ν = 1. (b) FQHE annulus at ν = 1/m formed by injecting a large number n of ﬂux quanta at the origin to create n quasiholes. There are thus two edge modes of opposite chirality. Changing n by one unit transfers fractional charge νe from one edge to the other by expanding or shrinking the size of the central hole. Thus the edge modes have topological sectors labeled by the “winding number” n and one can view the gapless edge excitations as a gas of fractionally charged Laughlin quasiparticles.

116

Topological Aspects of Low Dimensional Systems

10-7

10-6

10-7

10

10-5

10-4

10-3

I-V Theory Resistor model

10-2

10-1

11 T

-9

I (A)

10-11 10-13 10-15 10-7

10-9 I (A)

10-11

7T 0.5 T increments 9T 11 T -7

10

10-13

15 T

10

-6

10

13 T -5

10-4

10-3

10-2

10-15 10-1

V (V)

Fig. 1.21. Non-linear current voltage response for tunneling an electron into a FQHE edge state. Because the electron must break up into m fractionally charged quasiparticles, there is an orthogonality catastrophe leading to a power-law density of states. The ﬂattening at low currents is due to the ﬁnite temperature. The upper panel shows the ν = 1/3 Hall plateau. The theory [17, 39] works extremely well on the 1/3 quantized Hall plateau, but the unexpectedly smooth variation of the exponent with magnetic ﬁeld away from the plateau shown in the lower panel is not yet fully understood. (After M. Grayson, et al., reference [41].)

tunnel junction. The non-linear tunneling for the m = 3 state is shown in Figure 1.21. 1.15 Quantum hall ferromagnets Naively one might imagine that electrons in the QHE have their spin dynamics frozen out by the Zeeman splitting gµB B. In free space with g = 2 (neglecting QED corrections) the Zeeman splitting is exactly equal to the cyclotron splitting h ¯ ωc ∼ 100 K as illustrated in Figure 1.22a. Thus at low

S.M. Girvin: The Quantum Hall Eﬀect

117

4+ 4 43+ 3

11 1+

32+

2 21+ 1 10+

00 0+

0 0-

(a)

(b)

Fig. 1.22. (a) Landau energy levels for an electron in free space. Numbers label the Landau levels and +(−) refers to spin up (down). Since the g factor is 2, the Zeeman splitting is exactly equal to the Landau level spacing, ¯ hωc and there are extra degeneracies as indicated. (b) Same for an electron in GaAs. Because the eﬀective mass is small and g ≈ −0.4, the degeneracy is strongly lifted and the spin assignments are reversed.

temperatures we would expect for ﬁlling factors ν < 1 all the spins would be fully aligned. It turns out however that this naive expectation is incorrect in GaAs for two reasons. First, the small eﬀective mass (m∗ = 0.068) in the conduction band of GaAs increases the cyclotron energy by a factor of m/m∗ ∼ 14. Second, spin-orbit scattering tumbles the spins around in a way which reduces their eﬀective coupling to the external magnetic ﬁeld by a factor of −5 making the g factor −0.4. The Zeeman energy is thus some 70 times smaller than the cyclotron energy and typically has a value of about 2 K, as indicated in Figure 1.22b. This decoupling of the scales of the orbital and spin energies means that it is possible to be in a regime in which the orbital motion is fully quantized hωc ) but the low-energy spin ﬂuctuations are not completely frozen (kB T ¯ out (kB T ∼ g ∗ µB B). The spin dynamics in this regime are extremely unusual and interesting because the system is an itinerant magnet with a

118

Topological Aspects of Low Dimensional Systems

quantized Hall coeﬃcient. As we shall see, this leads to quite novel physical eﬀects. The introduction of the spin degree of freedom means that we are dealing with the QHE in multicomponent systems. This subject has a long history going back to an early paper by Halperin [42] and has been reviewed extensively [4, 43, 44]. In addition to the spin degree of freedom there has been considerable recent interest in other multicomponent systems in which spin is replaced by a pseudo-spin representing the layer index in double well QHE systems or the electric subband index in wide single well systems. Experiments on these systems are discussed by Shayegan in this volume [45] and have also been reviewed in [44]. Our discussion will focus primarily on ferromagnetism near ﬁlling factor ν = 1. In the subsequent section we will address analogous eﬀects for pseudo-spin degrees of freedom in multilayer systems. 1.16 Coulomb exchange We tend to think of the integer QHE as being associated with the gap due to the kinetic energy and ascribe importance to the Coulomb interaction only in the fractional QHE. However study of ferromagnetism near integer ﬁlling factor ν = 1 has taught us that Coulomb interactions play an important role there as well [46]. Magnetism occurs not because of direct magnetic forces, but rather because of a combination of electrostatic forces and the Pauli principle. In a fully ferromagnetically aligned state all the spins are parallel and hence the spin part of the wave function is exchange symmetric |ψ = Φ(z1 , . . . , zN ) | ↑↑↑↑↑ . . . ↑·

(1.183)

The spatial part Φ of the wave function must therefore be fully antisymmetric and vanish when any two particles approach each other. This means that each particle is surrounded by an “exchange hole” which thus lowers the Coulomb energy per particle as shown in equation (1.113). For ﬁlling factor ν = 1 π e2 V =− ∼ 200 K. (1.184) N 8 This energy scale is two orders of magnitude larger than the Zeeman splitting and hence strongly stabilizes the ferromagnetic state. Indeed at ν = 1 the ground state is spontaneously fully polarized at zero temperature even in the absence of the Zeeman term. Ordinary ferromagnets like iron are generally only partially polarized because of the extra kinetic energy cost of raising the fermi level for the majority carriers. Here however the kinetic energy has been quenched by the magnetic ﬁeld and all states in the lowest Landau level are degenerate. For ν = 1 the large gap to the next Landau

S.M. Girvin: The Quantum Hall Eﬀect

119

level means that we know the spatial wave function Φ essentially exactly. It is simply the single Slater determinant representing the fully ﬁlled Landau level. That is, it is m = 1 Laughlin wave function. This simple circumstance makes this perhaps the world’s best understood ferromagnet.

1.17 Spin wave excitations It turns out that the low-lying “magnon” (spin wave) excited states can also be obtained exactly. Before doing this for the QHE system let us remind ourselves how the calculation goes in the lattice Heisenberg model for N local moments in an insulating ferromagnet H

i · S j − ∆ S

Sjz

=

−J

=

1 + − − + z z Si Sj + Si Sj −J Sjz . (1.185) Si Sj + −∆ 2 j

j

ij

ij

The ground state for J > 0 is the fully ferromagnetic state with total spin S = N/2. Let us choose our coordinates in spin space so that Sz = N/2. Because the spins are fully aligned the spin-ﬂip terms in H are ineﬀective and (ignoring the Zeeman term) J H | ↑↑↑ . . . ↑ = − Nb | ↑↑↑ . . . ↑ 4

(1.186)

¯ = 1. where Nb is the number of near-neighbor bonds and we have set h There are of course 2S + 1 = N + 1 other states of the same total spin which will be degenerate in the absence of the Zeeman coupling. These are generated by successive applications of the total spin lowering operator S−

≡

N

j=1

Sj−

S − | ↑↑↑ . . . ↑ = | ↓↑↑ . . . ↑ + | ↑↓↑ . . . ↑ + | ↑↑↓ . . . ↑ + . . .

(1.187)

(1.188)

It is not hard to show that the one-magnon excited states are created by a closely related operator Sq− =

N

j=1

e−iq·Rj Sj−

(1.189)

120

Topological Aspects of Low Dimensional Systems

where q lies inside the Brillouin zone and is the magnon wave vector17 . Denote these states by (1.190) |ψq = Sq− |ψ0 where |ψ0 is the ground state. Because there is one ﬂipped spin in these states the transverse part of the Heisenberg interaction is able to move the ﬂipped spin from one site to a neighboring site Jz E0 + ∆ + H|ψq = |ψq 2 −

N J −iq·R j − e S |ψ0 j+δ 2 j=1

(1.191)

δ

H|ψq =

(E0 + q) |ψq

(1.192)

where z is the coordination number, δ is summed over near neighbor lattice vectors and the magnon energy is   1 −iq·δ  Jz  q ≡ + ∆. (1.193) 1− e  2  z δ

For small q the dispersion is quadratic and for a 2D square lattice q ∼

Ja2 2 q +∆ 4

(1.194)

where a is the lattice constant. This is very diﬀerent from the result for the antiferromagnet which has a linearly dispersing collective mode. There the ground and excited states can only be approximately determined because the ground state does not have all the spins parallel and so is subject to quantum ﬂuctuations induced by the transverse part of the interaction. This physics will reappear when we study non-collinear states in QHE magnets away from ﬁlling factor ν = 1. The magnon dispersion for the ferromagnet can be understood in terms of bosonic “particle” (the ﬂipped spin) hopping on the lattice with a tightbinding model dispersion relation. The magnons are bosons because spin operators on diﬀerent sites commute. They are not free bosons however because of the hard core constraint that (for spin 1/2) there can be no more than one ﬂipped spin per site. Hence multi-magnon excited states can not be computed exactly. Some nice renormalization group arguments about magnon interactions can be found in [47].

q ·R j 17 We use the phase factor e−i here rather than e+iq·Rj simply to be consistent with Sq− being the Fourier transform of Sj− .

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121

The QHE ferromagnet is itinerant and we have to develop a somewhat diﬀerent picture. Nevertheless there will be strong similarities to the lattice Heisenberg model. The exact ground state is given by equation (1.183) with 2 1 (zi − zj ) e− 4 k |zk | . (1.195) Φ(z1 , . . . , zN ) = i<j

To ﬁnd the spin wave excited states we need to ﬁnd the analog of equation (1.190). The Fourier transform of the spin lowering operator for the continuum system is Sq−

≡

N

j=1

e−iq·rj Sj−

(1.196)

where rj is the position operator for the jth particle. Recall from equation (1.144) that we had to modify Feynman’s theory of the collective mode in superﬂuid helium by projecting the density operator onto the Hilbert space of the lowest Landau level. This suggests that we do the same in equation (1.196) to obtain the projected spin ﬂip operator. In contrast to the good but approximate result we obtained for the collective density mode, this procedure actually yields the exact one-magnon excited state (much like we found for the lattice model). Using the results of Appendix A, the projected spin lowering operator is N

− − 14 |q|2 ¯ Sq = e τq (j) Sj− (1.197) j=1

where q is the complex number representing the dimensionless wave vector q and τq (j) is the magnetic translation operator for the j-th particle. The commutator of this operator with the Coulomb interaction Hamiltonian is [H, S¯q− ] =

) * 1 v(k) ρ¯−k ρ¯k , S¯q− 2 k=0

=

) * ) * ' & 1 v(k) ρ¯−k ρ¯k , S¯q− + ρ¯−k , S¯q− ρ¯k . 2

(1.198)

k=0

We will shortly be applying this to the fully polarized ground state |ψ. As discussed in Appendix A, no density wave excitations are allowed in this state and so it is annihilated by ρ¯k . Hence we can without approximation drop the second term above and replace the ﬁrst one by ** ) ) 1 v(k) ρ¯−k , ρ¯k , S¯q− |ψ· [H, S¯q− ] |ψ = 2 k=0

(1.199)

122

Topological Aspects of Low Dimensional Systems

Evaluation of the double commutator following the rules in Appendix A yields (1.200) [H, S¯q− ] |ψ = q S¯q− |ψ where q ≡ 2

2

1

e− 2 |k| v(k) sin2

k=0

1 q∧k . 2

(1.201)

Since |ψ is an eigenstate of H, this proves that S¯q− |ψ is an exact excited state of H with excitation energy q . In the presence of the Zeeman coupling q → q + ∆. This result tells us that, unlike the case of the density excitation, the single-mode approximation is exact for the case of the spin density excitation. The only assumption we made is that the ground state is fully polarized and has ν = 1. For small q the dispersion starts out quadratically q ∼ Aq 2 with A≡

(1.202)

1 − 1 |k|2 e 2 v(k) |k|2 4

(1.203)

k=0

as can be seen by expanding the sine function to lowest order. For very large q sin2 can be replaced by its average value of 12 to yield q ∼

1

2

v(k) e− 2 |k| .

(1.204)

k=0

Thus the energy saturates at a constant value for q → ∞ as shown in Figure 1.23. (Note that in the lattice model the wave vectors are restricted to the ﬁrst Brillouin zone, but here they are not.) While the derivation of this exact result for the spin wave dispersion is algebraically rather simple and looks quite similar (except for the LLL projection) to the result for the lattice Heisenberg model, it does not give a very clear physical picture of the nature of the spin wave collective mode. This we can obtain from equation (1.197) by noting that τq (j) translates the particle a distance q × zˆ 2 . Hence the spin wave operator S¯q− ﬂips the spin of one of the particles and translates it spatially leaving a hole behind and creating a particle-hole pair carrying net momentum proportional to their separation as illustrated in Figure 1.24. For large separations the excitonic Coulomb attraction between the particle and hole is negligible and the energy cost saturates at a value related to the Coulomb exchange energy of the ground state given in equation (1.113). The exact dispersion relation can also be obtained by noting that scattering processes of the type

S.M. Girvin: The Quantum Hall Eﬀect

123

ε

∆x ∆z

k Fig. 1.23. Schematic illustration of the QHE ferromagnet spinwave dispersion. There is a gap at small k equal to the Zeeman splitting, ∆Z . At large wave vectors, the energy saturates at the Coulomb exchange energy scale ∆x + ∆Z ∼ 100 K.

Fig. 1.24. Illustration of the fact that the spin ﬂip operator causes translations when projected into the lowest Landau level. For very large wave vectors the particles is translated completely away from the exchange hole and loses all its favorable Coulomb exchange energy.

illustrated by the dashed lines in Figure 1.24 mix together Landau gauge states c†k−qy ,↓ ck,↑ | ↑↑↑↑↑↑

(1.205)

with diﬀerent wave vectors k. Requiring that the state be an eigenvector of translation uniquely restricts the mixing to linear combinations of the form

k

2

e−ikqx c†k−qy ,↓ ck,↑ | ↑↑↑↑↑↑·

(1.206)

Evaluation of the Coulomb matrix elements shows that this is indeed an exact eigenstate.

124

Topological Aspects of Low Dimensional Systems

1.18 Eﬀective action It is useful to try to reproduce these microscopic results for the spin wave excitations within an eﬀective ﬁeld theory for the spin degrees of freedom. Let m( r ) be a vector ﬁeld obeying m ·m = 1 which describes the local orientation of the order parameter (the magnetization). Because the Coulomb forces are spin independent, the potential energy cost can not depend on the orientation of m but only on its gradients. Hence we must have to leading order in a gradient expansion 1 1 (1.207) U = ρs d2 r ∂µ mν ∂µ mν − n∆ d2 r mz 2 2 where ρs is a phenomenological “spin stiﬀness” which in two dimensions has ν units of energy and n ≡ 2π 2 is the particle density. We will learn how to evaluate it later. We can think of this expression for the energy as the leading terms in a functional Taylor series expansion. Symmetry requires that (except for the Zeeman term) the expression for the energy be invariant under uniform global rotations of m. In addition, in the absence of disorder, it must be translationally invariant. Clearly the expression in (1.207) satisﬁes these symmetries. The only zero-derivative term of the appropriate symmetry is mµ mµ which is constrained to be unity everywhere. There exist terms with more derivatives but these are irrelevant to the physics at very long wavelengths. (Such terms have been discussed by Read and Sachdev [47].) To understand how time derivatives enter the eﬀective action we have to recall that spins obey a ﬁrst-order (in time) precession equation under the inﬂuence of the local exchange ﬁeld18 . Consider as a toy model a single spin in an external ﬁeld ∆. H = −¯ h∆α S α .

(1.208)

The Lagrangian describing this toy model needs to contain a ﬁrst order time derivative and so must have the form (see discussion in Appendix B) L=h ¯ S {−m ˙ µ Aµ [m] + ∆µ mµ + λ(mµ mµ − 1)}

(1.209)

where S = 12 is the spin length and λ is a Lagrange multiplier to enforce can be determined the ﬁxed length constraint. The unknown vector A by requiring that it reproduce the correct precession equation of motion. 18 That

is, the Coulomb exchange energy which tries to keep the spins locally parallel. In a Hartree-Fock picture we could represent this by a term of the form −h( r) · s( r) where h( r ) is the self-consistent ﬁeld.

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125

The precession equation is d µ S dt

i [H, S µ ] = −i∆α [S α , S µ ] h ¯ = αµβ ∆α S β

(1.210)

×S = −∆

(1.211)

=

˙ S

which corresponds to counterclockwise precession around the magnetic ﬁeld. We must obtain the same equation of motion from the Euler-Lagrange equation for the Lagrangian in equation (1.209) d δL δL − =0 dt δ m ˙µ δmµ

(1.212)

∆µ + 2λmµ = F µν m ˙ν

(1.213)

F µν ≡ ∂µ Aν − ∂ν Aµ

(1.214)

which may be written as

where and ∂µ means nate). Since F

∂ ∂mµ µν

(not the derivative with respect to some spatial coordiis antisymmetric let us guess a solution of the form F µν = αµν mα .

(1.215)

Using this in equation (1.213) yields ∆µ + 2λmµ = αµν mα m ˙ ν.

(1.216)

Applying γβµ mβ to both sides and using the identity

we obtain

ναβ νλη = δαλ δβη − δαη δβλ

(1.217)

× m) −(∆ γ =m ˙ γ − mγ (m ˙ β mβ ).

(1.218)

The last term on the right vanishes due to the length constraint. Thus we ﬁnd that our ansatz in equation (1.215) does indeed make the EulerLagrange equation correctly reproduce equation (1.211). Equation (1.215) is equivalent to m] m × A[ =m ∇

(1.219)

is the vector potential of a unit magnetic monopole sitindicating that A ting at the center of the unit sphere on which m lives as illustrated in Figure 1.25. Note (the always confusing point) that we are interpreting m

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Topological Aspects of Low Dimensional Systems

Fig. 1.25. Magnetic monopole in spin space. Arrows indicate the curl of the Berry ×A emanating from the origin. Shaded region indicates closed path connection ∇ m(t) taken by the spin order parameter during which it acquires a Berry phase proportional to the monopole ﬂux passing through the shaded region.

as the coordinate of a ﬁctitious particle living on the unit sphere (in spin space) surrounding the monopole. Recalling equation (1.20), we see that the Lagrangian for a single spin in equation (1.209) is equivalent to the Lagrangian of a massless object of charge −S, located at position m, moving on the unit sphere containing a magnetic monopole. The Zeeman term represents a constant electric ﬁeld producing a force ∆S on the particle. The Lorentz force caused by −∆ the monopole causes the particle to orbit the sphere at constant “latitude”. ˙ α enters the Lagrangian, the Because no kinetic term of the form m ˙ αm charged particle is massless and so lies only in the lowest Landau level of the monopole ﬁeld. Note the similarity here to the previous discussion of the high ﬁeld limit and the semiclassical percolation picture of the integer Hall eﬀect. For further details the reader is directed to Appendix B and to Haldane’s discussion of monopole spherical harmonics [48]. If the “charge” moves slowly around a closed counterclockwise path m(t) during the time interval [0, T ] as illustrated in Figure 1.25, the quantum amplitude i

e h¯

T

0

dtL

(1.220)

contains a Berry’s phase [49] contribution proportional to the “magnetic ﬂux” enclosed by the path T + −iS dtm ˙ ν Aν 0 e = e−iS A·dm . (1.221)

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127

As discussed in Appendix B, this is a purely geometric phase in the sense that it depends only on the geometry of the path and not the rate at which the path is traversed (since the expression is time reparameterization invariant). Using Stokes theorem and equation (1.219) we can write the contour integral as a surface integral + = e−iS dΩ·∇×A = e−iSΩ (1.222) e−iS A·dm = mdΩ where dΩ is the directed area (solid angle) element and Ω is the total solid angle subtended by the contour as viewed from the position of the monopole. Note from Figure 1.25 that there is an ambiguity on the sphere as to which is the inside and which is the outside of the contour. Since the total solid angle is 4π we could equally well have obtained19 e+iS(4π−Ω) .

(1.223)

Thus the phase is ambiguous unless S is an integer or half-integer. This constitutes a “proof” that the quantum spin length must be quantized. Having obtained the correct Lagrangian for our toy model we can now readily generalize it to the spin wave problem using the potential energy in equation (1.207) , L = −¯ hSn 1 − ρs 2

d2 r 2

m ˙ µ (r ) Aµ [m] − ∆mz (r ) ν

ν

d r ∂µ m ∂µ m +

d2 r λ(r ) (mµ mµ − 1). (1.224)

The classical equation of motion can be analyzed just as for the toy model, however we will take a slightly diﬀerent approach here. Let us look in the low energy sector where the spins all lie close to the zˆ direction. Then we can write √

m = m x , my , 1 − mx mx − my my 1 1 (1.225) ≈ mx , my , 1 − mx mx − my my . 2 2 Now choose the “symmetric gauge” ≈ 1 (−my , mx , 0) A 2

(1.226)

19 The change in the sign from +i to −i is due to the fact that the contour switches from being counterclockwise to clockwise if viewed as enclosing the 4π − Ω area instead of the Ω area.

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Topological Aspects of Low Dimensional Systems

which obeys equation (1.219) for m close to zˆ. Keeping only quadratic terms in the Lagrangian we obtain 1 y x 2 (m ˙ m −m ˙ x my ) L = −¯ hSn d r 2 1 x x 1 y y −∆ 1 − m m − m m 2 2 1 − ρs d2 r (∂µ mx ∂µ mx + ∂µ my ∂µ my ). 2

(1.227)

This can be conveniently rewritten by deﬁning a complex ﬁeld ψ ≡ mx + imy L =

. / 1 ∗ ∂ ∂ −Sn¯ h d2 r ψ −i ψ − ψ −i ψ∗ 4 ∂t ∂t 1 1 −∆ 1 − ψ ∗ ψ − ρs d2 r ∂µ ψ ∗ ∂µ ψ. (1.228) 2 2

The classical equation of motion is the Schr¨odinger like equation +i¯ h

∂ψ ρs 2 ¯ ∆ψ. =− ∂ ψ+h ∂t nS µ

(1.229)

This has plane wave solutions with quantum energy k = h ¯∆ +

ρs 2 k . nS

(1.230)

We can ﬁt the phenomenological stiﬀness to the exact dispersion relation in equation (1.202) to obtain ρs =

nS − 1 |k|2 e 2 v(k)|k|2 . 4

(1.231)

k=0

Exercise 1.20. Derive equation (1.231) from ﬁrst principles by evaluating the loss of exchange energy when the Landau gauge ν = 1 ground state is distorted to make the spin tumble in the x direction θk θk |ψ = cos c†k↑ + sin c†k↓ |0 (1.232) 2 2 k

∂θ is the (constant) spin rotation angle gradiwhere θk = −γk 2 and γ = ∂x 2 ent (since x = −k in this gauge).

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129

Fig. 1.26. Illustration of a skyrmion spin texture. The spin is down at the origin and gradually turns up at inﬁnite radius. At intermediate distances, the XY components of the spin exhibit a vortex-like winding. Unlike a U (1) vortex, there is no singularity at the origin.

1.19 Topological excitations So far we have studied neutral collective excitations that take the form of spin waves. They are neutral because as we have seen from equation (1.197) they consist of a particle-hole pair. For very large momenta the spin-ﬂipped particle is translated a large distance q × zˆ 2 away from its original position as discussed in Appendix A. This looks locally like a charged excitation but it is very expensive because it loses all of its exchange energy. It is sensible to inquire if it is possible to make a cheaper charged excitation. This can indeed be done by taking into account the desire of the spins to be locally parallel and producing a smooth topological defect in the spin orientation [46,50–56] known as a skyrmion by analogy with related objects in the Skyrme model of nuclear physics [57]. Such an object has the beautiful form exhibited in Figure 1.26. Rather than having a single spin suddenly ﬂip over, this object gradually turns over the spins as the center is approached. At intermediate distances the spins have a vortex-like conﬁguration. However unlike a U (1) vortex, there is no singularity in the core region because the spins are able to rotate downwards out of the xy plane. In nuclear physics the Skyrme model envisions that the vacuum is a “ferromagnet” described by a four component ﬁeld Φµ subject to the constraint Φµ Φµ = 1. There are three massless (i.e. linearly dispersing) spin wave excitations corresponding to the three directions of oscillation about the ordered direction. These three massless modes represent the three (nearly) massless pions π + , π 0 , π − . The nucleons (proton and neutron) are

130

Topological Aspects of Low Dimensional Systems

represented by skyrmion spin textures. Remarkably, it can be shown (for an appropriate form of the action) that these objects are fermions despite the fact that they are in a sense made up of a coherent superposition of (an inﬁnite number of) bosonic spin waves. We shall see a very similar phenomenology in QHE ferromagnets. At ﬁlling factor ν, skyrmions have charge ±νe and fractional statistics much like Laughlin quasiparticles. For ν = 1 these objects are fermions. Unlike Laughlin quasiparticles, skyrmions are extended objects, and they involve many ﬂipped (and partially ﬂipped) spins. This property has profound implications as we shall see. Let us begin our analysis by understanding how it is that spin textures can carry charge. It is clear from the Pauli principle that it is necessary to ﬂip at least some spins to locally increase the charge density in a ν = 1 ferromagnet. What is the suﬃcient condition on the spin distortions in order to have a density ﬂuctuation? Remarkably it turns out to be possible, as we shall see, to uniquely express the charge density solely in terms of gradients of the local spin orientation. Consider a ferromagnet with local spin orientation m( r ) which is static. As each electron travels we assume that the strong exchange ﬁeld keeps the spin following the local orientation m. If the electron has velocity x˙ µ , the rate of change of the local spin orientation it sees is m ˙ ν = x˙ µ ∂x∂ µ mν . This in turn induces an additional Berry’s phase as the spin orientation varies. Thus the single-particle Lagrangian contains an additional ﬁrst order time derivative in addition to the one induced by the magnetic ﬁeld coupling to the orbital motion e ¯Sm ˙ ν Aν [m]. (1.233) L0 = − x˙ µ Aµ + h c Here Aµ refers to the electromagnetic vector potential and Aν refers to the monopole vector potential obeying equation (1.219) and we have set the mass to zero (i.e. dropped the 12 M x˙ µ x˙ µ term). This can be rewritten e L0 = − x˙ µ (Aµ + aµ ) c where (with Φ0 being the ﬂux quantum) ∂ µ ν a ≡ −Φ0 S m Aν [m] ∂xµ

(1.234)

(1.235)

represents the “Berry connection”, an additional vector potential which reproduces the Berry phase. The additional fake magnetic ﬂux due to the curl of the Berry connection is b

=

αβ

∂ β a ∂xα

S.M. Girvin: The Quantum Hall Eﬀect

= =

∂ ν m Aν [m] ∂xβ ∂ ∂ ν m −Φ0 Sαβ Aν [m] ∂xα ∂xβ ∂ ∂mγ ∂Aν ν + m · ∂xβ ∂xα ∂mγ

−Φ0 Sαβ

∂ ∂xα

131

(1.236)

The ﬁrst term vanishes by symmetry leaving b = −Φ0 Sαβ

∂mν ∂mγ 1 νγ F ∂xβ ∂xα 2

(1.237)

where F νγ is given by equation (1.215) and we have taken advantage of the fact that the remaining factors are antisymmetric under the exchange ν ↔ γ. Using equation (1.215) and setting S = 12 we obtain b = −Φ0 ρ˜

(1.238)

where ρ˜ ≡ =

1 αβ abc a m ∂α mb ∂β mc 8π 1 αβ m · ∂α m × ∂β m 8π

(1.239)

is (for reasons that will become clear shortly) called the topological density or the Pontryagin density. Imagine now that we adiabatically deform the uniformly magnetized spin state into some spin texture state. We see from equation (1.238) that the orbital degrees of freedom see this as adiabatically adding additional ﬂux b(r ). Recall from equation (1.171) and the discussion of the charge of the Laughlin quasiparticle, that extra charge density is associated with extra ﬂux in the amount δρ

=

δρ

=

1 σxy b c νeρ˜.

(1.240) (1.241)

Thus we have the remarkable result that the changes in the electron charge density are proportional to the topological density. Our assumption of adiabaticity is valid as long as the spin ﬂuctuation frequency is much lower than the charge excitation gap. This is an excellent approximation for ν = 1 and still good on the stronger fractional Hall plateaus. It is interesting that the fermionic charge density in this model can be expressed solely in terms of the vector boson ﬁeld m( r ), but there is

132

Topological Aspects of Low Dimensional Systems

something even more signiﬁcant here. The skyrmion spin texture has total topological charge 1 · ∂α m × ∂β m (1.242) Qtop ≡ d2 r αβ m 8π which is always an integer. In fact for any smooth spin texture in which the spins at inﬁnity are all parallel, Qtop is always an integer. Since it is impossible to continuously deform one integer into another, Qtop is a topological invariant. That is, if Qtop = ±1 because a skyrmion (antiskyrmion) is present, Qtop is stable against smooth continuous distortions of the ﬁeld m. For example a spin wave could pass through the skyrmion and Qtop would remain invariant. Thus this charged object is topologically stable and has fermion number (i.e., the number of fermions (electrons) that ﬂow into the region when the object is formed) N = νQtop .

(1.243)

For ν = 1, N is an integer (±1 say) and has the fermion number of an electron. It is thus continuously connected to the single ﬂipped spin example discussed earlier. We are thus led to the remarkable conclusion that the spin degree of freedom couples to the electrostatic potential. Because skyrmions carry charge, we can aﬀect the spin conﬁguration using electric rather than magnetic ﬁelds! To understand how Qtop always turns out to be an integer, it is useful to consider a simpler case of a one-dimensional ring. We follow here the discussion of [58]. Consider the unit circle (known to topologists as the onedimensional sphere S1 ). Let the angle θ [0, 2π] parameterize the position along the curve. Consider a continuous, suitably well-behaved, complex function ψ(θ) = eiϕ(θ) deﬁned at each point on the circle and obeying |ψ| = 1. Thus associated with each point θ is another unit circle giving the possible range of values of ψ(θ). The function ψ(θ) thus deﬁnes a trajectory on the torus S1 × S1 illustrated in Figure 1.27. The possible functions ψ(θ) can be classiﬁed into diﬀerent homotopy classes according to their winding number n∈Z 2π d 1 ∗ dθ ψ −i n ≡ ψ 2π 0 dθ 2π 1 dϕ 1 = [ϕ(2π) − ϕ(0)] . (1.244) dθ = 2π 0 dθ 2π Because the points θ = 0 and θ = 2π are identiﬁed as the same point ψ(0) = ψ(2π) ⇒ ϕ(2π) − ϕ(0) = 2π × integer

(1.245)

S.M. Girvin: The Quantum Hall Eﬀect

ϕ

133

ϕ

θ

θ

Fig. 1.27. Illustration of mappings ϕ(θ) with: zero winding number (left) and winding number +2 (right).

and so n is an integer. Notice the crucial role played by the fact that 1 dϕ the “topological density” 2π dθ is the Jacobian for converting from the coordinate θ in the domain to the coordinate ϕ in the range. It is this fact that makes the integral in equation (1.244) independent of the detailed local form of the mapping ϕ(θ) and depend only on the overall winding number. As we shall shortly see, this same feature will also turn out to be true for the Pontryagin density. Think of the function ϕ(θ) as deﬁning the path of an elastic band wrapped around the torus. Clearly the band can be stretched, pulled and distorted in any smooth way without any eﬀect on n. The only way to change the winding number from one integer to another is to discontinuously break the elastic band, unwind (or wind) some extra turns, and then rejoin the cut pieces. Another way to visualize the homotopy properties of mappings from S1 to S1 is illustrated in Figure 1.28. The solid circle represents the domain θ and the dashed circle represents the range ϕ. It is useful to imagine the θ circle as being an elastic band (with points on it labeled by coordinates running from 0 to 2π) which can be “lifted up” to the ϕ circle in such a way that each point of θ lies just outside the image point ϕ(θ). The ﬁgure illustrates how the winding number n can be interpreted as the number of times the domain θ circle wraps around the range ϕ circle. (Note: even though the elastic band is “stretched” and may wrap around the ϕ circle more than once, its coordinate labels still only run from 0 to 2π.) This interpretation is the one which we will generalize for the case of skyrmions in 2D ferromagnets. We can think of the equivalence class of mappings having a given winding number as an element of a group called the homotopy group π1 (S1 ). The group operation is addition and the winding number of the sum of two

134

Topological Aspects of Low Dimensional Systems

n=0 ϕ

θ

n=+1 ϕ

θ

n=+2 ϕ

θ

Fig. 1.28. A diﬀerent representation of the mappings from θ to ϕ. The dashed line represents the domain θ and the solid line represents the range ϕ. The domain is “lifted up” by the mapping and placed on the range. The winding number n is the number of times the dashed circle wraps the solid circle (with a possible minus sign depending on the orientation).

functions, ϕ(θ) ≡ ϕ1 (θ) + ϕ2 (θ), is the sum of the two winding numbers n = n1 + n2 . Thus π1 (S1 ) is isomorphic to Z, the group of integers under addition. Returning now to the ferromagnet we see that the unit vector order parameter m deﬁnes a mapping from the plane R2 to the two-sphere S2 (i.e. an ordinary sphere in three dimensions having a two-dimensional surface). Because we assume that m = zˆ for all spatial points far from the location of the skyrmion, we can safely use a projective map to “compactify” R2 into a sphere S2 . In this process all points at inﬁnity in R2 are mapped into a single point on S2 , but since m( r ) is the same for all these diﬀerent points, no harm is done. We are thus interested in the generalization of the concept of the winding number to the mapping S2 → S2 . The corresponding homotopy group π2 (S2 ) is also equivalent to Z as we shall see.

S.M. Girvin: The Quantum Hall Eﬀect

dω m(x,y+dy)

135

m(x+dx,y)

m(x,y)

Fig. 1.29. Inﬁnitesimal circuit in spin space associated with an inﬁnitesimal circuit in real space via the mapping m( r ).

Consider the following four points in the plane and their images (illustrated in Fig. 1.29) under the mapping (x, y) −→ m(x, y) (x + dx, y) −→ m(x + dx, y) (x, y + dy) −→ m(x, y + dy) (x + dx, y + dy) −→ m(x + dx, y + dy).

(1.246)

The four points in the plane deﬁne a rectangle of area dxdy. The four points on the order parameter (spin) sphere deﬁne an approximate parallelogram whose area (solid angle) is dω

≈ ≈ =

[m(x + dx, y) − m(x, y)] × [m(x, y + dy) − m(x, y)] · m(x, y) 1 µν m · ∂µ m × ∂ν m dxdy 2 4π ρ˜ dxdy. (1.247)

Thus the Jacobian converting area in the plane into solid angle on the sphere is 4π times the Pontryagin density ρ˜. This means that the total topological charge given in equation (1.242) must be an integer since it counts the number of times the compactiﬁed plane is wrapped around the order parameter sphere by the mapping. The “wrapping” is done by lifting each point r in the compactiﬁed plane up to the corresponding point m( r) on the sphere just as was described for π1 (S1 ) in Figure 1.28.

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Topological Aspects of Low Dimensional Systems

For the skyrmion illustrated in Figure 1.26 the order parameter function m( r ) was chosen to be the standard form that minimizes the gradient energy [58] mx

=

my

=

mz

=

2λr cos (θ − ϕ) λ2 + r2 2λr sin (θ − ϕ) λ2 + r2 2 r − λ2 λ2 + r2

(1.248a) (1.248b) (1.248c)

where (r, θ) are the polar coordinates in the plane, λ is a constant that controls the size scale, and ϕ is a constant that controls the XY spin orientation. (Rotations about the Zeeman axis leave the energy invariant.) From the ﬁgure it is not hard to see that the skyrmion mapping wraps the compactiﬁed plane around the order parameter sphere exactly once. The sense is such that Qtop = −1. Exercise 1.21. Show that the topological density can be written in polar spatial coordinates as ρ˜ =

∂m ∂m 1 m · × · 4πr ∂r ∂θ

Use this result to show 1 ρ˜ = − 4π

2λ λ2 + r2

2

and hence Qtop = −1 for the skyrmion mapping in equations (1.248a–1.248c). It is worthwhile to note that it is possible to write down simple microscopic variational wave functions for the skyrmion which are closely related to the continuum ﬁeld theory results obtained above. Consider the following state in the plane [51] zj Ψ , (1.249) ψλ = λ j 1 j

where Ψ1 is the ν = 1 ﬁlled Landau level state (·)j refers to the spinor for the jth particle, and λ is a ﬁxed length scale. This is a skyrmion because it has its spin purely down at the origin (where zj = 0) and has spin purely up at inﬁnity (where |zj | λ). The parameter λ is simply the size scale of the skyrmion [46, 58]. At radius λ the spinor has equal weight for up

S.M. Girvin: The Quantum Hall Eﬀect

137

and down spin states (since |zj | = λ) and hence the spin lies in the XY plane just as it does for the solution in equation (1.248c). Notice that in the limit λ −→ 0 (where the continuum eﬀective action is invalid but this microscopic wave function is still sensible) we recover a fully spin polarized ﬁlled Landau level with a charge-1 Laughlin quasihole at the origin. Hence the number of ﬂipped spins interpolates continuously from zero to inﬁnity as λ increases. In order to analyze the skyrmion wave function in equation (1.249), we use the Laughlin plasma analogy. Recall from our discussion in Section 1.11 that in this analogy the norm of ψλ , T r{σ} D[z] |Ψ[z]|2 is viewed as the partition function of a Coulomb gas. In order to compute the density distribution we simply need to take a trace over the spin Z=

&

D[z] e

−2

i>j

− log |zi −zj |− 12

k

log (|zk |2 +λ2 )+ 14

k

|zk |2

' . (1.250)

This partition function describes the usual logarithmically interacting Coulomb gas with uniform background charge plus a spatially varying impurity background charge ∆ρb (r),

∆ρb (r) V (r)

1 2 λ2 ∇ V (r) = + , 2 2π π(r + λ2 )2 1 = − log (r2 + λ2 ). 2

≡ −

(1.251) (1.252)

For large enough scale size λ , local neutrality of the plasma [59] forces the electrons to be expelled from the vicinity of the origin and implies that the excess electron number density is precisely −∆ρb (r), so that equation (1.251) is in agreement with the standard result [58] for the topological density given in Exercise 1.21. Just as it was easy to ﬁnd an explicit wave function for the Laughlin quasi-hole but proved diﬃcult to write down an analytic wave function for the Laughlin quasi-electron, it is similarly diﬃcult to make an explicit wave function for the anti-skyrmion. Finally, we note that by replacing λz by

zn λn , we can generate a skyrmion with a Pontryagin index n.

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Topological Aspects of Low Dimensional Systems

Exercise 1.22. The argument given above for the charge density of the microscopic skyrmion state wave function used local neutrality of the plasma and hence is valid only on large length scales and thus requires λ . Find the complete microscopic analytic solution for the charge density valid for arbitrary λ, by using the fact that the proposed manybody wave function is nothing but a Slater determinant of the single particle states φm (z), z − |z2 | zm φm (z) = (1.253) e 4 .

2 λ 2π2m+1 m! m + 1 + λ2 Show that the excess electron number density is then ∆n(1) (z) ≡

N −1

|φm (z)|2 −

m=0

1 , 2π

(1.254)

which yields ∆n

(1)

1 (z) = 2π

1 |z|2 1 λ2 − 2 (1−α) 2 2 2 dα α e (|z| + λ ) − 1 . 2 0

(1.255)

Similarly, ﬁnd the spin density distribution S z (r) and show that it also agrees with the ﬁeld-theoretic expression in equation (1.248c) in the large λ limit. The skyrmion solution in equations (1.248a–1.248c) minimizes the gradient energy 1 (1.256) E0 = ρs d2 r ∂µ mν ∂µ mν . 2 Notice that the energy cost is scale invariant since this expression contains two integrals and two derivatives. Hence the total gradient energy is independent of the scale factor λ and for a single skyrmion is given by [46, 58] E0 = 4πρs =

1 ∞ 4

(1.257)

where ∞ is the asymptotic large q limit of the spin wave energy in equation (1.201). Since this spin wave excitation produces a widely separated particle-hole pair, we

see that the energy of a widely separated skyrmionantiskyrmion pair 14 + 14 ∞ is only half as large. Thus skyrmions are considerably cheaper to create than simple ﬂipped spins20 . 20 This energy advantage is reduced if the ﬁnite thickness of the inversion layer is taken into account. The skyrmion may in some cases turn out to be disadvantageous in higher Landau levels.

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139

Notice that equation (1.257) tells us that the charge excitation gap, while only half as large as naively expected, is ﬁnite as long as the spin stiﬀness ρs is ﬁnite. Thus we can expect a dissipationless Hall plateau. Therefore, as emphasized by Sondhi et al. [46], the Coulomb interaction plays a central role in the ν = 1 integer Hall eﬀect. Without the Coulomb interaction the charge gap would simply be the tiny Zeeman gap. With the Coulomb interaction the gap is large even in the limit of zero Zeeman energy because of the spontaneous ferromagnetic order induced by the spin stiﬀness. At precisely ν = 1 skyrmion/antiskyrmion pairs will be thermally activated and hence exponentially rare at low temperatures. On the other hand, because they are the cheapest way to inject charge into the system, there will be a ﬁnite density of skyrmions even in the ground state if ν = 1. Skyrmions also occur in ordinary 2D magnetic ﬁlms but since they do not carry charge (and are energetically expensive since ρs is quite large) they readily freeze out and are not particularly important. The charge of a skyrmion is sharply quantized but its number of ﬂipped spins depends on its area ∼ λ2 . Hence if the energy were truly scale invariant, the number of ﬂipped spins could take on any value. Indeed one of the early theoretical motivations for skyrmions was the discovery in numerical work by Rezayi [46, 60] that adding a single charge to a ﬁlled Landau level converted the maximally ferromagnetic state into a spin singlet. In the presence of a ﬁnite Zeeman energy the scale invariance is lost and there is a term in the energy that scales with ∆λ2 and tries to minimize the size of the skyrmion. Competing with this however is a Coulomb term which we now discuss. The Lagrangian in equation (1.224) contains the correct leading order terms in a gradient expansion. There are several possible terms which are fourth order in gradients, but a particular one dominates over the others at long distances. This is the Hartree energy associated with the charge density of the skyrmion δρ(r ) δρ(r ) 1 d2 r (1.258) VH = d2 r 2 |r − r | where δρ =

νe αβ m · ∂α m × ∂β m 8π

(1.259)

and is the dielectric constant. The long range of the Coulomb interaction makes this eﬀectively a three gradient term that distinguishes it from the other possible terms at this order. Recall that the Coulomb interaction already entered in lower order in the computation of ρs . That however was the exchange energy while the present term is the Hartree energy. e2 The Hartree energy scales like λ and so prefers to expand the skyrmion size. The competition between the Coulomb and Zeeman energies yields an

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Topological Aspects of Low Dimensional Systems

Fig. 1.30. Illustration of the spin conﬁgurations for non-interacting electrons at ﬁlling factor ν = 1 in the presence of a hole (top) and an extra electron (bottom).

optimal number of approximately four ﬂipped spins according to microscopic Hartree Fock calculations [61]. Thus a signiﬁcant prediction for this model is that each charge added (or removed) from a ﬁlled Landau level will ﬂip several (∼ 4) spins. This is very diﬀerent from what is expected for non-interacting electrons. As illustrated in Figure 1.30 removing an electron leaves the non-interacting system still polarized. The Pauli principle forces an added electron to be spin reversed and the magnetization drops from unity at ν = 1 to zero at ν = 2 where both spin states of the lowest Landau level are fully occupied. Direct experimental evidence for the existence of skyrmions was ﬁrst obtained by Barrett et al. [62] using a novel optically pumped NMR technique. The Hamiltonian for a nucleus is [63] HN = −∆N I z + ΩI · s

(1.260)

where I is the nuclear angular momentum, ∆N is the nuclear Zeeman frequency (about 3 orders of magnitude smaller than the electron Zeeman frequency), Ω is the hyperﬁne coupling and s is the electron spin density at the nuclear site. If, as a ﬁrst approximation we replace s by its average value (1.261) HN ≈ (−∆N + Ωsz ) I z we see that the precession frequency of the nucleus will be shifted by an amount proportional to the magnetization of the electron gas. The magnetization deduced using this so-called Knight shift is shown in Figure 1.31. The electron gas is 100% polarized at ν = 1, but the polarization drops oﬀ sharply (and symmetrically) as charge is added or subtracted. This is in sharp disagreement with the prediction of the free electron model

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141

S

K (kHz)

20 15 10 5 0

0.6

0.8

1.0

1.2

ν

1.4

1.6

1.8

Fig. 1.31. NMR Knight shift measurement of the electron spin polarization near ﬁlling factor ν = 1. Circles are the data of Barrett et al. [62]. The dashed line is a guide to the eye. The solid line is the prediction for non-interacting electrons. The peak represents 100% polarization at ν = 1. The steep slope on each side indicates that many (∼ 4) spins ﬂip over for each charge added (or subtracted). The observed symmetry around ν = 1 is due to the particle-hole symmetry between skyrmions and antiskyrmions not present in the free-electron model.

as shown in the ﬁgure. The initial steep slope of the data allows one to deduce that 3.5–4 spins reverse for each charge added or removed. This is in excellent quantitative agreement with Hartree-Fock calculations for the skyrmion model [61]. Other evidence for skyrmions comes from the large change in Zeeman energy with ﬁeld due to the large number of ﬂipped spins. This has been observed in transport [64] and in optical spectroscopy [65]. Recall that spinorbit eﬀects in GaAs make the electron g factor −0.4. Under hydrostatic pressure g can be tuned towards zero which should greatly enhance the skyrmion size. Evidence for this eﬀect has been seen [66].

1.20 Skyrmion dynamics NMR [62] and nuclear speciﬁc heat [67] data indicate that skyrmions dramatically enhance the rate at which the nuclear spins relax. This nuclear spin relaxation is due to the transverse terms in the hyperﬁne interaction

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Topological Aspects of Low Dimensional Systems

1/T1 x 100 (sec-1)

5 H=9.39 T H=7.05 T

4 3 2 1 0 0.6

0.8

1.0

1.2 ν

1.4

1.6

1.8

Fig. 1.32. NMR nuclear spin relaxation rate 1/T1 as a function of ﬁlling factor. After Tycko et al. [68]. The relaxation rate is very small at ν = 1, but rises dramatically away from ν = 1 due to the presence of skyrmions.

which we neglected in discussing the Knight shift    

1 1 Ω (I + s− + I − s+ ) = Ω I + Sq− + h.c. ·  2 2 

(1.262)

q

The free electron model would predict that it would be impossible for an electron and a nucleus to undergo mutual spin ﬂips because the Zeeman energy would not be conserved. (Recall that ∆N ∼ 10−3 ∆.) The spin wave model shows that the problem is even worse than this. Recall from equation (1.201) that the spin Coulomb interaction makes spin wave energy much larger than the electron Zeeman gap except at very long wavelengths. The lowest frequency spin wave excitations lie above 20 − 50 GHz while the nuclei precess at 10 − 100 MHz. Hence the two sets of spins are unable to couple eﬀectively. At ν = 1 this simple picture is correct. The nuclear relaxation time T1 is extremely long (tens of minutes to many hours depending on the temperature) as shown in Figure 1.32. However the ﬁgure also shows that for ν = 1 the relaxation rate 1/T1 rises dramatically and T1 falls to ∼ 20 s. In order to understand this dramatic variation we need to develop a theory of spin dynamics in the presence of skyrmions. The 1/T1 data is telling us that for ν = 1 at least some of the electron spin ﬂuctuations are orders of magnitude lower in frequency than the Zeeman splitting and these low frequency modes can couple strongly to the nuclei. One way this might occur is through the presence of disorder. We see from equation (1.262) that NMR is a local probe which couples to spin ﬂip excitations at all wave vectors. Recall from equation (1.197) that lowest Landau level projection implies that Sq− contains a translation operator τq .

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In the presence of strong disorder the Zeeman and exchange cost of the spin ﬂips could be compensated by translation to a region of lower potential energy. Such a mechanism was studied in [69] but does not show sharp features in 1/T1 around ν = 1. We are left only with the possibility that the dynamics of skyrmions somehow involves low frequency spin ﬂuctuations. For simplicity we will analyze this possibility ignoring the eﬀects of disorder, although this may not be a valid approximation. Let us begin by considering a ferromagnetic ν = 1 state containing a single skyrmion of the form parameterized in equations (1.248a–1.248c). There are two degeneracies at the classical level in the eﬀective ﬁeld theory: The energy does not depend on the position of the skyrmion and it does not depend on the angular orientation ϕ. These continuous degeneracies are known as zero modes [58] and require special treatment of the quantum ﬂuctuations about the classical solution. In the presence of one or more skyrmions, the quantum Hall ferromagnet is non-colinear. In an ordinary ferromagnet where all the spins are parallel, global rotations about the magnetization axis only change the quantum phase of the state – they do not produce a new state21 . Because the skyrmion has distinguishable orientation, each one induces a new U (1) degree of freedom in the system. In addition because the skyrmion has a distinguishable location, each one induces a new translation degree of freedom. As noted above, both of these are zero energy modes at the classical level suggesting that they might well be the source of low energy excitations which couple so eﬀectively to the nuclei. We shall see that this is indeed the case, although the story is somewhat complicated by the necessity of correctly quantizing these modes. Let us begin by ﬁnding the eﬀective Lagrangian for the translation mode [8]. We take the spin conﬁguration to be (1.263) m( r , t) = m 0 r − R(t) where m 0 is the static classical skyrmion solution and R(t) is the position degree of freedom. We ignore all other spin wave degrees of freedom since they are gapped. (The gapless U (1) rotation mode will be treated separately below.) Equation (1.224) yields a Berry phase term L0 = −¯ hS d2 r m ˙ µ Aµ [m] n(r ) (1.264)

i

z

about the Zeeman alignment axis is accomplished by R = e− h¯ ϕS . But a colinear ferromagnet ground state is an eigenstate of S z , so rotation leaves the state invariant up to a phase. 21 Rotation

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Topological Aspects of Low Dimensional Systems

where

∂ (1.265) m ˙ µ = −R˙ ν ν mµ0 (r − R) ∂r and unlike in equation (1.224) we have taken into account our new-found knowledge that the density is non-uniform n(r ) = n0 +

1 µν m · ∂µ m × ∂ν m. 8π

(1.266)

The second term in equation (1.266) can be shown to produce an extra Berry phase when two skyrmions are exchanged leading to the correct minus sign for Fermi statistics (on the ν = 1 plateau) but we will not treat it further. Equation (1.264) then becomes L0 = +¯ hR˙ ν aν (r )

(1.267)

where the “vector potential” aν (r ) ≡ Sn0

d2 r (∂ν mµ )Aµ

(1.268)

has curl λν

∂ ν a ∂Rλ

= −λν

∂ ν a ∂rλ

= −Sn0 λν

d2 r ∂λ {(∂ν mµ )Aµ }

∂Aµ = −Sn0 λν d2 r (∂ν mµ ) (∂λ mγ ) ∂mγ Sn0 = − d2 r λν ∂ν mµ ∂λ mγ F γµ 2 = −2πn0 Qtop .

(1.269)

Thus equation (1.267) corresponds to the kinetic Lagrangian for a massless particle of charge −eQtop moving in a uniform magnetic ﬁeld of strength Φ0 B = 2π 2 . But this of course is precisely what the skyrmion is [8]. We have kept here only the lowest order adiabatic time derivative term in the action22 . This is justiﬁed by the existence of the spin excitation gap and the fact that we are interested only in much lower frequencies (for the NMR). If we ignore the disorder potential then the kinetic Lagrangian simply leads to a Hamiltonian that yields quantum states in the lowest Landau 22 There may exist higher-order time-derivative terms which give the skyrmion a mass and there will also be damping due to radiation of spin waves at higher velocities [70].

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145

level, all of which are degenerate in energy and therefore capable of relaxing the nuclei (whose precession frequency is extremely low on the scale of the electronic Zeeman energy). Let us turn now to the rotational degree of freedom represented by the coordinate ϕ in equations (1.248a–1.248c). The full Lagrangian is complicated and contains the degrees of freedom of the continuous ﬁeld m( r ). We need to introduce the collective coordinate ϕ describing the orientation of the skyrmion as one of the degrees of freedom and then carry out the Feynman path integration over the quantum ﬂuctuations in all the inﬁnite number of remaining degrees of freedom23 . This is a non-trivial task, but fortunately we do not actually have to carry it out. Instead we will simply write down the answer. The answer is some functional of the path for the single variable ϕ(t). We will express this functional (using a functional Taylor series expansion) in the most general form possible that is consistent with the symmetries in the problem. Then we will attempt to identify the meaning of the various terms in the expansion and evaluate their coeﬃcients (or assign them values phenomenologically). After integrating out the high frequency spin wave ﬂuctuations, the lowest-order symmetry-allowed terms in the action are h2 2 ¯ ϕ˙ + . . . ¯ K ϕ˙ + (1.270) Lϕ = h 2U Again, there is a ﬁrst-order term allowed by the lack of time-reversal symmetry and we have included the leading non-adiabatic correction. The full action involving m( r , t) contains only a ﬁrst-order time derivative but a second order term is allowed by symmetry to be generated upon integrating out the high frequency ﬂuctuations. We will not perform this explicitly but rather treat U as a phenomenological ﬁtting parameter. The coeﬃcient K can be computed exactly since it is simply the Berry phase term. Under a slow rotation of all the spins through 2π the Berry phase is (using Eq. (B.22) in Appendix B)

2

d r n(r ) (−S2π) [1 −

mz0 (r )]

1 = h ¯

0

T

Lϕ = 2πK.

(1.271)

(The non-adiabatic term gives a 1/T contribution that vanishes in the adiabatic limit T → ∞.) Thus we arrive at the important conclusion that K is the expectation value of the number of overturned spins for the classical solution m 0 (r ). We emphasize that this is the Hartree-Fock (i.e., “classical”) skyrmion solution and therefore K need not be an integer. 23 Examples of how to do this are discussed in various ﬁeld theory texts, including Rajaraman [58].

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Topological Aspects of Low Dimensional Systems

The canonical angular momentum conjugate to ϕ in equation (1.270) is Lz =

¯2 h δLϕ =h ¯ K + ϕ˙ δ ϕ˙ U

(1.272)

and hence the Hamiltonian is Hϕ

= Lz ϕ˙ − Lϕ ¯2 2 h h2 ¯ hK − ϕ˙ = hK + ϕ˙ ϕ˙ − ¯ ¯ U 2U = +

¯2 2 h U ϕ˙ = 2 (Lz − ¯ hK)2 . 2U 2¯ h

(1.273)

Having identiﬁed the Hamiltonian and expressed it in terms of the coordinate and the canonical momentum conjugate to that coordinate, we quantize Hϕ by simply making the substitution ∂ ∂ϕ

(1.274)

2 ∂ −K . −i ∂ϕ

(1.275)

h Lz −→ −i¯ to obtain Hϕ = +

U 2

This can be interpreted as the Hamiltonian of a (charged) XY quantum rotor with moment of inertia h ¯ 2 /U circling a solenoid containing K ﬂux quanta. (The Berry phase term in Eq. (1.270) is then interpreted as the Aharonov-Bohm phase.) The eigenfunctions are 1 ψm (ϕ) = √ eimϕ 2π

(1.276)

U (m − K)2 . 2

(1.277)

and the eigenvalues are m =

The angular momentum operator Lz is actually the operator giving the number of ﬂipped spins in the skyrmion. Because of the rotational symmetry about the Zeeman axis, this is a good quantum number and therefore takes on integer values (as required in any quantum system of ﬁnite size with rotational symmetry about the z axis). The ground state value of m is the nearest integer to K. The ground state angular velocity is $ % ∂Hϕ U (1.278) ϕ˙ = = (m − K). ∂Lz h ¯

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147

Hence if K is not an integer the skyrmion is spinning around at a ﬁnite velocity. In any case the actual orientation angle ϕ for the skyrmion is completely uncertain since from equation (1.276) 2

|ψm (ϕ)| =

1 2π

(1.279)

ϕ has a ﬂat probability distribution (due to quantum zero point motion). We interpret this as telling us that the global U (1) rotation symmetry broken in the classical solution is restored in the quantum solution because of quantum ﬂuctuations in the coordinate ϕ. This issue will arise again in our study of the Skyrme lattice where we will ﬁnd that for an inﬁnite array of skyrmions, the symmetry can sometimes remain broken. Microscopic analytical [71] and numerical [61] calculations do indeed ﬁnd a family of low energy excitations with an approximately parabolic relation between the energy and the number of ﬂipped spins just as is predicted by equation (1.277). As mentioned earlier, K ∼ 4 for typical parameters. Except for the special case where K is a half integer the spectrum is nondegenerate and has an excitation gap on the scale of U which is in turn some fraction of the Coulomb energy scale ∼ 100 K. In the absence of disorder even a gap of only 1 K would make these excitations irrelevant to the NMR. We shall see however that this conclusion is dramatically altered in the case where many skyrmions are present. 1.21 Skyrme lattices For ﬁlling factors slightly away from ν = 1 there will be a ﬁnite density of skyrmions or antiskyrmions (all with the same sign of topological charge) in the ground state [56,72,73]. Hartree-Fock calculations [72] indicate that the ground state is a Skyrme crystal. Because the skyrmions are charged, the Coulomb potential in equation (1.258) is optimized for the triangular lattice. This is indeed the preferred structure for very small values of |ν − 1| where the skyrmion density is low. However at moderate densities the square lattice is preferred. The Hartree-Fock ground state has the angular variable ϕj shifted by π between neighboring skyrmions as illustrated in Figure 1.33. This “antiferromagnetic” arrangement of the XY spin orientation minimizes the spin gradient energy and would be frustrated on the triangular lattice. Hence it is the spin stiﬀness that stabilizes the square lattice structure. The Hartree-Fock ground state breaks both global translation and global U (1) spin rotation symmetry. It is a kind of “supersolid” with both diagonal

and oﬀ-diagonal

Gz ≡ sz (r ) sz (r )

(1.280)

G⊥ ≡ s+ (r ) s− (r )

(1.281)

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Topological Aspects of Low Dimensional Systems

Fig. 1.33. Electronic structure of the skyrmion lattice as determined by numerical 2 Hartree-Fock calculations for ﬁlling factor ν = 1.1 and Zeeman energy 0.015 e . (a) Excess charge density (in units of 1/(2π2 )) and (b) Two-dimensional vector representation of the XY components of the spin density. The spin stiﬀness makes the square lattice more stable than the triangular lattice at this ﬁlling factor and Zeeman coupling. Because of the U (1) rotational symmetry about the Zeeman axis, this is simply one representative member of a continuous family of degenerate Hartree-Fock solutions. After Brey et al. [71].

long-range order. For the case of a single skyrmion we found that the U (1) symmetry was broken at the Hartree-Fock (classical) level but fully restored by quantum ﬂuctuations of the zero mode coordinate ϕ. In the thermodynamic limit of an inﬁnite number of skyrmions coupled together, it is possible for the global U (1) rotational symmetry breaking to survive quantum ﬂuctuations24 . If this occurs then an excitation gap is not produced. Instead we have a new kind of gapless spin wave Goldstone mode [74, 75]. This mode is gapless despite the presence of the Zeeman ﬁeld and hence has a profound eﬀect on the NMR relaxation rate. The gapless Goldstone mode associated with the broken translation symmetry is the ordinary magnetophonon of the Wigner crystal. This too contributes to the nuclear relaxation rate. In actual practice, disorder will be important. In addition, the NMR experiments have so far been performed at temperatures which are likely well above the lattice melting temperature. Nevertheless the zero temperature lattice calculations to be discussed below probably capture the essential physics of this non co-linear magnet. Namely, there exist spin ﬂuctuations at frequencies orders of magnitude below the Zeeman gap. 24 Loosely speaking this corresponds to the inﬁnite system having an inﬁnite moment of inertia (for global rotations) which allows a quantum wave packet which is initially localized at a particular orientation ϕ not to spread out even for long times.

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149

At zero temperature these are coherent Goldstone modes. Above the lattice melting temperature they will be overdamped diﬀusive modes derived from the Goldstone modes. The essential physics will still be that the spin ﬂuctuations have strong spectral density at frequencies far below the Zeeman gap. It turns out that at long wavelengths the magnetophonon and U (1) spin modes are decoupled. We will therefore ignore the positional degrees of freedom when analyzing the new U (1) mode. We have already found the U (1) Hamiltonian for a single skyrmion in equation (1.275). The simplest generalization to the Skyrme lattice which is consistent with the symmetries of the problem is H=

U ˆ (Kj − K)2 − J cos (ϕi − ϕj ) 2 j

(1.282)

ij

ˆ j ≡ −i ∂ is the angular momentum operator. The global U (1) where K ∂ϕj symmetry requires that the interactive term be invariant if all of the ϕj ’s are increased by a constant. In addition H must be invariant under ϕj → ϕj + 2π for any single skyrmion. We have assumed the simplest possible near-neighbor coupling, neglecting the possibility of longer range higherorder couplings of the form cos n(ϕi − ϕj ) which are also symmetry allowed. The phenomenological coupling J must be negative to be consistent with the “antiferromagnetic” XY order found in the Hartree-Fock ground state illustrated in Figure 1.33. However we will ﬁnd it convenient to instead make J positive and compensate for this by a “gauge” change ϕj → ϕj + π on one sublattice. This is convenient because it makes the coupling “ferromagnetic” rather than “antiferromagnetic”. Equation (1.282) is the Hamiltonian for the quantum XY rotor model, closely related to the boson Hubbard model [76–78]. Readers familiar with superconductivity will recognize that this model is commonly used to describe the superconductor-insulator transition in Josephson arrays [76, 77]. ˆ j operator represents the numThe angular momentum eigenvalue of the K ber of bosons (Cooper pairs) on site j and the U term describes the charging energy cost when this number deviates from the electrostatically optiˆ j has negative mal value of K. The boson number is non-negative while K eigenvalues. However we assume that K 1 so that the negative angular momentum states are very high in energy. The J term in the quantum rotor model is a mutual torque that transfers units of angular momentum between neighboring sites. In the boson language the wave function for the state with m bosons on site j contains a factor ψm (ϕj ) = eimϕj .

(1.283)

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Topological Aspects of Low Dimensional Systems

The raising and lowering operators are thus25 e±iϕj . This shows us that the cosine term in equation (1.282) represents the Josephson coupling that hops bosons between neighboring sites. For U J the system is in an insulating phase well-described by the wave function ψ(ϕ1 , ϕ2 , . . . , ϕN ) = eimϕj (1.284) j

where m is the nearest integer to K. In this state every rotor has the same ﬁxed angular momentum and thus every site has the same ﬁxed particle number in the boson language. There is a large excitation gap ∆ ≈ U (1 − 2|m − K|)

(1.285)

and the system is insulating26 . Clearly |ψ|2 ≈ 1 in this phase and it is therefore quantum disordered. That is, the phases {ϕj } are wildly ﬂuctuating because every conﬁguration is equally likely. The phase ﬂuctuations are nearly uncorrelated eiϕj e−iϕk ∼ e−|rj −rk |/ξ .

(1.286)

For J U the phases on neighboring sites are strongly coupled together and the system is a superconductor. A crude variational wave function that captures the essential physics is λ cos (ϕi −ϕj ) ij ψ(ϕ1 , ϕ2 , . . . , ϕN ) ∼ e (1.287) where λ is a variational parameter [79]. This is the simplest ansatz consistent with invariance under ϕj → ϕj + 2π. For J U , λ 1 and |ψ|2 is large only for spin conﬁgurations with all of the XY spins locally parallel. Expanding the cosine term in equation (1.282) to second order gives a harmonic Hamiltonian which can be exactly solved. The resulting gapless “spin waves” are the Goldstone modes of the superconducting phase. For simplicity we work with the Lagrangian rather than the Hamiltonian L=

. j

hK ϕ˙ j + ¯

/

¯2 2 h ϕ˙ j + J cos (ϕi − ϕj ). 2U

(1.288)

ij

25 These operators have matrix elements ψ +iϕ |ψ = 1 whereas a boson raising m m+1 |e √ operator would have matrix element m + 1. For K 1, m ∼ K and this is nearly a constant. Arguments like this strongly suggest that the boson Hubbard model and the quantum rotor model are essentially equivalent. In particular their order/disorder transitions are believed to be in the same universality class. 26 An exception occurs if |m − K| = 1 where the gap vanishes. See [78]. 2

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151

The Berry phase term is a total derivative and can not aﬀect the equations of motion27 . Dropping this term and expanding the cosine in the harmonic approximation yields L=

¯2 2 J h ϕ˙ − (ϕi − ϕj )2 . 2U j j 2

(1.289)

ij

This “phonon” model has linearly dispersing gapless collective modes at small wavevectors √ hωq = U J qa ¯ (1.290) where a is the lattice constant. The parameters U and J can be ﬁxed by ﬁtting to microscopic Hartree-Fock calculations of the spin wave velocity and the magnetic susceptibility (“boson compressibility”) [61, 75]. This in turn allows one to estimate the regime of ﬁlling factor and Zeeman energy in which the U (1) symmetry is not destroyed by quantum ﬂuctuations [75]. Let us now translate all of this into the language of our non-colinear QHE ferromagnet [74,75]. Recall that the angular momentum (the “charge”) conjugate to the phase angle ϕ is the spin angular momentum of the overturned spins that form the skyrmion. In the quantum disordered “insulating” phase, each skyrmion has a well deﬁned integer-valued “charge” (number of overturned spins) much like we found when we quantized the U (1) zero mode for the plane angle ϕ of a single isolated skyrmion in equation (1.276). There is an excitation gap separating the energies of the discrete quantized values of the spin. The “superﬂuid” state with broken U (1) symmetry is a totally new kind of spin state unique to non-colinear magnets [74, 75]. Here the phase angle is well-deﬁned and the number of overturned spins is uncertain. The oﬀdiagonal long-range order of a superﬂuid becomes b†j bk → eiϕj e−iϕk or in the spin language28

s+ (r )s− (r ) .

(1.291)

(1.292)

Thus in a sense we can interpret a spin ﬂip interaction between an electron and a nucleus as creating a boson in the superﬂuid. But this boson has a ﬁnite probability of “disappearing” into the superﬂuid “condensate” and 27 In fact in the quantum path integral this term has no eﬀect except for time histories hβ) = ϕj (0) ± 2π. We in which a “vortex” encircles site j causing the phase to wind ϕj (¯ explicitly ignore this possibility when we make the harmonic approximation. 28 There is a slight complication here. Because the XY spin conﬁguration of the skyrmion has a vortex-like structure s+ ≡ sx +isy winds in phase around the skyrmion so the “bose condensation” is not at zero wave vector.

152

Topological Aspects of Low Dimensional Systems

hence the system does not have to pay the Zeeman price to create the ﬂipped spin. That is, the superﬂuid state has an uncertain number of ﬂipped spins z commutes with H) and so the Zeeman energy cost is (even though Stot uncertain. In classical language the skyrmions locally have ﬁnite (slowly varying) x and y spin components which act as eﬀective magnetic ﬁelds around which the nuclear spins precess and which thus cause I z to change with time. The key here is that sx and sy can, because of the broken U (1) symmetry, ﬂuctuate very slowly (i.e. at MHz frequencies that the nuclei can follow rather than just the very high Zeeman precession frequency). Detailed numerical calculations [75] show that the Skyrme lattice is very eﬃcient at relaxing the nuclei and 1/T1 and is enhanced by a factor of ∼ 103 over the corresponding rate at zero magnetic ﬁeld. We expect this qualitative distinction to survive even above the Skryme lattice melting temperature for the reasons discussed earlier. Because the nuclear relaxation rate increases by orders of magnitude, the equilibration time at low temperatures drops from hours to seconds. This means that the nuclei come into thermal equilibrium with the electrons and hence the lattice. The nuclei therefore have a well-deﬁned temperature and contribute to the speciﬁc heat. Because the temperature is much greater than the nuclear Zeeman energy scale ∆ ∼ 1 mK, each nucleus contributes 2 only a tiny amount ∼ kB ∆ T 2 to the speciﬁc heat. On the other hand, the T electronic speciﬁc heat per particle ∼ kB Tfermi is low and the electron density 6 is low. In fact there are about 10 nuclei per quantum well electron and the nuclei actually enhance the speciﬁc heat more than 5 orders of magnitude [67]! Surprisingly, at around 30 mK there is a further enhancement of the speciﬁc heat by an additional order of magnitude. This may be a signal of the Skyrme lattice melting transition [67, 75, 80], although the situation is somewhat murky at the present time. The peak can not possibly be due to the tiny amount of entropy change in the Skyrme lattice itself. Rather it is due to the nuclei in the thick AlAs barrier between the quantum wells29 . 1.22 Double-layer quantum hall ferromagnets We learned in our study of quantum Hall ferromagnets that the Coulomb interaction plays an important role at Landau level ﬁlling factor ν = 1 because it causes the electron spins to spontaneously align ferromagnetically 29 For somewhat complicated reasons it may be that the barrier nuclei are eﬃciently dipole coupled to the nuclei in the quantum wells (and therefore in thermal equilibrium) only due to the critical slowing down of the electronic motion in the vicinity of the Skyrme lattice melting transition.

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153

W

W

d Fig. 1.34. Schematic conduction band edge proﬁle for a double-layer twodimensional electron gas system. Typical widths and separations are W ∼ d ∼ 100 ˚ A and are comparable to the spacing between electrons within each inversion layer.

and this in turn profoundly alters the charge excitation spectrum by producing a gap30 . A closely related eﬀect occurs in double-layer systems in which layer index is analogous to spin [43, 44, 81]. Building on our knowledge of the dynamics of ferromagnets developed in the last section, we will use this analogy to explore the rich physics of double-layer systems. Novel fractional quantum Hall eﬀects due to correlations [82] in multicomponent systems were anticipated in early work by Halperin [42] and the now extensive literature has been reviewed in [43]. There have also been recent interesting studies of systems in which the spin and layer degrees of freedom are coupled in novel ways [83, 84]. As described in this volume by Shayegan [45], modern MBE techniques make it possible to produce double-layer (and multi-layer) two-dimensional electron gas systems of extremely low disorder and high mobility. As illustrated schematically in Figure 1.34, these systems consist of a pair of 2D electron gases separated by a distance d so small (d ∼ 100 ˚ A) as to be comparable to the typical spacing between electrons in the same layer. A second type of system has also recently been developed to a high degree 30 Because the charged excitations are skyrmions, this gap is not as large as naive estimates would suggest, but it is still ﬁnite as long as the spin stiﬀness is ﬁnite.

154

Topological Aspects of Low Dimensional Systems

of perfection [85]. These systems consist of single wide quantum wells in which strong mixing of the two lowest electric subbands allows the electrons to localize themselves on opposites sides of the well to reduce their correlation energy. We will take the point of view that these systems can also be approximately viewed as double-well systems with some eﬀective layer separation and tunnel barrier height. As we have already learned, correlations are especially important in the strong magnetic ﬁeld regime because all electrons can be accommodated within the lowest Landau level and execute cyclotron orbits with a common kinetic energy. The fractional quantum Hall eﬀect occurs when the system has a gap for making charged excitations, i.e. when the system is incompressible. Theory has predicted [42, 82, 86] that at some Landau level ﬁlling factors, gaps occur in double-layer systems only if interlayer interactions are suﬃciently strong. These theoretical predictions have been conﬁrmed [87]. More recently work from several diﬀerent points of view [88–93] has suggested that inter-layer correlations can also lead to unusual broken symmetry states with a novel kind of spontaneous phase coherence between layers which are isolated from each other except for inter-layer Coulomb interactions. It is this spontaneous interlayer phase coherence which is responsible [43, 51, 73, 94] for a variety of novel features seen in the experimental data to be discussed below [44, 81]. 1.23 Pseudospin analogy We will make the simplifying assumption that the Zeeman energy is large enough that ﬂuctuations of the (true) spin order can be ignored, leaving out the possibility of mixed spin and pseudospin correlations [83, 84]. We will limit our attention to the lowest electric subband of each quantum well (or equivalently, the two lowest bands of a single wide well). Hence we have a two-state system that can be labeled by a pseudospin 1/2 degree of freedom. Pseudospin up means that the electron is in the (lowest electric subband of the) upper layer and pseudospin down means that the electron is in the (lowest electric subband of the) lower layer. Just as in our study of ferromagnetism we will consider states with total ﬁlling factor ν ≡ ν↑ + ν↓ = 1. A state exhibiting interlayer phase coherence and having the pseudospins ferromagnetically aligned in the direction deﬁned by polar angle θ and azimuthal angle ϕ can be written in the Landau gauge just as for ordinary spin 1 0 cos(θ/2)c†k↑ + sin(θ/2)eiϕ c†k↓ |0· (1.293) |ψ = k

Every k state contains one electron and hence this state has ν = 1 as desired. Note however that the layer index for each electron is uncertain.

S.M. Girvin: The Quantum Hall Eﬀect

155

The amplitude to ﬁnd a particular electron in the upper layer is cos(θ/2) and the amplitude to ﬁnd it in the lower layer is sin(θ/2)eiϕ . Even if the two layers are completely independent with no tunneling between them, quantum mechanics allows for the somewhat peculiar possibility that we are uncertain which layer the electron is in. For the case of ordinary spin we found that the Coulomb interaction produced an exchange energy which strongly favored having the spins locally parallel. Using the fact that the Coulomb interaction is completely spin independent (it is only the Pauli principle that indirectly induces the ferromagnetism) we wrote down the spin rotation invariant eﬀective theory in equation (1.224). Here we do not have full SU (2) invariance because the interaction between electrons in the same layer is clearly stronger than the interaction between electrons in opposite layers. Thus for example, if all the electrons are in the upper (or lower) layer, the system will look like a charged capacitor and have higher energy than if the layer occupancies are equal. Hence to leading order in gradients we expect the eﬀective action to be modiﬁed slightly hSn m ˙ µ (r )Aµ [m] − λ(r )(mµ mµ − 1)} L = − d2 r {¯ 1 − d2 r ρs ∂µ mν ∂µ mν +βmz mz −∆mz −ntmx · (1.294) 2 The spin stiﬀness ρs represents the SU (2) invariant part of the exchange energy and is therefore somewhat smaller than the value computed in equation (1.231). The coeﬃcient β is a measure of the capacitive charging energy31 . The analog of the Zeeman energy ∆ represents an external electric ﬁeld applied along the MBE growth direction which unbalances the charge densities in the two layers. The coeﬃcient t represents the amplitude for the electrons to tunnel between the two layers. It prefers the pseudospin to be aligned in the x ˆ direction because this corresponds to the spinor 1 1 √ (1.295) 1 2 which represents the symmetric (i.e. bonding) linear combination of the two well states. The state with the pseudospin pointing in the −ˆ x direction represents the antisymmetric (i.e. antibonding) linear combination which is higher in energy. 31 We have taken the charging energy to be a local quantity characterized by a ﬁxed, wave vector independent capacitance. This is appropriate only if mz ( r ) represents the local charge imbalance between the layers coarse-grained over a scale larger than the layer separation. Any wave vector dependence of the capacitance will be represented by higher derivative terms which we will ignore.

156

Topological Aspects of Low Dimensional Systems

For the moment we will assume that both t and ∆ vanish, leaving only the β term which breaks the pseudospin rotational symmetry. The case β < 0 would represent “Ising anisotropy”. Clearly the physically realistic case for the capacitive energy gives β > 0 which represents so-called “easy plane anisotropy”. The energy is minimized when mz = 0 so that the order parameter lies in the XY plane giving equal charge densities in the two layers. Thus we are left with an eﬀective XY model which should exhibit long-range oﬀ-diagonal order32 Ψ(r ) = mx (r ) + imy (r ).

(1.296)

The order is “oﬀ-diagonal” because it corresponds microscopically to an operator Ψ(r ) = s+ (r ) = ψ↑† (r )ψ↓ (r ) (1.297) which is not diagonal in the sz basis, much as in a superﬂuid where the ﬁeld operator changes the particle number and yet it condenses and acquires a ﬁnite expectation value. One other comment worth making at this point is that equation (1.297) shows that, unlike the order parameter in a superconductor or superﬂuid, this one corresponds to a charge neutral operator. Hence it will be able to condense despite the strong magnetic ﬁeld (which ﬁlls charged condensates with vortices and generally destroys the order). In the next subsection we review the experimental evidence that longrange XY correlations exist and that as a result, the system exhibits excitations which are highly collective in nature. After that we will return to further analysis and interpretation of the eﬀective Lagrangian in equation (1.294) to understand those excitations. 1.24 Experimental background As illustrated by the dashed lines in Figure 1.34, the lowest energy eigenstates split into symmetric and antisymmetric combinations separated by an energy gap ∆SAS = 2t which can, depending on the sample, vary from essentially zero to many hundreds of Kelvins. The splitting can therefore be much less than or greater than the interlayer interaction energy scale, Ec ≡ e2 /d. Thus it is possible to make systems which are in either the weak or strong correlation limits. When the layers are widely separated, there will be no correlations between them and we expect no dissipationless quantum Hall state since each layer has [95] ν = 1/2. For smaller separations, it is observed experimentally 32 At ﬁnite temperatures Ψ( r ) will vanish but will have long-range algebraically decaying correlations. Above the Kosterlitz-Thouless phase transition temperature, the correlations will fall oﬀ exponentially.

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157

4.0 NO QHE

d/lB

3.0

QHE

2.0

1.0 0.00

0.02

0.04

2

0.06

0.08

0.10

∆SAS/(e /εlB) Fig. 1.35. Phase diagram for the double layer QHE system (after Murphy et al. [81]). Only samples whose parameters lie below the dashed line exhibit a quantized Hall plateau and excitation gap.

that there is an excitation gap and a quantized Hall plateau [81,85,96]. This has either a trivial or a highly non-trivial explanation, depending on the ratio ∆SAS /Ec . For large ∆SAS the electrons tunnel back and forth so rapidly that it is as if there is only a single quantum well. The tunnel splitting ∆SAS is then analogous to the electric subband splitting in a (wide) single well. All symmetric states are occupied and all antisymmetric states are empty and we simply have the ordinary ν = 1 integer Hall eﬀect. Correlations are irrelevant in this limit and the excitation gap is close to the single-particle ¯ ωc , whichever is smaller). What is highly non-trivial about gap ∆SAS (or h this system is the fact that the ν = 1 quantum Hall plateau survives even when ∆SAS Ec . In this limit the excitation gap has clearly changed to become highly collective in nature since the observed [81, 85] gap can be on the scale of 20 K even when ∆SAS ∼ 1 K. Because of the spontaneously broken XY symmetry [51,73,88,89,92], the excitation gap actually survives the limit ∆SAS −→ 0! This cross-over from single-particle to collective gap is quite analogous to that for spin polarized single layers. There the excitation gap survives the limit of zero Zeeman splitting so long as the Coulomb interaction makes the spin stiﬀness non-zero. This eﬀect in double-layer systems is visible in Figure 1.35 which shows the QHE phase diagram obtained by Murphy et al. [44,81] as a function of layer-separation and tunneling energy. A ν = 1 quantum Hall plateau and gap is observed in the regime below the

158

Topological Aspects of Low Dimensional Systems

Fig. 1.36. The charge activation energy gap, ∆, as a function of tilt angle in a weakly tunneling double-layer sample (∆SAS = 0.8 K). The solid circles are for ﬁlling ν = 1, open triangles for ν = 2/3. The arrow indicates the critical angle θc . The solid line is a guide to the eye. The dashed line refers to a simple estimate of the renormalization of the tunneling amplitude by the parallel magnetic ﬁeld. Relative to the actual decrease, this one-body eﬀect is very weak and we have neglected it. Inset: Arrhenius plot of dissipation. The low temperature activation energy is ∆ = 8.66 K and yet the gap collapses at a much lower temperature scale of about 0.4 K (1/T ≈ 2.5). (After Murphy et al. [81].)

dashed line. Notice that far to the right, the single particle tunneling energy dominates over the coulomb energy and we have essentially a one-body integer QHE state. However the QHE survives all the way into ∆SAS = 0 provided that the layer separation is below a critical value d/ B ≈ 2. In this limit there is no tunneling and the gap is purely many-body in origin and, as we will show, is associated with the remarkable “pseudospin ferromagnetic” quantum state exhibiting spontaneous interlayer phase coherence. A second indication of the highly collective nature of the excitations can be seen in the Arrhenius plots of thermally activated dissipation [81] shown in the inset of Figure 1.36 The low temperature activation energy ∆ is, as already noted, much larger than ∆SAS . If ∆ were nevertheless somehow a single-particle gap, one would expect the Arrhenius law to be valid up to temperatures of order ∆. Instead one observes a fairly sharp leveling oﬀ in the dissipation as the temperature increases past values as low as ∼ 0.05∆. This is consistent with the notion of a thermally induced collapse of the order that had been producing the collective gap. The third signiﬁcant feature of the experimental data pointing to a highly-ordered collective state is the strong response of the system to relatively weak magnetic ﬁelds B applied in the plane of the 2D electron gases.

S.M. Girvin: The Quantum Hall Eﬀect

d

159

Φo

L ||

Fig. 1.37. A process in a double-layer two-dimensional electron gas system which encloses ﬂux from the parallel component of the magnetic ﬁeld. One interpretation of this process is that an electron tunnels from the upper layer to the lower layer (near the left end of the ﬁgure). The resulting particle-hole pair then travels coherently to the right and is annihilated by a subsequent tunneling event in the reverse direction. The quantum amplitude for such paths is sensitive to the parallel component of the ﬁeld.

In Figure 1.36 we see that the charge activation gap drops dramatically as the magnetic ﬁeld is tilted (keeping B⊥ constant). Within a model that neglects higher electric subbands, we can treat the electron gases as strictly two-dimensional. This is important since B can aﬀect the system only if there are processes that carry electrons around closed loops containing ﬂux. A prototypical such process is illustrated in Figure 1.37. An electron tunnels from one layer to the other at point A, and travels to point B. Then it (or another indistinguishable electron) tunnels back and returns to the starting point. The parallel ﬁeld contributes to the quantum amplitude for this process (in the 2D gas limit) a gauge-invariant Aharonov-Bohm phase factor exp (2πiΦ/Φ0 ) where Φ is the enclosed ﬂux and Φ0 is the quantum of ﬂux. Such loop paths evidently contribute signiﬁcantly to correlations in the system since the activation energy gap is observed to decrease very rapidly with B , falling by factors of order two or more until a critical ﬁeld, B ∗ ∼ 0.8 T, is reached at which the gap essentially ceases changing [81]. To understand how remarkably small B ∗ is, consider the following. We can deﬁne a length L from the size of the loop needed to enclose one quantum of ﬂux: L B ∗ d = Φ0 . (L [˚ A] = 4.137 × 105 /d[˚ A]B ∗ [T].) For B ∗ = 0.8 T A which is approximately twenty times the and d = 150 ˚ A, L = 2700 ˚ spacing between electrons in a given layer and thirty times larger than the quantized cyclotron orbit radius ≡ (¯ hc/eB⊥ )1/2 within an individual layer. Signiﬁcant drops in the excitation gap are already seen at ﬁelds of 0.1 T implying enormous phase coherent correlation lengths must exist. Again this shows the highly-collective long-range nature of the ordering in this system.

160

Topological Aspects of Low Dimensional Systems

In the next subsection we shall brieﬂy outline a detailed model which explains all these observed eﬀects. 1.25 Interlayer phase coherence The essential physics of spontaneous inter-layer phase coherence can be examined from a microscopic point of view [51, 73, 90–92] or a macroscopic Chern-Simons ﬁeld theory point of view [51, 73, 88, 89], but it is perhaps most easily visualized in the simple variational wave function which places the spins purely in the XY plane [51] 1 0 † ck↑ + c†k↓ eiϕ |0. (1.298) |ψ = k

Note for example, that if ϕ = 0 then we have precisely the non-interacting single Slater determinant ground state in which electrons are in the symmetric state which, as discussed previously in the analysis of the eﬀective Lagrangian in equation (1.294), minimizes the tunneling energy. This means that the system has a deﬁnite total number of particles (ν = 1 exactly) but an indeﬁnite number of particles in each layer. In the absence of inter-layer tunneling, the particle number in each layer is a good quantum number. Hence this wave function represents a state of spontaneously broken symmetry [51,88,89] in the same sense that the BCS state for a superconductor has indeﬁnite (total) particle number but a deﬁnite phase relationship between states of diﬀerent particle number. In the absence of tunneling (t = 0) the energy can not depend on the phase angle ϕ and the system exhibits a global U (1) symmetry associated with conservation of particle number in each layer [88]. One can imagine allowing ϕ to vary slowly with position to produce excited states. Given the U (1) symmetry, the eﬀective Hartree-Fock energy functional for these states is restricted to have the leading form 1 (1.299) H = ρs d2 r|∇ϕ|2 + . . . . 2 The origin of the ﬁnite “spin stiﬀness” ρs is the loss of exchange energy which occurs when ϕ varies with position. Imagine that two particles approach each other. They are in a linear superposition of states in each of the layers (even though there is no tunneling!). If they are characterized by the same phase ϕ, then the wave function is symmetric under pseudospin exchange and so the spatial wave function is antisymmetric and must vanish as the particles approach each other. This lowers the Coulomb energy. If a phase gradient exists then there is a larger amplitude for the particles to be near each other and hence the energy is higher. This loss of exchange energy is the source of the ﬁnite spin stiﬀness and is what causes the system to spontaneously “magnetize”.

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161

We see immediately that the U (1) symmetry leads to equation (1.299) which deﬁnes an eﬀective XY model which will contain vortex excitations which interact logarithmically [97, 98]. In a superconducting ﬁlm the vortices interact logarithmically because of the kinetic energy cost of the supercurrents circulating around the vortex centers. Here the same logarithm appears, but it is due to the potential energy cost (loss of exchange) associated with the phase gradients (circulating pseudo-spin currents). Hartree-Fock estimates [51] indicate that the spin stiﬀness ρs and hence the Kosterlitz-Thouless (KT) critical temperature are on the scale of 0.5 K in typical samples. Vortices in the ϕ ﬁeld are reminiscent of Laughlin’s fractionally charged quasiparticles but in this case carry charges ± 21 e and can be left- or right-handed for a total of four “ﬂavors” [51, 73]. It is also possible to show [51, 94] that the presence of spontaneous magnetization due to the ﬁnite spin stiﬀness means that the charge excitation gap is ﬁnite (even though the tunnel splitting is zero). Thus the QHE survives [51] the limit ∆SAS −→ 0. Since the “charge” conjugate to the phase ϕ is the z component of the pseudo spin S z , the pseudospin “supercurrent”

J = ρs ∇ϕ

(1.300)

represents oppositely directed charge currents in each layer. Below the KT transition temperature, such current ﬂow will be dissipationless (in linear response) just as in an ordinary superﬂuid. Likewise there will be a linearly dispersing collective Goldstone mode as in a superﬂuid [51,73,88–90] rather than a mode with quadratic dispersion as in the SU (2) symmetric ferromagnet. (This is somewhat akin to the diﬀerence between an ideal bose gas and a repulsively interacting bose gas.) If found, this Kosterlitz-Thouless transition would be the ﬁrst example of a ﬁnite-temperature phase transition in a QHE system. The transition itself has not yet been observed due to the tunneling amplitude t being signiﬁcant in samples having the layers close enough together to have strong correlations. As we have seen above however, signiﬁcant eﬀects which imply the existence of long-range XY order correlations have been found. Whether or not an appropriate sample can be constructed to observe the phase transition is an open question at this point.

162

Topological Aspects of Low Dimensional Systems

Exercise 1.23. Following the method used to derive equation (1.230), show that the collective mode for the Lagrangian in equation (1.294) has linear rather than quadratic dispersion due to the presence of the β term. (Assume ∆ = t = 0.) Hint: Consider small ﬂuctuations of the magnetization away from m = (1, 0, 0) and choose an appropriate gauge for A for this circumstance. Present a qualitative argument that layer imbalance caused by ∆ does not fundamentally change any of the results described in this section but rather simply renormalizes quantities like the collective mode velocity. That is, explain why the ν = 1 QHE state is robust against charge imbalance. (This is an important signature of the underlying physics. Certain other interlayer correlated states (such as the one at total ﬁlling ν = 1/2) are quite sensitive to charge imbalance [43].) 1.26 Interlayer tunneling and tilted ﬁeld eﬀects As mentioned earlier, a ﬁnite tunneling amplitude t between the layers breaks the U (1) symmetry . / 1 (1.301) Heﬀ = d2 r ρs |∇ϕ|2 − nt cos ϕ 2 by giving a preference to symmetric tunneling states. This can be seen from the tunneling Hamiltonian 0 1 (1.302) HT = −t d2 r ψ↑† (r)ψ↓ (r) + ψ↓† (r)ψ↑ (r) which can be written in the spin representation as HT = −2t d2 rS x (r).

(1.303)

(Recall that the eigenstates of S x are symmetric and antisymmetric combinations of up and down.) As the separation d increases, a critical point d∗ is reached at which the magnetization vanishes and the ordered phase is destroyed by quantum ﬂuctuations [51, 73]. This is illustrated in Figure 1.35. For ﬁnite tunneling t, we will see below that the collective mode becomes massive and quantum ﬂuctuations will be less severe. Hence the phase boundary in Figure 1.35 curves upward with increasing ∆SAS . The introduction of ﬁnite tunneling amplitude destroys the U (1) symmetry and makes the simple vortex-pair conﬁguration extremely expensive. To lower the energy the system distorts the spin deviations into a domain wall or “string” connecting the vortex cores as shown in Figure 1.38.

S.M. Girvin: The Quantum Hall Eﬀect

163

Fig. 1.38. Meron pair connected by a domain wall. Each meron carries a charge e/2 which tries to repel the other one.

The spins are oriented in the x ˆ direction everywhere except in the central domain wall region where they tumble rapidly through 2π. The domain wall has a ﬁxed energy per unit length and so the vortices are now conﬁned by a linear “string tension” rather than logarithmically. We can estimate the string tension by examining the energy of a domain wall of inﬁnite length. The optimal form for a domain wall lying along the y axis is given by ϕ(r) = 2 arcsin [tanh (λx)],

(1.304)

where the characteristic width of the string is

−1

λ

.

2π 2 ρs = t

/ 12 .

(1.305)

164

Topological Aspects of Low Dimensional Systems

The resulting string tension is .

tρs T0 = 8 2π 2

/ 12 .

(1.306)

Provided the string is long enough (Rλ 1), the total energy of a segment of length R will be well-approximated by the expression Epair = 2Emc +

e2 + T0 R. 4R

(1.307)

This is minimized at R∗ = e2 /4T0. The linear conﬁnement brings the charged vortices closer together and rapidly increases the Coulomb energy. In the limit of very large tunneling, the meron pair shrinks and the singleparticle excitation (hole or extra spin-reversed electron) limit must be recovered. The presence of parallel ﬁeld B ﬁeld can be conveniently described with the gauge choice = xB zˆ A (1.308) where zˆ is the growth direction. In this gauge the tunneling amplitude transforms to t → t eiQx (1.309) and the energy becomes . / t 1 2 2 H = d r ρs |∇ϕ| − cos (ϕ − Qx) 2 2π 2

(1.310)

where Q = 2π/L and L is the length associated with one quantum of ﬂux for the loops shown in Figure 1.37. This is the so-called Pokrovsky-Talopov model which exhibits a commensurate-incommensurate phase transition. At low B , Q is small and the low energy state has ϕ ≈ Qx; i.e. the local spin orientation “tumbles”. In contrast, at large B the gradient cost is too large and we have ϕ ≈ constant. It is possible to show [51, 94] that this phase transition semiquantitatively explains the rapid drop and subsequent leveling oﬀ of the activation energy vs. B seen in Figure 1.36. Exercise 1.24. Derive equation (1.304) for the form of the “soliton” that minimizes the energy cost for the Hamiltonian in equation (1.301). Much of my work on the quantum Hall eﬀect has been in collaboration with Allan MacDonald. The more recent work on quantum Hall ferromagnets has also been done in collaboration with M. Abolfath, L. Belkhir, L. Brey, R. Cˆ ot´e, H. Fertig, P. Henelius, K. Moon, H. Mori, J. J. Palacios, A. Sandvik, H. Stoof, C. Timm, K. Yang, D. Yoshioka, S. C. Zhang and L. Zheng. It is a pleasure to acknowledge

S.M. Girvin: The Quantum Hall Eﬀect

165

many useful conversations with S. Das Sarma, M.P.A. Fisher, N. Read and S. Sachdev. It is a pleasure to thank Ms. Daphne Klemme for her expert typesetting of my scribbled notes and Jairo Sinova for numerous helpful comments on the manuscript. This work was supported by NSF DMR-9714055.

Appendix A

Lowest Landau level projection

A convenient formulation of quantum mechanics within the subspace of the Lowest Landau Level (LLL) was developed by Girvin and Jach [26], and was exploited by Girvin, MacDonald and Platzman in the magnetoroton theory of collective excitations of the incompressible states responsible for the fractional quantum Hall eﬀect [29]. Here we brieﬂy review this formalism. See also reference [8]. We ﬁrst consider the one-body case and choose the symmetric gauge. The single-particle eigenfunctions of kinetic energy and angular momentum in the LLL are given in equation (1.76) 1 |z|2 m z exp − φm (z) = , (A.1) 4 (2π2m m!)1/2 where m is a non-negative integer, and z = (x + iy)/ . From (A.1) it is clear that any wave function in the LLL can be written in the form ψ(z) = f (z) e−

|z|2 4

(A.2)

where f (z) is an analytic function of z, so the subspace in the LLL is isomorphic to the Hilbert space of analytic functions [8, 26, 99]. Following Bargman [26, 99], we deﬁne the inner product of two analytic functions as (f, g) = dµ(z) f ∗ (z) g(z), (A.3) where dµ(z) ≡ (2π)−1 dxdy e−

|z|2 2

.

(A.4)

Now we can deﬁne bosonic ladder operators that connect φm to φm±1 (and which act on the polynomial part of φm only): a† a

z √ , 2 √ ∂ , = 2 ∂z =

(A.5a) (A.5b)

166

Topological Aspects of Low Dimensional Systems

so that a† ϕm a ϕm

= =

(f, a† g) = (f, a g) =

√ m + 1 ϕm+1 , √ m ϕm−1 ,

(A.6a) (A.6b)

(a f, g), (a† f, g).

(A.6c) (A.6d)

All operators that have non-zero matrix elements only within the LLL can † be expressed in√terms of a and to notice that the adjoint √ a . It is essential † ∗ of a is not z / 2 but a ≡ 2∂/∂z, because z ∗ connects states in the LLL √ to higher Landau levels. Actually a is the projection of z ∗ / 2 onto the LLL as seen clearly in the following expression: z∗ z (f, √ g) = ( √ f, g) = (a† f, g) = (f, a g). 2 2 So we ﬁnd

∂ , (A.7) ∂z where the overbar indicates projection onto the LLL. Since z ∗ and z do not commute, we need to be very careful to properly order the operators before projection. A little thought shows that in order to project an operator which is a combination of z ∗ and z, we must ﬁrst normal order all the z ∗ ’s to the left of the z’s, and then replace z ∗ by z ∗ . With this rule in mind and (A.7), we can easily project onto the LLL any operator that involves space coordinates only. For example, the one-body density operator in momentum space is z∗ = 2

∗ ∗ ∗ i i i ∗ 1 1 1 ρq = √ e−iq·r = √ e− 2 (q z+qz ) = √ e− 2 qz e− 2 q z , A A A

where A is the area of the system, and q = qx + iqy . Hence |q|2 ∂ i ∗ 1 1 ρq = √ e−iq ∂z e− 2 q z = √ e− 4 τq , A A

where

∂

i

τq = e−iq ∂z − 2 q

∗

z

(A.8)

(A.9)

is a unitary operator satisfying the closed Lie algebra τq τk

=

[τq , τk ] =

i

τq+k e 2 q∧k , 2i τq+k

q∧k , sin 2

(A.10a) (A.10b)

S.M. Girvin: The Quantum Hall Eﬀect

167

^z x (-q)

^z x (-k)

φ = 2π Φ Φ0

^z x k

^z x q Fig. 1.39. Illustration of magnetic translations and phase factors. When an electron travels around a parallelogram (generated by τq τk τ−q τ−k ) it picks up a phase φ = 2π ΦΦ0 = q ∧ k, where Φ is the ﬂux enclosed in the parallelogram and Φ0 is the ﬂux quantum.

ˆ. We also have τq τk τ−q τ−k = eiq∧k . This is a where q ∧ k ≡ 2 (q × k) · z familiar feature of the group of translations in a magnetic ﬁeld, because q ∧k is exactly the phase generated by the ﬂux in the parallelogram generated by q 2 and k 2 . Hence the τ ’s form a representation of the magnetic translation group (see Fig. 1.39). In fact τq translates the particle a distance 2 zˆ × q. This means that diﬀerent wave vector components of the charge density do not commute. It is from here that non-trivial dynamics arises even though the kinetic energy is totally quenched in the LLL subspace. This formalism is readily generalized to the case of many particles with spin, as we will show next. In a system with area A and N particles the projected charge and spin density operators are ρq

=

N N 1 −iq·ri 1 − |q|2 √ e = √ e 4 τq (i) A i=1 A i=1

(A.11a)

Sqµ

=

N N 1 −iq·ri µ 1 − |q|2 √ e Si = √ e 4 τq (i) Siµ , A i=1 A i=1

(A.11b)

where τq (i) is the magnetic translation operator for the ith particle and Siµ is the µth component of the spin operator for the ith particle. We immediately ﬁnd that unlike the unprojected operators, the projected spin and charge density operators do not commute: k∧q 2i |k+q|2 −|k|2 −|q|2 µ 4 [¯ ρk , S¯qµ ] = √ e Sk+q sin = 0. (A.12) 2 A

168

Topological Aspects of Low Dimensional Systems

This implies that within the LLL, the dynamics of spin and charge are entangled, i.e., when you rotate spin, charge gets moved. As a consequence of that, spin textures carry charge as discussed in the text. B

B Berry’s phase and adiabatic transport

Consider a quantum system with a Hamiltonian HR which depends on a set Assume of externally controlled parameters represented by the vector R. there is always a ﬁnite excitation gap separating that for some domain of R the ground state energy from the rest of the spectrum of HR . Consider now the situation where the parameters R(t) are slowly varied around a closed loop in parameter space in a time interval T ). R(0) = R(T

(B.1)

If the circuit is transversed suﬃciently slowly so that h/T ∆min where ∆min is the minimum excitation gap along the circuit, then the state will evolve adiabatically. That is, the state will always be the local ground state (0) ΨR(t) of the instantaneous Hamiltonian HR(t) . Given the complete set of energy eigenstates for a given R (j)

(j)

(j)

HR ΨR = R ΨR ,

(B.2)

the solution of the time-dependent Schr¨ odinger equation i¯ h

∂ψ(r, t) = HR(t) ψ(r, t) ∂t

is ψ(r, t) =

(0) (r ) ΨR(t)

+

j=0

e

iγ(t)

e

i −h ¯

t 0

(B.3)

(0)

dt

R(t )

(j)

aj (t) ΨR(t) .

(B.4)

The adiabatic approximation consists of neglecting the admixture of excited states represented by the second term. In the limit of extremely slow varia tion of R(t), this becomes exact as long as the excitation gap remains ﬁnite. The only unknown at this point is the Berry Phase [49] γ(t) which can be found by requiring that ψ(r, t) satisfy the time-dependent Schr¨odinger equation. The LHS of equation (B.3) is i¯ h

∂ψ(r, t) (0) ψ(r, t) = −¯ hγ(t) ˙ + R(t) ∂t / . t (0) i ∂ dt (0) iγ(t) − h ¯ R(t ) 0 +i¯ hR˙ µ Ψ ( r ) e e ∂Rµ R(t)

(B.5)

S.M. Girvin: The Quantum Hall Eﬀect

169

if we neglect the aj (t) for j > 0. The RHS of equation (B.3) is (0)

HR(t) ψ(r, t) = R(t) ψ(r, t)

(B.6)

within the same approximation. Now using the completeness relation % % ∞ $ ∂ (j) (0) (j) ∂ (0) = (B.7) ΨR µ ΨR · ΨR ∂Rµ ΨR ∂R j=0 In the adiabatic limit we can neglect the excited state contributions so equation (B.5) becomes . $ % / ∂ψ (0) ∂ (0) (0) = −¯ hγ(t) ˙ + i¯ hR˙ µ ΨR µ ΨR(t) i¯ h + ψ. (B.8) R(t) ∂t ∂R This matches equation (B.6) provided $ % (0) ∂ (0) µ ˙ γ(t) ˙ = iR (t) ΨR(t) · (B.9) ∂Rµ ΨR(t) (0) (0) = 1 guarantees that γ˙ is purely real. The constraint ΨR ΨR we have a Notice that there is a kind of gauge freedom here. For each R diﬀerent set of basis states and we are free to choose their phases independently. We can think of this as a gauge choice in the parameter space. Hence γ˙ and γ are “gauge dependent” quantities. It is often possible to choose a gauge in which γ˙ vanishes. The key insight of Berry [49] however was that this is not always the case. For some problems involving a closed-circuit Γ in parameter space the gauge invariant phase $ % T ( (0) ∂ (0) dt γ˙ = i dRµ ΨR µ ΨR (B.10) γBerry ≡ ∂R 0 Γ is non-zero. This is a gauge invariant quantity because the system returns to its starting point in parameter space and the arbitrary phase choice drops out of the answer. This is precisely analogous to the result in electrodynamics that the line integral of the vector potential around a closed loop is gauge invariant. In fact it is useful to deﬁne the “Berry connection” A on the parameter space by $ % ) = i Ψ(0) ∂ Ψ(0) (B.11) Aµ (R R R ∂Rµ which gives the suggestive formula ( γBerry =

Γ

· A(r ). dR

(B.12)

170

Topological Aspects of Low Dimensional Systems

Notice that the Berry’s phase is a purely geometric object independent of the particular velocity R˙ µ (t) and dependent solely on the path taken in parameter space. It is often easiest to evaluate this expression using Stokes theorem since the curl of A is a gauge invariant quantity. As a simple example [49] let us consider the Aharonov-Bohm eﬀect where A will turn out to literally be the electromagnetic vector potential. Let there be an inﬁnitely long solenoid running along the z axis. Consider a particle with charge q trapped inside a box by a potential V H=

q 2 1 p− A + V r − R(t) . 2m c

(B.13)

The position of the box is moved along a closed path R(t) which encircles the solenoid but keeps the particle outside the region of magnetic ﬂux. Let be the adiabatic wave function in the absence of the vector χ(0) r − R(t) potential. Because the particle only sees the vector potential in a region is readily where it has no curl, the exact wave function in the presence of A constructed r i q r ) d r ·A( (0) h ¯ c R(t) (0) r − R(t) (B.14) ΨR(t) ( r ) = e χ where the precise choice of integration path is immaterial since it is interior has no curl. It is straightforward to verify that Ψ(0) exto the box where A R(t) actly solves the Schr¨ odinger equation for the Hamiltonian in equation (B.13) in the adiabatic limit. The arbitrary decision to start the line integral in equation (B.14) at R constitutes a gauge choice in parameter space for the Berry connection. Using equation (B.11) the Berry connection is easily found to be ) = + q Aµ (R ) Aµ (R hc ¯

(B.15)

and the Berry phase for the circuit around the ﬂux tube is simply the Aharonov-Bohm phase ( Φ γBerry = dRµ Aµ = 2π (B.16) Φ0 where Φ is the ﬂux in the solenoid and Φ0 ≡ hc/q is the ﬂux quantum. As a second example [49] let us consider a quantum spin with Hamiltonian H = −∆(t) · S. (B.17) and so the circuit in parameter The gap to the ﬁrst excited state is h ¯ |∆| = 0 where the spectrum has a degeneracy. space must avoid the origin ∆

S.M. Girvin: The Quantum Hall Eﬀect

171

Clearly the adiabatic ground state has

∆ (0) (0) Ψ∆ · =h ¯S S Ψ∆ |∆|

(B.18)

is deﬁned by polar angle θ and azimuthal angle ϕ, If the orientation of ∆ An appropriate set of states obeying this the same must be true for S. for the case S = 12 is cos θ2 |ψθ,ϕ = (B.19) sin θ2 eiϕ since these obey θ θ ψθ,ϕ |S z | ψθ,ϕ = h ¯ S cos2 − sin2 =h ¯ S cos θ 2 2

(B.20)

¯ S sin θ eiϕ . ψθ,ϕ |S x + iS y | ψθ,ϕ = ψθ,ϕ S + ψθ,ϕ = h

(B.21)

and

rotates slowly about the z Consider the Berry’s phase for the case where ∆ axis at constant θ $ % 2π ∂ ψθ,ϕ dϕ ψθ,ϕ γBerry = i ∂ϕ 0 2π θ −iϕ θ 0 dϕ cos sin e = i i sin θ2 eiϕ 2 2 0 2π = −S dϕ (1 − cos θ) 0

= −S

2π

1

dϕ 0

cos θ

d cos θ = −SΩ

(B.22)

where Ω is the solid angle subtended by the path as viewed from the origin of the parameter space. This is precisely the Aharonov-Bohm phase one expects for a charge −S particle traveling on the surface of a unit sphere surrounding a magnetic monopole. It turns out that it is the degeneracy in the spectrum at the origin which produces the monopole [49]. Notice that there is a singularity in the connection at the “south pole” θ = π. This can be viewed as the Dirac string (solenoid containing one quantum of ﬂux) that is attached to the monopole. If we had chosen the basis (B.23) e−iϕ |ψθ,ϕ the singularity would have been at the north pole. The reader is directed to Berry’s original paper [49] for further details.

172

Topological Aspects of Low Dimensional Systems

In order to correctly reproduce the Berry phase in a path integral for the spin whose Hamiltonian is given by equation (B.17), the Lagrangian must be (B.24) L=h ¯ S {−m ˙ µ Aµ + ∆µ mµ + λ(mµ mµ − 1)} where m is the spin coordinate on a unit sphere, λ enforces the length constraint, and =m m×A (B.25) ∇ is the monopole vector potential. As discussed in the text in Section 1.15, this Lagrangian correctly reproduces the spin precession equations of motion. References [1] Girvin S.M. in Chap. 10 and App. I of Ref. [3]; Girvin S.M. and MacDonald A.H., Phys. Rev. Lett. 58 (1987) 1252; Zhang S.-C., Hansson H. and Kivelson S., Phys. Rev. Lett. 62 (1989) 82; Read N., Phys. Rev. Lett. 62 (1989) 86; Lee D.-H. and Fisher M.P.A., Phys. Rev. Lett. 63 (1989) 903. [2] For reviews and extensive references see the Chapters by Halperin B.I. and by Jain J.K. in reference [6]. [3] The Quantum Hall Eﬀect, 2nd Ed., edited by Prange R.E. and Girvin S.M. (Springer-Verlag, New York, 1990). [4] Chakraborty T. and Pietil¨ ainen P., The Fractional Quantum Hall Eﬀect (SpringerVerlag, Berlin, New York, 1988). [5] MacDonald A.H., Quantum Hall Eﬀect: A Perspective (Kluwer Academic Publishers, 1989). [6] Perspectives in Quantum Hall Eﬀects, Edited by Das Sarma S. and Pinczuk A. (Wiley, New York, 1997). [7] Introduction to the Theory of the Integer Quantum Hall Eﬀect, edited by Janßen M., Viehweger O., Fastenrath U. and Hajdu J. (VCH, Weinheim, New York, 1994). [8] Quantum Hall Eﬀect, edited by Stone M. (World Scientiﬁc, Singapore, 1992). [9] Kivelson S., Lee D.-H. and Zhang S.-C., Scientiﬁc American (1996) p. 86. [10] Shou Cheng Zhang, Int. J. Mod. Phys. B 6 (1992) 25. [11] MacDonald A.H., in Mesoscopic Quantum Physics, Les Houches, Session LXI, edited by Akkermans E., Montambaux G., Pichard J.-L. and Zinn-Justin J. (North Holland, Amsterdam, 1995). [12] Lee P.A. and Ramakrishnan T.V., Rev. Mod. Phys. 57 (1985) 287. [13] Sondhi S.L., Girvin S.M., Carini J.P. and Shahar D., Rev. Mod. Phys. Colloq. 69 (1997) 315. [14] Bergmann G., Phys. Rep. 107 (1984) 1–58. [15] St¨ ormer H.L., Physica B177 (1992) 401. [16] Eisenstein J.P., St¨ ormer H.L., Narayanamurti V., Cho A.Y., Gossard A.C. and Tu C.W., Phys. Rev. Lett. 55 (1985) 875. [17] Kane C.L. and Fisher M.P.A., Phys. Rev. B 46 (1992) 7268; Op. Cit. (1992) 15233; Phys. Rev. B 51 (1995) 13449; Kane C.L., Fisher M.P.A. and Polchinksi J., Phys. Rev. Lett. 72 (1994) 4129. [18] Buttiker M., Phys. Rev. B 38 (1988) 9375. [19] Huckestein B., Rev. Mod. Phys. 67 (1995) 357 and numerous references therein.

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COURSE 3

ASPECTS OF CHERN-SIMONS THEORY

G.V. DUNNE Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.

Contents 1 Introduction 2 Basics of planar field theory 2.1 Chern-Simons coupled to matter ﬁelds - “anyons” . . 2.2 Maxwell-Chern-Simons: Topologically massive gauge 2.3 Fermions in 2 + 1-dimensions . . . . . . . . . . . . . 2.4 Discrete symmetries: P, C and T . . . . . . . . . . . 2.5 Poincar´e algebra in 2 + 1-dimensions . . . . . . . . . 2.6 Nonabelian Chern-Simons theories . . . . . . . . . .

179

. . . . . .

. . . . . .

. . . . . .

. . . . . .

182 182 186 189 190 192 193

3 Canonical quantization of Chern-Simons theories 3.1 Canonical structure of Chern-Simons theories . . . . . . . . . 3.2 Chern-Simons quantum mechanics . . . . . . . . . . . . . . . 3.3 Canonical quantization of abelian Chern-Simons theories . . . 3.4 Quantization on the torus and magnetic translations . . . . . 3.5 Canonical quantization of nonabelian Chern-Simons theories . 3.6 Chern-Simons theories with boundary . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

195 195 198 203 205 208 212

4 Chern-Simons vortices 4.1 Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices 4.2 Relativistic Chern-Simons vortices . . . . . . . . . . . . . . 4.3 Nonabelian relativistic Chern-Simons vortices . . . . . . . . 4.4 Nonrelativistic Chern-Simons vortices: Jackiw-Pi model . . 4.5 Nonabelian nonrelativistic Chern-Simons vortices . . . . . . 4.6 Vortices in the Zhang-Hansson-Kivelson model for FQHE . 4.7 Vortex dynamics . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

214 214 219 224 225 228 231 234

5 Induced Chern-Simons terms 5.1 Perturbatively induced Chern-Simons terms: Fermion loop . . . . 5.2 Induced currents and Chern-Simons terms . . . . . . . . . . . . . 5.3 Induced Chern-Simons terms without fermions . . . . . . . . . . 5.4 A ﬁnite temperature puzzle . . . . . . . . . . . . . . . . . . . . . 5.5 Quantum mechanical ﬁnite temperature model . . . . . . . . . . 5.6 Exact ﬁnite temperature 2 + 1 eﬀective actions . . . . . . . . . . 5.7 Finite temperature perturbation theory and Chern-Simons terms

. . . . . . .

237 238 242 243 246 248 253 256

. . . . theory . . . . . . . . . . . . . . . .

. . . . . . .

ASPECTS OF CHERN-SIMONS THEORY G.V. Dunne

Abstract Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar ﬁeld theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.

1

Introduction

Planar physics – physics in two spatial dimensions – presents many interesting surprises, both experimentally and theoretically. The behaviour of electrons and photons [or more generally: fermions and gauge ﬁelds) diﬀers in interesting ways from the standard behaviour we are used to in classical and quantum electrodynamics. For example, there exists a new type of gauge theory, completely diﬀerent from Maxwell theory, in 2+1 dimensions. This new type of gauge theory is known as a “Chern-Simons theory” (the origin of this name is discussed below in Section 2.6 on nonabelian theories). These Chern- Simons theories are interesting both for their theoretical novelty, and for their practical application for certain planar condensed matter phenomena, such as the fractional quantum Hall eﬀect (see Steve Girvin’s lectures at this School). In these lectures I concentrate on ﬁeld theoretic properties of ChernSimons theories. I have attempted to be relatively self-contained, and accessible to someone with a basic knowledge of ﬁeld theory. Actually, several important new aspects of Chern-Simons theory rely only on quantum mechanics and classical electrodynamics. Given the strong emphasis of this Summer School on condensed matter phenomena, I have chosen, wherever possible, to phrase the discussion in terms of quantum mechanical and solid state physics examples. For example, in discussing the canonical quantization of Chern-Simons theories, rather than delving deeply into conformal ﬁeld theory, instead I have expressed things in terms of the Landau problem (quantum mechanical charged particles in a magnetic ﬁeld) and the magnetic translation group. c EDP Sciences, Springer-Verlag 1999

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Topological Aspects of Low Dimensional Systems

In Section 2, I introduce the basic kinematical and dynamical features of planar ﬁeld theories, such as anyons, topologically massive gauge ﬁelds and planar fermions. I also discuss the discrete symmetries P, C and T , and nonabelian Chern-Simons gauge theories. Section 3 is devoted to the canonical structure and canonical quantization of Chern-Simons theories. This is phrased in quantum mechanical language using a deep analogy between Chern-Simons gauge theories and quantum mechanical Landau levels (which are so important in the understanding of the fractional quantum Hall eﬀect). For example, this connection gives a very simple understanding of the origin of massive gauge excitations in Chern-Simons theories. In Section 4, I consider the self-dual vortices that arise when Chern-Simons gauge ﬁelds are coupled to scalar matter ﬁelds, with either relativistic or nonrelativistic dynamics. Such vortices are interesting examples of self-dual ﬁeld theoretic structures with anyonic properties, and also arise in models for the fractional quantum Hall eﬀect where they correspond to Laughlin’s quasipartcle excitations. The ﬁnal Section concerns Chern-Simons terms that are induced radiatively by quantum eﬀects. These can appear in fermionic theories, in Maxwell-Chern-Simons models and in Chern-Simons models with spontaneous symmetry breaking. The topological nature of the induced term has interesting consequences, especially at ﬁnite temperature. We begin by establishing some gauge theory notation. The familiar Maxwell (or, in the nonabelian case, Yang-Mills) gauge theory is deﬁned in Here A0 is terms of the fundamental gauge ﬁeld (connection) Aµ = (A0 , A). the scalar potential and A is the vector potential. The Maxwell Lagrangian 1 LM = − Fµν F µν − Aµ J µ 4

(1)

is expressed in terms of the ﬁeld strength tensor (curvature) Fµν = ∂µ Aν − ∂ν Aµ , and a matter current J µ that is conserved: ∂µ J µ = 0. This Maxwell Lagrangian is manifestly invariant under the gauge transformation Aµ → Aµ + ∂µ Λ; and, correspondingly, the classical Euler-Lagrange equations of motion ∂µ F µν = J ν

(2)

are gauge invariant. Observe that current conservation ∂ν J ν = 0 follows from the antisymmetry of Fµν . Now note that this Maxwell theory could easily be deﬁned in any spacetime dimension d simply by taking the range of the space-time index µ on the gauge ﬁeld Aµ to be µ = 0, 1, 2, . . . , (d − 1) in d-dimensional spacetime. The ﬁeld strength tensor is still the antisymmetric tensor Fµν = ∂µ Aν − ∂ν Aµ , and the Maxwell Lagrangian (1) and the equations of motion (2) do not change their form. The only real diﬀerence is that the number of independent ﬁelds contained in the ﬁeld strength tensor Fµν is diﬀerent in

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181

diﬀerent dimensions. (Since Fµν can be regarded as a d × d antisymmetric matrix, the number of ﬁelds is equal to 12 d(d − 1).) So at this level, planar (i.e. 2 + 1 dimensional) Maxwell theory is quite similar to the familiar 3 + 1 dimensional Maxwell theory. The main diﬀerence is simply that the magnetic ﬁeld is a (pseudo-) scalar B = ij ∂i Aj in 2 + 1 dimensions, rather =∇ ×A in 3 + 1 dimensions. This is just because than a (pseudo-) vector B is a two-dimensional vector, and in 2 + 1 dimensions the vector potential A the curl in two dimensions produces a scalar. On the other hand, the electric ˙ is a two dimensional vector. So the antisymmetric 3×3 = −∇A 0 −A ﬁeld E ﬁeld strength tensor has three nonzero ﬁeld components: two for the electric and one for the magnetic ﬁeld B. ﬁeld E The real novelty of 2 + 1 dimensions is that instead of considering this ‘reduced’ form of Maxwell theory, we can also deﬁne a completely diﬀerent type of gauge theory: a Chern-Simons theory. It satisﬁes our usual criteria for a sensible gauge theory – it is Lorentz invariant, gauge invariant, and local. The Chern- Simons Lagrangian is LCS =

κ µνρ Aµ ∂ν Aρ − Aµ J µ . 2

(3)

There are several comments to make about this Chern-Simons Lagrangian. First, it does not look gauge invariant, because it involves the gauge ﬁeld Aµ itself, rather than just the (manifestly gauge invariant) ﬁeld strength Fµν . Nevertheless, under a gauge transformation, the Chern-Simons Lagrangian changes by a total space-time derivative δLCS =

κ ∂µ (λ µνρ ∂ν Aρ ) . 2

(4)

Therefore, if we can neglect boundary terms (later we shall encounter important examples where this is not true) then the corresponding Chern-Simons action, SCS = d3 x LCS , is gauge invariant. This is reﬂected in the fact that the classical Euler-Lagrange equations κ µνρ Fνρ = J µ ; 2

or equivalently :

Fµν =

1 µνρ J ρ κ

(5)

are clearly gauge invariant. Note that the Bianchi identity, µνρ ∂µ Fνρ = 0, is compatible with current conservation: ∂µ J µ = 0. A second important feature of the Chern-Simons Lagrangian (3) is that it is first-order in space-time derivatives. This makes the canonical structure of these theories signiﬁcantly diﬀerent from that of Maxwell theory. A related property is that the Chern-Simons Lagrangian is particular to 2 + 1 dimensions, in the sense that we cannot write down such a term in 3 + 1 dimensions – the indices simply do not match up. Actually, it is possible to write down a “Chern-Simons theory” in any odd space-time dimension,

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but it is only in 2 + 1 dimensions that the Lagrangian is quadratic in the gauge ﬁeld. For example, the Chern-Simons Lagrangian in ﬁve-dimensional space-time is L = µνρστ Aµ ∂ν Aρ ∂σ Aτ . At ﬁrst sight, pure Chern-Simons theory looks rather boring, and possibly trivial, because the source-free classical equations of motion (5) reduce to Fµν = 0, the solutions of which are just pure gauges or “ﬂat connections”. This is in contrast to pure Maxwell theory, where even the source–free theory has interesting, and physically important, solutions: plane-waves. Nevertheless, Chern-Simons theory can be made interesting and nontrivial in a number of ways: (i) coupling to dynamical matter ﬁelds (charged scalars or fermions); (ii) coupling to a Maxwell term; (iii) taking the space-time to have nontrivial topology; (iv) nonabelian gauge ﬁelds; (v) gravity.

I do not discuss 2 + 1 dimensional gravity in these lectures as it is far from the topic of this School, but I stress that it is a rich subject that has taught us a great deal about both classical and quantum gravity [1]. 2

Basics of planar field theory

2.1 Chern-Simons coupled to matter ﬁelds - “anyons” In order to understand the signiﬁcance of coupling a matter current J µ = to a Chern-Simons gauge ﬁeld, consider the Chern-Simons equations (ρ, J) (5) in terms of components: ρ Ji

= =

κB κij Ej .

(6)

The ﬁrst of these equations tells us that the charge density is locally proportional to the magnetic ﬁeld – thus the eﬀect of a Chern-Simons ﬁeld is to tie magnetic ﬂux to electric charge. Wherever there is one, there is the other, and they are locally proportional, with the proportionality constant given by the Chern-Simons coupling parameter κ. This is illustrated in Figure 1 for a collection of point charges. The second equation in (6) ensures that this charge-ﬂux relation is preserved under time evolution because the time derivative of the ﬁrst equation ρ˙ = κB˙ = κij ∂i A˙ j

(7)

G.V. Dunne: Aspects of Chern-Simons Theory

e_ κ

e_ κ e_ κ

183

e_ κ

e

e

e

e_ κ e

e_ κ

e

e

Fig. 1. A collection of point anyons with charge e, and with magnetic ﬂux lines of strength κe tied to the charges. The charge and ﬂux are tied together throughout the motion of the particles as a result of the Chern-Simons equations (6).

together with current conservation, ρ˙ + ∂i J i = 0, implies that J i = −κij A˙ j + ij ∂j χ

(8)

which is just the second equation in (6), with the transverse piece χ identiﬁed with κA0 . Thus, the Chern-Simons coupling at this level is pure constraint – we can regard the matter ﬁelds as having their own dynamics, and the eﬀect of the Chern-Simons coupling is to attach magnetic ﬂux to the matter charge density in such a way that it follows the matter charge density wherever it goes. Clearly, this applies either to relativistic or nonrelativistic dynamics for the matter ﬁelds. (A word of caution here–although the Chern-Simons term is Lorentz invariant, we can regard this simply as a convenient shorthand for expressing the constraint Eqs. (6), in much the same way as we can always express a continuity equation ρ˙ + ∂i J i = 0 in a relativistic-looking way as ∂µ J µ = 0. Thus, there is no problem mixing nonrelativistic dynamics for the matter ﬁelds with a “relativistic-looking” Chern-Simons term. The actual dynamics is always inherited from the matter ﬁelds.) This tying of ﬂux to charge provides an explicit realization of “anyons” [2, 3]. (For more details on anyons, see Jan Myrheim’s lectures at this school). Consider, for example, nonrelativistic point charged particles moving in the plane, with magnetic ﬂux lines attached to them. The charge

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Topological Aspects of Low Dimensional Systems

density ρ(x, t) = e

N

δ(x − xa (t))

(9)

a=1

describes N such particles, with the ath particle following Nthe trajectory xa (t). The corresponding current density is j(x, t) = e a=1 x˙ a (t)δ(x − xa (t)). The Chern-Simons equations (6) attach magnetic ﬂux (see Fig. 1) B(x, t) =

N 1 e δ(x − xa (t)) κ a=1

(10)

which follows each point particle throughout its motion. If each particle has mass m, the net action is N

m S= 2 a=1

dt va2

κ + 2

3

d x

µνρ

Aµ ∂ν Aρ −

d3 x Aµ J µ .

(11)

The Chern-Simons equations of motion (5) determine the gauge ﬁeld Aµ (x, t) in terms of the particle current. The gauge freedom may be ﬁxed ·A = 0. in a Hamiltonian formulation by taking A0 = 0 and imposing ∇ Then N (xj − y j ) 1 e ij (xj − xja (t)) Ai (x, t) = ρ( y , t) = (12) d2 y ij 2πκ |x − y |2 2πκ a=1 |x − xa (t)|2 where we have used the two dimensional Green’s function 1 2 log |x − y | = δ (2) (x − y ). ∇ 2π

(13)

As an aside, note that using the identity ∂i arg(x) = −ij xj /|x|2 , where the argument function is arg(x) = arctan( xy ), we can express this vector potential (12) as N

Ai (x) =

e ∂i arg(x − xa ). 2πκ a=1

(14)

Naively, this looks like a pure gauge vector potential, which could presumably therefore be removed by a gauge transformation. However, under such a gauge transformation the corresponding nonrelativistic ﬁeld ψ(x) would acquire a phase factor N e2 ˜ ψ(x) → ψ(x) = exp −i arg(x − xa ) ψ(x) (15) 2πκ a=1

G.V. Dunne: Aspects of Chern-Simons Theory

e_ κ

185

e_ κ

e

e

Fig. 2. Aharonov-Bohm interaction between the charge e of an anyon and the ﬂux e of another anyon under double-interchange. Under such an adiabatic transport, κ the multi-anyon wavefunction acquires an Aharonov-Bohm phase (19).

which makes the ﬁeld non-single-valued for general values of the ChernSimons coupling parameter κ. This lack of single-valuedness is the nontrivial remnant of the Chern-Simons gauge ﬁeld coupling. Thus, even though it looks as though the gauge ﬁeld has been gauged away, leaving a “free” system, the complicated statistical interaction is hidden in the nontrivial ˜ boundary conditions for the non-single-valued ﬁeld ψ. Returning to the point-anyon action (11), the Hamiltonian for this system is N

H=

N

1 m 2 xa )]2 va = [ pa − eA( 2 a=1 2m a=1

(16)

where Ai (xa ) =

N

e ij (xja − xjb ) · 2πκ |xa − xb |2

(17)

b=a

The corresponding magnetic ﬁeld is N

e B(xa ) = δ(xa − xb ) κ

(18)

b=a

so that each particle sees each of the N − 1 others as a point vortex of ﬂux Φ = κe , as expected. Note that the gauge ﬁeld in (17) excludes the self-interaction a = b term, with suitable regularization [3, 4]. An important consequence of this charge-ﬂux coupling is that it leads to new Aharonov-Bohm-type interactions. For example, when one such

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Topological Aspects of Low Dimensional Systems

particle moves adiabatically around another (as shown in Fig. 2), in addition to whatever electrical interactions mediate between them, at the quantum level the nonrelativistic wavefunction acquires an Aharonov-Bohm phase 2 · dx = exp ie A exp ie · (19) κ C If this adiabatic excursion is interpreted as a double interchange of two such identical particles (each with ﬂux attached), then this gives an “anyonic” exchange phase e2 (20) 2κ which can be tuned to any value by specifying the value of the ChernSimons coupling coeﬃcient κ. This is the origin of anyonic statistics in point-particle language. This is a ﬁrst-quantized description of anyons as point particles. However, N-anyon quantum mechanics can be treated, in the usual manner of nonrelativistic many-body quantum mechanics, as the N-particle sector of a nonrelativistic quantum ﬁeld theory [3, 4]. In this case, a Chern-Simons ﬁeld is required to ensure that the appropriate magnetic ﬂux is always attached to the (smeared-out) charged particle ﬁelds ϕ(x, t). This, together with the above-mentioned statistics transmutation, explains the appearance of Chern- Simons ﬁelds in the “composite boson” or “composite fermion” models for the fractional quantum Hall eﬀect, which involve quasiparticles that have magnetic ﬂuxes attached to charged particles [5–8]. In such ﬁeld theories there is a generalized spin-statistics relation similar to (20) – see later in equation (100). By choosing κ appropriately, the anyonic exchange phase (20) can be chosen so that the particles behave either as fermions or as bosons. An explicit example of this statistical transmutation will be used in Section 4.6 on the Zhang- Hansson-Kivelson model [5] for the fractional quantum Hall eﬀect. 2π∆θ =

2.2 Maxwell-Chern-Simons: Topologically massive gauge theory Since both the Maxwell and Chern-Simons Lagrangians produce viable gauge theories in 2 + 1 dimensions, it is natural to consider coupling them together. The result is a surprising new form of gauge ﬁeld mass generation. Consider the Lagrangian 1 κ Fµν F µν + µνρ Aµ ∂ν Aρ . 2 4e 2 The resulting classical ﬁeld equations are LMCS = −

∂µ F µν +

κe2 ναβ Fαβ = 0 2

(21)

(22)

G.V. Dunne: Aspects of Chern-Simons Theory

187

which describe the propagation of a single (transverse) degree of freedom with mass (note that e2 has dimensions of mass in 2 + 1 dimensions, while κ is dimensionless): mMCS = κe2 .

(23)

This has resulted in the terminology “topologically massive gauge theory” [9], where the term “topological” is motivated by the nonabelian ChernSimons theory (see Sect. 2.6). The most direct way to see the origin of this mass is to re-write the equation of motion (22) in terms of the pseudovector “dual” ﬁeld F˜ µ ≡ 1 µνρ Fνρ : 2

∂µ ∂ µ + (κe2 )2 F˜ ν = 0.

(24)

Note that this dual ﬁeld F˜ µ is manifestly gauge invariant, and it also satisﬁes ∂µ F˜ µ = 0. The MCS mass can also be identiﬁed from the corresponding representation theory of the Poincar´e algebra in 2 + 1-dimensions, which also yields the spin of the massive excitation as sMCS =

κ = ±1. |κ|

(25)

We shall discuss these mass and spin properties further in Section 2.5.

Exercise 2.2.1: Another useful way to understand the origin of the massive gauge excitation is to compute the gauge ﬁeld propagator in (for example) a 1 µ 2 covariant gauge with gauge ﬁxing term Lgf = − 2ξe 2 (∂µ A ) . By inverting the quadratic part of the momentum space lagrangian, show that the gauge ﬁeld propagator is 2 p gµν − pµ pν − iκe2 µνρ pρ pµ pν 2 +ξ 2 2 · (26) ∆µν = e p2 (p2 − κ2 e4 ) (p ) This clearly identiﬁes the gauge ﬁeld mass via the pole at p2 = (κe2 )2 .

I emphasize that this gauge ﬁeld mass (23) is completely independent of the standard Higgs mechanism for generating masses for gauge ﬁelds through a nonzero expectation value of a Higgs ﬁeld. Indeed, we can also consider the Higgs mechanism in a Maxwell-Chern-Simons theory, in which case we ﬁnd two independent gauge ﬁeld masses. For example, couple this Maxwell-Chern-Simons theory to a complex scalar ﬁeld φ with a symmetry

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Topological Aspects of Low Dimensional Systems

breaking potential V (|φ|) LMCSH = −

1 κ Fµν F µν + µνρ Aµ ∂ν Aρ + (Dµ φ)∗ Dµ φ − V (|φ|) 2 4e 2

(27)

where V (|φ|) has some nontrivial minimum with < φ >= v. In this broken vacuum there is an additional quadratic term v 2 Aµ Aµ in the gauge ﬁeld Lagrangian which leads to the momentum space propagator (with a covariant gauge ﬁxing term) [10]

κe2 µνρ pρ e2 (p2 − m2W ) pµ pν ∆µν = − i − g µν (p2 − m2+ )(p2 − m2− ) (p2 − ξm2W ) (p2 − m2W ) +e2 ξ

(p2

pµ pν (p2 − κ2 e4 − m2W ) · − m2+ )(p2 − m2− )(p2 − ξm2W )

(28)

where m2W = 2e2 v 2 is the usual Higgs mechanism W -mass scale (squared) and the other masses are κe2 (κe2 )2 ± m2± = m2W + κ2 e4 + 4m2W (29) 2 2 or mMCS m± = 2

4m2 1+ 2 W ±1 mMCS

·

(30)

From the propagator (28) we identify two physical mass poles at p2 = m2± . The counting of degrees of freedom goes as follows. In the unbroken vacuum, the complex scalar ﬁeld has two real massive degrees of freedom and the gauge ﬁeld has one massive excitation (with mass coming from the Chern-Simons term). In the broken vacuum, one component of the scalar ﬁeld (the “Goldstone boson”) combines with the longitudinal part of the gauge ﬁeld to produce a new massive gauge degree of freedom. Thus, in the broken vacuum there is one real massive scalar degree of freedom (the “Higgs boson”) and two massive gauge degrees of freedom. The Higgs mechanism also occurs, albeit somewhat diﬀerently, if the gauge ﬁeld has just a Chern-Simons term, and no Maxwell term [11]. The Maxwell term can be decoupled from the Maxwell-Chern-Simons-Higgs Lagrangian (27) by taking the limit e2 → ∞

κ = ﬁxed

(31)

which leads to the Chern-Simons-Higgs Lagrangian LCSH =

κ µνρ Aµ ∂ν Aρ + (Dµ φ)† Dµ φ − V (|φ|). 2

(32)

G.V. Dunne: Aspects of Chern-Simons Theory

189

Exercise 2.2.2: Show that the MCSH propagator (28) reduces in the limit (31) to

∆µν

2

2v 1 1 i ρ gµν − 2 pµ pν + µνρ p = 2 2v κ p2 − ( 2vκ )2 κ

(33)

2

which has a single massive pole at p2 = ( 2vκ )2 .

The counting of degrees of freedom is diﬀerent in this Chern-SimonsHiggs model. In the unbroken vacuum the gauge ﬁeld is nonpropagating, and so there are just the two real scalar modes of the scalar ﬁeld φ. In the broken vacuum, one component of the scalar ﬁeld (the “Goldstone boson”) combines with the longitudinal part of the gauge ﬁeld to produce a massive gauge degree of freedom. Thus, in the broken vacuum there is one real massive scalar degree of freedom (the “Higgs boson”) and one massive gauge degree of freedom. This may also be deduced from the mass formulae (30) for the Maxwell-Chern-Simons-Higgs model, which in the limit (31) tend to m+ → ∞

m− →

2v 2 κ

(34)

so that one mass m+ decouples to inﬁnity, while the other mass m− agrees with the mass pole found in (33). In Section 3.2 we shall see that there is a simple way to understand these various gauge masses in terms of the characteristic frequencies of the familiar quantum mechanical Landau problem.

2.3 Fermions in 2 + 1-dimensions Fermion ﬁelds also have some new and interesting features when restricted to the plane. The most obvious diﬀerence is that the irreducible set of Dirac matrices consists of 2 × 2 matrices, rather than 4 × 4. Correspondingly, the irreducible fermion ﬁelds are 2-component spinors. The Dirac equation is (iγ µ ∂µ − eγ µ Aµ − m) ψ = 0,

or

i

∂ + mβ ψ (35) ψ = −i α·∇ ∂t

where α = γ 0γ and β = γ 0 . The Dirac gamma matrices satisfy the anticommutation relations: {γ µ , γ ν } = 2g µν , where we use the Minkowski metric g µν = diag(1, −1, −1). One natural representation is a “Dirac”

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Topological Aspects of Low Dimensional Systems

representation: γ0

=

γ1

=

γ2

=

1 0 0 −1 0 i iσ 1 = i 0 0 1 iσ 2 = · −1 0 σ3 =

(36)

while a “Majorana” representation (in which β is imaginary while the α are real) is: 0 −i γ 0 = σ2 = i 0 i 0 γ 1 = iσ 3 = 0 −i 0 i · (37) γ 2 = iσ 1 = i 0 These 2 × 2 Dirac matrices satisfy the identities: γ µ γ ν = g µν 1 − iµνρ γρ

(38)

tr(γ µ γ ν γ ρ ) = −2i µνρ .

(39)

Note that in familiar 3 + 1 dimensional theories, the trace of an odd number of gamma matrices vanishes. In 2 + 1 dimensions, the trace of three gamma matrices produces the totally antisymmetric µνρ symbol. This fact plays a crucial role in the appearance of induced Chern-Simons terms in quantized planar fermion theories, as will be discussed in detail in Section 5. Another important novel feature of 2 + 1 dimensions is that there is no “γ 5 ” matrix that anticommutes with all the Dirac matrices–note that iγ 0 γ 1 γ 2 = 1. Thus, there is no notion of chirality in the usual sense. 2.4 Discrete symmetries: P, C and T The discrete symmetries of parity, charge conjugation and time reversal act very diﬀerently in 2 + 1-dimensions. Our usual notion of a parity transformation is a reﬂection x → −x of the spatial coordinates. However, in the plane, such a transformation is equivalent to a rotation (this Lorentz transformation has det(Λ) = (−1)2 = +1 instead of det(Λ) = (−1)3 = −1). So the improper discrete “parity” ansformation should be taken to be reﬂection in just one of the spatial axes (it doesn’t matter which we choose): x1 x2

→ −x1 → x2 .

(40)

G.V. Dunne: Aspects of Chern-Simons Theory

191

From the kinetic part of the Dirac Lagrangian we see that the spinor ﬁeld ψ transforms as ψ → γ1ψ

(41)

(where we have suppressed an arbitrary unimportant phase). But this means that a fermion mass term breaks parity ¯ → −ψψ. ¯ ψψ

(42)

Under P, the gauge ﬁeld transforms as A1 → −A1 ,

A2 → A2 ,

A0 → A0

(43)

which means that while the standard Maxwell kinetic term is P-invariant, the Chern-Simons term changes sign under P: µνρ Aµ ∂ν Aρ → −µνρ Aµ ∂ν Aρ .

(44)

Charge conjugation converts the “electron” Dirac equation (35) into the “positron” equation: (iγ µ ∂µ + eγ µ Aµ − m) ψc = 0.

(45)

As is standard, this is achieved by the deﬁnition ψc ≡ Cγ 0 ψ ∗ , where the charge conjugation matrix C must satisfy (γ µ )T = −C −1 γ µ C.

(46)

In the Dirac representation (36) we can choose C = γ 2 . Note then that the fermion mass term is invariant under C (recall the anticommuting nature of the fermion ﬁelds), as is the Chern-Simons term for the gauge ﬁeld. Time reversal is an anti-unitary operation (T : i → −i) in order to implement x0 → −x0 without taking P 0 → −P 0 . The action on spinor and gauge ﬁelds is (using the Dirac representation (36)) ψ → γ 2 ψ,

→ −A, A

A0 → A0 .

(47)

From this we see that both the fermion mass term and the gauge ﬁeld ChernSimons term change sign under time reversal. The fact that the fermion mass term and the Chern-Simons term have the same transformation properties under the discrete symmetries of P, C and T will be important later in Section 5 when we consider radiative corrections in planar gauge and fermion theories. One way to understand this connection is that these two terms are supersymmetric partners in 2 + 1 dimensions [13].

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Topological Aspects of Low Dimensional Systems

2.5 Poincar´e algebra in 2 + 1-dimensions The novel features of fermion and gauge ﬁelds in 2 + 1-dimensions, as well as the anyonic ﬁelds, can be understood better by considering the respresentation theory of the Poincar´e algebra. Our underlying guide is Wigner’s Principle: that in quantum mechanics the relativistic single-particle states should carry a unitary, irreducible representation of the universal covering group of the Poincar´e group [14]. The Poincar´e group ISO(2, 1) combines the proper Lorentz group SO(2, 1) with space-time translations [15, 16]. The Lorentz generators Lµν and translation generators P µ satisfy the standard Poincar´e algebra commutation relations, which can be re-expressed in 2 + 1-dimensions as [J µ , J ν ] = [J µ , P ν ] = [P µ , P ν ] =

iµνρ Jρ iµνρ Pρ 0

where the pseudovector generator J µ is J µ = 12 µνρ Lνρ . Irreducible representations of this algrebra may be characterized by the eigenvalues of the two Casimirs: P 2 = Pµ P µ ,

W = Pµ J µ .

(48)

Here, W is the Pauli-Lubanski pseudoscalar, the 2 + 1 dimensional analogue of the familiar Pauli-Lubanski pseudovector in 3 + 1 dimensions. We deﬁne single-particle representations Φ by P 2 Φ = m2 Φ

W Φ = −smΦ

(49)

deﬁning the mass m and spin s. For example, a spin 0 scalar ﬁeld may be represented by a momentum space ﬁeld φ(p) on which P µ acts by multiplication and J µ as an orbital angular momentum operator: P µ φ = pµ φ

J µ φ = −iµνρ pν

∂ φ. ∂pρ

(50)

Then the eigenvalue conditions (49) simply reduce to the Klein-Gordon equation (p2 − m2 )φ = 0 for a spin 0 ﬁeld since P · Jφ = 0. For a two-component spinor ﬁeld ψ we take J µ = −iµνρ pν

∂ 1 1 − γµ ∂pρ 2

(51)

so the eigenvalue conditions (49) become a Dirac equation of motion (iγ µ ∂µ − m)ψ = 0, corresponding to spin s = ± 21 .

G.V. Dunne: Aspects of Chern-Simons Theory

193

For a vector ﬁeld Aµ , whose gauge invariant content may be represented through the pseudovector dual F˜ µ = 12 µνρ Fνρ , we take ∂ δαβ + iµαβ . ∂pρ

(52)

(P · J)αβ F˜ β = iµαβ pµ F˜ β = −smF˜α

(53)

(J µ )αβ = −iµνρ pν Then the eigenvalue condition

has the form of the topologically massive gauge ﬁeld equation of motion (22). We therefore deduce a mass m = κe2 and a spin s = sign(κ) = ±1. This agrees with the Maxwell-Chern-Simons mass found earlier in (23), and is the source of the Maxwell-Chern-Simons spin quoted in (25). In general, it is possible to modify the standard “orbital” form of J µ appearing in the scalar ﬁeld case (50) without aﬀecting the Poincar´e algebra: µ p + mη µ ∂ (54) J µ = −iµνρ pν ρ − s , η µ = (1, 0, 0). ∂p p·η+m It is easy to see that this gives W = P · J = −sm, so that the spin can be arbitrary. This is one way of understanding the possibility of anyonic spins in 2+1 dimensions. Actually, the real question is how this form of J µ can be realized in terms of a local equation of motion for a ﬁeld. If s is an integer or a half-integer then this can be achieved with a (2s + 1)-component ﬁeld, but for arbitrary spin s we require inﬁnite component ﬁelds [17]. 2.6 Nonabelian Chern-Simons theories It is possible to write a nonabelian version of the Chern-Simons Lagrangian (3): 2 µνρ (55) LCS = κ tr Aµ ∂ν Aρ + Aµ Aν Aρ . 3 The gauge ﬁeld Aµ takes values in a ﬁnite dimensional representation of the (semi-simple) gauge Lie algebra G. In these lectures we take G = su(N ). In an abelian theory, the gauge ﬁelds Aµ commute, and so the trilinear term in (55) vanishes due to the antisymmetry of the µνρ symbol. In the nonabelian case (just as in Yang-Mills theory) we write Aµ = Aaµ T a where the T a are the generators of G (for a = 1, . . . dim(G)), satisfying the commutation relations [T a , T b ] = f abc T c , and the normalization tr(T a T b ) = − 12 δ ab .

Exercise 2.6.1: Show that under inﬁnitesimal variations δAµ of the gauge ﬁeld the change in the nonabelian Chern-Simons Lagrangian is δLCS = κµνρ tr (δAµ Fνρ )

(56)

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Topological Aspects of Low Dimensional Systems

where Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ] is the nonabelian ﬁeld strength.

From the variation (56) we see that the nonabelian equations of motion have the same form as the abelian ones: κµνρ Fνρ = J µ . Note also that the Bianchi identity, µνρ Dµ Fνρ = 0, is compatible with covariant current conservation: Dµ J µ = 0. The source-free equations are once again Fµν = 0, for which the solutions are pure gauges (ﬂat connections) Aµ = g −1 ∂µ g, with g in the gauge group. An important diﬀerence, however, lies in the behaviour of the nonabelian Chern- Simons Lagrangian (55) under a gauge transformation. The nonabelian gauge transformation g (which is an element of the gauge group) tarnsforms the gauge ﬁeld as Aµ → Agµ ≡ g −1 Aµ g + g −1 ∂µ g.

(57)

Exercise 2.6.2: Show that under the gauge transformation (57), the Chern-Simons Lagrangian LCS in (55) transforms as κ LCS → LCS − κµνρ ∂µ tr ∂ν g g −1 Aρ − µνρ tr g −1 ∂µ gg −1 ∂ν gg −1 ∂ρ g . 3 (58) We recognize, as in the abelian case, a total space-time derivative term, which vanishes in the action with suitable boundary conditions. However, in the nonabelian case there is a new term in (58), known as the winding number density of the group element g: w(g) =

1 µνρ −1 tr g ∂µ gg −1 ∂ν gg −1 ∂ρ g . 24π 2

(59)

With appropriate boundary conditions, the integral of w(g) is an integer - see Exercise 2.6.3. Thus, the Chern-Simons action changes by an additive constant under a large gauge transformation (i.e., one with nontrivial winding number N ): SCS → SCS − 8π 2 κN.

(60)

This has important implications for the development of a quantum nonabelian Chern-Simons theory. To ensure that the quantum amplitude exp(i S) remains gauge invariant, the Chern-Simons coupling parameter κ must assume discrete values [9] κ=

integer · 4π

(61)

G.V. Dunne: Aspects of Chern-Simons Theory

195

This is analogous to Dirac’s quantization condition for a magnetic monopole [18]. We shall revisit this Chern-Simons discreteness condition in more detail in later sections.

Exercise 2.6.3: In three dimensional Euclidean space, take the SU (2) group element x · σ (62) g = exp iπN √ x2 + R2 where σ are the Pauli matrices, and R is an arbitrary scale parameter. Show that the winding number for this g is equal to N . Why must N be an integer?

To conclude this brief review of the properties of nonabelian ChernSimons terms, I mention the original source of the name “Chern-Simons”. Chern and Simons were studying a combinatorial approach to the Pontryagin density µνρσ tr (Fµν Fρσ ) in four dimensions and noticed that it could be written as a total derivative: 2 µνρσ µνρσ tr (Fµν Fρσ ) = 4 ∂σ tr Aµ ∂ν Aρ + Aµ Aν Aρ . (63) 3 Their combinatorial approach “got stuck by the emergence of a boundary term which did not yield to a simple combinatorial analysis. The boundary term seemed interesting in its own right, and it and its generalizations are the subject of this paper” [19]. We recognize this interesting boundary term as the Chern-Simons Lagrangian (55). 3

Canonical quantization of Chern-Simons theories

There are many ways to discuss the quantization of Chern-Simons theories. Here I focus on canonical quantization because it has the most direct relationship with the condensed matter applications which form the primary subject of this School. Indeed, the well-known Landau and Hofstadter problems of solid state physics provide crucial physical insight into the canonical quantization of Chern-Simons theories. 3.1 Canonical structure of Chern-Simons theories In this Section we consider the classical canonical structure and Hamiltonian formulation of Chern-Simons theories, in preparation for a discussion of their

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Topological Aspects of Low Dimensional Systems

quantization. We shall discover an extremely useful quantum mechanical analogy to the classic Landau problem of charged electrons moving in the plane in the presence of an external uniform magnetic ﬁeld perpendicular to the plane. I begin with the abelian theory because it contains the essential physics, and return to the nonabelian case later. The Hamiltonian formulation of Maxwell (or Yang-Mills) theory is standard. In the Weyl gauge (A0 = 0) the spatial components of the gauge are canonically conjugate to the electric ﬁeld components E, and ﬁeld A Gauss’s law ∇ · E = ρ appears as a constraint, for which the nondynamical ﬁeld A0 is a Lagrange multiplier. (If you wish to remind yourself of the Maxwell case, simply set the Chern-Simons coupling κ to zero in the following Eqs. (64-69)). Now consider instead the canonical structure of the Maxwell-ChernSimons theory with Lagrangian (21): LMCS =

1 2 1 κ Ei − 2 B 2 + ij A˙ i Aj + κA0 B. 2 2e 2e 2

(64)

The A0 ﬁeld is once again nondynamical, and can be regarded as a Lagrange multiplier enforcing the Gauss law constraint ∂i F i0 + κe2 ij ∂i Aj = 0.

(65)

This is simply the ν = 0 component of the Euler-Lagrange equations (22). In the A0 = 0 gauge we identify the Ai as “coordinate” ﬁelds, with corresponding “momentum” ﬁelds Πi ≡

∂L 1 κ = 2 A˙ i + ij Aj . ˙ e 2 ∂ Ai

(66)

The Hamiltonian is obtained from the Lagrangian by a Legendre transformation HMCS

= =

Πi A˙ i − L e2 i κ ij 2 1 Π − Aj + 2 B 2 + A0 ∂i Πi + κB . 2 2 2e

(67)

At the classical level, the ﬁelds Ai (x, t) and Πi (x, t) satisfy canonical equaltime Poisson brackets. These become equal-time canonical commutation relations in the quantum theory: [Ai (x), Πj (y )] = i δij δ(x − y ).

(68)

Notice that this implies that the electric ﬁelds do not commute (for κ = 0) [Ei (x), Ej (y )] = −i κe4 ij δ(x − y ).

(69)

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197

2 + The Hamiltonian (67) still takes the standard Maxwell form H = 2e12 (E 2 B ) when expressed in terms of the electric and magnetic ﬁelds. This is because the Chern-Simons term does not modify the energy – it is, after all, ﬁrst order in time derivatives. But it does modify the relation between momenta and velocity ﬁelds. This is already very suggestive of the eﬀect of an external magnetic ﬁeld on the dynamics of a charged particle. Now consider a pure Chern-Simons theory, with no Maxwell term in the Lagrangian. LCS =

κ ij ˙ Ai Aj + κA0 B. 2

(70)

Once again, A0 is a Lagrange multiplier ﬁeld, imposing the Gauss law: B = 0. But the Lagrangian is ﬁrst order in time derivatives, so it is already in the Legendre transformed form L = px˙ − H, with H = 0. So there is no dynamics – indeed, the only dynamics would be inherited from coupling to dynamical matter ﬁelds. Another way to see this is to notice that the pure Chern-Simons energy momentum tensor T µν ≡ √

2 δSCS detg δgµν

(71)

vanishes identically because the Chern-Simons action is independent of the metric, since the Lagrange density is a three-form L = tr(AdA + 23 AAA). Another important fact about the pure Chern-Simons system (70) is that the components of the gauge ﬁeld are canonically conjugate to one another: [Ai (x), Aj (y )] =

i ij δ(x − y ). κ

(72)

This is certainly very diﬀerent from the Maxwell theory, for which the components of the gauge ﬁeld commute, and it is the Ai and Ei ﬁelds that are canonically conjugate. So pure Chern-Simons is a strange new type of gauge theory, with the components Ai of the gauge ﬁeld not commuting with one another. We can recover this noncommutativity property from the MaxwellChern-Simons case by taking the limit e2 → ∞, with κ kept ﬁxed. Then, from the Hamiltonian (67) we see that we are forced to impose the constraint Πi =

κ ij Aj 2

(73)

then the Maxwell-Chern-Simons Hamiltonian (67) vanishes and the Lagrangian (64) reduces to the pure Chern-Simons Lagrangian (70). The canonical commutation relations (72) arise because of the constraints (73), noting that these are second-class constraints so we must use Dirac brackets to ﬁnd the canonical relations between Ai and Aj [23].

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Topological Aspects of Low Dimensional Systems

3.2 Chern-Simons quantum mechanics To understand more deeply this somewhat unusual projection from a Maxwell-Chern-Simons theory to a pure Chern-Simons theory we appeal to the following quantum mechanical analogy [23, 24]. Consider the long wavelength limit of the Maxwell-Chern-Simons Lagrangian, in which we drop all spatial derivatives. (This is suﬃcient for identifying the masses of excitations.) Then the resulting Lagrangian 1 ˙ 2 κ ij ˙ A + Ai Aj 2e2 i 2

L=

(74)

has exactly the same form as the Lagrangian for a nonrelativistic charged particle moving in the plane in the presence of a uniform external magnetic ﬁeld b perpendicular to the plane L=

1 b mx˙ 2i + ij x˙ i xj . 2 2

(75)

The canonical analysis of this mechanical model is a simple undergraduate physics exercise. The momenta pi =

∂L b = mx˙ i + ij xj ∂ x˙ i 2

are shifted from the velocities and the Hamiltonian is 2 1 b ij m H = pi x˙ i − L = = vi2 · pi − xj 2m 2 2

(76)

(77)

At the quantum level the canonical commutation relations, [xi , pj ] = iδij , imply that the velocities do not commute: [vi , vj ] = −i mb2 ij . It is clear that these features of the Landau problem mirror precisely the canonical structure of the Maxwell-Chern-Simons system, for both the Hamiltonian (67) and the canonical commutation relations (68) and (69). MCS ﬁeld theory ←→ Landau problem 1 e2 ←→ m κ ←→ b b mMCS = κe2 ←→ ωc = · m This correspondence is especially useful because the quantization of the Landau system is well understood. The quantum mechanical spectrum consists of equally spaced energy levels (Landau levels), spaced by ωc where the b . (see Fig. 3). Each Landau level is inﬁnitely cyclotron frequency is ωc = m

G.V. Dunne: Aspects of Chern-Simons Theory

199

_ h ωc

Fig. 3. The energy spectrum for charged particles in a uniform magnetic ﬁeld consists of equally spaced “Landau levels”, separated by ωc where ωc is the cyclotron frequency. Each Landau level has degeneracy given by the total magnetic ﬂux through the sample.

degenerate in the open plane, while for a ﬁnite area the degeneracy is related to the net magnetic ﬂux Ndeg =

bA 2π

(78)

where A is the area [20, 21]. The pure Chern-Simons limit is when e2 → ∞, with κ ﬁxed. In the quantum mechanical case this corresponds to taking the mass m → 0, with b becomes inﬁnite and so the b ﬁxed. Thus, the cyclotron frequency ωc = m energy gap between Landau levels becomes inﬁnite, isolating each level from the others. We therefore have a formal projection onto a highly degenerate ground state–the lowest Landau level (LLL). Interestingly, this is exactly the type of limit that is of physical interest in quantum Hall systems – see Steve Girvin’s lectures at this School for more details on the importance of the lowest Landau level. In the limit m → 0 of projecting to the lowest Landau level, the Lagrangian (75) becomes L = 2b ij x˙ i xj . This is ﬁrst order in time derivatives, so the two coordinates x1 and x2 are in fact canonically

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Topological Aspects of Low Dimensional Systems

conjugate to one another, with commutation relations (compare with (72)): i [xi , xj ] = ij . b

(79)

Thus, the two-dimensional coordinate space has become (in the LLL projection limit) a two-dimensional phase space, with 1b playing the role of “”. Applying the Bohr-Sommerfeld estimate of the number of quantum states in terms of the area of phase space, we ﬁnd Ndeg ≈

bA A = “h” 2π

(80)

which is precisely Landau’s estimate (78) of the degeneracy of the lowest Landau level. This projection explains the physical nature of the pure Chern-Simons theory. The pure Chern-Simons theory can be viewed as the e2 → ∞ limit of the topologically massive Maxwell-Chern-Simons theory, in which one truncates the Hilbert space onto the ground state by isolating it from the rest of the spectrum by an inﬁnite gap. The Chern-Simons analogue of the cyclotron frequency ωc is the Chern- Simons mass κe2 . So the inclusion of a Chern-Simons term in a gauge theory Lagrangian is analogous to the inclusion of a Lorentz force term in a mechanical system. This explains how it was possible to obtain a mass (23) for the gauge ﬁeld in the Maxwell-ChernSimons theory without the Higgs mechanism–in the mechanical analogue, the Higgs mechanism corresponds to introducing a harmonic binding term 1 2 2 x , which gives a characteristic frequency in the most obvious way. 2 mω But the Landau system shows how to obtain a characteristic frequency (the cyclotron frequency) without introducing a harmonic binding term. We can view the Chern-Simons theory as a gauge ﬁeld realization of this mechanism. To clarify the distinction between these two diﬀerent mass generation mechanisms for the gauge ﬁeld, consider them both acting together, as we did in Section 2.2. That is, consider the broken (Higgs) phase of a Maxwell-Chern-Simons theory coupled to a scalar ﬁeld (27). If we are only interested in the masses of the excitations it is suﬃcient to make a zerothorder (spatial) derivative expansion, neglecting all spatial derivatives, in which case the functional Schr¨ o dinger representation reduces to the familiar Schr¨ o dinger representation of quantum mechanics. Physical masses of the ﬁeld theory appear as physical frequencies of the corresponding quantum mechanical system. In the Higgs phase, the quadratic Lagrange density becomes L=

1 ˙ 2 κ ij ˙ A + Ai Aj − v 2 Ai Ai . 2e2 i 2

(81)

This is the Maxwell-Chern-Simons Lagrangian with a Proca mass term v 2 A2i . In the analogue quantum mechanical system this corresponds to a

G.V. Dunne: Aspects of Chern-Simons Theory

201

charged particle of mass e12 moving in a uniform magnetic ﬁeld of strength √ κ, and a harmonic potential well of frequency ω = 2ev. Such a quantum mechanical model is exactly solvable, and is well-known (see Exercice 3.2.1) to separate into two distinct harmonic oscillator systems of characteristic frequencies ωc 4ω 2 ω± = 1+ 2 ±1 (82) 2 ωc where ωc is the cyclotron frequency corresponding to the magnetic √ ﬁeld and ω is the harmonic well frequency. Taking ωc = κe2 and ω = 2ev, we see that these characteristic frequencies are exactly the mass poles m± in (30) of the Maxwell-Chern-Simons Higgs system, identiﬁed from the covariant gauge propagator. The pure Chern-Simons Higgs limit corresponds to the physical limit in which the cyclotron frequency dominates, so that ω− →

ω2 2v 2 = m− = ωc κ

ω+ → ∞.

(83)

The remaining ﬁnite frequency ω− is exactly the mass m− found in the covariant propagator (33) for the Higgs phase of a pure Chern-Simons Higgs theory. So we see that in 2+1 dimensions, the gauge ﬁeld can acquire one massive mode via the standard Higgs mechanism (no Chern-Simons term), or via the Chern- Simons-Higgs mechanism (no Maxwell term); or the gauge ﬁeld can acquire two massive modes (both Chern- Simons and Maxwell term).

Exercise 3.2.1: Consider the planar quantum mechanical system with 1 (pi + 2b ij xj )2 + 12 mω 2x2 . Show that the deﬁnitions Hamiltonian H = 2m ω± 1 mΩω± 2 p± = p ± x 2mΩ 2 mΩ 1 1 x± = x ∓ p2 (84) 2ω± 2mΩω± b2 b where Ω = ω 2 + 4m 2 , and ω± = Ω ± 2m are as in (82), separate H into two distinct harmonic oscillators of frequency ω± . A natural way to describe the lowest Landau level (LLL) projection is in terms of coherent states [23, 25]. To see how these enter the picture, consider the quantum mechanical Lagrangian, which includes a harmonic binding term L=

1 b 1 mx˙ 2i + ij x˙ i xj − mω 2 x2i . 2 2 2

(85)

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Topological Aspects of Low Dimensional Systems

This quantum mechanics problem can be solved exactly. Converting to polar coordinates, the wavefunctions can be labelled by two integers, N and n, with 1+|n| 2 N! 1 |n| (mΩ) 2 r|n| einθ e− 2 mΩr LN mΩr2 (86)

x|N, n = π(N + |n|)! |n| b2 2 where LN is an associated Laguerre polynomial, Ω = 4m 2 + ω , and the

b n. In the m → 0 limit, the energy is E(N, n) = (2N + |n| + 1)Ω − 2m N = 0 and n ≥ 0 states decouple from the rest, and the corresponding wavefunctions behave as 1+n 2 1 1 (mΩ) 2 rn einθ e− 2 mΩr

x|0, n = √ πn! b z n − 1 |z|2 √ e 2 → (87) 2π n! b where we have deﬁned the complex coordinate z = 2 (x1 + ix2 ). The norms of these states transform under this LLL projection limit as dzdz ∗ −|z|2 d2 x | x|0, n|2 → e | z|n|2 . (88) 2πi

We recognize the RHS as the norm in the coherent state representation of a one-dimensional quantum system. Thus, the natural description √ of the LLL is in terms of coherent state wavefunctions < z|n >= z n / n!. 2 1 The exponential factor e− 2 mΩr becomes, in the m → 0 limit, part of the coherent state measure factor. This explains how the original twodimensional system reduces to a one-dimensional system, and how it is possible to have z and z ∗ being conjugate to one another, as required by the commutation relations (79). Another way to ﬁnd the lowest Landau level wavefunctions is to express the single-particle Hamiltonian as H=−

1 2 D1 + D22 2m

(89)

where D1 = ∂1 + i 2b x2 and D2 = ∂2 − i 2b x1 . Then, deﬁne the complex combinations D± = D1 ± iD2 as: b D+ = 2∂z¯ + z, 2

b D− = 2∂z − z¯. 2

(90)

The Hamiltonian (89) factorizes as H=−

1 b D− D+ + 2m 2m

(91)

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203

so that the lowest Landau level states (which all have energy satisfy D+ ψ = 0,

b

2

ψ = f (z)e− 4 |z| .

or

1 2 ωc

=

b 2m )

(92)

We recognize the exponential factor (after absorbing 2b into the deﬁnition of z as before) as the factor in (87) which contributes to the coherent state measure factor. Thus, the lowest Landau level Hilbert space consists of holomorphic wavefunctions f (z), with coherent state norm as deﬁned in (88) [25]. This is a standard feature of the analysis of the fractional quantum Hall eﬀect – see Steve Girvin’s lectures for further applications. When the original Landau Hamiltonian contains also a potential term, this leads to interesting eﬀects under the LLL projection. With ﬁnite m, a potential V (x1 , x2 ) depends on two commuting coordinates. But in the LLL limit (i.e., m → 0 limit) the coordinates become non-commuting (see (79)) and V (x1 , x2 ) becomes the projected Hamiltonian on the projected phase space. Clearly, this leads to possible operator-ordering problems. However, these can be resolved [23, 25] by insisting that the projected Hamiltonian is ordered in such a way that the coherent state matrix elements computed within the LLL agree with the m → 0 limit of the matrix elements of the potential, computed with m nonzero. 3.3 Canonical quantization of abelian Chern-Simons theories Motivated by the coherent state formulation of the lowest Landau level projection of the quantum mechanical systems in the previous section, we now formulate the canonical quantization of abelian Chern-Simons theories in terms of functional coherent states. Begin with the Maxwell-Chern-Simons Lagrangian in the A0 = 0 gauge: LMCS =

1 ˙ 2 κ ij ˙ 1 A + Ai Aj − 2 (ij ∂i Aj )2 . 2e2 i 2 2e

(93)

This is a quadratic Lagrangian, so we expect we can ﬁnd the groundstate wavefunctional. Physical states must also satisfy the Gauss law constraint: ·Π − κB = 0. This Gauss law is satisﬁed by functionals of the form ∇ κ

Ψ[A1 , A2 ] = e−i 2

Ê

Bλ

Ψ[AT ]

(94)

into its longitudinal and transverse parts: where we have decomposed A Ai = ∂i λ+ATi . Using the Hamiltonian (67), the ground state wavefunctional is [9] κ

Ψ0 [A1 , A2 ] = e−i 2

Ê

Bλ − 2e12

e

Ê

AT i

√ κ2 e4 −∇2 AT i

.

(95)

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Topological Aspects of Low Dimensional Systems

The pure Chern-Simons limit corresponds to taking e2 → ∞, so that the wavefunctional becomes Ψ0 [A1 , A2 ] → e

− 12

Ê

∂

A ∂+ A − 1 − 2

e

Ê

|A|2

.

(96)

κ where we have deﬁned A = 2 (A1 + iA2 ) (in analogy to the deﬁnition of z in the previous section), and ∂± = (∂1 ∓ i∂2 ). From this form of the groundstate wavefunctional we recognize the functional coherent state 1

Ê

−1

2

Ê

A

∂+

A

∂− measure factor e− 2 |A| , multiplying a functional Ψ[A] = e 2 that ∗ depends only on A, and not on A . This is the functional analogue of the 2 1 fact that the LLL wavefunctions have the form ψ = f (z)e− 2 |z| , as in (87) and (92). The fact that the Chern-Simons theory has a single ground state, rather than a highly degenerate LLL, is a consequence of the Gauss law constraint, for which there was no analogue in the quantum mechanical model. These pure Chern-Simons wavefunctionals have a functional coherent state inner product (compare with (88)) Ê 2 ∗ (97) < Ψ|Φ >= DADA∗ e− |A| (Ψ[A]) Φ[A].

Actually, we needn’t have gone through the process of taking the e2 → ∞ limit of the Maxwell-Chern-Simons theory. It is much more direct simply to adopt the functional coherent state picture. This is like going directly to the lowest Landau level using coherent states, instead of projecting down from the full Hilbert space of all the Landau levels. The canonical commutation relations (72) imply [A(z), A∗ (w)] = δ(z −w), so that we can represent A∗ as δ . Then the pure Chern-Simons a functional derivative operator: A∗ = − δA Gauss law constraint F12 = 0 acts on states as δ + ∂+ A Ψ[A] = 0 (98) ∂− δA with solution Ψ0 [A] = e

− 12

Ê

∂

A ∂+ A −

(99)

as in (96). If this pure Chern-Simons theory is coupled to some charged matter ﬁelds with a rotationally covariant current, then the physical state (99) is an eigenstate of the conserved angular momentum operator M = − κ2 xi ij (Aj B + BAj ): M Ψ0 [A] =

Q2 Ψ0 [A] 4πκ

(100)

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205

where Q = d2 x ρ. Comparing with the Aharonov-Bohm exchange phase e2 ∆θ = 4πκ in (20) we see that the statistics phase s coincides with the spin eigenvalue M . This is the essence of the generalized spin-statistics relation for extended (ﬁeld theoretic) anyons. 3.4 Quantization on the torus and magnetic translations The quantization of pure Chern-Simons theories on the plane is somewhat boring because there is just a unique physical state (99). To make this more interesting we could include external sources, which appear in the canonical formalism as point delta-function sources on the ﬁxed-time surface. The appearance of these singularities makes the projection to ﬂat connections satisfying Gauss’s law more intricate, and leads to important connections with knot theory and the braid group. Alternatively, we could consider the spatial surface to have nontrivial topology, rather than simply being the open plane R2 . For example, take the spatial manifold to be a Riemann surface Σ of genus g. This introduces extra degrees of freedom, associated with the nontrivial closed loops around the handles of Σ [29–33]. Interestingly, the quantization of this type of Chern-Simons theory reduces once again to an eﬀective quantum mechanics problem, with a new feature that has also been treated long ago in the solid state literature under the name of the “magnetic translation group”. To begin, it is useful to reconsider the case of R2 . To make connection with the coherent state representation, we express the longitudinaltransverse decomposition of the vector potential, Ai = ∂i ω + ij ∂j σ, in terms of the holomorphic ﬁelds A = 12 (A1 + iA2 ) and A∗ = 12 (A1 − iA2 ). Thus, with z = x1 + ix2 and Ai dxi = A∗ dz + Ad¯ z , we have A = ∂z¯χ,

A∗ = ∂z χ∗

(101)

where χ = ω − iσ is a complex ﬁeld. If χ were real, then A would be purely longitudinal–i.e. pure gauge. But with a complex ﬁeld χ, this representation spans all ﬁelds. A gauge transformation is realized as a shift in the real part of χ: χ → χ + λ, where λ is real. On a nontrivial surface this type of longitudinal-tranverse decomposition is not suﬃcient, as we know from elementary vector calculus on surfaces. The gauge ﬁeld is decomposed using a Hodge decomposition, which incorporates the windings around the 2g independent noncontractible loops on Σ. For simplicity, consider the g = 1 case: i.e., the torus. (The generalization to higher genus is quite straightforward). The torus can be parametrized as a parallelogram with sides 1 and τ , as illustrated in Figure 4. The area of the parallelogram is Im(τ ), and the ﬁeld A can be expressed as A = ∂z¯χ + i

π ω(z) a Im(τ )

(102)

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Topological Aspects of Low Dimensional Systems

β

τ

α 1 Fig. 4. The torus can be parametrized as a parallelogram with sides τ and 1. There are two cycles α and β representing the two independent non-contractible loops on the surface.

2 where ω(z) is a holomorphic one-form normalized according to |ω(z)| = Im(τ ). This holomorphic form has integrals α ω = 1 and β ω = τ around the homology basis cycles α and β. For the torus, we can simpy take ω(z) = 1. The complex parameter a appearing in (102) is just a function of time, independent of the spatial coordinates. Thus the A0 = 0 gauge ChernSimons Lagrangian decouples into two pieces iκπ 2 (aa ˙ ∗ − a˙ ∗ a) + iκ (∂z¯χ∂ ˙ z χ∗ − ∂z χ˙ ∗ ∂z¯χ) . (103) LCS = Im(τ ) Σ So the coherent state wavefunctionals factorize as Ψ[A] = Ψ[χ]ψ(a), with the χ dependence exactly as discussed in the previous section. On the other hand, the a dependence corresponds exactly to a quantum mechanical LLL 2 κ problem, with “magnetic ﬁeld” B = 4π Imτ . So the quantum mechanical wavefunctions ψ(a) have inner product −2π 2 κ 2

ψ|φ = dada∗ e− Im(τ ) |a| (ψ(a))∗ φ(a). (104) But we have neglected the issue of gauge invariance. Small gauge transformations, χ → χ + λ, do not aﬀect the a variables. But because of the nontrivial loops on the spatial manifold there are also “large” gauge transformations, which only aﬀect the a’s: a → a + p + qτ,

p, q ∈ Z.

(105)

To understand how these large gauge transformations act on the wavefunctions ψ(a), we recall the notion of the “magnetic translation group”. That is, in a uniform magnetic ﬁeld, while the magnetic ﬁeld is uniform, the corresponding vector potential, which is what appears in the Hamiltonian, is

G.V. Dunne: Aspects of Chern-Simons Theory

207

not! Take, for example, Ai = − B2 ij xj . Then there are magnetic translation operators

p−eA)

≡ e−iR·( T (R)

(106)

which commute with the particle Hamiltonian H = not commute with one another:

1 p 2m (

2 ) = T (R 2 )T (R 1 )e−ieB·(R1 ×R2 ) . 1 )T (R T (R

2 , but do + eA) (107)

The exponential factor here involves the magnetic ﬂux through the parallel 2 . In solid state applications, a crystal lattice 1 and R ogram spanned by R establishes a periodic potential for the electrons. If, in addition, there is a magnetic ﬁeld, then we can ask how the spectrum of Landau levels is modiﬁed by the periodic potential, or alternatively we can ask how the Bloch band structure of the periodic potential is modiﬁed by the presence of the magnetic ﬁeld [26]. The important quantity in answering this question is the magnetic ﬂux through one unit cell of the periodic lattice. It is known [27, 28] that the magnetic translation group has ﬁnite dimensional representations if the magnetic ﬁeld is related to a primitive lattice vector e by = 2π 1 N e B eΩ M

(108)

where Ω is the area of the unit cell, and N and M are integers. These representations are constructed by ﬁnding an invariant subgroup of magnetic translation operators; the rationality condition arises because all members of this invariant subgroup must commute, which places restrictions on the phase factors in (107). Since we are considering a two-dimensional system, with the magnetic ﬁeld perpendicular to the two-dimensional surface, the condition (108) simpliﬁes to: N eBΩ = · 2π M

(109)

The case M = 1 is special; here the magnetic translations act as onedimensional ray representations on the Hilbert space, transforming the wavefunction with a phase. Consistency of this ray representation gives the number of states as N = eBΩ 2π , which is just Landau’s estimate (78) of N the degeneracy of the LLL. But when M is rational, we still have a consistent ﬁnite dimensional action of the magnetic translation group on the wavefunctions. The invariant subgroup consists of “superlattice” translations, where the superlattice is obtained by enlarging each length dimension of the unit cell by a factor of M . This produces an enlarged unit cell with eﬀective ﬂux M N on which the magnetic translation group acts one-dimensionally.

208

Topological Aspects of Low Dimensional Systems

N is irrational, then the magThus the total dimension is M N . Finally, if M netic translation group has inﬁnite dimensional representations. These results can be mapped directly to the quantization of the abelian Chern-Simons theory on the torus. The quantum mechanical degrees of 2 κ freedom, a, have a LLL Lagrangian with magnetic ﬁeld eB = 4π Imτ . The large gauge transformations (105) are precisely magnetic translations across a parallelogram unit cell. The area of the unit cell is Ω = Imτ , the area of the torus. Thus eBΩ 1 4π 2 κ = Imτ = 2πκ (110) 2π 2π Imτ

and the condition for ﬁnite dimensional representations of the action of the large gauge transformations becomes 2πκ =

N · M

(111)

If we require states to transform as a one-dimensional ray representation under large gauge transformations then we must have 2πκ = integer. But if 2πκ is rational, then we still have a perfectly good quantization, provided we identify the physical states with irreducible representations of the global gauge transformations (i.e., the magnetic translations). These states transform according to a ﬁnite dimensional irreducible representation of the global gauge transformations, and any element of a given irreducible representation may be used to evaluate matrix elements of a gauge invariant operator, because physical gauge invariant operators commute with the generators of large gauge transformations. The dimension of the Hilbert space is M N . If 2πκ is irrational, there is still nothing wrong with the Chern-Simons theory – it simply means that there are an inﬁnite number of states in the Hilbert space. These results are consistent with the connection between abelian Chern-Simons theories and two dimensional conformal ﬁeld theories. Chern- Simons theories with rational 2πκ correspond to what are known as “rational CFT’s”, which have a ﬁnite number of conformal blocks, and these conformal blocks are in one-to-one correspondence with the Hilbert space of the Chern-Simons theory [29–32]. 3.5 Canonical quantization of nonabelian Chern-Simons theories The canonical quantization of the nonabelian Chern-Simons theory with Lagrangian (55) is similar in spirit to the abelian case discussed in the previous Section. There are, however, some interesting new features [22,29, 31, 32, 34]. As before, we specialize to the case where space-time has the form R × Σ, where Σ is a torus. With Σ = T 2 , the spatial manifold has two noncontractible loops and these provide gauge invariant holonomies.

G.V. Dunne: Aspects of Chern-Simons Theory

209

The problem reduces to an eﬀective quantum mechanics problem for these holonomies. Just as in the abelian case, it is also possible to treat holonomies due to sources (which carry a representation of the gauge algebra), and to consider spatial manifolds with boundaries. These two approaches lead to deep connections with two-dimensional conformal ﬁeld theories, which are beyond the scope of these lectures – the interested reader is referred to [29, 31, 32, 34] for details. We begin as in the abelian case by choosing a functional coherent state representation for the holomorphic wavefunctionals Ψ = Ψ[A], where A = 1 2 (A1 + iA2 ). The coherent state inner product is Ê ∗

Ψ|Φ = DADA∗ e4κ tr(AA ) (Ψ[A])∗ Φ[A]. (112) Note that with our Lie algebra conventions (see Sect. 2.6) tr(AA∗ ) = − 21 Aa (Aa )∗ . Physical states are annihilated by the Gauss law generator F12 = −2iFzz¯. Remarkably, we can solve this constraint explicitly using the properties of the Wess-Zumino-Witten (WZW) functionals: 1 i S ± [g] = tr(g −1 ∂z gg −1 ∂z¯g) ± µνρ tr(g −1 ∂µ gg −1 ∂ν gg −1 ∂ρ g) 2π Σ 12π (3) (113) where in the second term the integral is over a three dimensional manifold with a two dimensional boundary equal to the two dimensional space Σ.

Exercise 3.5.1: Show that the WZW functionals (113) have the fundamental variations 1 − π tr(g −1 δg∂z (g −1 ∂z¯g)] ± (114) δS [g] = − π1 tr(g −1 δg∂z¯(g −1 ∂z g)]· Consider ﬁrst of all quantization on the spatial manifold Σ = R2 . To solve the Gauss law constraint we express the holomorphic ﬁeld A, using Yang’s representation [35], as A = −∂z¯U U −1 ,

U ∈ GC.

(115)

This is the nonabelian analogue of the complexiﬁed longitudinal-transverse decomposition (101) A = ∂z¯χ for the abelian theory on the plane. U belongs to the complexiﬁcation of the gauge group, which, roughly speaking, is the exponentiation of the gauge algebra, with complex parameters. With A parametrized in this manner, the Gauss law constraint Fzz¯Ψ = 0 is solved by the functional Ψ0 [A] = e−4πκS

−

[U]

.

(116)

210

Topological Aspects of Low Dimensional Systems

To verify this, note that the results of Exercise 3.5.1 imply that

δΨ0 = 4κ tr(δA ∂z U U −1 ) Ψ0 .

(117)

From the canonical commutation relations (72), the ﬁeld Aaz = 12 (Aa1 − iAa2 ) acts on a wavefunctional Ψ[A] as a functional derivative operator Aaz =

1 δ · 2κ δAa

(118)

Thus, acting on the state Ψ0 [A] in (116): Aaz Ψ0 [A] = −(∂z U U −1 )a Ψ0 [A].

(119)

Since Aaz¯ acts on Ψ0 by multiplication, it immediately follows that Fzz¯Ψ0 [A] = 0, as required. The physical state (116) transforms with a cocycle phase factor under a gauge transformation. We could determine this cocycle from the variation (58) of the nonabelian Lagrangian [22]. But a more direct way here is to use the fundamental Polyakov-Wiegmann transformation property [36] of the WZW functionals: 1 S[g1 g2 ] = S[g1 ] + S[g2 ] + (120) tr(g1−1 ∂z g1 ∂z¯g2 g2−1 ). π With the representation A = −∂z¯U U −1 of the holomorphic ﬁeld A, the gauge transformation A → Ag = g −1 Ag + g −1 ∂z¯g is implemented by U → g −1 U , with g in the gauge group. Then Ψ0 [Ag ] = e−4πκS

−

[g−1 U]

= e−4πκS

+

[g]−4κ

Ê

tr(A∂z gg−1 )

Ψ0 [A].

(121)

Exercise 3.5.2: Check that the transformation law (121) is consistent under composition, and that it combines properly with the measure factor to make the coherent state inner product (112) gauge invariant.

Furthermore, note that the WZW factors in (116) and (121) are only well deﬁned provided 4πκ = integer. This is the origin of the discreteness condition (61) on the Chern-Simons coeﬃcient in canonical quantization. This describes the quantum pure Chern-Simons theory with spatial manifold being the open plane R2 . There is a unique physical state (116).

G.V. Dunne: Aspects of Chern-Simons Theory

211

To make things more interesting we can introduce sources, boundaries, or handles on the spatial surface. As in the abelian case, here we just consider the eﬀect of higher genus spatial surfaces, and for simplicity we concentrate on the torus. Then the nonabelian analogue of the abelian Hodge decomposition (102) is [31, 32] A = −∂z¯U U −1 +

iπ U aU −1 Imτ

(122)

where U ∈ G C , and a can be chosen to be in the Cartan subalgebra of the gauge Lie algebra. This is the nonabelian generalization of the abelian torus Hodge decomposition (102). To motivate this decomposition, we note that when U ∈ G (not G C !), this is the most general pure gauge (ﬂat connection) on the torus. The a degrees of freedom represent the nontrivial content of A that cannot be gauged away, due to the noncontractible loops on the spatial manifold. By a gauge transformation, a can be taken in the Cartan subalgebra (indeed, there is further redundancy due to the action of Weyl reﬂections on the Cartan subalgebra). Then, extending U from G to G C , the representation (122) spans out to cover all connections, just as in Yang’s representation (115) on R2 . Combining the representation (122) with the transformation law (121), we see that the physical state wavefunctionals on the torus are Ψ[A] = e−4πκS

−

[U]+ 4πiκ Imτ

Ê

tr(aU −1 ∂z U)

ψ(a).

(123)

In the inner product (112), we can change ﬁeld variables from A to U and a. But this introduces nontrivial Jacobian factors [31, 32]. The corresponding determinant is another Polyakov-Weigmann factor [37], with a coeﬃcient c arising from the adjoint representation normalization (c is called the dual Coxeter number of the gauge algebra, and for SU (N ) it is N ). The remaining functional integral over the gauge invariant combination U † U may be performed (it is the generating functional of the gauged WZW model on the torus [37]). The ﬁnal result is an eﬀective quantum mechanical model with coherent state inner product ∗ π

Ψ|Φ = dada∗ e Imτ (4πκ+c)tr(aa ) (ψ(a))∗ φ(a). (124) This looks like the abelian case, except for the shift of the Chern-Simons coeﬃcient κ by 4πκ → 4πκ + c. In fact, we can represent the quantum mechanical Cartan subalgebra degrees of freedom as r-component vectors a, where r is the rank of the gauge algebra. Then large gauge transformations act on these vectors as a → a + m + τn, where m and n belong to the root lattice ΛR of the gauge algebra. The wavefunction with the correct transformation properties under these large gauge transformation shifts is a

212

Topological Aspects of Low Dimensional Systems

generalized theta function, which is labelled by an element λ of the weight lattice ΛW of the algebra. These are identiﬁed under translations by root vectors, and also under Weyl reﬂections. Thus the physical Hilbert space of the nonabelian Chern-Simons theory on the torus corresponds to ΛW · W × (4πκ + c)ΛR

(125)

This parametrization of states is a familiar construction in the theory of Kac-Moody algebras and conformal ﬁeld theories. 3.6 Chern-Simons theories with boundary We conclude this review of basic facts about the canonical structure of Chern- Simons theories by commenting brieﬂy on the manifestation of boundary degrees of freedom in Chern-Simons theories deﬁned on spatial manifolds which have a boundary. We have seen in the previous sections that the canonical quantization of pure Chern-Simons theory on the spacetime Σ × R, where Σ is a compact Riemann surface, leads to a Hilbert space that is in one-to-one correspondence with the conformal blocks of a conformal ﬁeld theory deﬁned on Σ. But there is another important connection between Chern-Simons theories and CFT – namely, if the spatial manifold Σ has a boundary ∂Σ, then the Hilbert space of the Chern- Simons theory is inﬁnite dimensional, and provides a representation of the chiral current algebra of the CFT deﬁned on ∂Σ × R [29, 31, 32]. The source of these boundary eﬀects is the fact that when we checked the variation of the Chern-Simons action in (56) we dropped a surface term. Retaining the surface term, the variation of the Chern-Simons action splits naturally into a bulk and a surface piece [29, 32]: 3 µνρ (126) δSCS = κ d x tr(δAµ Fνρ ) + κ d3 x∂ν [µνρ tr(Aµ δAρ )] . The boundary conditions must be such that bndy tr(A δA) = 0. When it is the spatial manifold Σ that has a boundary ∂Σ, we can impose the boundary condition that A0 = 0. The remaining local symmetry corresponds to gauge transformations that reduce to the identity on ∂Σ, while the time independent gauge transformations on the boundary are global gauge transformations. With this boundary condition we can write 3 ij ˙ d x tr(Ai Aj ) + κ d3 x ij tr(A0 Fij ). (127) SCS = −κ Σ×R

Σ×R

Variation with respect to the Lagrange multiplier ﬁeld A0 imposes the constraint Fij = 0, which has as its solution the pure gauges Ai = g −1 ∂i g.

G.V. Dunne: Aspects of Chern-Simons Theory

Then it follows that the Chern-Simons action becomes S = −κ dθ dt tr(g −1 ∂θ gg −1 ∂0 g) ∂Σ×R κ + µνρ tr g −1 ∂µ gg −1 ∂ν gg −1 ∂ρ g . 3 Σ×R

213

(128)

This is the chiral WZW action. The quantization of this system leads to a chiral current algerba of the gauge group, with the boundary values of the gauge ﬁeld Aθ = g −1 ∂θ g being identiﬁed with the chiral Kac-Moody currents. This relation gives another important connection between ChernSimons theories (here, deﬁned on a manifold with a spatial boundary) and conformal ﬁeld theories [29, 32]. Boundary eﬀects also play an important role in the theory of the quantum Hall eﬀect [38, 39], where there are gapless edge excitations which are crucial for explaining the conduction properties of a quantum Hall liquid. Consider the variation of the abelian Chern-Simons action δ d3 xµνρ Aµ ∂ν Aρ = 2 d3 xµνρ δAµ ∂ν Aρ + d3 xµνρ ∂ν (Aµ δAρ ) . (129) For an inﬁnitesimal gauge variation, δAµ = ∂µ λ, this becomes a purely surface term (130) δ d3 xµνρ Aµ ∂ν Aρ = d3 xµνρ ∂µ (λ∂ν Aρ ) . For a space-time D × R, where D is a disc with boundary S 1 δ d3 xµνρ Aµ ∂ν Aρ = λ(∂0 Aθ − ∂θ A0 ).

(131)

S 1 ×R

Thus, the Chern-Simons action is not gauge invariant. Another way to say κ µνρ CS this is that the current J µ = δS Fνρ is conserved within the bulk, δAµ = 2 but not on the boundary. For a disc-like spatial surface, this noninvariance leads to an accumulation of charge density at the boundary at a rate given by the radial current: Jr = κEθ

(132)

where Eθ is the tangential electric ﬁeld at the boundary. However, we recognize this noninvariance as exactly that of a 1 + 1 dimensional Weyl fermion theory deﬁned on the boundary S 1 × R. Due to the 1 + 1 dimensional chiral anomaly, an electric ﬁeld (which must of course point along the boundary)

214

Topological Aspects of Low Dimensional Systems

leads to the anomalous creation of charge at the rate (with n ﬂavours of fermions): n ∂ Q= E. ∂t 2π

(133)

Therefore, when 2πκ is an integer (recall the abelian discreteness condition (111)) the noninvariance of the Chern-Simons theory matches precisely the noninvariance of the anomalous boundary chiral fermion theory. This corresponds to a ﬂow of charge from the bulk to the edge and vice versa. This gives a beautiful picture of a quantum Hall droplet, with integer ﬁlling fraction, as an actual physical realization of the chiral anomaly phenomenon. Indeed, when 2πκ = n, we can view the Hall droplet as an actual coordinate space realization of the Dirac sea of the edge fermions [40]. This also provides a simple eﬀective description of the integer quantum Hall eﬀect as a quantized ﬂow of charge onto the edge of the Hall droplet. For the fractional quantum Hall eﬀect we need more sophisticated treatments on the edge, such as bosonization of the 1 + 1 dimensional chiral fermion edge theory in terms of chiral boson ﬁelds [38], or representations of W1+∞ , the quantum algebra of area preserving diﬀeomorphisms associated with the incompressibility of the quantum Hall droplet [41]. 4

Chern-Simons vortices

Chern-Simons models acquire dynamics via coupling to other ﬁelds. In this section we consider the dynamical consequences of coupling ChernSimons ﬁelds to scalar ﬁelds that have either relativistic or nonrelativistic dynamics. These theories have vortex solutions, similar to (in some respects) but diﬀerent from (in other respects) familiar vortex models such as arise in Landau-Ginzburg theory or the Abelian Higgs model. The notion of Bogomol’nyi self-duality is ubiquitous, with some interesting new features owing to the Chern-Simons charge-ﬂux relation ρ = κB. 4.1 Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices I begin by reviewing brieﬂy the Abelian-Higgs model in 2 + 1 dimensions. This model describes a charged scalar ﬁeld interacting with a U (1) gauge ﬁeld, and exhibits vortex solutions carrying magnetic ﬂux, but no electric charge. These vortex solutions are important in the Landau-Ginzburg theory of superconductivity because the static energy functional (see (135) below) for the relativistic Abelian-Higgs model coincides with the nonrelativistic Landau-Ginzburg free energy in the theory of Type II superconductors, for which vortex solutions were ﬁrst studied by Abrikosov [42].

G.V. Dunne: Aspects of Chern-Simons Theory

Fig. 5. The self-dual quartic potential The vacuum manifold is |φ| = v.

λ 4

|φ|2 − v 2

2

215

for the Abelian-Higgs model.

Consider the Abelian-Higgs Lagrangian [43] 2 1 λ 2 |φ| − v 2 LAH = − Fµν F µν + |Dµ φ|2 − 4 4

(134)

where the covariant derivative is Dµ φ = ∂µ φ + ieAµ φ, and the quartic potential has the standard symmetry breaking form as shown in Figure 5. The static energy functional of the Abelian-Higgs model is

1 2 + λ |φ|2 − v 2 2 (135) EAH = d2 x B 2 + |Dφ| 2 4 where B = F12 . The potential minimum has constant solutions φ = eiα v, where α is a real phase. Thus the vacuum manifold is isomorphic to the circle S 1 . Furthermore, any ﬁnite energy solution must have φ(x) tending to an element of this vacuum manifold at inﬁnity. Therefore, ﬁnite energy solutions are classiﬁed by their winding number or vorticity N , which counts the number of times the phase of φ winds around the circle at spatial inﬁnity: φ(x)|| x|=∞ = v eiN θ .

(136)

The vorticity is also related to the magnetic ﬂux because ﬁnite energy so → 0 as |x| → ∞. This implies that lutions also require |Dφ| eAi ∼ −i∂i ln φ ∼ N ∂i θ

as |x| → ∞.

Therefore, the dimensionless magnetic ﬂux is 2 Φ = e d xB = e Ai dxi = 2πN. | x|=∞

(137)

(138)

216

Topological Aspects of Low Dimensional Systems

A brute-force approach to vortex solutions would be to make, for example in the 1-vortex case, a radial ansatz: φ(x) = f (r)eiθ ,

ˆ x) = a(r)θ. A(

(139)

The ﬁeld equations then reduce to coupled nonlinear ordinary diﬀerential equations for f (r) and a(r). One can seek numerical solutions with the 1 as r → ∞; appropriate boundary conditions: f (r) → v and a(r) → er and f (r) → 0 and a(r) → 0 as r → 0. No exact solutions are known, but approximate solutions can be found numerically. The solutions are localized vortices in the sense that the ﬁelds approach their asymptotic vacuum values exponentially, with characteristic decay lengths set by the mass scales of the theory. Note that λ, e2 and v 2 each has dimensions of mass; and the Lagrangian (134) has a Higgs phase with a massive gauge ﬁeld of √ mass √ mg = 2ev, together with a massive real scalar ﬁeld of mass ms = λv. In general, these two mass scales are independent, but the Abelian-Higgs model displays very diﬀerent behavior depending on the relative magnitude of these two mass scales. Numerically, it has been shown that two vortices (or two antivortices) repel if ms > mg , but attract if ms < mg . When the masses are equal ms = mg

(140)

then the forces betwen vortices vanish and it is possible to ﬁnd stable static multivortex conﬁgurations. When translated back into the LandauGinzburg model for superconductivity, this critical point, ms = mg , corresponds to the boundary between type-I and type-II superconductivity. In terms of the Abelian-Higgs model (134), this critical point is known as the Bogomol’nyi [44] self-dual point where λ = 2e2 .

(141)

With this relation between the charge e and the potential strength λ, special things happen. To proceed, we need a fundamental identity – one that will appear many times throughout our study of vortex solutions in planar gauge theories. 2 = |(D1 ± iD2 )φ|2 ∓ eB|φ|2 ± ij ∂i Jj |Dφ|

(142)

1 [φ∗ Dj φ − φ(Dj φ)∗ ]. Using this identity, the energy functional where Jj = 2i (135) becomes [44] 2 1 2 EAH = d x B ∓ e(|φ|2 − v 2 ) + |D± φ|2 2 (143)

λ e2 2 2 2 2 − |φ| − v + ∓ ev B 4 2

G.V. Dunne: Aspects of Chern-Simons Theory

217

where D± ≡ (D1 ± iD2 ), and we have dropped a surface term. At the self-dual point (141) the potential terms cancel, and we see that the energy is bounded below by a multiple of the magnitude of the magnetic ﬂux (for positive ﬂux we choose the lower signs, and for negative ﬂux we choose the upper signs): EAH ≥ v 2 |Φ|.

(144)

This bound is saturated by ﬁelds satisfying the ﬁrst-order Bogomol’nyi selfduality equations [44]: D± φ = B =

0 ±e(|φ|2 − v 2 ).

(145)

The self-dual point (141) is also the point at which the 2 + 1 dimensional Abelian-Higgs model (134) can be extended to an N = 2 supersymmetric (SUSY) model [45, 46]. That is, ﬁrst construct an N = 1 SUSY Lagrangian of which (134) is the bosonic part. This SUSY can then be extended to N = 2 SUSY only when the φ potential is of the form in (134) and the selfduality condition (141) is satisﬁed. This is clearly related to the mass degeneracy condition (140) because for N = 2 SUSY we need pairs of bosonic particles with equal masses (in fact, the extension to N = 2 SUSY requires an additional neutral scalar ﬁeld to pair with the gauge ﬁeld Aµ ). This feature of N = 2 SUSY corresponding to the self-dual point is a generic property of self-dual models [47, 48], and we will see it again in our study of Chern-Simons vortices. The self-duality equations (145) are not solvable, or even integrable, but a great deal is known about the solutions. To bring them to a more manageable form, we decompose the scalar ﬁeld φ into its phase and magnitude: 1

φ = eiω ρ 2 .

(146)

Then the ﬁrst of the self-duality equations (145) determines the gauge ﬁeld 1 eAi = −∂i ω ∓ ij ∂j ln ρ 2

(147)

everywhere away from the zeros of the scalar ﬁeld. The second self-duality equation in (145) then reduces to a nonlinear elliptic equation for the scalar ﬁeld density ρ: (148) ∇2 ln ρ = 2e2 ρ − v 2 . No exact solutions are known for this equation, even when reduced to an ordinary diﬀerential equation by the condition of radial symmetry. However,

218

Topological Aspects of Low Dimensional Systems

it is easy to ﬁnd (numerically) vortex-like solutions with φ = f (r)e±iN θ where f (r) satisﬁes 1 d d (149) r f 2 (r) = 2e2 (f 2 − v 2 ). r dr dr Many interesting theorems have been proved concerning the general solutions to the self-dual Abelian-Higgs equations (145). These are paraphrased below. Readers interested in all the ﬁne-print should consult [49] and the original papers. be a smooth ﬁnite energy solution Existence and Uniqueness: Let (φ, A) to the Abelian-Higgs self-duality equations (145). Then (i) φ has a ﬁnite number of zeros z1 , . . . , zm ; (ii) around each zero, φ ∼ (z − zk )nk hk (z), where hk (z) is smooth and hk (zk ) = 0; m (iii) the vorticity is given by the net multiplicity of zeros: N = k=1 nk ; (iv) given any set of zeros, z1 , . . . , zm , the solution is unique, up to gauge equivalence; (v) |φ| < v on R2 . Furthermore, it has been shown that all ﬁnite energy solutions to the full second-order static equations of motion are solutions to the ﬁrst-order selfduality equations. Thus, the solutions described in the above theorem cover all ﬁnite energy static solutions. These results mean that the moduli space of static multivortex solutions is 2N dimensional, and these 2N parameters can be associated with the locations of the zeros of the Higgs ﬁeld φ. This counting is conﬁrmed by an index-theorem ﬂuctuation analysis [50]. We shall return to this moduli space later in Section 4.7 when we discuss the dynamics of vortices. To conclude this review of the Abelian-Higgs model I mention that this model has also been studied on spatial manifolds that are compact Riemann surfaces. This is of interest for making comparisons with numerical simulations, which are necessarily ﬁnite, and also for studying the thermodynamics of vortices [51]. The main new feature is that there is an upper limit, known as Bradlow’s bound [52], on the vorticity for a given area of the surface. The appearance of such a bound is easy to see by integrating the second of the self-duality equations (145) over the surface (assuming positive ﬂux, we take the lower signs): 2 2 2 2 2 (150) d x−e d2 x|φ|2 . d xeB = e v Since

d2 xeB = 2πN , and

d2 x|φ|2 is positive, this implies that N≤

e2 v 2 area. 2π

(151)

G.V. Dunne: Aspects of Chern-Simons Theory

219

(In the mathematics literature v 2 and λ are usually scaled to 1, so that the self-dual value of e2 is 12 , in which case the bound reads: 4πN ≤ area.) A similar bound applies when considering the Abelian-Higgs vortex solutions with periodic ’t Hooft boundary conditions deﬁned on a unit cell of ﬁnite area [53]. 4.2 Relativistic Chern-Simons vortices A natural generalization of the Abelian-Higgs model of the previous section is to consider the eﬀect of taking the gauge ﬁeld to be governed by a Chern- Simons Lagrangian rather than a Maxwell Lagrangian. The name “relativistic” Chern- Simons vortices comes from the fact that a ChernSimons gauge ﬁeld inherits its dynamics from the matter ﬁelds to which it is coupled, and here it is coupled to a relativistic scalar ﬁeld – later we shall consider vortices arising from a Chern-Simons gauge ﬁeld coupled to matter ﬁelds with nonrelativistic dynamics. Numerous studies were made of vortex solutions in models with Chern-Simons and/or Maxwell terms, with symmetry breaking scalar ﬁeld potentials [54, 55]. However, no analogue of the Bogomol’nyi self-dual structure of the Abelian-Higgs model was found until a particular sixth-order scalar potential was chosen in a model with a pure Chern-Simons term [56, 57]. Consider the Lagrangian LRCS =

κ µνρ Aµ ∂ν Aρ + |Dµ φ|2 − V (|φ|) 2

(152)

where V (|φ|) is the scalar ﬁeld potential, to be speciﬁed below. The associated energy functional is 2 + V (|φ|) . ERCS = d2 x |D0 φ|2 + |Dφ| (153) Before looking for self-dual vortices we note a fundamental diﬀerence between vortices in a Chern-Simons model and those in the Abelian-Higgs model, where the gauge ﬁeld is governed by a Maxwell term. The AbelianHiggs vortices carry magnetic ﬂux but are electrically neutral. In contrast, in a Chern-Simons model the Chern-Simons Gauss law constraint relates the magnetic ﬁeld B to the conserved U (1) charge density as B=

1 i J0 = (φ∗ D0 φ − (D0 φ)∗ φ) . κ κ

Thus, if there is magnetic ﬂux there is also electric charge: Q = d2 xJ 0 = κ d2 xB = κΦ.

(154)

(155)

220

Topological Aspects of Low Dimensional Systems

Fig. 6. The self-dual potential κ12 |φ|2 (|φ|2 − v 2 )2 for the relativistic self-dual Chern-Simons system. Note the existence of two degenerate vacua: φ = 0 and |φ| = v.

So solutions of vorticity N necessarily carry both magnetic ﬂux Φ and electric charge Q. They are therefore excellent candidates for anyons. To uncover the Bogomol’nyi-style self-duality, we use the factorization identity (142), together with the Chern-Simons Gauss law constraint (154), to express the energy functional as i 2 ERCS = d x |D0 φ ± (|φ|2 − v 2 )φ|2 + |D± φ|2 κ

(156) 1 2 2 2 2 2 +V (|φ|) − 2 |φ| (|φ| − v ) ∓ v B . κ Thus, if the potential is chosen to take the self-dual form V (|φ|) =

1 |φ|2 (|φ|2 − v 2 )2 κ2

(157)

then the energy is bounded below (choosing signs depending on the sign of the ﬂux) ERCS ≥ v 2 |Φ|.

(158)

The bound (158) is saturated by solutions to the ﬁrst-order equations i D0 φ = ∓ (|φ|2 − v 2 )φ κ

D± φ = 0,

(159)

which, when combined with the Gauss law constraint (154), become the self-duality equations: D± φ

= 0

G.V. Dunne: Aspects of Chern-Simons Theory

B

= ±

2 |φ|2 (|φ|2 − v 2 ). κ2

221

(160)

These are clearly very similar to the self-duality equations (145) obtained in the Abelian-Higgs model. However, there are some signiﬁcant diﬀerences. Before discussing the properties of solutions, a few comments are in order. First, as is illustrated in Figure 6, the self-dual potential (157) is sixth-order, rather than the more commonly considered case of fourth-order. Such a potential is still power-counting renormalizable in 2 + 1 dimensions. Furthermore, the potential is such that the minima at φ = 0 and at |φ|2 = v 2 are degenerate. Correspondingly, there are domain wall solutions that interpolate between the two vacua [58]. In the Higgs vacuum, the Chern-Simons-Higgs mechanism leads to a massive gauge ﬁeld (recall (33)) and a massive real scalar ﬁeld. With the particular form of the self-dual potential (157) these masses are equal: ms =

2v 2 = mg . κ

(161)

Just as in the Abelian-Higgs case, the relativistic Chern-Simons vortex model has an associated N = 2 SUSY, in the sense that the Lagrangian (152), with scalar potential (157), is the bosonic part of a SUSY model with extended N = 2 SUSY [59].

Exercise 4.2.1: The N = 2 SUSY extension of the relativistic ChernSimons vortex system (152) has Lagrangian LSUSY =

κ µνρ ¯ /ψ Aµ ∂ν Aρ + |Dµ φ|2 + iψD 2 1 1 ¯ − 2 |φ|2 (|φ|2 − v 2 )2 + (3|φ|2 − v 2 )ψψ. κ κ

(162)

Show that there are pairs of bosonic ﬁelds degenerate with pairs of fermionic ﬁelds, in both the symmetric and asymmetric phases.

To investigate vortex solutions, we decompose the scalar ﬁeld φ into √ its magnitude ρ and phase ω as in (146). The gauge ﬁeld is once again determined by the ﬁrst self-duality equation to be Ai = −∂i ω ∓ 12 ij ∂j ln ρ, as in (147), away from the zeros of the scalar ﬁeld. The second self-duality equation then reduces to a nonlinear elliptic equation: ∇2 ln |φ|2 =

4 |φ|2 (|φ|2 − v 2 ). κ2

(163)

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Topological Aspects of Low Dimensional Systems

Just as in the Abelian-Higgs case (148), this equation is neither solvable nor integrable. However, numerical solutions can be found using a radial vortex-like ansatz. A signiﬁcant diﬀerence from the Abelian-Higgs case is that while the Abelian-Higgs vortices have magnetic ﬂux strings located at the zeros of the scalar ﬁeld φ, in the Chern-Simons case we see from (160) that the magnetic ﬁeld vanishes at the zeros of φ. The magnetic ﬁeld actually forms rings centred on the zeros of φ. Numerical studies lead to two diﬀerent types of solutions, distinguished by their behaviour at spatial inﬁnity: 1. Topological solutions: |φ| → v as |x| → ∞. 2. Nontopological solutions: |φ| → 0 as |x| → ∞. In case 1, the solutions are topologically stable because they interpolate between the unbroken vacuum φ = 0 at the origin and the broken vacuum |φ| = v at inﬁnity. For these solutions, existence has been proven using similar complex analytic and variational techniques to those used for the Ablian-Higgs model [60]. to the relaExistence: There exist smooth ﬁnite energy solutions (φ, A) tivistic Chern-Simons self-duality equations (160) such that (i) |φ| → v as |x| → ∞; (ii) φ has a ﬁnite number of zeros z1 , . . . , zm ; (iii) around each zero, φ ∼ (z − zk )nk hk (z), where hk (z) is smooth and hk (zk ) = 0; (iv) the vorticity is given by the net multiplicity of zeros: N = m k=1 nk . Interestingly, the uniqueness of these solutions has not been rigorously proved. Nor has the equivalence of these self-dual solutions to all ﬁnite energy solutions of the full second-order equations of motion. The topological vortex solutions have ﬂux, charge, energy: Φ = 2πN,

Q = κΦ,

E = v 2 |Φ|.

(164)

Furthermore, they have nonzero angular momentum. For N superimposed vortices, the angular momentum can be evaluated as J = −πκN 2 = − which is the anyonic relation (20).

Q2 4πκ

(165)

G.V. Dunne: Aspects of Chern-Simons Theory

223

The nontopological solutions, with asymptotic behaviour |φ| → 0 as |x| → ∞, are more complicated. The only existence proof so far is for superimposed solutions [61]. However, numerical studies are quite convincing, and show that [58] Φ = 2π(N + α),

Q = κΦ,

E = v 2 |Φ|

(166)

where α is a continuous parameter. They have nonzero angular momentum, and for N superimposed vortices J = −πκ(N 2 − α2 ) = −

Q + N Q. 4πκ2

(167)

There is an analogue of Bradlow’s bound (151) for the relativistic ChernSimons vortices. Integrating the second self-duality equation in (160), we get 2 v4 2 v2 2 2 2 2 (168) d x |φ| − d xB = 2 d x − 2 2κ κ 2 which implies that the vorticity is bounded above by N≤

v4 area. 4πκ2

(169)

A related bound has been found in the study of periodic solutions to the relativistic Chern-Simons equations [62, 64].

Exercise 4.2.2: The self-dual model (152) may be generalized to include also a Maxwell term for the gauge ﬁeld, but this requires an additional neutral scalar ﬁeld N [45]: LMCS = −

1 κ 1 Fµν F µν + µνρ Aµ ∂ν Aρ + |Dµ φ|2 + 2 (∂µ N )2 − V (|φ|, N ) 4e2 2 2e (170)

with self-dual potential 2 e2 v2 V = |φ|2 N − + (|φ|2 − κN )2 . κ 2

(171)

Show that in the symmetric phase the neutral scalar ﬁeld N is degenerate with the massive gauge ﬁeld. Show that in the asymmetric phase the N ﬁeld and the real part of φ have masses equal to the two masses of the gauge ﬁeld. Check that (i) the limit e2 → ∞ reduces to the relativistic Chern-Simons vortex model of (152); (ii) the limit κ → 0 reduces to the Abelian-Higgs model (134).

224

Topological Aspects of Low Dimensional Systems

4.3 Nonabelian relativistic Chern-Simons vortices The self-dual Chern-Simons vortex systems studied in the previous section can be generalized to incorporate nonabelian local gauge symmetry [65, 66]. This can be done with the matter ﬁelds and gauge ﬁelds in diﬀerent representations, but the most natural and interesting case seems to be with adjoint coupling, with the matter ﬁelds and gauge ﬁelds in the same Lie algebra representation. Then the gauge covariant derivative is Dµ φ = ∂µ φ+ [Aµ , φ] and the Lagrangian is 2 µνρ L = κ tr Aµ ∂ν Aρ + Aµ Aν Aρ + tr |Dµ φ|2 3 (172) 1 − 2 tr |[[φ, φ† ], φ] − v 2 φ|2 4κ where we have used the short-hand notation |Dµ φ|2 = (Dµ φ)† Dµ φ. There is a nonabelian version of the factorization identity (142) which with adjoint coupling reads 2 = tr |D± φ|2 ± i tr φ† [F12 , φ] ± ij ∂i tr φ† Dj φ − (Dj φ)† φ . tr |Dφ| (173) By the same argument as in the abelian case, we can show that with the potential as in (172), the associated energy functional is bounded below by an abelian magnetic ﬂux. This Bogomol’nyi bound is saturated by solutions to the nonabelian self-duality equations D± φ = F+−

=

0 1 2 [v φ − [[φ, φ† ], φ], φ† ]. κ2

(174)

Once again, the self-dual point is the point at which the model becomes the bosonic part of an N = 2 SUSY model. The self-dual potential has an intricate pattern of degenerate minima, given by solutions of the embedding equation [[φ, φ† ], φ] = v 2 φ.

(175)

This equation describes the embedding of SU (2) into the gauge Lie algebra, as can be seen by making the identiﬁcations: 1 φ = √ J+ ; v

1 φ† = √ J− ; v

[φ, φ† ] =

1 1 [J+ , J− ] = J3 v v

(176)

in which case the vacuum condition (175) reduces to the standard SU (2) commutation relations. Therefore, for SU (N ), the number of gauge inequivalent vacua is given by the number of inequivalent ways of embedding

G.V. Dunne: Aspects of Chern-Simons Theory

225

SU (2) into SU (N ). This number is in fact equal to the number P (N ) of partitions of the integer N . In each of these vacua, the masses of the gauge and scalar ﬁelds pair up in degenerate pairs, reﬂecting the N = 2 SUSY of the extended model including fermions. The masses are given by universal formulae in terms of the exponents of the gauge algebra [66]. Not many rigorous mathematical results are known concerning solutions to the nonabelian self-duality equations, although partial results have been found [63]. Physically, we expect many diﬀerent classes of solutions, with asymptotic behaviour of the solutions corresponding to the various gauge inequivalent vacua. 4.4 Nonrelativistic Chern-Simons vortices: Jackiw-Pi model As mentioned before, Chern-Simons gauge ﬁelds acquire their dynamics from the matter ﬁelds to which they couple, and so they can be coupled to either relativistic or nonrelativistic matter ﬁelds. The nonrelativistic couplings discussed in this and subsequent sections are presumably more immediately relevant for applications in condensed matter systems. We shall see that Bogomol’nyi self-duality is still realizable in the nonrelativistic systems. We begin with the abelian Jackiw-Pi model [67] κ µνρ 1 2 g 4 Aµ ∂ν Aρ + iψ ∗ D0 ψ − |Dψ| + |ψ| . (177) 2 2m 2 The quartic term represents a self-coupling contact term of the type commonly found in nonlinear Schr¨ o dinger systems. The Euler-Lagrange equations are 1 2 2 D ψ − g |ψ| ψ iD0 ψ = − 2m 1 µνρ J ρ (178) Fµν = κ is a Lorentz covariant short-hand notation for the conwhere J µ ≡ (ρ, J) LJP =

2

served nonrelativistic charge and current densities: ρ = |ψ| , and J j = j ∗ i ∗ j − 2m ψ D ψ − D ψ ψ . This system is Galilean invariant, and there are corresponding conserved quantities: energy, momentum, angular momentum and Galilean boost generators. There is, in fact, an addition dynamical symmetry [67] involving dilations, with generator 1 D = tE − (179) d2 x x · P 2 and special conformal transformations, with generator m d2 x x2 ρ. K = −t2 E + 2tD + 2

(180)

226

Topological Aspects of Low Dimensional Systems

is the momentum density. Here E is the energy and P The static energy functional for the Jackiw-Pi Lagrangian (177) is

1 2 g 4 EJP = d2 x |Dψ| − |ψ| . (181) 2m 2 Using the factorization identity (142), together with the Chern-Simons Gauss law constraint F12 = κ1 |ψ|2 , the energy becomes EJP =

1 |D± ψ|2 − d x 2m 2

g 1 ± 2 2mκ

4

|ψ|

.

(182)

Thus, with the self-dual coupling g=∓

1 mκ

(183)

the energy is bounded below by zero, and this lower bound is saturated by solutions to the ﬁrst-order self-duality equations D± ψ

=

B

=

0 1 2 |ψ| . κ

(184)

Note that with the self-dual coupling (183), the original quartic interaction 1 term, − g2 |ψ|4 = ± 2mκ |ψ|4 , can be understood as a Pauli interaction term B 2 ± 2m |ψ| , owing to the Chern-Simons constraint |ψ|2 = κB. The self-duality equations (184) can be disentangled as before, by decomposing the scalar ﬁeld ψ into a phase and a magnitude (146), resulting in a nonlinear elliptic equation for the density ρ: 2 ∇2 ln ρ = ± ρ. κ

(185)

Surprisingly (unlike the previous nonlinear elliptic equations (148, 163) in the Abelian-Higgs and relativistic Chern-Simons vortex models), this elliptic equation is exactly solvable! It is known as the Liouville equation [68], and has the general real solution ρ = κ∇2 ln 1 + |f |2 (186) where f = f (z) is a holomorphic function of z = x1 + ix2 only.

Exercise 4.4.1: Verify that the density ρ in (186) satisﬁes the Liouville equation (185). Show that only one sign is allowed for physical solutions,

G.V. Dunne: Aspects of Chern-Simons Theory

227

Fig. 7. Density ρ for a radially symmetric solution (188) representing one vortex with N = 2.

and show that this corresponds to an attractive quartic potential in the original Lagrangian (177).

As a consequence of the Chern-Simons Gauss law, these vortices carry both magnetic and electric charge: Q = κΦ. The net matter charge Q is Q=κ

∞ d 2 2 d x ∇ ln 1 + |f | = 2πκ r ln 1 + |f | · dr 0 2

2

(187)

Explicit radially symmetric solutions may be obtained by taking f (z) = ( zz0 )N . The corresponding charge density is 2

ρ=

2(N −1) r

r0 4κN 2 2N 2 · r0 1 + rr0

(188)

As r → 0, the charge density behaves as ρ ∼ r2(N −1) , while as r → ∞, ρ ∼ r−2−2N . At the origin, the vector potential behaves as Ai (r) ∼ −∂i ω ∓ j (N −1)ij xr2 . We can therefore avoid singularities in the the vector potential at the origin if we choose the phase of ψ to be ω = ±(N − 1)θ. Thus the

228

Topological Aspects of Low Dimensional Systems

self-dual ψ ﬁeld is  N −1  r √ 2N κ  r0  ±i(N −1)θ ψ= .  2N  e r0 r 1 + r0

(189)

Requiring that ψ be single-valued we ﬁnd that N must be an integer, and for ρ to decay at inﬁnity we require that N be positive. For N > 1 the ψ solution has vorticity N − 1 at the origin and ρ goes to zero at the origin. See Figure 7 for a plot of the density for the N = 2 case. Note the ring-like form of the magnetic ﬁeld for these Chern-Simons vortices, as the magnetic ﬁeld is proportional to ρ and so B vanishes where the ﬁeld ψ does. For the radial solution (188) the net matter charge is Q = d2 x ρ = 4πκN ; and the corresponding ﬂux is Φ = 4πN , which represents an even number of ﬂux units. This quantized character of the ﬂux is a general feature and is not particular to the radially symmetric solutions. The radial solution (188) arose from choosing the holomorphic function f (z) = ( zz0 )N , and corresponds to N vortices superimposed at the origin. A solution corresponding to N separated vortices may be obtained by taking

f (z) =

N

ca · z − za a=1

(190)

There are 4N real parameters involved in this solution: 2N real parameters za (a = 1, . . . N ) describing the locations of the vortices, and 2N real parameters ca (a = 1, . . . N ) corresponding to the scale and phase of each vortex. See Figure 8 for a plot of the two vortex case. The solution in (190) is in fact the most general ﬁnite multi-soliton solution on the plane. Solutions with a periodic matter density ρ may be obtained by choosing the function f in (186) to be a doubly periodic function [69]. I conclude this section by noting that the dynamical symmetry of the Jackiw-Pi system guarantees that static solutions are necessarily self-dual. This follows from the generators (179) and (180). Consider the dilation for static solutions. This generator D in (179). It is conserved, but so is P implies that E must vanish, which is only true for self-dual solutions.

4.5 Nonabelian nonrelativistic Chern-Simons vortices Just as the relativistic Chern-Simons vortices of Section 4.2 could be generalized to incorporate local nonabelian gauge symmetry, so too can the nonrelativistic models discussed in the previous section. We consider the

G.V. Dunne: Aspects of Chern-Simons Theory

229

Fig. 8. Density ρ for a solution (190) representing two separated vortices.

case of adjoint coupling, with Lagrangian 2 µνρ L = κ tr Aµ ∂ν Aρ + Aµ Aν Aρ + i tr ψ † D0 ψ 3 1 2 + 1 tr [ψ, ψ † ]2 . tr |Dψ| − 2m 4mκ

(191)

Using the nonabelian factorization identity (173), together with the Gauss law constraint, F+− = κ1 [ψ, ψ † ], the static energy functional can be written as 1 (192) E= d2 x tr |D± ψ|2 2m which is clearly bounded below by 0. The solutions saturating this lower bound satisfy the ﬁrst-order self-duality equations D± ψ F+−

= 0 1 [ψ, ψ † ]. = κ

(193)

These self-duality equations have been studied before in a diﬀerent context, as they are the dimensional reduction of the four-dimensional self-dual YangMills equations 1 F µν = ± µνρσ Fρσ . 2

(194)

230

Topological Aspects of Low Dimensional Systems

Exercise 4.5.1: Show that the self-dual Yang-Mills equations, with signature (2, 2), reduce to the self-dual Chern-Simons equations (193) if we take ﬁelds independent of two of the coordinates, say x3 and x4 , and combine the gauge ﬁelds A3 and A4 to form the ﬁelds ψ and ψ † .

The self-duality equations (193) are integrable, as they can be expressed as a zero curvature condition in the following way. Deﬁne a spectral connection (with spectral paramter λ) 1 1 1 † ψ, A− = A− + ψ . (195) A+ = A+ − λ κ λ κ Then the corresponding curvature is F+−

= =

∂+ A− − ∂− A+ + [A+ , A− ] 1 1 11 † λD− ψ − D+ ψ † . F+− − [ψ, ψ ] + κ κ κλ

(196)

Therefore, the condition of zero curvature, F+− = 0, for arbitrary spectral parameter λ, encodes the self-dual Chern-Simons equations (193). Explicit exact solutions can also be obtained by making simplifying algebraic ans¨atze which reduce the self-duality equations to the Toda equations, which are coupled analogues of the Liouville equation (185) and which are still integrable [70, 71]. In fact, all ﬁnite charge solutions can be found by mapping the selfduality equations (193) into the chiral model equations, which can then be integrated exactly in terms of unitons. To see this, set the spectral parameter λ = 1 in (195) and use the zero curvature F+− = 0 to deﬁne A± = g −1 ∂± g.

(197)

Then the conjugation χ = √1κ gψg −1 transforms the self-duality equations (193) into a single equation ∂− χ = [χ† , χ].

(198)

Furthermore, if we deﬁne χ = 12 h−1 ∂+ h, with h in the gauge group, then (198) becomes the chiral model equation ∂+ (h−1 ∂− h) + ∂− (h−1 ∂+ h) = 0.

(199)

All solutions to the chiral model equations with ﬁnite tr(h−1 ∂− hh−1 ∂+ h) can be constructed in terms of Uhlenbeck’s unitons [72, 73]. These are solutions of the form h = 2p − 1

(200)

G.V. Dunne: Aspects of Chern-Simons Theory

231

where p is a holomorphic projector satisfying: (i) p† = p, (ii) p2 = p, and (iii) (1 − p)∂+ p = 0. This means that all ﬁnite charge solutions of the selfdual Chern-Simons vortex equations (193) can be constructed in terms of unitons [66].

Exercise 4.5.2: Show that a holomorphic projector p can be expressed as p = M (M † M )−1 M † , where M = M (x− ) is any rectangular matrix. For SU (2) show that the uniton solution leads to a charge density [ψ, ψ † ] which, when diagonalized, is just the Liouville solution (186) times the Pauli matrix σ3 . 4.6 Vortices in the Zhang-Hansson-Kivelson model for FQHE There have been many applications of Chern-Simons theories to the description of the quantum Hall eﬀect, and the fractional quantum Hall eﬀect in particular (see e.g. [7,8,38,39,74]). In this section I describe one such model, and show how it is related to our discussion of Chern-Simons vortices. Zhang, et al. [5] reformulated the problem of interacting fermions in an external magnetic ﬁeld as a problem of interacting bosons with an extra Chern-Simons interaction describing the statistical transmutation of the fermions into bosons. This transmutation requires a particular choice for the Chern-Simons coupling constant, as we shall see below. The ChernSimons coupling is such that an odd number of ﬂux quanta are “tied” to the fermions (recall Fig. 1); thus the fermions acquire an additional statistics parameter (given by (20)) and so become eﬀective bosons. The ZHK model is basically a Landau-Ginzburg eﬀective ﬁeld theory description of these boson ﬁelds, coupled to a Chern-Simons ﬁeld that takes care of the statistical transmutation. It looks like a fairly innocent variation on the Jackiw-Pi model of Section 4.4, but the minor change makes a big diﬀerence to the vortex solutions. The ZHK Lagrangian is 2 κ 1 | ∂i + i(ai + Aext LZHK = − µνρ aµ ∂ν aρ + iψ ∗ (∂0 + ia0 ) ψ − i ) ψ| 2 2m 1 2 2 d x |ψ(x)| − n V (x − x ) |ψ(x )|2 − n − 2 (201) where we have adopted the notation that the statistical Chern-Simons gauge ﬁeld is aµ , while the external gauge ﬁeld descrbing the external magnetic ﬁeld is Aext i . We have also, for convenience in some of the subsequent equations, written the Chern-Simons coupling as −κ. The constant n appearing in the potential term denotes a uniform condensate charge density.

232

Topological Aspects of Low Dimensional Systems

Normally a complex scalar ﬁeld ψ is used to describe bosons. But when the Chern-Simons coupling takes the values κ=

1 ; 2π(2k − 1)

k≥1

(202)

1 the anyonic statistics phase (20) of the ψ ﬁelds is 2κ = (2k − 1)π; that is, the ﬁelds are antisymmetric under interchange. Thus the ﬁelds are actually fermionic. We can alternatively view this as the condensing of the fundamental fermionic ﬁelds into bosons by the attachment of an odd number of ﬂuxes through the Chern-Simons coupling [5]. Consider a delta-function contact interaction with

V (x − x ) =

1 δ(x − x ) mκ

(203)

in which case we can simply express the potential as V (ρ) =

1 (ρ − n)2 . 2mκ

(204)

The static energy functional for this model is EZHK =

d2 x

2 1 1 2 | ∂i + i(ai + Aext (ρ − n) ) ψ| + . i 2m 2mκ

(205)

Clearly, the minimum energy solution corresponds to the constant ﬁeld solutions ψ=

√ n,

ai = −Aext i ,

a0 = 0

(206)

for which the Chern-Simons gauge ﬁeld opposes and cancels the external ﬁeld. Since the Chern-Simons constraint is b = − κ1 ρ, we learn that these minimum energy solutions have density ρ = n = κB ext .

(207)

With the values of κ in (202), these are exactly the conditions for the uniform Laughlin states of ﬁlling fraction ν=

1 · 2k − 1

(208)

To describe excitations about these ground states, we re-express the energy

G.V. Dunne: Aspects of Chern-Simons Theory

233

using the factorization identity (142).

1 1 1 1 |D± ψ|2 ∓ (ρ − n)2 B ext − ρ ρ + 2m 2m κ 2mκ 1 1 2 |D± ψ|2 ± ρ − κB ext = d2 x 2m 2mκ

κ ext 1 2 B B+ (ρ − n) ∓ 2m 2mκ

1 κ ext 2 2 |D− ψ| + B B · = d x (209) 2m 2m

EZHK =

d2 x

In the last step we have chosen the lower sign, and used the relation n = κB ext to cancel the potential terms. Note that in the last line, B is the total magnetic ﬁeld B = B ext + b, where b is the Chern-Simons magnetic ﬁeld. Thus, the energy is bounded below by a multiple of the total magnetic ﬂux. This bound is saturated by solutions to the ﬁrst-order equations D− ψ

=

0

B

=

B ext −

1 ρ. κ

(210)

As before, the ﬁrst equation allows us to express the total gauge ﬁeld Ai = ai + Aext in terms of the phase and the density, and the second equation i reduces to a nonlinear elliptic equation for the density: ∇2 ln ρ =

2 (ρ − n). κ

(211)

Comparing this with the corresponding equation (185) in the Jackiw-Pi model, we see that the eﬀect of the external ﬁeld and the modiﬁed potential (204) is to include a constant term on the RHS. But this converts the Liouville equation back into the vortex equation (148) for the Abelian-Higgs model! This can be viewed as both good and bad news – bad in the sense that we no longer have the explicit exact solutions to the Liouville equation (185) that we had in the Jackiw-Pi model, but good because we know a great deal about the Abelian-Higgs models vortices, even though we do not have any explicit exact solutions. First, we learn that there are indeed well-behaved vortex solutions in the ZHK model, and that their magnetic charge is related to their vorticity. But now, because of the Chern-Simons relation, these vortices also have electric charge, proportional to their magnetic charge. In particular these vortices have the correct quantum numbers for the quasi-particles in the Laughlin model for the FQHE [5].

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Topological Aspects of Low Dimensional Systems

Exercise 4.6.1: Show that if we modify the Jackiw-Pi model by including a background charge density ρ0 (instead of an external magnetic ﬁeld) [75] L=

κ µνρ 1 2 Aµ ∂ν Aρ + iψ ∗ D0 ψ − |Dψ| − V (ρ) + ρ0 A0 2 2m

(212)

1 then with the potential V (ρ) = 2mκ (ρ − ρ0 )2 , the self-dual vortex equations also reduce to a nonlinear elliptic equation of the Abelian-Higgs form (148):

∇2 ln ρ =

2 (ρ − ρ0 ). κ

(213)

4.7 Vortex dynamics So far, we have only dealt with static properties of vortices in various 2 + 1dimensional ﬁeld theories. However, the more interesting question concerns their dynamics; and beyond that, we are ultimately interested in their quantization. Various diﬀerent approaches have been developed over the years for studying vortex dynamics. Particle physicists and ﬁeld theorists, motivated largely by Manton’s work [76] on the low energy dynamics of solitons (of which these planar Bogomol’nyi vortices are an example), have studied the dynamics of vortices in the Abelian-Higgs model, which is governed by relativistic dynamics for the scalar ﬁeld. Condensed matter physicists have developed techniques for studying vortices in superconductors and in Helium systems, where the dynamics is nonrelativistic [77]. The Chern-Simons vortices are particularly interesting, because in addition to introducing the new feature of anyon statistics of vortices, they appear to require methods from both the particle physics and condensed matter physics approaches. Having said that, there is, as yet, no clear and detailed understanding of the dynamics of Chern-Simons vortices. This is a major unsolved problem in the ﬁeld. Consider ﬁrst of all the dynamics of vortices in the Abelian-Higgs model of Section 4.1. Since no exact vortex solutions are known, even for the static case, we must be content with approximate analytic work and/or numerical simulations. As mentioned earlier, it is known from numerical work [78] that the vortices in the Abelian-Higgs model repel one another when the scalar mass exceeds the gauge mass, and attract when the gauge mass exceeds the scalar mass. When these two mass scales are equal (140) we are in the self-dual case, and there are no forces between static vortices. Manton’s approach to the dynamics of solitons provides an eﬀective description of the dynamics at low energies when most of the ﬁeld theoretic degrees of freedom are frozen out. Suppose we have static multi-soliton solutions parametrized by a ﬁnite dimensional “moduli space” – the space consisting of the minima of the static energy functional (135). We assume that the true dynamics of the full ﬁeld theory is in some sense “close to” this moduli space of

G.V. Dunne: Aspects of Chern-Simons Theory

235

static solutions. Then the full dynamics should be approximated well by a projection onto a ﬁnite dimensional problem of dynamics on the moduli space. This is an adiabatic approximation in which one assumes that at each moment of time the ﬁeld is a static solution, but that the parameters of the static solution (in the vortex case we can loosely think of these parameters as the locations of the vortices) vary slowly with time. This approach has been applied successfully to the Abelian-Higgs vortices [79], with the N -vortex parameters taken to be the zeros z1 , . . . , zN of the scalar φ ﬁeld (recall the theorem in Sect. 4.1). For well separated zeros we can think of these zeros as specifying the locations of the vortices. Indeed, the exponential approach of the ﬁelds to their asymptotic values motivates and supports the approximation of well separated vortices as a superposition of single vortices, with only exponentially small errors. (Actually, to be a bit more precise, the N -vortex moduli space is not really CN ; we need to take into account the identical nature of the vortices and factor out by the permutation group SN . Thus the true N -vortex moduli space is CN /SN , for which a good set of global coordinates is given by the symmetric polynomials in the zeros z1 , . . . , zN .) The total energy functional is H =T +V

(214)

where the kinetic energy is T =

1 ˙2 d x A˙ i A˙ i + |φ| 2 2

(215)

and the potential energy V is the static energy functional (135). There is ·E = J 0 , to be imposed. In the adiabatic also the Gauss law constraint, ∇ approximation, the potential energy remains ﬁxed at v 2 |Φ|, given by the saturated Bogomol’nyi bound (144). But when the moduli space parameters become time dependent, we can insert these adiabatic ﬁelds φ = φ(x; z1 (t), . . . , zN (t)),

= A( x; z1 (t), . . . , zN (t)) A

(216)

into the kinetic energy (215), integrate over position x, and obtain an effective kinetic energy for the moduli parameters za (t), for a = 1, . . . N . In terms of real coordinates xa on the plane, this kinetic energy takes the form T =

1 gab x˙ a · x˙ b 2

(217)

where the metric gab is a (complicated) function depending on the positions and properties of all the vortices. Samols [79] has shown that this construction has a beautiful geometric interpretation, with the metric gab

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Topological Aspects of Low Dimensional Systems

being hermitean and K¨ ahler. Furthermore, the dynamics of the slowly moving vortices corresponds to geodesic motion on the moduli space. While the metric cannot be derived in closed form, much is known about it, and it can be expressed solely in terms of the local properties of the vortices. It should be mentioned that the step of performing the spatial integrations to reduce the ﬁeld theoretic kinetic energy (215) to the ﬁnite dimensional moduli space kinetic energy (217) involves some careful manipulations due to the nature of vortex solutions in the neighbourhood of the zeros of the scalar ﬁeld φ. The essential procedure is ﬁrst to excise small discs surrounding the zeros. The contributions from the interior of the discs can be shown to be negligible as the size of the disc shrinks to zero. The contribution from the outside of the discs can be projected onto a line integral around each disc, using Stokes’s theorem and the linearized Bogomol’nyi self-duality equations. These line integrals may then be expressed in terms of the local data of the scalar ﬁelds in the neighbourhood of each disc: ln |φ|2 ≈ ln |z − zk |2 + ak +

1 {bk (z − zk ) + b∗k (z ∗ − zk∗ )} + . . . 2

(218)

There are several important diﬀerences complicating the direct application of this “geodesic approximation” to the dynamics of the relativistic ChernSimons vortices described in Section 4.2. While it is still true that the static multi-vortex solutions can be characterized by the zeros of the scalar ﬁeld (although no rigorous proof of uniqueness has been given so far), the fact that the vortices appear to be anyonic means that we cannot simply factor out by the symmetric group SN to obtain the true moduli space. Presumably the true moduli space would need to account for braidings of the vortex zeros. Second, the gauge ﬁeld makes no contribution to the kinetic energy in the case of Chern-Simons vortices – all the dynamics comes from the scalar ﬁeld. Correspondingly, even though there are no repulsive or attractive forces between the static self-dual vortices, there may still be velocity dependent forces that we do not see in the completely static limit. Thus, it is more convenient to consider the eﬀective action (rather than the energy) for motion on the moduli space. Both these considerations suggest that we should expect a term linear in the velocities, in addition to a quadratic kinetic term like that in (217). To see how these velocity dependent forces might arise, consider the relativistic Chern-Simons vortex model (152): LRCS = |D0 φ|2 + κA0 B −

κ ij 2 − 1 |φ|2 (|φ|2 − v 2 )2 . (219) Ai A˙ j − |Dφ| 2 κ2

√ Decomposing φ = ρeiω as in (146), Gauss’s law determines the nondynamical ﬁeld A0 to be: A0 = −ω˙ − κB 2ρ . Then the Lagrangian (219) can be

G.V. Dunne: Aspects of Chern-Simons Theory

237

re-expressed as LRCS =

1 ρ˙ 2 4 ρ

κ − κB ω˙ − ij Ai A˙ j 2 ! 2 " κ2 2 2 2 − |D± φ| + ± v 2 B. (220) B ∓ 2 ρ(ρ − v ) 4ρ κ

To implement Manton’s procedure, we take ﬁelds that solve the static selfdual equations (160), but with adiabatically time-dependent parameters. As moduli parameters we take the zeros qa (t) of the φ ﬁeld. Then the term in the square brackets in (220) vanishes for self-dual solutions. Furthermore, N for an N vortex solution the vorticity is such that ω = a=1 arg(x − qa (t)). Then we can integrate over x to obtain an eﬀective quantum mechanical Lagrangian for the vortex zeros: 1 ij ia (q)q˙ai ± 2πv 2 N. L(t) = d2 xL = gab (q) q˙ai q˙bj + A (221) 2 The term linear in the velocities comes from the B ω˙ term in (220), while the ij Ai A˙ j term integrates to zero [81]. The coeﬃcient of the linear term is Aia = 2πκij

qj − qj a b + local | qa − qb |2

(222)

b=a

where the ﬁrst term is responsible for the anyonic nature of the vortices, while the “local” term is only known approximately in terms of the local expansion (218) in the neighbourhood of each vortex, and is a complicated function of the positions of all the vortices. The linear coeﬃcient Aia is interpreted as a linear connection on the moduli space. But, despite a number of attempts [80, 81], we still do not have a good understanding of the quadratic metric term g ij in the eﬀective Lagrangian (221). This is an interesting outstanding problem. Another important problem concerns the implementation of this adiabatic approximation for the description of vortex dynamics in nonrelativistic Chern-Simons theories, such as the Jackiw-Pi model or the Zhang-HanssonKivelson model. In these cases the ﬁeld Lagrangian has only ﬁrst-order time derivatives, so the nature of the adiabatic approximation is somewhat diﬀerent [82, 83].

5

Induced Chern-Simons terms

An important feature of Chern-Simons theories is that Chern-Simons terms can be induced by radiative quantum eﬀects, even if they are not present

238

Topological Aspects of Low Dimensional Systems

as bare terms in the original Lagrangian. The simplest manifestation of this phenomenon occurs in 2 + 1 dimensional QED, where a Chern-Simons term is induced in a simple one-loop computation of the fermion eﬀective action [84]. Such a term breaks parity and time-reversal symmetry, as does a ¯ There are two complementary ways to investigate fermion mass term mψψ. this eﬀective action – the ﬁrst is a direct perturbative expansion in powers of the coupling for an arbitrary background gauge ﬁeld, and the second is based on a Schwinger-style calculation of the induced current J µ (from which the form of the eﬀective action may be deduced) in the presence of a special background with constant ﬁeld strength Fµν . Chern-Simons terms can also be induced in gauge theories without fermions, and in the broken phases of Chern-Simons-Higgs theories. Interesting new features arise when we consider induced Chern-Simons terms at ﬁnite temperature. 5.1 Perturbatively induced Chern-Simons terms: Fermion loop We begin with the perturbative eﬀective action. To facilitate later comparison with the ﬁnite temperature case, we work in Euclidean space. The one fermion loop eﬀective action is /+A / + m) Seﬀ [A, m] = Nf log det(i∂

(223)

where m is a fermion mass. The physical signiﬁcance of this fermion mass will be addressed below. We have also included the overall factor of Nf corresponding to the number of fermion ﬂavours. This, too, will be important later. For now, simply regard Nf and m as parameters. A straightforward perturbative expansion yields 1 A / / + m) + Nf tr Seﬀ [A, m] = Nf tr log(i∂ i∂ /+m (224) 1 1 Nf tr A / A / + ... + 2 i∂ / + m i∂ /+m The ﬁrst term is just the free (A = 0) case, which is subtracted, while the second term is the tadpole. Since we are seeking an induced ChernSimons term, and the abelian Chern-Simons term is quadratic in the gauge ﬁeld Aµ , we restrict our attention to the quadratic term in the eﬀective action (interestingly, we shall see later that this step is not justiﬁed at ﬁnite temperature) Nf d3 p quad Seﬀ [A, m] = [Aµ (−p)Γµν (p)Aν (p)] (225) 2 (2π)3 where the kernel is µν

Γ (p, m) =

/p + /k − m k− m d3 k µ ν / tr γ γ 2 (2π)3 (p + k)2 + m2 k + m2

(226)

G.V. Dunne: Aspects of Chern-Simons Theory

239

corresponding to the one-fermion-loop self-energy diagram shown in Figure 9a. Furthermore, since the Chern-Simons term involves the parityodd Levi-Civita tensor µνρ , we consider only the µνρ contribution to the fermion self-energy. This can arise because of the special property of the gamma matrices (here, Euclidean) in 2 + 1 dimensions tr(γ µ γ ν γ ρ ) = −2µνρ .

(227)

(Note that this may be somewhat unfamiliar because in 3 + 1 dimensions we are used to the fact that the trace of an odd number of gamma matrices is zero). It is then easy to see from (226) that the parity odd part of the kernel has the form µνρ pρ Πodd (p2 , m) Γµν odd (p, m) =

where Πodd (p2 , m) = =

(228)

1 d3 k 3 2 (2π) [(p + k) + m2 ][k 2 + m2 ] |p| 1 m arcsin · 2π |p| p2 + 4m2

2m

(229)

In the long wavelength (p → 0) and large mass (m → ∞) limit we ﬁnd 2 1 m µνρ p µν pρ + O Γodd (p, m) ∼ · (230) 4π |m| m2 Inserting the leading term into the quadratic eﬀective action (225) and returning to coordinate space, we ﬁnd an induced Chern-Simons term Nf 1 m CS = −i (231) Seﬀ d3 xµνρ Aµ ∂ν Aρ . 2 4π |m|

Exercise 5.1.1: Consider the three-photon leg diagram in Figure 9b, and show that in the large mass limit (m p1 , p2 ): 2 p 1 m µνρ (p , p , m) ∼ −i + O · (232) Γµνρ 1 2 odd 4π |m| m2 Hence show that in the nonabelian theory a nonabelian Chern-Simons term is induced at one-loop (note that the Chern-Simons coeﬃcient is imaginary in Euclidean space): Nf 1 m 2 CS 3 µνρ Seﬀ = −i (233) d x tr Aµ ∂ν Aρ + Aµ Aν Aρ . 2 4π |m| 3

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Topological Aspects of Low Dimensional Systems

(a)

(b)

Fig. 9. The one-loop Feynman diagrams used in the calculation of the induced Chern-Simons term (at zero temperature). The self-energy diagram (a) is computed in (226), while the three-photon-leg diagram (b) is treated in Exercise 5.1.1.

We now come to the physical interpretation of these results [84]. Consider the evaluation of the QED eﬀective action (223) at zero fermion mass. The computation of Seﬀ [A, m = 0] requires regularization because of ultraviolet (p → ∞) divergences. This regularization may be achieved, for example, by the standard Pauli-Villars method: reg [A, m = 0] = Seﬀ [A, m = 0] − lim Seﬀ [A, M ]. Seﬀ M→∞

(234)

The Pauli-Villars technique respects gauge invariance. But the M → ∞ limit of the second term in (234) produces an induced Chern-Simons term, because of the perturbative large mass result (230). Therefore, in the process of maintaining gauge invariance we have broken parity symmetry – this ¯ which is initiated by the introduction of the Pauli-Villars mass term M ψψ breaks parity, and survives the M → ∞ limit in the form of an induced Chern-Simons term (231). This is the “parity anomaly” of 2 + 1 dimensional QED [84]. It is strongly reminiscent of the well known axial anomaly in 3 + 1 dimensions, where we can maintain gauge invariance only at the expense of the discrete axial symmetry. There, Pauli-Villars regularization introduces a fermion mass which violates the axial symmetry. Recall that there is no analogous notion of chirality in 2 + 1 dimensions because of the diﬀerent Dirac gamma matrix algebra; in particular, there is no “γ 5 ” matrix that anticommutes with all the gamma matrices γ µ . Nevertheless, there is a parity anomaly that is similar in many respects to the 3 + 1 dimensional axial anomaly. In the nonabelian case, the induced Chern-Simons term (233) violates parity but restores invariance under large gauge transformations. It is

G.V. Dunne: Aspects of Chern-Simons Theory

241

known from a nonperturbative spectral ﬂow argument [84] that Seﬀ [A, m = 0] for a single ﬂavour of fermion (Nf = 1) is not gauge invariant, because the determinant (of which Seﬀ is the logarithm) changes by a factor (−1)N under a large gauge transformation with winding number N . Thus Seﬀ [A, m = 0] is shifted by N πi. But the induced Chern-Simons term (233) also shifts by N πi, when Nf = 1, under a large gauge transformation with winding number N . These two shifts cancel, and the regulated eﬀective action (234) is gauge invariant. This is reminiscent of Witten’s “SU(2) anomaly” in 3 + 1 dimensions [85]. This is a situation where the chiral fermion determinant changes sign under a large gauge transformation with odd winding number, so that the corresponding eﬀective action is not invariant under such a gauge transformation. As is well known, this anomaly is avoided in theories having an even number Nf of fermion ﬂavours, because the shift in the effective action is Nf N πi, which is always an integer multiple of 2πi if Nf is even (here N is the integer winding number of the large gauge transformation). The same is true here for the parity anomaly in the nonabelian 2 + 1 dimensional case; if Nf is even then both Seﬀ [A, m = 0] and the induced Chern-Simons term separately shift by a multiple of 2πi under any large gauge transformation. These results are from one-loop calculations. Nevertheless, owing to the topological origin of the Chern-Simons term, there is a strong expectation that the induced Chern-Simons terms should receive no further corrections at higher loops. This expectation is based on the observation that in a nonabelian theory the Chern-Simons coeﬃcient must take discrete quantized values in order to preserve large gauge invariance. At one loop we have seen that the induced coeﬃcient is N2f , which is an integer for even numbers of fermion ﬂavours, and reﬂects the parity anomaly in theories with an odd number of fermion ﬂavours. At higher loops, if there were further corrections they would necessarily destroy the quantized nature of the one-loop coeﬃcient. This suggests that there should be no further corrections at higher loops. This expectation has strong circumstantial evidence from various higher order calculations. Indeed, an explicit calculation [86] of the two-loop induced Chern-Simons coeﬃcient for fermions showed that the two-loop contribution vanishes, in both the abelian and nonabelian theories. This is a highly nontrivial result, with the zero result arising from cancellations between diﬀerent diagrams. This led to a recursive diagrammatic proof by Coleman and Hill [87] that in the abelian theory there are no contributions to the induced Chern-Simons term beyond those coming from the one fermion loop self-energy diagram. This has come to be known as the “Coleman-Hill theorem”. There is, however, some important ﬁne-print – the Coleman-Hill proof only applies to abelian theories (and zero temperature) because it relies on manifest Lorentz invariance and the absence of massless particles.

242

Topological Aspects of Low Dimensional Systems

5.2 Induced currents and Chern-Simons terms Another way to compute the induced Chern-Simons term in the fermionic eﬀective action (223) is to use Schwinger’s proper time method to calculate the induced current J µ , and deduce information about the eﬀective action from the relation

J µ =

δ Seﬀ [A]. δAµ

(235)

Schwinger’s famous “proper-time” computation [88] showed that the 3 + 1 dimensional QED eﬀective action can be computed exactly for the special case of a background gauge ﬁeld Aµ whose corresponding ﬁeld strength Fµν is constant. The corresponding calculation in 2 + 1 dimensions [84] is actually slightly easier because there is only a single Lorentz invariant combination of Fµν , namely Fµν F µν . (In 3 + 1 dimensions there is also Fµν F˜ µν .) Schwinger’s “proper-time” technique is also well-suited for computing the induced current J µ in the presence of a constant background ﬁeld strength. A constant ﬁeld strength may be represented by a gauge ﬁeld linear in the space-time coordinates: Aµ = 12 xν Fνµ , with Fµν being the constant ﬁeld strength. Since A is linear in x, ﬁnding the spectrum of the Dirac operator /∂ + iA / reduces to ﬁnding the spectrum of a harmonic oscillator. This spectrum is simple and discrete, thereby permitting an explicit exact solution. This computation does, however, require the introduction of a regulator mass m for the fermions. The result for the induced current is [84]

J µ =

1 m 1 µνρ Fνρ . 2 |m| 4π

(236)

By Lorentz invariance, we conclude that this result should hold for nonconstant background ﬁelds, at least to leading order in a derivative expansion. This is the result for a single ﬂavour of fermions. For Nf ﬂavours the result is simply multiplied by Nf . Integrating back to get the eﬀective action, we deduce that the eﬀective action must have the form NA [A] + Seﬀ [A] = Seﬀ

Nf m 1 SCS 2 |m| 4π

(237)

NA [A] is parity even but nonanalytic in the background ﬁeld. This where Seﬀ agrees with the perturbative calculation described in the previous section. Furthermore, we can also do this same computation of the induced current for special nonabelian backgrounds with constant ﬁeld strength (note that a constant nonabelian ﬁeld strength, Fµν = ∂µ Aν − ∂µ Aν + [Aµ , Aν ], can be obtained by taking commuting gauge ﬁelds that are linear in the spacetime coordinates, as in the abelian case, or by taking constant but noncommuting gauge ﬁelds).

G.V. Dunne: Aspects of Chern-Simons Theory

243

Exercise 5.2.1: Illustrate the appearance of terms in the 2 + 1 dimensional eﬀective action that are parity preserving but nonanalytic in the background ﬁeld strength, by computing the eﬀective energy of 2 + 1 dimensional fermions in a constant background magnetic ﬁeld B. Make things explicitly parity preserving by computing 12 (Seﬀ [B, m] + Seﬀ [B, −m]). 5.3 Induced Chern-Simons terms without fermions The issue of induced Chern-Simons terms becomes even more interesting when bare Chern-Simons terms are present in the original Lagrangian. Then Chern-Simons terms may be radiatively induced even in theories without fermions. In a classic calculation, Pisarski and Rao [10] showed that a gauge theory of 2 + 1 dimensional SU (N ) Yang-Mills coupled to a Chern-Simons term has, at one-loop order, a ﬁnite additive renormalization of the bare Chern-Simons coupling coeﬃcient: 4πκren = 4πκbare + N

(238)

where the N corrsponds to the N of the SU (N ) gauge group. This radiative correction is consistent with the discretization condition (recall (61)) that the Chern-Simons coeﬃcient 4πκ must be an integer for consistency with large gauge invariance at the quantum level. As such, this integer-valued ﬁnite shift is a startling result, since it arises from a one-loop perturbative computation, which a priori we would not expect to “know” anything about the nonperturbative large gauge transformations. Here I brieﬂy outline the computation of the renormalized Chern-Simons coeﬃcient in such a Chern-Simons-Yang-Mills (CSYM) theory [10]. The Euclidean space bare Lagrangian is 1 2 (239) LCSYM = − tr(Fµν F µν ) − im µνρ tr Aµ ∂ν Aρ + eAµ Aν Aρ 2 3 where Fµν = ∂µ Aν − ∂µ Aν + e[Aµ , Aν ]. Note that the Chern-Simons coeﬃcient is imaginary in Euclidean space. The discreteness condition (61) requires 4π

m = integer e2

(240)

where m is the mass of the gauge ﬁeld. The bare gauge propagator (with covariant gauge ﬁxing) is ∆bare µν (p)

1 = 2 p + m2

δµν

pρ pµ pν − 2 − mµνρ 2 p p

+ξ

pµ pν · (p2 )2

(241)

244

Topological Aspects of Low Dimensional Systems

bare −1 + Πµν , The gauge self-energy Πµν comes from the relation ∆−1 µν = (∆µν ) and may be decomposed as

Πµν (p) = (δµν p2 − pµ pν )Πeven (p2 ) + mµνρ pρ Πodd (p2 ).

(242)

Then the renormalized gauge propagator is deﬁned as pρ 1 pµ pν pµ pν 2 ∆µν (p) = − − m (p ) +ξ 2 2 δ µν ren µνρ 2 Z(p2 )[p2 + m2ren (p2 )] p2 p (p ) (243) where Z(p2 ) is a wavefunction renormalization factor and the renormalized mass is mren (p2 ) =

Zm (p2 ) m Z(p2 )

(244)

Zm (p2 ) = 1 + Πodd (p2 ).

(245)

with Z(p2 ) = 1 + Πeven (p2 ),

The important divergences are in the infrared (p2 → 0), and we deﬁne the renormalized Chern-Simons mass to be mren = mren (0) =

Zm (0) m. Z(0)

(246)

There is also, of course, charge renormalization to be considered. The renormalized charge is e2ren =

e2 2 ˜ Z(0)(Z(0))

(247)

˜ 2 ) comes from the renormalization of the ghost propagator. In where Z(p writing this expression for the renormalized charge we have used the standard perturbative Ward-Takahashi identities for (inﬁnitesimal) gauge invariance (note, however, that the Chern-Simons term introduces new vertices; but this is only a minor change). The important thing is that none of the Ward-Takahashi identities places any constraint on Zm (0), which comes from the odd part of the gauge self-energy at zero external momen˜ tum (245). The renormalization factors Z(0) and Z(0) are ﬁnite in Landau gauge, and a straightforward (but messy) one-loop calculation [10] leads to the results Zm (0) = 1 +

e2 7 N , 12π m

1 e2 ˜ Z(0) =1− N · 6π m

(248)

G.V. Dunne: Aspects of Chern-Simons Theory

245

Putting these together with the renormalized mass (246) and charge (247) we ﬁnd that, to one-loop order: m m 2 ˜ Zm (0)(Z(0)) = e2 ren e2 m 1 e2 7 − )N = 2 1+( e 12π 3π m m N · (249) = 2 + e 4π But this is just the claimed result: 4πκren = 4πκbare + N

(250)

It is widely believed that this is in fact an all-orders result, although no rigorous proof has been given. This expectation is motivated by the obser˜ at two vation that if there were further contributions to Zm (0) and Z(0) loops, for example, m m N N2 +α (251) = 2 + 2 e ren e 4π (m/e2 ) (where α is some numerical coeﬃcient) then the renormalized combination 4π( em2 )ren could no longer be an integer. Explicit two-loop calculations have shown that there is indeed no two-loop contribution [89], and there has been much work done (too much to review here) investigating this ﬁnite renormalization shift to all orders. Nevertheless, from the point of view of perturbation theory, the result 4πκren = 4πκbare + N seems almost too good. We will acquire a deeper appreciation of the signiﬁcance of this result when we consider the computation of induced Chern-Simons terms using ﬁnite temperature perturbation theory in Section 5.4. I should also mention that there are nontrivial subtleties concerning regularization schemes in renormalizing these Chern-Simons theories [90], in part due to the presence of the antisymmetric µνρ tensor which does not yield easily to dimensional regularization. These issues are particularly acute in the renormalization of pure Chern-Simons theories (no Yang-Mills term). The story of induced Chern-Simons terms becomes even more interesting when we include scalar (Higgs) ﬁelds and spontaneous symmetry breaking. In a theory with a Higgs scalar coupled to a gauge ﬁeld with a bare ChernSimons term, there is a radiatively induced Chern-Simons term at one loop. If this Higgs theory has a nonabelian symmetry that is completely broken, say SU (2) → U (1), then the computation of the zero momentum limit of the odd part of the gauge self-energy suggests the shift mHiggs (252) 4πκren = 4πκbare + f mCS

246

Topological Aspects of Low Dimensional Systems

where f is some complicated (noninteger!) function of the dimensionless ratio of the Higgs and Chern-Simons masses [91]. So 4πκren is not integer valued. But this is not a problem here because there is no residual nonabelian symmetry in the broken phase, since the SU (2) symmetry has been completely broken. However, consider instead a partial breaking of the original nonabelian symmetry (say from SU (3) to SU (2)) so that the broken phase does have a residual nonabelian symmetry. Then, remarkably, we ﬁnd [92, 93] that the complicated function f reduces to an integer: 4πκren = 4πκbare + 2, (the 2 corresponds to the residual SU (2) symmetry in this case). This result indicates a surprising robustness at the perturbative level of the nonperturbative discreteness condition on the Chern-Simons coeﬃcient, when there is a nonabelian symmetry present. Actually, in the case of complete symmetry breaking, the shift (252) should really be interpreted as the appearance of “would be” Chern-Simons terms in the eﬀective action. For example, a term µνρ tr(Dµ φFνρ ) in the eﬀective action is gauge invariant, and in the Higgs phase in which φ → φ at large distances, this term looks exactly like a Chern-Simons term. This is because we extracted the Chern-Simons coeﬃcient in the large distance (p2 → 0) limit where φ could be replaced by its asymptotic expectation value

φ. This observation has led to an interesting extension of the ColemanHill theorem to include the case of spontaneous symmetry breaking [94]. However, in the partial symmetry breaking case no such terms can be written down with the appropriate symmetry behaviour, so this eﬀect does not apply in a phase with residual nonabelian symmetry. Correspondingly, we ﬁnd that the integer shift property does hold in such a phase.

5.4 A ﬁnite temperature puzzle In this section we turn to the question of induced Chern-Simons terms at nonzero temperature. All the results mentioned above are for T = 0. The case of T > 0 turns out to be signiﬁcant both for practical and fundamental reasons. In the study of anyon superconductivity [95] one of the key steps involves a cancellation between the bare Chern-Simons term and an induced Chern-Simons term. While this cancellation was demonstrated at T = 0, it was soon realised that at T > 0 this same cancellation does not work because the ﬁnite T induced Chern-Simons coeﬃcient is temperature dependent. The resolution of this puzzle is not immediately obvious. This strange T dependent Chern-Simons coeﬃcient has also caused signiﬁcant confusion regarding the Chern-Simons discreteness condition: 4πκ = integer. It seems impossible for a temperature dependent Chern-Simons coeﬃcient κ(T ) to satisfy this consistency condition. However, recent work [96–99] has led to a new understanding and appreciation of this issue, with some important lessons about ﬁnite temperature perturbation theory in general.

G.V. Dunne: Aspects of Chern-Simons Theory

247

We concentrate on the induced Chern-Simons terms arising from the fermion loop, as discussed in Sections 5.1 and 5.2, but now generalized to nonzero temperature. Recall from (229) and (230) that the induced ChernSimons coeﬃcient is essentially determined by κind

= = =

Nf Πodd (p2 = 0, m) 2 Nf 2m d3 k 2 (2π)3 (k 2 + m2 )2 Nf 1 m · 2 4π |m|

(253)

If we simply generalize this one loop calculation to ﬁnite temperature (using the imaginary time formalism) then we arrive at ∞ 2m Nf d2 k (T ) κind = T (254) 2 2 (2π) [((2n + 1)πT )2 + k 2 + m2 ]2 n=−∞ where we have used the fact that at ﬁnite temperature the “energy” k0 takes discrete values (2n + 1)πT , for all integers n ∈ Z.

Exercise 5.4.1: Take the expression (254) and do the k integrals and then the k0 summation, to show that (T )

κind

= = =

where β =

∞ 2m Nf T 2 4π n=−∞ [((2n + 1)πT )2 + m2 ] βm Nf 1 tanh 2 4π 2 β|m| Nf 1 m tanh 2 4π |m| 2

(255)

1 T.

Thus, it looks as though the induced Chern-Simons coeﬃcient is temperature dependent. Note that the result (255) reduces correctly to the zero → 1 as T → 0 (i.e., as β → ∞). Indeed, T result (253) because tanh β|m| 2 the T> 0 result is just the T = 0 result multiplied by the smooth function β|m| tanh . This result has been derived in many diﬀerent ways [100], in 2 both abelian and nonabelian theories, and in both the real time and imaginary time formulations of ﬁnite temperature ﬁeld theory. The essence of the calculation is as summarized above.

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Topological Aspects of Low Dimensional Systems

On the face of it, a temperature dependent induced Chern-Simons term would seem to violate large gauge invariance. However, the nonperturbative (spectral ﬂow) argument for the response of the fermion determinant to large gauge transformations at zero T [84] is unchanged when generalized to T > 0. The same is true for the hamiltonian argument for the discreteness of 4πκ in the canonical formalism. Thus the puzzle. Is large gauge invariance really broken at ﬁnite T , or is there something wrong with the application of ﬁnite T perturbation theory? We answer these questions in the next sections. The essential new feature is that at ﬁnite temperature, other parity violating terms (other than the Chern-Simons term) can and do appear in the eﬀective action; and if one takes into account all such terms to all orders (in the ﬁeld variable) correctly, the full eﬀective action maintains gauge invariance even though it contains a Chern-Simons term with a temperature dependent coeﬃcient. In fact, it is clear that if there are higher order terms present (which are not individually gauge invariant), one cannot ignore them in discussing the question of invariance of the eﬀective action under a large gauge transformation. Remarkably, this mechanism requires the existence of nonextensive terms (i.e., terms that are not simply space-time integrals of a density) in the ﬁnite temperature eﬀective action, although only extensive terms survive in the zero temperature limit. 5.5 Quantum mechanical ﬁnite temperature model The key to understanding this ﬁnite temperature puzzle can be illustrated with a simple exactly solvable 0 + 1 dimensional Chern-Simons theory [96]. This is a quantum mechanical model, which at ﬁrst sight might seem to be a drastic over-simpliﬁcation, but in fact it captures the essential points of the 2 + 1 dimensional computation. Moreover, since it is solvable we can test various perturbative approaches precisely. Consider a 0+1 dimensional ﬁeld theory with Nf ﬂavours of fermions ψj , j = 1 . . . Nf , minimally coupled to a U (1) gauge ﬁeld A. It is not possible to write a Maxwell-like kinetic term for the gauge ﬁeld in 0 + 1 dimensions, but we can write a Chern-Simons term–it is linear in A. (Recall that it is possible to deﬁne a Chern-Simons term in any odd dimensional spacetime). We formulate the theory in Euclidean space (i.e., imaginary time τ , with τ ∈ [0, β]) so that we can go smoothly between nonzero and zero temperature using the imaginary time formalism. The Lagrangian is L=

Nf j=1

ψj† (∂τ − iA + m) ψj − iκA.

(256)

There are many similarities between this model and the 2 + 1 dimensional model of fermions coupled to a nonabelian Chern-Simons gauge ﬁeld. First, this model supports gauge transformations with nontrivial winding number.

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249

This may look peculiar since it is an abelian theory, but under the U (1) gauge transformation ψ → eiλ ψ, A → A + ∂τ λ, the Lagrange density changes by a total derivative and the action changes by ∆S = −iκ

0

β

dτ ∂τ λ = −2πiκN

(257)

β 1 dτ ∂τ λ is the integer-valued winding number of the topowhere N ≡ 2π 0 logically nontrivial gauge transformation.

Exercise 5.5.1: Show that, in the imaginary time formalism, such a nontrivial gauge transformation is λ(τ ) = 2Nβ π (τ − β2 ); while, in real time, a nontrivial gauge transformation is λ(t) = 2N arctan(t). In each case, explain why the winding number N must be an integer.

From (257) we see that choosing κ to be an integer, the action changes by an integer multiple of 2πi, so that the Euclidean quantum path integral e−S is invariant. This is the analogue of the discreteness condition (61) on the Chern-Simons coeﬃcient in three dimensional nonabelian Chern-Simons theories. (The extra 4π factor in the 2 + 1 dimensional case is simply a solid angle normalization factor.) Another important similarity of this quantum mechanical model to its three dimensional counterpart is its behaviour under discrete symmetries. Under naive charge conjugation C: ψ → ψ † , A → −A, both the fermion mass term and the Chern-Simons term change sign. This mirrors the situation in three dimensions where the fermion mass term and the Chern-Simons term each change sign under a discrete parity transformation. In that case, introducing an equal number of fermions of opposite sign mass, the fermion mass term can be made invariant under a generalized parity transformation. Similarly, with an equal number of fermion ﬁelds of opposite sign mass, one can generalize charge conjugation to make the mass term invariant in our 0 + 1 dimensional model. Induced Chern-Simons terms appear when we compute the fermion effective action for this theory: S[A] = log

det (∂τ − iA + m) det (∂τ + m)

N f (258)

The eigenvalues of the operator ∂τ − iA + m are ﬁxed by imposing the boundary condition that the fermion ﬁelds be antiperiodic on the imaginary time interval, ψ(0) = −ψ(β), as is standard at ﬁnite temperature. Since the

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Topological Aspects of Low Dimensional Systems

eigenfunctions are ψ(τ ) = e(Λ−m)τ +i

Êτ

A(τ )dτ

.

(259)

the antiperiodicity condition determines the eigenvalues to be Λn = m − i

a (2n − 1)πi + , β β

n = −∞, . . . , +∞

(260)

where we have deﬁned a≡

β

dτ A(τ )

(261)

0

which is just the 0 + 1 dimensional Chern-Simons term. Given the eigenvalues (260), the determinants in (258) are simply ! " ∞ # m − i βa + (2n−1)πi det (∂τ − iA + m) β = det (∂τ + m) m + (2n−1)πi n=−∞ β βm a cosh 2 − i 2 = cosh βm 2

(262)

where we have used the standard inﬁnite product representation of the cosh function. Thus, the exact ﬁnite temperature eﬀective action is a a βm − i tanh sin . (263) S[A] = Nf log cos 2 2 2 Several comments are in order. First, notice that the eﬀective action S[A] is not an extensive quantity (i.e., it is not an integral of a density). Rather, it is a complicated function of the Chern-Simons action: a = dτ A. We will have more to say about this later. Second, in the zero temperature limit, the eﬀective action reduces to Nf m S[A]T =0 = −i dτ A(τ ) (264) 2 |m| which [compare with (233)] is an induced Chern-Simons term, with coeﬃcient ± N2f . This mirrors precisely the zero T result (231) for the induced 1 is irrelevant beChern-Simons term in three dimensions (the factor of 4π cause with our 2 + 1 dimensional normalizations it is 4πκ that should be an integer, while in the 0 + 1 dimensional model it is κ itself that should be an integer. This extra 4π is just a solid angle factor).

G.V. Dunne: Aspects of Chern-Simons Theory

251

At nonzero temperature the eﬀective action is much more complicated. A formal perturbative expansion of the exact result (263) in powers of the gauge ﬁeld yields βm βm Nf i S[A] = −i tanh a − sech2 a2 2 2 4 2

(265) βm βm 1 + tanh sech2 a3 + . . . . 12 2 2 The ﬁrst term in this perturbative expansion βm Nf tanh S (1) [A] = −i A 2 2

(266)

is precisely the Chern-Simons action, but with a temperature dependent coeﬃcient. Moreover, this T dependent coeﬃcient is simply thezero T . Once coeﬃcient from (264), multiplied by the smooth function tanh β|m| 2 again, this mirrors exactly what we found in the 2 + 1 dimensional case in the previous section – see (253) and (255). If the computation stopped here, then we would arrive at the apparent contradiction mentioned earlier – namely, the “renormalized” Chern-Simons coeﬃcent βm Nf tanh (267) κren = κbare − 2 2 would be temperature dependent, and so could not take discrete values. Thus, it would seem that the eﬀective action cannot be invariant under large gauge transformations. The ﬂaw in this argument is clear. At nonzero temperature there are other terms in the eﬀective action, besides the Chern-Simons term, which cannot be ignored; and these must all be taken into account when considering the question of the large gauge invariance of the eﬀective action. Indeed, it is easy to check that the exact eﬀective action (263) shifts by (Nf N )πi, independent of the temperature, under a large gauge transformation, for which a → a + 2πN . But if the perturbative expansion (265) is truncated to any order in perturbation theory, then the result cannot be invariant under large gauge transformations: large gauge invariance is only restored once we resum all orders. The important point is that the full ﬁnite T effective action transforms under a large gauge transformation in exactly the same way as the zero T eﬀective action. When Nf N is odd, this is just the familiar global anomaly, which can be removed (for example) by taking an even number of ﬂavours, and is not directly related to the issue of the temperature dependence of the Chern-Simons coeﬃcient. The clearest way

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to understand this global anomaly is through zeta function regularization of the theory [97], as is illustrated in the following exercise.

Exercise 5.5.2: Recall the zeta function regularization deﬁnition of the fermion determinant, det(O) = exp(−ζ (0)), where the zeta function ζ(s) for the operator O is ζ(s) = (λ)−s (268) λ

where the sum is over the entire spectrum of O. Using the eigenvalues in (260), express this zeta function for the 0 + 1 dimensional Dirac operator in ∞ terms of the Hurwitz zeta function ζH (s, v) ≡ n=0 (n + v)−s . Hence show that the zeta function regularized eﬀective action is a a Nf βm − i tanh sin . (269) Szeta [A] = ±i a + Nf log cos 2 2 2 2 (You will need the Hurwitz zeta function properties: ζH (0, v) = 12 − v, and ζH (0, v) = log Γ(v) − 12 log(2π)). The sign ambiguity in the ﬁrst term corresponds to the ambiguity in deﬁning (λ)−s . The eﬀect of this additional term is that the zeta function regularized eﬀective action (269) changes by an integer multiple of 2πi under the large gauge transformation a → a + 2πN , even when Nf is odd. Show that this is consistent with the fact that this large gauge transformation simply permutes the eigenvalues in (260) and so should not aﬀect the determinant. (Note that this explanation of the global anomaly [101] is independent of the temperature, so it is somewhat beside the point for the resolution of the problem of an apparently T dependent Chern-Simons coeﬃcient.)

To conclude this section, note that only the ﬁrst term in the perturbative expansion (265) survives in the zero temperaturelimit. The higher order 2 βm . This is signiﬁcant terms all vanish because they have factors of sech 2 because all these higher order terms are nonextensive – they are powers of the Chern-Simons action. We therefore do not expect to see them at zero temperature. Indeed, the corresponding Feynman diagrams vanish identically at zero temperature. This is usually understood by noting that they must vanish because there is no gauge invariant (even under inﬁnitesimal gauge transformations) term involving more than one factor of A(τ ) that can be written down. This argument, however, assumes that we only look for extensive terms; at nonzero temperature, this assumption breaks down and correspondingly we shall see that our notion of perturbation theory

G.V. Dunne: Aspects of Chern-Simons Theory

253

must be enlarged to incorporate nonextensive contributions to the eﬀective action. For example, let us consider an action quadratic in the gauge ﬁelds which can have the general form 1 (2) dτ1 dτ2 A(τ1 )F (τ1 − τ2 )A(τ2 ) S [A] = (270) 2 where, by symmetry, F (τ1 − τ2 ) = F (τ2 − τ1 ). Under an inﬁnitesimal (2) gauge transformation, A → A + ∂τ λ, this action changes by: δS [A] = − dτ1 dτ2 λ(τ1 )∂τ1 F (τ1 − τ2 )A(τ2 ). Clearly, the action (270) will be invariant under an inﬁnitesimal gauge transformation if F = 0. This corresponds to excluding such a quadratic term from the eﬀective action. But the action can also be invariant under inﬁnitesimal gauge transformations if F = constant, which would make the quadratic action (270) nonextensive, and in fact proportional to the square of the Chern-Simons action. The origin of such nonextensive terms will be discussed in more detail in Section 5.7 in the context of ﬁnite temperature perturbation theory. 5.6 Exact ﬁnite temperature 2 + 1 eﬀective actions Based on the results for the 0+1 dimensional model described in the previous section, it is possible to compute exactly the parity violating part of the 2+1 dimensional QED eﬀective action when the backgound gauge ﬁeld Aµ (x, τ ) takes the following special form: A0 (x, τ ) = A0 ,

x, τ ) = A( x) A(

(271)

x) has quantized ﬂux: and the static background vector potential A( N ∈ Z. (272) d2 xij ∂i Aj = d2 x B = 2πN , Under these circumstances, the three dimensional ﬁnite temperature effective action breaks up into an inﬁnite sum of two dimensional eﬀective x). To see this, choose actions for the two dimensional background A( Euclidean gamma matrices in three dimensions to be: γ 0 = iσ 3 , γ 1 = iσ 1 , γ 2 = iσ 2 . Then the Dirac operator appearing in the three dimensional eﬀective action is ∂0 − iA0 + m D− (273) −i(∂ / − iA /) + m = D+ −∂0 + iA0 + m where D± = D1 ± iD2 are independent of τ by virtue of the ansatz (271). , for n ∈ Recalling that at ﬁnite T the operator ∂0 has eigenvalues (2n+1)πi β Z, we see that the problem is reduced to an inﬁnite set of Euclidean two dimensional problems.

254

Topological Aspects of Low Dimensional Systems

To proceed, consider the eigenfunctions

f , and eigenvalues µ, of the g

massless two dimensional Dirac operator f f 0 D− =µ · D+ 0 g g

(274)

It is a straightforward (but messy) algebraic exercise to show that given such an eigenfunction corresponding to a nonzero eigenvalue, µ = 0, it is possible to construct two independent eigenfunctions φ± of the three dimensional Dirac operator [97]: [m − iA0 + (2n+1)πi ] D− β φ =λ φ (275) ± ± ± D+ m + iA0 − (2n+1)πi β where

λ± = m ± i µ2 + (A0 −

(2n + 1)π 2 ) β

(276)

and φ± = i α± = µ

f , with α± g

2 (2n + 1)π 1 (2n + 1)π · A0 − ± i 1 + 2 A0 − β 4µ β

(277)

So, for each nonzero eigenvalue µ of the two dimensional problem, there are two eigenvalues λ± of the three dimensional Dirac operator. But from the form (276) of these eigenvalues, we see that their contribution to the three dimensional determinant is even in the mass m; and therefore these eigenvalues (coming from nonzero eigenvalues of the two-dimensional problem) do not contribute to the parity odd part of the three dimensional eﬀective action. In fact, the only contribution to the parity odd part comes from the zero eigenvalues of the two dimensional problem. From the work of Landau 2[20] 1 d xB (and Aharonov and Casher [21]) we know that there are N = 2π of these zero eigenvalues. This “lowest Landau level” can be deﬁned by the condition D− g = 0, so that the eigenfunctions of the three dimensional Dirac operator are 0 , where D− g = 0. (278) φ0 = g Thus the relevant eigenvalues of the three dimensional Dirac operator are (n)

λ0

= m + iA0 −

(2n + 1)πi , β

n∈Z

(279)

G.V. Dunne: Aspects of Chern-Simons Theory

255

each with degeneracy N . There is no paired eigenvalue, so to compute the parity odd part of the ﬁnite temperature three dimensional eﬀective action we simply trace over these eigenvalues, and multiply by N . But this is exactly the same problem that we solved in the last section (see (260)), with N playing the role of Nf , the number of fermion ﬂavours. Thus, we see immediately that a βm N a odd [A] = Seﬀ log cos( ) − i tanh sin 2 2 2 2 a a βm + i tanh sin − log cos 2 2 2 a βm = −iN arctan tanh tan (280) 2 2 β where a ≡ βA0 = 0 A0 . This is simply the imaginary part of the 0 + 1 exact eﬀective action (263). A more rigorous zeta function analysis of this problem has been given in [97], along the lines outlined in the Exercise from the last section. But the key idea is the same – when the three dimensional gauge background has the restricted static form of (271), the problem reduces to a set of two dimensional problems; and moreover, only the zero modes of this two dimensional system contribute to the parity odd part of the three dimensional eﬀective action. This can also be phrased in terms of chiral Jacobians of the two dimensional system [98]. The background in (271) supports large gauge transformations at ﬁnite temperature asa consequence of the S 1 of the Euclidean time direction. So, β 2N π if λ(τ ) = β τ − 2 , independent of x, then the gauge transformation but A0 → A0 + 2N π . In the notation Aµ → Aµ + ∂µ λ, does not aﬀect A, β

of (280) this means a → a + 2N π. Thus our discussion of these large gauge transformations reduces exactly to the discussion of the previous section for the 0 + 1 dimensional model. While this is a nice result, it is still a bit unsatisfying because these are not the nonabelian large gauge transformations in three dimensions that we were originally considering. In fact, if we adopt the static ansatz (271) then the abelian Chern-Simons term reduces to β 3 µνρ A0 (281) d x Aµ ∂ν Aρ = 4πN 0

which is just the 0 + 1 dimensional Chern-Simons term. So transformations that stay within this ansatz are simply the nontrivial winding number transformations of the 0 + 1 dimensional model. We can make a similar static ansatz in the nonabelian case. For static ﬁelds, the nonabelian

256

Topological Aspects of Low Dimensional Systems

Chern-Simons term simpliﬁes to

3

d x

µνρ

β 2 2 ij tr Aµ ∂ν Aρ + Aµ Aν Aρ = dτ tr A0 d x Fij 3 0 (282)

where ij Fij is the (Lie algebra valued) nonabelian covariant anomaly in two dimensions. It is possible to make gauge transformations that shift this appropriately (for Chern-Simons action by a constant, and by choosing A example, in terms of unitons) this constant shift can be made integer mulitple of 2πi. But this constant shift is not due to the winding number term in the change (58) of the nonabelian Chern-Simons Lagrangian under a gauge transformation – rather, it is due to the total derivtive term. Therefore, the simple nonabelian generalization of (280), with a static nonabelian ansatz, does not really answer the question of what happens to the discreteness condition (61) at ﬁnite temperature. 5.7 Finite temperature perturbation theory and Chern-Simons terms These results for the ﬁnite temperature eﬀective action contain some interesting lessons concerning ﬁnite temperature perturbation theory. The exact results of the previous sections are clearly very special. For general 2 + 1 dimensional backgrounds we cannot compute the eﬀective action exactly. Nor can we do so in truly nonabelian backgrounds that support large gauge transformations with nonvanishing winding number. Furthermore, ChernSimons terms may be induced not only in fermionic systems, but also in Chern-Simons-Yang-Mills [10] and in gauge-Higgs models with spontaneous symmetry breaking [91, 92]. In such models there are no known exact results, even at zero temperature. At ﬁnite T , perturbation theory is one of the few tools we have. An important lesson we learn is that there is an inherent incompatibility between large gauge invariance and ﬁnite temperature perturbation theory. We are accustomed to perturbation theory being gauge invariant order-by-order in the coupling e, but this is not true for large gauge invariance at ﬁnite temperature. We see this explicitly in the perturbative expansion (265) (note that since we had absorbed e into the gauge ﬁeld A, the order of perturbation is eﬀectively counted by the number of A factors). If we truncate this expansion at any ﬁnite order, then the result is invariant under small gauge transformations, but it transforms under a large gauge transformation in a T dependent manner. It is only when we re-sum all orders, to obtain the exact eﬀective action (263), that the response of the eﬀective action to a large gauge transformation becomes T independent, as it should be. There is actually a simple way to understand this breakdown of large gauge invariance at any ﬁnite order of perturbation theory [97].

G.V. Dunne: Aspects of Chern-Simons Theory

257

A gauge transformation (with factors of e restored) is 1 Aµ → Aµ + ∂µ Λ. e

(283)

For an inﬁnitesimal gauge transformation, the 1e factor can be absorbed harmlessly into a redeﬁnition of the gauge function Λ. But such a rescaling does not remove the e dependence for a large gauge transformation, because such a gauge transformation must satisfy special boundary conditions at τ = 0 and τ = β (in the imaginary time formalism). A rescaling of Λ simply moves e into the boundary conditions. The eﬀect is that a large gauge transformation can mix all orders in a perturbative expansion in powers of e, thus destroying the large gauge invariance order-by-order. Diagrammatically, the appearance of higher order terms, other than the Chern- Simons term, in the perturbative expansion (265) means that at ﬁnite temperature the diagrams with many external “photon” legs contribute to the parity odd part of the eﬀective action. This is in contrast to the case at T = 0 where only a single graph contributes – in 0+1 dimensions it is the one-leg graph, and in 2 + 1 dimensions it is the two-leg self-energy graph. Actually, these higher-leg graphs are perfectly compatible with inﬁnitesimal gauge invariance, but they violate the zero temperature requirement of only have extensive quantities in the eﬀective action. In the 0 + 1 dimensional model, the standard Ward identities for inﬁnitesimal gauge invariance [pµ Γµν... = 0, etc.] simplify (because there is no contraction of indices) to imply that the diagram is proportional to a product of delta functions in the external energies. In position space this simply means that each term is proportional to a nonextensive term like ( A)n . But at zero temperature such nonextensive terms are excluded for n > 1, and indeed one ﬁnds, reassuringly, that the corresponding diagrams vanish identically. At ﬁnite temperature we cannot exclude terms that are nonextensive in time, and so these terms can appear; and correspondingly we discover that these diagrams are indeed nonvanishing at T > 0. Accepting the possibility of nonextensive terms, the requirement that the fermion determinant change by at most a sign under a large gauge transformation, a → a + 2πN , leads to the general form: ∞ (2j + 1)a (2j + 1)a ) + fj sin( ) · dj cos( exp [−Γ(a)/Nf ] = i 2 2 j=0

(284)

The actual answer (263) gives as the only nonzero coeﬃcients: d0 = 1 and f0 = i tanh( βm 2 ). This fact can only be deduced by computation, not solely from gauge invariance requirements. These same comments apply to the 2 + 1 dimensional case when the background is restricted by the static ansatz (271). The static nature of the

258

Topological Aspects of Low Dimensional Systems

background once again makes a multi-leg diagram proportional to a product of delta functions in the external energies. For the answer to be extensive in space (but possibly nonextensive in time) we can only have one external spatial index, say i, and then invariance under inﬁnitesimal static gauge transformations requires this diagram to be proportional to ij pj . Factoring this out, the remaining diagrams are just like the multi-leg diagrams of the 0 + 1 dimensional model, and can be computed exactly. (There is a slight infrared subtlety due to the diﬃculty in Fourier transforming a ﬁnite ﬂux static background, but this is easily handled.) So, not surprisingly, the perturbative computation in the static anstaz reduces to that of the 0 + 1 case, just as happens in the exact evaluation. As soon as we attempt to go beyond the static ansatz, or consider induced Chern- Simons terms in non-fermionic theories, we strike some critical problems. The most signiﬁcant is that the zero momentum limit (230), via which we identiﬁed the induced Chern-Simons terms, is no longer well deﬁned at ﬁnite temperature. This is a physics problem, not just a mathematical complication. At ﬁnite T , Lorentz invariance is broken by the thermal bath and so a self-energy function Π(p) = Π(p0 , p) is separately a function of energy p0 and momentum p. Thus, as is well known even in scalar ﬁeld theories [102], the limits of p0 → 0 and p → 0 do not commute. The original computations of the ﬁnite temperature induced Chern- Simons coeﬃcient (see (255)) explicitly employed the “static limit” ). lim Π(p0 = 0, p

| p|→0

(285)

p| = 0) gives a It is easy to see that the “opposite” limit limp0 →0 Π(p0 , | diﬀerent answer at ﬁnite T [103]. This ambiguity simply does not arise in the 0 + 1 dimensional model, and the exact 2 + 1 dimensional results of the previous section avoided this ambiguity because the static ansatz (271) corresponds explicitly to the static limit (285). Finally, another important issue that is not addressed by our 0 + 1 dimensional model, or the corresponding static 2 + 1 dimensional results, is the Coleman-Hill theorem [87], which essentially states that only one-loop graphs contribute to the induced Chern-Simons term. This is an explicitly zero temperature result, as the proof assumes manifest Lorentz covariance. But the question of higher loops does not even come up in the 0 + 1 dimensional model, or the static 2 + 1 dimensional backgrounds, because the “photon” does not propagate; thus, there are no higher loop diagrams to consider. It would be interesting to learn more about ﬁnite temperature eﬀective actions whose zero temperature forms have induced Chern-Simons terms. There is undoubtedly more to discover.

G.V. Dunne: Aspects of Chern-Simons Theory

259

I thank the Les Houches organizers, A. Comtet, Th. Jolicoeur, S. Ouvry and F. David, for the opportunity to participate in this Summer School. This work has been supported by the U.S. Department of Energy grant DE-FG02-92ER40716.00, and by the University of Connecticut Research Foundation.

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d2 logλ dudv

±

λ 2a2

= 0, J. Math.

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COURSE 4

ANYONS

J. MYRHEIM Department of Physics, The Norwegian University of Science and Technology (NTNU), N–7034 Trondheim, Norway

Contents 1 Introduction 1.1 The concept of particle statistics . . . . . . . . . . 1.2 Statistical mechanics and the many-body problem 1.3 Experimental physics in two dimensions . . . . . . 1.4 The algebraic approach: Heisenberg quantization . 1.5 More general quantizations . . . . . . . . . . . . .

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configuration space The Euclidean relative space for two particles Dimensions d = 1, 2, 3 . . . . . . . . . . . . . Homotopy . . . . . . . . . . . . . . . . . . . . The braid group . . . . . . . . . . . . . . . .

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3 Schr¨ odinger quantization in one dimension

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4 Heisenberg quantization in one dimension 290 4.1 The coordinate representation . . . . . . . . . . . . . . . . . . . . . 291 5 Schr¨ odinger quantization in dimension 5.1 Scalar wave functions . . . . . . 5.2 Homotopy . . . . . . . . . . . . . 5.3 Interchange phases . . . . . . . . 5.4 The statistics vector potential . . 5.5 The N -particle case . . . . . . . 5.6 Chern–Simons theory . . . . . .

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Feynman path integral for anyons Eigenstates for position and momentum . . . . . . The path integral . . . . . . . . . . . . . . . . . . . Conjugation classes in SN . . . . . . . . . . . . . . The non-interacting case . . . . . . . . . . . . . . . Duality of Feynman and Schr¨ odinger quantization

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harmonic oscillator The two-dimensional harmonic oscillator . . . Two anyons in a harmonic oscillator potential More than two anyons . . . . . . . . . . . . . The three-anyon problem . . . . . . . . . . .

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anyon gas The cluster and virial expansions . . . . . . . . . First and second order perturbative results . . . Regularization by periodic boundary conditions . Regularization by a harmonic oscillator potential Bosons and fermions . . . . . . . . . . . . . . . . Two anyons . . . . . . . . . . . . . . . . . . . . . Three anyons . . . . . . . . . . . . . . . . . . . .

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The Monte Carlo method . . . . . . . The path integral representation of the Exact and approximate polynomials . The fourth virial coefficient of anyons Two polynomial theorems . . . . . . .

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356 358 362 364 368

9 Charged particles in a constant magnetic field 373 9.1 One particle in a magnetic field . . . . . . . . . . . . . . . . . . . . 374 9.2 Two anyons in a magnetic field . . . . . . . . . . . . . . . . . . . . 377 9.3 The anyon gas in a magnetic field . . . . . . . . . . . . . . . . . . . 380 10 Interchange phases and geometric phases 10.1 Introduction to geometric phases . . . . . . . . . . . . 10.2 One particle in a magnetic field . . . . . . . . . . . . . 10.3 Two particles in a magnetic field . . . . . . . . . . . . 10.4 Interchange of two anyons in potential wells . . . . . . 10.5 Laughlin’s theory of the fractional quantum Hall effect

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383 383 385 387 390 392

ANYONS

J. Myrheim

Abstract Fractional statistics of identical particles is a theoretical possibility both in one and two dimensions. Two-dimensional particles of this kind are called anyons. The most important application so far is in the theory of the fractionally quantized Hall effect, where the quasiparticle excitations can be described as anyons. The theory of identical particles, in particular the theory of anyons, is discussed here from the points of view of Schr¨ odinger and Heisenberg quantization, as well as the Feynman path integral quantization. Two topics discussed in some detail are the equation of state of a gas of anyons, and the relation between particle interchange phases and geometric phases (Berry phases).

1

Introduction

The subject of these notes is the non-relativistic quantum theory of identical particles, and in particular the fractional statistics allowed in one- or twodimensional systems. The concept of fractional statistics has now both theoretical and experimental interest, and may serve as an example, among many others, to illustrate that quantum mechanics is still a very active field of research, one hundred years after Planck’s constant. At least three different formulations of the quantum theory exist, mostly but not entirely equivalent, to be identified here by the names of Heisenberg, Schr¨ odinger and Feynman quantization. Most attention is given to the last two, and to the conclusion drawn from both that “anyons” may exist in two dimensions having “any” statistics in between Bose–Einstein and Fermi– Dirac. Fractional statistics in one dimension is discussed in less detail here but is also the subject of other lectures. It is a pleasure and a great honour to talk at Les Houches about Feynman path integrals for systems of identical particles, since the founder of the School, DeWitt, is also the founder of this particular branch of quantum theory [1,2]. By means of path integrals, Laidlaw and DeWitt gave the first topological proof of the symmetrization postulate in the quantum theory of c EDP Sciences, Springer-Verlag 1999

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identical particles, and at the same time showed the fundamental difference between two- and three-dimensional space. As an application of the general theory, the gas of anyons is discussed, in the two special cases of free anyons and of anyons in a magnetic field. Especially the magnetic field case should be of interest for applications in the fractionally quantized Hall effect. The final topic is the connection between the statistics phases related to the topology of the configuration space of a system of identical particles, and the geometric phases related to the geometry of the quantum mechanical state space. This connection has been used as a tool for investigating the statistics of quasi-particle excitations in the fractional quantum Hall system. I hope that the list of references is useful and representative. It is certainly incomplete, and I want to apologize for omissions. Some review articles and books are e.g. the references [3–20]. Several of the books are reprint collections, containing reprints of many articles cited here, and many more. 1.1 The concept of particle statistics Immediately after Heisenberg and Schr¨ odinger formulated quantum mechanics as it is known today, Heisenberg and Dirac extended the theory to systems of identical particles [21–23]. They noted that the operators representing observables in such a system must be symmetric under any interchange of particle labels, since non-symmetric observables would allow an observer to distinguish between particles. This rather obvious statement was the key to the correct quantum theory, because symmetric operators preserve the symmetry properties of the wave functions. For example, if the operator A and the wave function ψ are both totally symmetric, then the wave function Aψ is also totally symmetric. And similarly, if A is symmetric but ψ is totally antisymmetric, then Aψ is totally antisymmetric. Consequently, there exists a complete quantum theory of identical particles using only the totally symmetric wave functions, and there exists a different complete theory using only the totally antisymmetric wave functions. The symmetry or antisymmetry of the allowed wave functions is a characteristic property of a given system of identical particles, called the statistics of the particles. Particles described by symmetric wave functions satisfy Bose–Einstein statistics and are called bosons. Particles described by antisymmetric wave functions satisfy Fermi–Dirac statistics, they are fermions, and because of the antisymmetry they obey the Pauli exclusion principle, that two particles can not occupy the same quantum state. The symmetry or antisymmetry results in an effective attraction between bosons and an effective repulsion between fermions, both of a purely quantum mechanical nature. We may refer to this kind of attraction or repulsion as a statistics interaction. The mutual repulsion between fermions is quite literally a

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tangible fact: we can walk on the earth because matter consists of a small number of different species of fermions. In fact, the stability of matter (at least the proof of stability) depends on the fermionic nature of matter [24, 25]. Since the theory of Heisenberg and Dirac predicted that identical particles had to be either bosons or fermions, and since this prediction was verified experimentally, there was not much need for a better theory. However, the theory could be questioned on philosophical rather than experimental grounds. One possible objection is the vagueness of the concept of particle interchange. The most obvious way to make it precise is perhaps to define it simply as an interchange of particle labels. Then it becomes a purely mathematical operation with no physical content, meaningful in the mathematical description of a system of identical particles, but with no counterpart in the physical reality. It simply reflects the fact that the correspondence between physics and mathematics is not one to one. One may argue, however, that such an interpretation is too superficial. Let us consider particles that are so far apart that they can not be physically interchanged. Then it is intuitively obvious, and indeed true, that it does not matter whether we symmetrize or antisymmetrize our wave functions, or do neither of the two. This example suggests that the symmetrization or antisymmetrization postulate is not truly fundamental, but is rather a consequence of some more fundamental principle. It also indicates that this new fundamental principle must somehow give meaning to the concept of physical interchange of particles. Regardless of whether an interchange of identical particles is regarded as a mathematical or a physical operation, it is obviously an identity transformation from the physical point of view. In quantum mechanics it is not unusual that a physical identity transformation is represented mathematically by a phase factor, since two wave functions represent the same physical state if they differ only by an overall phase factor. Any permutation of bosons is represented by the trivial phase factor +1, whereas even and odd permutations of fermions are represented by +1 and −1, respectively. A natural question is then, why only ±1 and not more general phase factors? Laidlaw and DeWitt answered this question in the context of nonrelativistic quantum mechanics when they applied the Feynman path integral formalism to systems of identical particles [1]. In their formalism the interchange of identical particles has a clear physical meaning as a continuous process in which each particle moves along a continuous path. The path dependence of the interchange is all important, since it relates the quantum mechanical concept of particle statistics to the topology of the classical configuration space. The phase factors associated with different interchange paths must define a representation of the first homotopy group

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(the fundamental group) of the configuration space [26]. This requirement leads to the conclusion that only bosons and fermions can exist in Euclidean space of dimension three or higher, whereas more general possibilities open up in the two-dimensional case. The formalism does not apply in one dimension. Using a more traditional approach to quantization, Leinaas and I derived the same relation between particle statistics and topology [27]. Our approach was based on the geometrical interpretation of wave functions which is the basis of gauge theories, and which goes back to Weyl and Dirac [28–32]. We studied in some detail the more general kinds of statistics allowed in one- and two-dimensional systems. In either case there exists a continuously variable parameter defining the statistics, interpolating continuously between Bose–Einstein and Fermi–Dirac statistics. In one dimension the parameter may be interpreted as the strength of a δ-function potential between bosons, and when the strength becomes infinite, the bosons become fermions [33–36]. In two dimensions the parameter may be chosen as a phase angle θ which is 0 for bosons and π for fermions, and we showed by the example of the two-dimensional harmonic oscillator that the continuous variation of the phase angle gives a continuous interpolation between the boson and fermion energy spectra. The intermediate statistics, as we called it, is now usually called fractional statistics. In the two-dimensional case, the word “fractional” refers to interchange phases that are arbitrary rational or irrational fractions of π. A third approach leading to the same results is that of Goldin et al. [6, 37–41]. They studied the representations of the commutator algebra of particle density and current operators. This algebra has commutation relations that are independent of the particle statistics, but has inequivalent representations corresponding to the different statistics. Wilczek arrived at the concept of fractional statistics by considering the fact that the spin of two-dimensional particles is theoretically allowed to take arbitrary values, not just integer or half-integer multiples of ~. The relation between spin and statistics would require particles of fractional spin to have fractional statistics as well [42, 43]. He introduced the name anyons for two-dimensional identical particles having an interchange phase of “any” fixed value, not necessarily 0 or π, and also proposed a model for them as particles carrying both electric charge and magnetic flux, so that the interchange phase could be understood as an Aharonov–Bohm effect [44–46]. The fundamental group of the configuration space of identical particles in the plane plays a fundamental role in the theory of anyons. This group is called the braid group [47, 48], and its role was emphasized especially by Wu [49, 50]. It is interesting that mathematicians have arrived at exactly the same configuration space concept from the opposite direction, namely

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as a useful tool for studying the braid group [51–54]. For reviews of braids and knots and some applications in physics, see e.g. [55, 56]. The concept of the geometric phase, discovered by Berry [15, 57–61], was immediately applied to the calculation of interchange phases by Arovas et al. [62–64]. We will return to the question of how these phases are related, in Section 10 below. In quantum field theory the symmetry or antisymmetry of many-particle wave functions results from the canonical commutation or anticommutation relations of the field operators [65]. It is not obvious how to interpolate continuously between commutation and anticommutation relations so as to get a quantum field theory for anyons, but a solution to this problem is to use either a “boson” or a “fermion” gauge and then describe the deviation from Bose–Einstein or Fermi–Dirac statistics as due to a “statistics” field, which is then a vector potential analoguous to the electromagentic vector potential [3, 4, 66–69]. The statistics vector potential is an example of a Chern–Simons field [70–76]. There are many other developments in the theory of fractional statistics about which little, or nothing, will be said here. Among those are statistics in one dimension [5, 77], or on two dimensional surfaces of a more complicated topology than the Euclidean plane. Thouless and Wu considered identical particles on the sphere, and found restrictions on the statistics angle dependent on the number of particles [78, 79]. Einarsson showed how to implement fractional statistics on a torus [80, 81], and more general discussions can be found e.g. in references [82, 83]. Certainly one of the most interesting topics is the connection between spin and statistics. The spin of the statistics field plays an essential part in establishing a connection [42, 43, 84–86]. However, it seems impossible to exclude for example the possibility that non-relativistic spin zero particles could be fermions, unless some extra assumptions are introduced [82, 87]. General topological arguments have been put forward, in which the existence of antiparticles is a crucial assumption [5, 88–92]. Particles and antiparticles are just one example of interacting anyons of different kinds. This is closely related to the possibility of interactions between distinguishable particles resembling the statistics interaction of identical particles [41, 93–95]. 1.2 Statistical mechanics and the many-body problem The statistical mechanics of bosons and fermions, i.e. the Bose–Einstein and Fermi–Dirac statistics, existed even before quantum mechanics received its final form [22, 96–100]. The theory is no more difficult than the corresponding theory of distinguishable particles, since the only effect of the indistinguishability of bosons or fermions is to forbid wave functions of the wrong symmetry type, thereby reducing the degeneracy of each energy level.

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The statistics interaction between bosons or between fermions does not change the energies of individual levels. The ideal gas, i.e. a gas of particles with no interaction apart from the statistics interaction, is a simple model which is useful for many purposes. In order to calculate the energy levels for a system of many non-interacting particles that are either bosons or fermions, one need only distribute the particles among the one-particle energy levels, counting degeneracies according to the Bose–Einstein or Fermi–Dirac statistics. Knowing the many-particle energy levels and their degeneracies, one may proceed to calculate the equation of state. The statistical mechanics of anyons is more difficult. It has to be, simply because the bosonic and fermionic energy spectra are different, and the bosonic spectrum is supposed to change continuously into the fermion spectrum when the statistics angle θ changes from the boson value 0 to the fermion value π. The only way this can happen is that the energy levels move, either upwards or downwards. Thus, the statistics interaction of anyons affects not only the state counting, but also the energy eigenvalues. The harmonic oscillator problem is the standard exercise in quantum mechanics, but even this is difficult for more than two anyons. The centre of mass motion in an external harmonic oscillator potential is separable, even for anyons, and the relative motion is governed by a two-body harmonic oscillator interaction potential. A slightly more general class of quadratic Hamiltonians, including that of a constant magnetic field, can be treated just as easily. The two-anyon harmonic oscillator problem was solved in reference [27]. Wu made the first attempt to solve the three-anyon problem, and found a class of exact solutions [101]. However, the ground state close to Fermi statistics was not among his exact solutions, and it is still not exactly known. More general exact solutions in harmonic oscillator potentials and magnetic fields, alone or together, have been found, but all have energies that depend linearly on the statistics angle [11, 102–115]. In the three-anyon problem, approximations to the wave functions corresponding to non-linear variation of energy have been found [116], and an almost complete separation of variables has been achieved [117]. The lowest part of the energy spectrum of three or four anyons in a harmonic oscillator potential has been calculated numerically [118–122]. Another line of attack is to use perturbation theory, starting from the known boson and fermion spectra [123–125]. The Hartree–Fock approximation has also been used [126, 127]. Arovas et al. made the first step towards determining the equation of state for a gas of non-interacting anyons when they calculated the second virial coefficient [66,128]. Their result is exact, since it is obtained from the exactly soluble two-anyon problem. To have a finite density with only two particles, they put them in a box with hard walls. Comtet et al., and also

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Johnson and Canright, simplified the calculation by confining the particles in an external harmonic oscillator potential [102, 129–131], in the same way as Fermi did for fermions [98]. The calculation of the third virial coefficient involves the three-anyon problem, which is not yet completely solved for any potential. Some exact results are nevertheless known. In particular, Sen has shown that the third virial coefficient is symmetric under a “supersymmetry” transformation which transforms bosons into fermions and vice versa, and more generally transforms θ into π − θ [132, 133]. Other exact results are the first and second order perturbation expansions about the boson and fermion values θ = 0 and θ = π, not only for the third virial coefficient, but for the full cluster expansion [134–141]. The equation of state for anyons in a magnetic field can be computed exactly, in the strong field limit where all particles are in the ground state [142]. Numerical results exist for the third and fourth virial coefficients at general values of θ [122, 143–146]. See also [147] for a summary and general discussion. 1.3 Experimental physics in two dimensions There are three examples of physical systems that are studied experimentally, where it has been suggested that the theory of fractional statistics may be relevant. One of these applications, in the fractional quantum Hall effect, seems rather well established, whereas the other two, in high temperature superconductivity and in superfluid helium, are doubtful, at best. The last example, vortex motion in superfluid helium, will not be discussed any further here [148–153]. The statistics of vortices is discussed in more general contexts e.g. in [154–156]. It is a surprising fact that zero-, one- and two-dimensional experimental physics is possible in our three-dimensional world [157–163] (Ref. [157] is a review with nearly 2000 references). The strict confinement of electrons to surfaces, or even to lines or points, is possible thanks to the third law of thermodynamics, which states that all degrees of freedom freeze out in the limit of zero temperature. Thus, in a strongly confining potential at low enough temperature it may happen that the excitation energy in one or more directions is much higher than the average thermal energy of the particles, so that those dimensions are effectively frozen out. Fowler, Fang, Howard and Stiles performed the first experiment with a two-dimensional electron gas in 1966, and later experiments use essentially the same technique [164]. The electrons are confined to the surface of a semiconductor by a strong electric field, and they move freely along the surface, whereas the energy ∆E needed to excite motion in the direction perpendicular to the surface is typically several millielectronvolt [165]. At a temperature of, for example, T = 1 K, the thermal energy is kB T ≈ 0.1 meV, where kB is Boltzmann’s constant. Hence, assuming for example

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a transverse excitation energy of ∆E = 10 meV, the fraction of electrons in the lowest excited transverse energy level is given by the Boltzmann factor e

− k∆ET B

= e−100 ≈ 10−44 ,

(1)

which is zero for all practical purposes. Thus the electron gas is truly twodimensional. Two-dimensional physics is no longer an exotic field since von Klitzing et al. discovered the quantized Hall effect (QHE) in 1980 [166, 167]. The discovery was totally unanticipated, and a new surprise was the discovery of the fractionally quantized Hall effect (FQHE) by Tsui et al. in 1982 [168, 169]. The effect observed is that, under certain conditions, the Hall resistance for a two-dimensional electron gas in a magnetic field is quantized as RH =

h 25 812.807 Ω , = νe2 ν

(2)

where h is Planck’s constant and e is the elementary charge. ν is either an integer or a rational fraction, which can be interpreted as the filling fraction, i.e. the number of degenerate energy levels (Landau levels) filled by conduction electrons, in the simple picture of a two-dimensional gas of free electrons. Thus, the fact that ν is not just inversely proportional to the magnetic field, but may stay constant while the field is changed by a finite amount, means that the number of conduction electrons varies with the field within certain limits. The universality of the quantized Hall effect has been tested to a precision of 10−10 in an experiment comparing two different integer quantization levels in two different materials [170]. Thus, in spite of the fact that it involves an extremely complicated many-body problem, the integer quantum Hall effect seems to provide a precise method for measuring the fine structure constant α = e2 /(4π0 ~c) (in MKSA units), where 2π~ = h, and c is the speed of light. It is independent of other methods, such as the measurement of the anomalous magnetic moments of electrons and muons, and gives a comparable precision. The same effect also provides a very accurate and stable standard resistor, easily realizable in the laboratory, and of a convenient magnitude. The conventional value of 25 812.807 Ω is fixed by international agreement from January 1, 1990. Laughlin proposed to explain the observed fractional quantization of the Hall resistance as the manifestation of a new state of matter, the incompressible quantum fluid, with elementary excitations that could be described as quasiparticles, or quasiholes, with fractional electric charge [171–174]. Halperin suggested that the fractional charge was associated with fractional statistics as well, and Arovas et al. verified by calculation the fractional values for both the charge and statistics phase angle of the quasiparticles

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in Laughlin’s theory [62, 175]. Jain et al. have tried to treat the integer and fractional quantum Hall effects in a more unified way [176–181]. Furthermore, L¨ utken and Ross have emphasized the universal character of the transition between different quantum Hall plateaux, and have suggested that the complete structure of the phase diagram, including the plateaux and the transition regions, can be understood as resulting from a discrete SL(2, Z) symmetry [182–186]. But none of these theories change the prediction of quasiparticle excitations having fractional charge and statistics. Different experiments seem to confirm the existence of fractionally charged excitations [187–193]. Thus, if fractional charge can be taken as a signature of fractional statistics [194], anyons may be said to have been directly observed in the fractional quantum Hall system. Other examples of two-dimensional systems experimentally available are the high temperature superconductors, discovered by Bednorz and M¨ uller [195–197]. The conduction takes place in two-dimensional layers, and Laughlin suggested a connection with the fractional quantum Hall effect [198, 199]. This idea raises two questions, discussed e.g. in references [14, 200]. First, whether systems of anyons show superfluidity and superconductivity, and second, whether such effects have anything to do with the observed high temperature superconductivity. The second question must be answered experimentally, and some attempts have been made. The experiments are based on the general, but not very quantitative, prediction that anyons violate both time reversal and parity invariance, and that these effects are likely to arise because of local magnetic fields. The fields in adjacent layers might point in opposite directions, so as to cancel, or they might add up to a global field. Three experimental groups have tried to measure the effects of such global fields on transmitted or reflected polarized light, but with conflicting results [201–204]. A fourth group has probed the local magnetic field by means of muons, and set a rather small upper limit of 0.8 G [205]. Since no effect is seen either in this experiment or in the most sensitive of the optical experiments [203], the experimental evidence is clearly against the anyon theory for high temperature superconductivity. 1.4 The algebraic approach: Heisenberg quantization The various approaches to the quantum theory for systems of identical particles mentioned so far, are closely related and may be grouped together under the heading of Schr¨ odinger quantization. There exists an alternative approach, which we may call Heisenberg quantization, leading to somewhat different results, especially in one and two dimensions [152, 206–208]. Note that Schr¨ odinger and Heisenberg quantization are not unique and detailed prescriptions for how to quantize, but rather two different general strategies. Schr¨ odinger quantization is a configuration space approach, emphasizing the

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role of the wave functions defined on the configuration space. Heisenberg quantization is a phase space approach, emphasizing the algebraic relations between observables, which in the classical theory are real valued functions defined on the phase space. For example, the most general classical observable for one point particle on a line is a function A = A(x, p) of the coordinate x and the momentum p. From two such observables A and B we may form the linear combination C = αA + βB, where α and β are arbitrary real numbers, as well as two different bilinear products, the pointwise product D = AB = BA, and the Poisson bracket E = {A, B} = −{B, A}. By definition, C(x, p) = αA(x, p) + βB(x, p),

D(x, p) = A(x, p) B(x, p),

(3)

and E=

∂B ∂A ∂A ∂B − · ∂x ∂p ∂x ∂p

(4)

In Heisenberg quantization one tries to represent the classical observables as linear, Hermitean operators on some complex Hilbert space, preserving as many as possible of the algebraic relations. The pointwise product is replaced by the operator product, and the Poisson bracket by the commutator product, E=

1 1 [A, B] = (AB − BA). i~ i~

(5)

Since it is impossible to preserve all the algebraic relations exactly, one has to select some relations to be treated as more fundamental than the rest. Thus, in the example with one particle on a line, the relation {x, p} = 1

(6)

is considered fundamental, and is replaced by the canonical commutation relation [x, p] = i~.

(7)

However, for two or more identical particles this simple prescription does not work, and one has to find alternatives. We will return to this point of view, although our main concern here is with the Schr¨odinger quantization. Briefly stated, the results are as follows, when the Heisenberg quantization is performed so as to respect the full symmetry between position and momentum variables. In one dimension fractional statistics is possible, described by one continuously variable statistics parameter. It is different from the fractional statistics obtained

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by Schr¨ odinger quantization in one dimension, and resembles more the two-dimensional fractional statistics of Schr¨odinger quantization. In two dimensions only the standard Bose–Einstein and Fermi–Dirac statistics are obtained. Thus, anyons are not included in this maximally symmetric version of Heisenberg quantization. In fact, anyons respect the rotational symmetry which involves coordinates only or momenta only, but break the phase space symmetry between coordinate and momentum. 1.5 More general quantizations The basic philosophy behind both Schr¨ odinger and Heisenberg quantization, as discussed above, is that the quantum theory of indistinguishable particles should resemble as much as possible the theory of distinguishable particles, that only such modifications are permitted as are necessary because the particles are indistinguishable. A number of different theories have been proposed departing more radically from the standard theory. They may allow interpolation between Bose–Einstein and Fermi–Dirac statistics independent of the configuration space dimension. One possibility is to consider quantum field theories with fields that do not commute according to the canonical commutation or anticommutation relations. An example is the so-called parastatistics proposed by Green [209–211]. It allows not only the completely symmetric or antisymmetric representations of the symmetric group, but also more general symmetry classes [23,212]. Thus, parastatistics of order p allows Young tableaux of up to p rows in the para-Bose case, or up to p columns in the para-Fermi case, while infinite order parastatistics allows all symmetry classes. Doplicher et al. deduced precisely these three possibilities in local relativistic quantum theory without long range forces [213, 214]. A number of proposed generalizations of the canonical commutation or anticommutation relations, starting with Wigner [215], are summarized in reference [216]. A simple example, leading to infinite statistics, is the socalled “q-mutation relations”, aj a†k − qa†k aj = δjk ,

(8)

where a and a† are annihilation and creation operators, j, k label the degrees of freedom of the field, and q is a number [217–225]. See also [226]. A vacuum state |0i is postulated with the property that aj |0i = 0 for all j, and the Fock space is generated from it by repeated applications of creation operators. The scalar product in the Fock space is uniquely defined by the q-mutation relations, and Fivel and others have shown that the condition −1 ≤ q ≤ 1 is necessary and sufficient to ensure that the scalar product is positive definite [220,223,224]. No rules exist relating the products aj ak and ak aj , or a†j a†k and a†k a†j , except that it is possible to prove commutativity

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in the boson case q = 1 and anticommutativity in the fermion case q = −1. The simple form Nj = a†j aj for the number operators is no longer valid in general. In the special case of one degree of freedom the number operator is [221] ∞ X (1 − q)n †n n a a . N= 1 − qn n=1

(9)

An entirely different approach, suggested by Haldane, is to modify directly the Pauli exclusion principle [227–237]. Johnson and Canright have applied this so called fractional exclusion statistics in the fractional quantum Hall system [238, 239]. 2

The configuration space

We will now discuss in more detail the quantum theory of identical particles. Our basic principle here is that an interchange of identical particles gives rise to a phase factor depending on the type of particles and on a continuous interchange path. The path dependence of the phase factor suggests immediately a path integral approach [1], but we will discuss first the description by means of wave functions, which is usually more suitable for calculations [27]. There are two steps in our quantization scheme. The first step, discussed in the present section, is to identify the configuration space of the system of identical particles, and the different classes of possible interchange paths. The second step, discussed in the Sections 3 and 5, is to introduce wave functions on the configuration space. In two or higher dimensions the wave functions must be treated as geometrical objects. Let X be the configuration space of a system of one particle. The configuration space of a system of N distinguishable particles moving in X is the Cartesian product space XN , defined as the set of all ordered N -tuples of the form x = (x1 , x2 , . . . , xN ) with

xj ∈ X for

j = 1, 2, . . . , N.

(10)

If p is a permutation of the particle labels 1, 2, . . . , N , then we define p(x) = (xp−1 (1) , xp−1 (2) , . . . , xp−1 (N ) ).

(11)

The set of all permutations of N objects is the symmetric group SN . It acts as a group of transformations on XN , by the above definition. If the particles are indistinguishable, then a configuration of N particles is simply a set of N points in X, x = {x1 , x2 , . . . , xN } ⊂ X.

(12)

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The order in which we list the particle positions is now arbitrary, so that the two points x and p(x) in XN represent the same configuration of the N particle system. Thus, the configuration space of the system of N identical particles is the Cartesian product XN with the identification p(x) ≡ x for any x ∈ XN and any p ∈ SN . A natural name for this identification space is XN /SN . We will consider here only the Euclidean one-particle spaces X = Rd , of dimension d = 1, 2, 3, since these are the simplest examples and the most useful for applications. An important simplification in the Euclidean case is that the centre of mass position splits off in a trivial way, so that XN /SN = X × (XN −1 /SN ),

(13)

where the factor X in the Cartesian product represents the centre of mass position, and XN −1 /SN represents the relative positions of the particles. The same factorization is not possible when the one-particle space is, e.g., a circle [27], a torus [80] or a sphere [78]. 2.1 The Euclidean relative space for two particles In the Euclidean case the interesting part of the configuration space is the relative space Rd(N −1) /SN . Let us examine the simplest case, N = 2. We have to label the particles arbitrarily as 1 and 2, in order to define the relative position as x = x1 − x2 .

(14)

If the two particles are at the positions a and b, then we get either x = a−b or x = b − a, depending on which one of the two possible labellings we choose. Thus, because the particles are identical, the relative positions x and −x describe the same configuration, and we see that the two-particle relative space Rd /S2 is Rd with the identification x ≡ −x. An immediate consequence of the identification x ≡ −x in Rd is that any time dependent curve x(t) is identified with y(t) = −x(t). Hence the tangent (or velocity) vector v = dx/dt at x is identified with w = dy/dt = −v at y = −x. The two spaces Rd /S2 and Rd are locally isometric, in fact the identification x ≡ −x is clearly irrelevant whenever we look at a small region / Ω for every x ∈ Ω. However, this isometry does Ω ⊂ Rd such that −x ∈ not hold at the origin, because any open region in Rd containing the origin must contain at least one pair of points x and −x. In other words, the origin is a singular point for the identification x ≡ −x. This local difference between Rd /S2 and Rd at the origin results also in a global difference. Perhaps the most dramatic manifestation of the global difference is the fact that Rd is flat, whereas Rd /S2 is globally curved when d ≥ 2.

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Topological Aspects of Low Dimensional Systems

To measure the global curvature one need not even approach the singularity at the origin. Curvature is defined in terms of the parallel transport of tangent vectors around closed curves, and the interesting curves are those starting at any given point x ∈ Rd and ending at −x. By definition, they are closed curves in Rd /S2 . Start with some vector v at x, and parallel transport it along any curve to −x. Because Rd is flat, the vector is moved unchanged, but, as we have seen, the vector v at −x is identified with the vector −v at x. Hence, the effect of the parallel transport around this kind of closed curve in Rd /S2 is to reverse the direction of every tangent vector. This reversion by parallel transport may lead to confusion as to whether or not a given vector field v = v(x) on Rd /S2 is single-valued. Let us write

v(x) =

d X

vj (x) exj ,

(15)

j=1

where each exj is a basis vector at x, and vj (x) is the j-th component of the vector v(x) located at x. In Euclidean space we are used to basis vectors that are parallel everywhere, so that exj is parallel to eyj for any two points x and y. Unfortunately, as we have seen, it is impossible to introduce parallel basis vectors in Rd /S2 , unless we place two sets of basis vectors, ex1 , ex2 , . . . , exd and −ex1 , −ex2 , . . . , −exd , at the same point x. It follows that if v = v(x) is a single-valued vector field on Rd /S2 , its components vj = vj (x) with respect to parallel basis vectors are doublevalued functions on Rd /S2 . We may of course introduce basis vectors that are single-valued functions of position, so that the components of a singlevalued vector field are also single-valued, but such basis vectors can not be parallel. As we shall see, similar problems arise when we introduce wave functions. The generalization to N identical particles, with N > 2, is straightforward. Let us ignore those configurations where two or more particle positions coincide. Then each point in the full configuration space RdN /SN , or in the relative space Rd(N −1) /SN , corresponds to N ! points either in RdN or in Rd(N −1) . In general, a closed curve in Rd(N −1) /SN connects a point x ∈ Rd(N −1) to the point p(x), where p is any one of the N ! permutations in the symmetric group SN . Parallel transport moves a vector v unchanged from x to p(x). However, the vector v at p(x) is not the same as v at x. Rather, v at x is identified with p(v) at p(x), hence v at p(x) is identified with p−1 (v) at x. Thus we see that the effect of the parallel transport of v is to transform it into p−1 (v). Given one vector v at x ∈ Rd(N −1) /SN , there are altogether N ! vectors at x that are parallel to it, by parallel transport around different closed curves in Rd(N −1) /SN .

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2.2 Dimensions d = 1, 2, 3 Let us consider a little more explicitly the simplest examples with two identical particles in Euclidean space. In one dimension the relative space R/S2 is the half-line x ≥ 0, where x = x1 − x2 is the single relative coordinate. Choosing x ≥ 0 instead of x ≤ 0 is of course pure convention, it simply means that we always label the rightmost particle as number 1. In two dimensions the identification x ≡ −x can be pictured as a folding of the plane into a cone of opening half-angle 30◦ . The points x and −x in the plane are folded onto the same point on the cone, and the origin of the plane becomes the top of the cone. Equivalently, writing the relative position as x = (x, y), we may define R2 /S2 as the upper half-plane y ≥ 0, but with the boundary points (x, 0) and (−x, 0) identified. The cone is locally flat everywhere except at the top point, since it is locally isometric to the plane. But it is globally curved, with infinite curvature at the top, so that parallel transport of a tangent vector once around the top point reverses its direction. In three dimensions, if we write the relative position as x = (x, y, z), then we may define R3 /S2 as the upper half-space z ≥ 0, with the boundary points (x, y, 0) and (−x, −y, 0) identified. Again the origin is a singular point of the identification space, and the space is locally flat everywhere except at the origin, since it is locally isometric to R3 . And again there is a global curvature, located at the origin, such that parallel transport of a tangent vector once around the origin reverses its direction. 2.3 Homotopy In order to classify the interchange paths, we have to examine the path connectivity of the configuration space. Again we consider only the Euclidean case, so that it is enough to examine the relative space Rd(N −1) /SN . Two curves from a point x to a point y are said to be homotopic if they can be continuously deformed one into the other [26]. A homotopy class consists of all the curves that are homotopic to one given curve. Concatenation of curves defines a natural product: two curves C1 and C2 can be spliced into one curve C2 C1 if C2 starts at the point where C1 ends. That is, if C1 goes from x to y and C2 from y to z, then C2 C1 is a curve going from x to z. This multiplication of curves is also a multiplication of homotopy classes. If we consider only the closed curves, or loops, starting and ending at one given point x, then the product of any two such loops is well-defined. The homotopy classes of loops at x form a group, called the first homotopy group, or fundamental group, of our space. In a connected space this definition does not depend on the point x, in the sense that groups defined at different points are isomorphic. The single point x is a degenerate kind of loop, the

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Topological Aspects of Low Dimensional Systems

corresponding homotopy class consists of all the loops from x back to x that can be continuously deformed into a point, and this class is the unit element of the group. The inverse of a loop is the same loop traversed in the opposite direction. By definition, a space is simply connected if every loop can be continuously deformed into a single point, or equivalently, if the fundamental group is the trivial group consisting of one element only. Similarly, it is doubly connected if the fundamental group has exactly two elements, and it is infinitely connected if the fundamental group is infinite, e.g. isomorphic to Z, the addition group of integers. The Euclidean space of any dimension is simply connected, and in particular the configuration space RdN for N distinguishable particles in d dimensions is simply connected. The path connectivity of the configuration space RdN /SN for N identical particles is a matter of definition. In the literal sense RdN /SN is simply connected, but we want to argue that a more natural definition of homotopy is such that RdN /SN is not simply connected when N ≥ 2. Note that the fundamental group is the same for RdN /SN as for the relative configuration space Rd(N −1) /SN , because the centre of mass position splits off as in equation (13). We have seen in the example with N = 2 that there exist two classes of loops in Rd /S2 with respect to the parallel transport of relative tangent vectors, transporting a vector v into +v or −v, respectively, and it is natural to define that a “+” and a “−” loop are not homotopic. If we want to deform a “+” loop continuously into a “−” loop, or vice versa, then one stage in the process must be a loop going through the singular point where the two particles collide. Such a loop is itself singular in the sense that the parallel transport of a vector is ambiguous. The natural solution is to simply exclude such singular paths, or equivalently, to exclude the singular point from the relative space, making it multiply connected. In the general N -particle case there will be at least N ! inequivalent classes of loops corresponding to the N ! possible permutations of particle labels in the local space of tangent vectors. This definition of homotopy means that we exclude all the singular points of the configuration space, i.e. all those configurations in which two or more particles are at the same position. This restriction implies that the one-dimensional case (d = 1) is uninteresting, because the relative space RN −1 /SN without its singular points is connected, but has no continuous paths that interchange particle positions. It implies further that in dimension two or higher (d ≥ 2) there is always a homomorphism from the fundamental group onto the symmetric group SN . In dimension three or higher (d ≥ 3) the homomorphism is in fact an isomorphism: the fundamental group is just the symmetric group

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SN . In two dimensions, however, the fundamental group is a non-trivial extension of the symmetric group, called the braid group. 2.4 The braid group For two particles in the plane (N = 2, d = 2), i.e. for the relative configuration space R2 /S2 , the fundamental group is Z. This is so because every loop has an integer winding number, which is the number of times it encircles the origin, and the winding number is additive under concatenation of loops. By arbitrary convention, we count anticlockwise winding as positive and clockwise winding as negative. Two loops are homotopic if and only if they have the same winding number, in other words, the winding number labels uniquely a homotopy class. A curve in R2 /S2 can also be regarded as a curve in R2 , and its winding number is even if the curve in R2 starts at x and returns to x, or odd if the curve in R2 goes from x to −x. Thus, parallel transport of a tangent vector v around a closed loop in R2 /S2 gives the vector (−1)Q v, where Q is the winding number of the loop. For N particles in the plane (d = 2), i.e. for the relative configuration space R2(N −1) /SN , the fundamental group is the braid group BN [47,48,5154]. We have seen that B2 = Z. In general, BN can be generated from N − 1 elements, in the following way. The j-th generating element Tj is the homotopy class of loops that do nothing more than interchange the particles j and j + 1 in the anticlockwise direction. It can be represented graphically as in Figure 1. Obviously, two such generators commute if they do not interfere, that is, Tj Tk = Tk Tj

if

|j − k| > 1.

(16)

Neighbouring generators do not commute, but satisfy the following relations, Tj Tj+1 Tj = Tj+1 Tj Tj+1

for

j = 1, 2, . . . , N − 2,

(17)

which can be proved graphically as in Figure 2. Note that Tj and Tj+1 are homotopy classes of loops, so that the equality sign here means homotopy of loops. It is easy to see that every one-dimensional representation of BN is given by one single number τ . In fact, if the generator Tj is represented by the number τj , then the relation τj τj+1 τj = τj+1 τj τj+1 means that τj = τj+1 = τ , independent of j. The general braid has the form b = Tjn11 Tjn22 · · · TjnKK ,

(18)

where each index jk is an integer from 1 to N − 1, and each power nk is a positive or negative integer. In the one-dimensional representation b is

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Topological Aspects of Low Dimensional Systems

j j+1 r r @ @

1 r ···

r

r j

1

@ @r j+1

N r ···

r N

Fig. 1. The braid group generator Tj , an anticlockwise continuous interchange of the particles j and j + 1. The horizontal axis represents space, R2 , the vertical axis represents “time”, i.e. the parameter of the curve.

j j+1 j+2 r r r @ @ r

@ @r @ @

r @ @ r j

r

j r

r @ @r

j+1 j+2 r r @ @

r @ @

=

@ @r r j+1 j+2

r

@ @r

r

@ @r @ @

r j

@ @r r j+1 j+2

r

Fig. 2. Graphical proof of the relation Tj Tj+1 Tj = Tj+1 Tj Tj+1 .

represented by τ Q , where Q is the winding number, defined as Q=

K X

nk .

(19)

k=1

The difference between the braid group BN and the symmetric group SN is that there is one more set of defining relations for the symmetric group, Tj−1 = Tj

for

j = 1, 2, . . . , N − 1.

(20)

This implies for the one-dimensional representations of the symmetric group that τ −1 = τ . Hence there are exactly two such representations, one with τ = 1 and one with τ = −1. In three or higher dimensions a clockwise continuous interchange of two particles is homotopic to an anticlockwise interchange. See Figure 3. Therefore equation (20) holds, so that the fundamental group of the configuration space RdN /SN in dimension d ≥ 3 is SN . 3

Schr¨ odinger quantization in one dimension

The one-dimensional case is rather special, since particles on the line can not be continuously interchanged without colliding. The mathematical

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287

'$

'$

? r

r

r

6 &%

? r

6 &%

Fig. 3. Interchange of two particles, either anticlockwise (left) or clockwise (right). In three or higher dimensions these two interchange loops are homotopic, by rotation an angle π about the line joining the particles.

expression of this fact is that the configuration space of a system of identical particles on the line has a boundary, consisting of those configurations where two or more particle positions coincide. In particular, as we have seen, the relative space of the two-particle system is a half-line, with the origin as a boundary. Therefore the quantization problem reduces to the problem of specifying the proper boundary conditions on the wave functions. The role of boundary conditions in quantum mechanics is to make certain operators Hermitean, and the most important operator is the Hamiltonian H. Hermiticity of H means that probability is conserved. Let us assume a standard two-particle Hamiltonian of the form ~2 ∂ 2 ~2 ∂ 2 − + V (x1 , x2 ) 2 2m ∂x1 2m ∂x22 ~2 ∂ 2 ~2 ∂ 2 − + V (X, x). =− 2 4m ∂X m ∂x2

H =−

(21)

Here m is the particle mass, X = (x1 +x2 )/2 the centre of mass position, and x = x1 −x2 the relative position. For identical particles the potential V must be symmetric, V (x2 , x1 ) = V (x1 , x2 ), or equivalently V (X, −x) = V (X, x), which implies that H is symmetric under interchange of particle labels, as an observable should be. For simplicity we will further assume here that V is non-singular as a function of x, or at least is no more singular than 1/x. We will discuss a 1/x2 potential below, in connection with Heisenberg quantization. More singular potentials lead to important complications. The Schr¨ odinger equation

i~

∂ψ = Hψ ∂t

(22)

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Topological Aspects of Low Dimensional Systems

for the wave function ψ = ψ(x1 , x2 , t) = ψ(X, x, t) implies the continuity equation ∂jx ∂ρ ∂jX + + = 0, ∂t ∂X ∂x

(23)

where ρ = |ψ|2 is the probability density, and ∗ ~ ∂ψ ∗ ~ ∂ψ , jx = Re ψ jX = Re ψ 4mi ∂X mi ∂x

(24)

are the X and x components of the probability current density. ψ ∗ is the complex conjugate of ψ. The physically acceptable way to impose conservation of probability is to require that the normal component of the probability current vanishes everywhere on the boundary. That is, in the two-particle case, jx (X, 0) = 0 for every X. However, this is a quadratic boundary condition for the wave function, whereas the superposition principle demands a linear condition. We therefore postulate that ! 1 ∂ψ ∂ψ ∂ψ = − ∂x 2 ∂x1 ∂x2 X=const.

x2 =const.

= ηψ

x1 =const.

at x = x1 − x2 = 0,

(25)

with η a real parameter, independent of ψ. This is a stronger condition, implying that jx = 0 at x = 0, and it is linear. η could in principle be a function of X, but that would break translation invariance. The particles are bosons if η = 0 and fermions if η = ±∞, but in principle η is a continuous variable that could take any intermediate value. Since the wave function ψ = ψ(X, x) is defined only for x ≥ 0, we are free to extend the domain of definition to x < 0, for example by imposing the bosonic symmetry ψ(X, −x) = ψ(X, x). The symmetric extension will make the partial derivative at x = 0 discontinuous if equation (25) holds with η 6= 0. The discontinuity of the partial derivative is then equivalent to a statistics interaction described by the δ-function potential Vs (x) =

2η~2 δ(x). m

(26)

As is well known, the δ-function potential has exactly one bound state if η < 0. We may use the external harmonic oscillator potential V =

1 1 mω 2 x21 + x22 = mω 2 X 2 + mω 2 x2 2 4

(27)

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as an example to illustrate how the parameter η defines a continuous interpolation between bosons and fermions. The Schr¨ odinger equation is separable, so that we need only solve the energy eigenvalue equation 2 2 1 ~ d 2 2 mω + x ψ = Eψ (28) − m dx2 4 for the relative wave function ψ = ψ(x). For a given relative energy E there exists a unique solution which is square integrable in the limit x → ∞, and it can be expressed in terms of the confluent hypergeometric function U = U (a, b, z) defined in Chapter 13 of reference [240], 1 E 1 x2 x2 − , , exp − 2 , (29) ψ(x) = c0 U 4 2~ω 2 a20 2a0 where c0 is a normalization constant, and a0 is a characteristic length, r 2~ · (30) a0 = mω The boundary condition at x = 0, equation (25), gives the following energy quantization condition, involving the Euler Γ-function [240], E 2 Γ 34 − 2~ω ψ 0 (0) = η. =− (31) E ψ(0) a0 Γ 14 − 2~ω In particular, with η = 0 we get the boson spectrum 1 ~ω, n = 0, 1, 2, . . . , E = 2n + 2 and with η = ±∞ we get the fermion spectrum 3 ~ω, n = 0, 1, 2, . . . E = 2n + 2

(32)

(33)

The level spacing is constant for bosons and fermions, but not for intermediate values of η. Figure 4 shows how the lowest energies vary with η. The obvious generalization to the N -particle case is the convention that the general wave function ψ = ψ(x1 , x2 , . . . , xN ) is defined for x1 ≥ x2 ≥ . . . ≥ xN and satisfies the boundary conditions ∂ψ ∂ψ − = 2ηψ ∂xj ∂xj+1

at xj = xj+1

(j = 1, 2, . . . , N − 1).

(34)

Lieb and Liniger have solved this particular N -particle problem in the case when η > 0 and there is no other external or interaction potential [34, 35].

290

4

Topological Aspects of Low Dimensional Systems

Heisenberg quantization in one dimension

The Schr¨ odinger quantization, as presented above, is not the only way to get intermediate statistics of particles in one dimension. In fact, the Heisenberg quantization leads just as naturally to a different type of intermediate statistics, equivalent to an inverse square statistics potential rather than a δ-function potential [152, 206]. The one-dimensional case is special in this respect. In higher dimensions only bosons and fermions emerge if we apply Heisenberg quantization in the most straightforward way. The indistinguishability of the particles implies extra freedom in the quantization for a system of two or more particles, because it restricts the class of observables. To see how, it is again convenient to discuss the twoparticle case as an example. The centre of mass position X = (x1 + x2 )/2 and the total momentum P = p1 +p2 are observables, since they are symmetric under interchange, but the relative position x = x1 − x2 and momentum p = (p1 − p2 )/2 are antisymmetric and therefore not observables. Thus, the canonical commutation relation, equation (7), between relative position and momentum is meaningless in a minimal theory which includes only such operators as represent observable quantities. If we can not use x and p as basic observables, then the next simplest choice are the quadratic polynomials x2 , p2 and xp, which are symmetric and therefore observables, at least in the classical theory. It is convenient to introduce an arbitrary length scale a0 and define the dimensionless observables A=

1 a2 1 1 a20 2 (xp + px) . p + 2 x2 , B = 02 p2 − 2 x2 , C = 2 4~ 4a0 4~ 4a0 4~

(35)

In the quantum theory they should satisfy the following commutation relations, which follow either from the Poisson brackets in the classical theory, or from the canonical commutation relation in the quantum theory, [A, B] = iC,

[A, C] = −iB,

[B, C] = −iA.

(36)

It is natural to adopt equation (36) as the basic set of commutation relations defining the quantum theory of two identical particles on the line. They define the Lie algebra sp(1, R) = sl(2, R) of the real symplectic group Sp(1, R) = SL(1, R), consisting of the area-preserving linear transformations in the plane1 . There exists a quadratic Casimir operator, Γ = A2 − B 2 − C 2 , 1 Unfortunately,

(37)

different conventions exist, and this group is often called Sp(2, R).

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commuting with all operators in the Lie algebra. It must take a constant value if we require the linear representation of the Lie algebra to be irreducible, implying that only two of the three observables A, B and C are independent. Clearly two independent variables are just what we need to describe the two-dimensional relative phase space. In the classical case, Γ = 0 identically, wheras equation (35) together with equation (7) imply that Γ = −3/16. However, if x and p do not exist as operators, then we have to give up both equation (35) and equation (7), and there is no obvious reason any more to require that either Γ = 0 or Γ = −3/16. There exists in fact a family of physically acceptable irreducible representations of sp(1, R), depending on one continuously variable parameter α0 > 0. If we denote the basis vectors of one such representation by |α0 , ni, with n = 0, 1, 2, . . . , then Γ |α0 , ni = α0 (α0 − 1) |α0 , ni, A |α0 , ni = (α0 + n) |α0 , ni, p (B + iC) |α0 , ni = (n + 1)(n + 2α0 ) |α0 , n + 1i, p (B − iC) |α0 , ni = n(n − 1 + 2α0 ) |α0 , n − 1i·

(38)

Note that if a0 is given by equation (30), then 2~ωA =

1 p2 + mω 2 x2 m 4

(39)

is just the harmonic oscillator relative Hamiltonian encountered earlier in equation (28). Thus, α0 = 1/4 corresponds to bosons and α0 = 3/4 to fermions, and the parameter α0 provides a continuous interpolation between these two special cases. When α0 changes, the whole harmonic oscillator spectrum is rigidly shifted with all level spacings constant, which proves that Schr¨ odinger and Heisenberg quantization lead to inequivalent types of intermediate statistics. Figure 4 shows the bottom part of the harmonic oscillator energy spectrum as a function of the statistics parameter, both for Schr¨ odinger and Heisenberg quantization. 4.1 The coordinate representation We may change basis from the harmonic oscillator eigenstates |α0 , ni to the eigenstates |xi of the relative position x, restricted to x ≥ 0. In this coordinate representation x2 is diagonal, whereas p2 is a differential operator containing the parameter α0 , p2 = −~2

λ~2 d2 + 2 , 2 dx x

with

1 3 λ = 4 α0 − α0 − · 4 4

(40)

292

Topological Aspects of Low Dimensional Systems

4 3 3 2 2 1 1 -1

1 -1

B

F

1

Fig. 4. E/(2~ω), where E is the energy of relative motion of two identical particles with harmonic oscillator interaction. The lowest energies are shown as functions of the statistics parameter η 0 = (2/π) arctan η (Schr¨ odinger quantization, left), or α0 (Heisenberg quantization, right). Bosons have η 0 = 0 and α0 = 1/4, fermions have η 0 = ±1 and α0 = 3/4. From [208], reprinted with permission.

x2 and p2 define the operators A and B, whereas C is given by the commutation relation, d 1 d i 2 2 + x . (41) x C = −i[A, B] = 2 [p , x ] = 8~ 4i dx dx When the above definition of p2 is inserted into the harmonic oscillator Hamiltonian, equation (39), the result is an extra inverse square statistics potential, λ~2 1 3 , with λ = 4 α − − α , (42) Vs (x) = 0 0 mx2 4 4 in the Schr¨ odinger equation, vanishing precisely in the boson case α0 = 1/4 and the fermion case α0 = 3/4. The modified eigenfunctions are of the form E x2 x2 (2α0 − 12 ) , 2α0 , 2 exp − 2 · M α0 − ψ(x) = c0 x (43) 2~ω a0 2a0 The main difference from equation (29) is that the confluent hypergeometric function M = M (a, b, z) replaces U , and the energy quantization condition

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293

is now that M reduces to a polynomial, which happens when [240] E = 2(n + α0 )~ω,

n = 0, 1, 2, . . .

(44)

The choice of eigenfunction in equation (43) is dictated by the boundary condition at x → 0+, and there is an argument behind the choice of boundary condition. The eigenvalue equation Hψ = Eψ, regarded as a second order ordinary differential equation, 2 2 λ~2 1 ~ d 2 2 + + mω x ψ = Eψ, (45) − m dx2 mx2 4 has two independent solutions behaving asymptotically as xν in the limit x → 0+, where 1 ± (2α0 − 1). 2

(46)

ψ(x) ∼ c+ xν+ + c− xν− ,

(47)

ν = ν± =

√ (The case α0 = 1/2 is special, then the asymptotic form is either x or √ x log x.) The general solution ψ = ψ(x) has the asymptotic form (for α0 6= 1/2),

for some constants c± , implying the asymptotic form of the probability current density, ~ dψ 2~(2α0 − 1) Im c∗− c+ . (48) ∼ jx = Re ψ ∗ mi dx m The condition that the wave function must be square integrable requires that c− = 0 for ν− ≤ −1/2, i.e. for α0 ≥ 1, but puts no restriction on the coefficients c± when 0 < α0 < 1. One possible linear condition which will make jx → 0 as x → 0+, is that c− = ηc+ ,

(49)

with η a real parameter. The superposition principle requires that η = c− /c+ must be the same for all wave functions. The parameter η here is of course related to the one introduced earlier, equation (25) is in fact just the special case α0 = 1/4. In the present case η can not vary continuously, however. The point is that we want all three operators A, B and C to have a common domain of definition, but C changes the asymptotic form of the eigenfunctions of A in the limit x → 0+, unless we impose one of the two conditions c+ = 0 or c− = 0. In fact, C (c+ xν+ + c− xν− ) =

1 ((2ν+ + 1)c+ xν+ + (2ν− + 1)c− xν− ) , 4i

(50)

294

Topological Aspects of Low Dimensional Systems

which means that C transforms η into ηe =

1 − α0 2ν− + 1 (2ν− + 1)c− η= = η. (2ν+ + 1)c+ 2ν+ + 1 α0

(51)

Which of the two conditions c+ = 0 or c− = 0 we impose, is only a matter of convention, since we may interchange ν+ and ν− by replacing α0 with 1 − α0 . We choose the condition c− = 0, so that wave functions have 1 the asymptotic form x(2α0 − 2 ) as x → 0+, and this convention selects the particular solution in equation (43). There is a somewhat more physical way to understand why only the values η = 0 or η = ±∞ are left invariant by the operator C. The reason is that C is the infinitesimal generator of scaling transformations, it scales x and hence η, since η has the same dimension as x(ν+ −ν− ) = x2(2α0 −1) . To see that C generates scaling of x, consider the transformed wave function ψe = (I − 2iC)ψ, where ψ is a general wave function, I is the identity operator and is an infinitesimal parameter. The functions ψe and ψ have the same shape, but ψe is expanded by the factor 1 + = 1/(1 − ) as compared to ψ, since 1 0 e ψ((1 − )x) . (52) ψ(x) = ψ(x) − xψ (x) + ψ(x) = 1 − 2 2 The Heisenberg quantization for systems of more than two identical particles is an unsolved problem. However, if the two-particle Heisenberg quantization in one dimension is regarded as a special kind of Schr¨ odinger quantization, involving an inverse square statistics potential, then it can be immediately generalized to the N -particle case [206]. The statistics potential becomes X

λ~2 , m(xj − xk )2 1≤j
Vs (x1 , x2 , . . . , xN ) = with

(53)

The general wave function ψ = ψ(x1 , x2 , . . . , xN ) is defined for x1 ≥ x2 ≥ . . . ≥ xN and satisfies the boundary conditions 1

ψ(x1 , x2 , . . . , xN ) ∼ (xj − xj+1 )(2α0 − 2 ) as xj − xj+1 → 0 + (j = 1, 2, . . . , N − 1). (54) The N -particle problem of this kind is again exactly soluble, when there is no external or interaction potential besides the statistics potential, or when there is a harmonic oscillator potential which is either external or defines a two-particle interaction [241–250].

J. Myrheim: Anyons

5

295

Schr¨ odinger quantization in dimension d ≥ 2

The geometrical interpretation of the wave function is too trivial to be meaningful in one dimension, but is non-trivial in higher dimensions, and indeed necessary when we want a theory that can describe fermions as well as bosons. Bosons are easy to describe, because the wave functions of a system of N bosons are symmetric functions on RdN and hence are singlevalued functions on the configuration space RdN /SN . The fermionic wave functions are also single-valued on RdN , but their antisymmetry implies that they are double-valued on RdN /SN . Many-valued wave functions find a natural place in the geometrical picture introduced by Weyl and Dirac around 1930 [28, 29]. The mathematical structures involved are called fibre bundles in modern terminology [26, 251–256]. To be more precise, in our case the fibre bundle is a vector bundle. It has the configuration space RdN /SN as its base space, and at every point x ∈ RdN /SN there is located a fibre, which is a finite dimensional complex Hilbert space hx . A wave function Ψ is a cross-section of the fibre bundle, that is, the function value Ψ(x) at the point x ∈ RdN /SN is a vector in the local Hilbert space hx . Let us assume that the complex vector space hx has dimension r, independent of x. Then we may choose, for every x ∈ RdN /SN , a set of basis vectors χx1 , χx2 , . . . , χxr ∈ hx , so that we may write Ψ(x) =

r X

ψk (x)χxk ,

(55)

k=1

where each component ψk (x) is a complex number. Each ψk is a complex valued function defined on the configuration space RdN /SN , it is one component of an r-component wave function   ψ1  ψ2    (56) ψ =  . .  ..  ψr Note that we use the term “wave function” here for two different, related objects. One is the cross-section Ψ of the fibre bundle, and the other is the column matrix ψ of r complex valued functions. The set of basis vectors {χxk | x ∈ RdN /SN , k = 1, 2, . . . , r}

(57)

may be called a gauge. It defines a translation between the language of fibre bundles and the language of multi-component complex valued wave functions.

296

Topological Aspects of Low Dimensional Systems

The wave function Ψ is assumed to be single-valued, in the sense that the function value Ψ(x) ∈ hx at any given point x is unique. It is a geometrical object which exists without any reference to local basis vectors, whereas the r-component complex wave function ψ is undefined until we have “chosen a gauge”, i.e. defined a set of local basis vectors. Clearly ψ is also singlevalued as long as we introduce a unique set of basis vectors at each point x. However, we may sometimes want to use simultaneously two or more different sets of local basis vectors, with the result that ψ becomes manyvalued. This is the natural way to introduce the double-valued fermion wave functions, as we shall see. 5.1 Scalar wave functions Let us examine the simplest case, when the particles have no spin or other internal degrees of freedom. Then there is only one basis vector χx ∈ hx , and equation (55) simplifies to Ψ(x) = ψ(x)χx .

(58)

A vector bundle with one-dimensional fibres is called a line bundle. It is natural to impose the normalization condition |χx | = 1. This still leaves us with the freedom to make a change of basis of the form e x = eiα(x) χx , χx 7→ χ

(59)

where α = α(x) is an x-dependent real phase (in addition it could be time dependent). Such a local change of basis is called a local gauge transformation. The complex wave function ψ must transform as follows, e ψ(x) 7→ ψ(x) = e−iα(x) ψ(x) ,

(60)

since Ψ is gauge independent, e χ . Ψ(x) = ψ(x)χx = ψ(x)e x

(61)

In order to define gauge invariant differentiation we need a connection on the fibre bundle, i.e. a rule for parallel transport between the fibres along continuous curves in the base space. The parallel transport along some curve C from a point x to a point y must define a linear and unitary operator PC (y, x): hx → hy . It is actually sufficient to define the infinitesimal parallel displacement from x to x + dx, and we postulate the following rule, (62) P (x + dx, x) χx = 1 + i dxj aj (x) χx+dx . Here we denote the local coordinates in RdN /SN by xj , j = 1, 2, . . . , dN , and we use the summation convention for repeated indices. a1 , a2 , . . . , adN

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are the components of a covariant vector field, a gauge potential, which is real in order to make P (x + dx, x) unitary. The infinitesimal rule implies the following rule for a finite curve C, Z dxj aj (x) χy . (63) PC (y, x) χx = exp i C

The gauge invariant differentiation Dj with respect to xj is defined by the relation (64) Ψ(x + dx) = P (x + dx, x) Ψ(x) + dxj [Dj Ψ](x) . By definition, both Ψ(x) and [Dj Ψ](x) are vectors in the local Hilbert space hx , whereas Ψ(x + dx) belongs to the neighbouring space hx+dx . We need the parallel displacement operator in order to compare the local vectors at two different, neighbouring points. Note that this definition is explicitly gauge independent, since it does not involve the local basis vectors. Once we have chosen a gauge, which defines ψ as the component of Ψ, it is very natural to define Dj ψ as the component of Dj Ψ, [Dj Ψ](x) = [Dj ψ](x)χx .

(65)

Comparing equation (64) with another formula,

Ψ(x + dx) = ψ(x + dx) χx+dx = ψ(x) + dxj [∂j ψ](x) χx+dx ,

(66)

in which ∂j = ∂/∂xj is the ordinary partial derivative, we see that Dj ψ = ∂j ψ − iaj ψ,

(67)

Dj = ∂j − iaj .

(68)

which we write simply as

By definition, the local gauge transformation in equation (59) transforms aj , such that aj into e e x+dx . e x = 1 + i dxj e aj (x) χ (69) P (x + dx, x) χ It follows that e aj = aj − ∂j α,

(70)

e j = ∂j − i e aj = Dj + i∂j α = e−iα(x) Dj eiα(x) . D

(71)

and hence that

This formula for the gauge transformation of the differentiation operator Dj implies that Dj ψ transforms in the same way as ψ under a gauge transformation, e j ψe = e−iα(x) Dj ψ. Dj ψ 7→ D

(72)

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Topological Aspects of Low Dimensional Systems

5.2 Homotopy The commutator fjk = i[Dj , Dk ] = ∂j ak − ∂k aj

(73)

is invariant under local gauge transformations, and measures the nontriviality, or curvature, of the connection. It is the field strength corresponding to the gauge potential aj , thus it corresponds to the magnetic flux density in electromagnetism (the electric field is included if we add time components), or to the Riemann curvature tensor in geometry. Here we want to discuss only the special case when fjk = 0 identically. One way to justify this restriction might be to say that we do not want to discuss the kind of interactions represented by a non-vanishing field strength. Another way might be to say that we want to study the ambiguity in the (1) (2) gauge potential aj for a given field strength fjk . In fact, if fjk = fjk , with (i)

(i)

(i)

fjk = ∂j ak − ∂k aj , (1)

(74)

(2)

then the difference aj = aj − aj is a gauge potential having fjk = 0. See however reference [257] for an example where the non-vanishing part of fjk is also important. The curvature, or field strength, fjk vanishes identically if and only if aj is the gradient of some function α, in other words, if and only if there exists a gauge transformation such that e aj = aj − ∂j α = 0 identically. The “if” part of this statement is trivial, that fjk = 0 when aj = ∂j α, The “only if” part is true with the important reservation that the function α is guaranteed to be single-valued only when the space is simply connected. Let us see how the relation between aj and α depends on the path connectivity. If aj = ∂j α, then α is obtained from aj by a line integral, Z dz j aj (z) , (75) α(y) = α(x) + C

where C is an arbitrary curve from x to y. This equation proves that the line integral is independent of C whenever aj is the gradient of a single-valued function α. Conversely, when aj is given, we may always try to solve the equation ∂j α = aj for α by choosing one point x, fixing α(x) arbitrarily, and using equation (75) to define α(y) for general y. If fjk = 0 identically, then the value of the line integral is unchanged by a continuous deformation of the curve C, and the function α defined by equation (75) has the desired gradient aj . The invariance of the line integral under continuous deformation means that equation (75) defines a unique value of α(y) for every homotopy class of curves from x to y. In particular, if we restrict ourselves to a simply

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connected region Ω, then by definition there exists exactly one homotopy class of curves inside Ω from x to y, and equation (75) defines a function α which is single-valued on Ω. In a doubly connected region where fjk = 0, α defined by equation (75) may become double-valued, and so on. Let us summarize our discussion so far. We assume that the wave function Ψ = Ψ(x) is single-valued, and that there exists a set of local basis vectors {χx } which is also single-valued (i.e., contains only one basis vector at each point x), such that the connection is given by equation (62). We assume further that the gauge potential aj is such that the curvature, or field strength, fjk vanishes. With a single-valued basis the complex wave function ψ = ψ(x) is also single-valued. We have shown how to introduce a local gauge transformation, as defined in equation (59), such that the transformed gauge potential vanishes, i.e. e aj = aj − ∂j α = 0. Such a gauge transformation is always single-valued in a simply connected region, otherwise it may be many-valued. For example, it may be double-valued in a doubly connected region. A many-valued gauge transformation will transform the single-valued complex wave function ψ into a many-valued complex wave function ψe = e−iα ψ. Thus, if the field strength vanishes, fjk = 0, we may choose a gauge such that the vector potential vanishes, aj = 0, but in a multiply connected space this may imply that the complex valued wave function becomes manyvalued. On the other hand, we may always work with single-valued wave function, but then, if the space is multiply connected, we may have to live with a vector potential which is not zero. 5.3 Interchange phases If we put y = x in equation (63), then the basis vector χx is the same on the two sides of the equality sign, so that the parallel transport operator becomes just a gauge independent phase factor, I j dx aj (x) . (76) PC (x, x) = exp i C

Obviously, the parallel transport around first one loop C1 and then another loop C2 gives a phase factor which is the product of the two individual phase factors, PC2 C1 (x, x) = PC2 (x, x)PC1 (x, x).

(77)

Our previous assumption that fjk = 0 implies that the line integral in equation (76), and therefore the whole phase factor, is invariant under any continuous deformation of the loop C. We may deform the loop continuously without moving the point x where it starts and ends, and such deformations produce loops in the same homotopy class as the original loop C. Thus, all

300

Topological Aspects of Low Dimensional Systems

loops belonging to the same homotopy class have the same phase factor. However, the phase factor PC (x, x) is unchanged even if we deform the loop C continuously in such a way that the point x moves. In summary, we have derived the following important result. – A connection with zero curvature on a complex line bundle is uniquely characterized by a one-dimensional unitary representation of the fundamental group. We have seen that the fundamental group for the configuration space RdN /SN of N identical particles in dimension d is SN if d ≥ 3. The symmetric group SN with N > 1 has exactly two one-dimensional representations: the completely symmetric representation defining bosons, and the completely antisymmetric representation defining fermions. We have also seen that the fundamental group in two dimensions is the braid group BN . Since SN is a homomorphic image of BN , any representation of SN defines a representation of BN , but BN has more general one-dimensional representations in addition to the symmetric and antisymmetric representations. In particular, the braid group B2 for two particles is isomorphic to Z, and its general representation by phase factors is characterized by one real number, a phase angle θ, such that Q 7→ e−iQθ ,

(78)

where Q is the winding number of the given homotopy class of loops. Obviously, this relation defines θ only up to an arbitrary multiple of 2π, since Q is an integer. Two-dimensional identical particles characterized by a general statistics angle θ are called anyons. Special cases are bosons, with θ = 0, and fermions, with θ = π. We see that for two anyons there is a phase factor e−iθ associated with a loop of winding number Q = 1. Let us introduce Cartesian coordinates x, y and polar coordinates r, φ such that the relative position of the two particles is x = x1 − x2 = (x, y) = (r cos φ, r sin φ).

(79)

The relative angle φ increases by π when we go through a loop of winding number one, and we symbolize this by saying that we go from the point (r, φ) to (r, φ + π). The parallel transport around this loop takes the basis vector χr,φ at (r, φ) into χr,φ+π = e−iθ χr,φ .

(80)

This corresponds to the following condition on the complex wave function ψ, in polar coordinates, ψ(r, φ + π) = eiθ ψ(r, φ).

(81)

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301

It is important to read this formula correctly. The implicit convention when we write φ + π, is that the angle φ is increased continuously by π. Note also that the phase factor eiθ appears when we use parallel basis vectors, so that the gauge potential aj vanishes. In every case, except for bosons, local basis vectors that are parallel, have to be many-valued on the relative configuration space R2 /S2 . For example, in the fermionic case e−iθ = −1, parallel basis vectors are double-valued. 5.4 The statistics vector potential As we have seen, it is not necessary to use parallel many-valued basis vectors and the corresponding many-valued complex wave functions. One alternative is to use single-valued basis vectors, with a non-vanishing gauge potential. The most general possibility, however, is to use many-valued basis vectors and at the same time a non-vanishing gauge potential. Thus, let ψ denote the many-valued wave function relative to a parallel basis, satisfying equation (81), and let the wave function ψν be defined by ψν (r, φ) = e−iνφ ψ(r, φ),

(82)

where ν is some arbitrary constant. The new wave function satisfies the following symmetry condition, ψν (r, φ + π) = ei(θ−νπ) ψν (r, φ).

(83)

The gauge invariant derivative is trivial in the “parallel” gauge, Dr = ∂r , Dφ = ∂φ , but is non-trivial in the “ν” gauge, Dr = ∂r ,

Dφ = ∂φ + iν.

(84)

The corresponding formulae in Cartesian coordinates are, sin φ y Dφ = ∂x − iν 2 , r r cos φ x Dφ = ∂y + iν 2 · Dy = sin φ Dr + r r

Dx = cos φ Dr −

(85)

Note that the gauge potential in the general gauge, ax = ν

y , r2

ay = −ν

x , r2

(86)

is singular at r = 0 when ν 6= 0. The reason is that the gauge transformation in equation (82) is singular, in the sense that the factor e−iνφ is discontinuous at r = 0. We see that the definition in equation (82), with the special choice ν = νB ≡

θ , π

(87)

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Topological Aspects of Low Dimensional Systems

gives a wave function ψν = ψB which is single-valued, since it has the bosonic symmetry ψB (r, φ + π) = ψB (r, φ). We may call this the bosonic gauge. fermionic gauge, with

(88)

The next simplest choice is the

θ−π , (89) π in which the wave function ψν = ψF is double-valued, having the fermionic symmetry ν = νF ≡ νB − 1 =

ψF (r, φ + π) = −ψF (r, φ).

(90)

Remember that ax and ay given here are only the relative components of the gauge potential, i.e. the components in the relative space R2 /S2 . If we transform from the centre of mass and relative coordinates x1 + x2 , x = x1 − x2 , X= 2 y1 + y2 , y = y1 − y2 , (91) Y = 2 back to the particle coordinates x1 , y1 , x2 , y2 , the gauge potential must transform in the same way as the partial derivatives ∂ 1 ∂ ∂ + , = ∂x1 2 ∂X ∂x ∂ 1 ∂ ∂ + , = ∂y1 2 ∂Y ∂y

∂ ∂ 1 ∂ − , = ∂x2 2 ∂X ∂x ∂ ∂ 1 ∂ − · = ∂y2 2 ∂Y ∂y

(92)

By assumption, the centre of mass components of the gauge potential vanish. Hence, the gauge potential expressed in particle coordinates is, with r2 = (x1 − x2 )2 + (y1 − y2 )2 , y1 − y2 y2 − y1 , a2x = −ax = ν , r2 r2 x1 − x2 x2 − x1 , a2y = −ay = −ν · (93) a1y = ay = −ν 2 r r2 A natural way to interpret equation (93) is that a particle at the position x ∈ R2 experiences a certain vector potential A = A(x). It does not experience its own field, only the one generated by the other particle. Thus, particle 1 at x1 = (x1 , y1 ) generates a vector potential at x = (x, y) with components y − y1 , Ax (x) = ν (x − x1 )2 + (y − y1 )2 x − x1 , (94) Ay (x) = −ν (x − x1 )2 + (y − y1 )2 a1x = ax =

ν

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and particle 2 at x2 experiences the vector potential a2 = A(x2 ). To the vector potential A corresponds the field strength, or flux density, B = ∂x Ay − ∂y Ax .

(95)

Green’s theorem gives the relation between B and A in integral form, I Z −2πν if x1 ∈ Ω, d2x B = dx · A = (96) 0 otherwise. Ω ∂Ω where Ω is a region with boundary ∂Ω, and where the direction of the line integral is anti-clockwise. This shows that the flux is located exactly at the position of particle 1, B(x) = −2πν δ(x − x1 ).

(97)

5.5 The N -particle case The generalization to N particles is quite straightforward. As always, we use a notation identical to or similar to the one introduced in equation (10) and equation (11). A closed loop in the configuration space RdN /SN , or in the relative space d(N −1) /SN , induces a permutation p ∈ SN of the N identical particles, R and is characterized by a winding number Q. If we work in the many-valued parallel gauge, where the gauge invariant differentiation is trivial, Dj = ∂j , then an interchange path of winding number Q is accompanied by a phase factor eiQθ in the wave function. If we work instead in the single-valued bosonic gauge, in the doublevalued fermionic gauge, or more generally in some many-valued “ν” gauge with a non-vanishing gauge potential, then the interchange phase factor in the wave function is eiQ(θ−νπ) . In addition there is a gauge potential which has the following components, as we can see by generalizing equation (93), ajx =

ν

X k6=j

ajy = −ν

X

k6=j

yj − yk , (xj − xk )2 + (yj − yk )2 xj − xk · (xj − xk )2 + (yj − yk )2

(98)

It is worth noting that the components as defined in equation (98) are manyvalued, but they define a single-valued vector field on RdN /SN . Like in the two-particle case, the special choice ν = νB = θ/π gives symmetric (i.e. bosonic) wave functions, whereas ν = νF = νB − 1 gives antisymmetric (fermionic) wave functions.

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Topological Aspects of Low Dimensional Systems

The N -particle Hamiltonian is 2 1 X 1 (p − ~a)2 + V = pj − ~aj + V. 2m 2m j=1 N

H=

(99)

In our notation particle j has the position xj = (xj , yj ) and the canonical momentum ∂ ~ ∂ ~ ∂ = , · (100) pj = i ∂xj i ∂xj ∂yj The statistics vector potential aj = (ajx , ajy ) is given by equation (98). The potential V = V (x) = V (x1 , x2 , . . . , xN ) may be the sum V = VE + VI of an external, one-particle part VE =

N X

V1 (xj ),

(101)

j=1

and an internal or interaction, two-particle part X V2 (xj , xk ), VI =

(102)

j
where V2 is a symmetric function, V2 (y, x) = V2 (x, y). There is usually no need to include more complicated interactions involving three or more particles, although it could of course easily be done. P Note however that the square of the statistics vector potential, a2 = j a2j , contains three-body terms. Note that the N -anyon problem for N ≥ 2 actually contains another continuously variable parameter, in addition to the phase angle θ [258–261]. The reason is that the eigenfunctions of the above Hamiltonian depend on the boundary conditions in the limits |xj − xk | → 0 for 1 ≤ j < k ≤ N . Usually the condition of quadratic integrability forbids singular solutions and thereby fixes uniquely the asymptotic form of the wave function for small distances, but there is some exception to this rule. 5.6 Chern–Simons theory An equivalent point of view in the N -particle case is that particle j, at xj ∈ R2 , experiences a vector potential aj = A(xj ) which is generated by all the other particles, X y − yk , Ax (x) = ν (x − xk )2 + (y − yk )2 k6=j X x − xk · (103) Ay (x) = −ν (x − xk )2 + (y − yk )2 k6=j

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305

It is easily seen by comparison with the equations (94, 95) and (97) that the statistics gauge potential A defined here satisfies the field equation ∂x Ay − ∂y Ax = −2πνρ, where ρ = ρ(x) is the particle density, X δ(x − xk ). ρ(x) =

(104)

(105)

k6=j

In a field theory we want to modify the source term ρ so that the particle j itself is included, even though this leads to the usual problems with selfinteractions. The field equation can be derived from a Lagrangian if we allow the field components Ax and Ay to become explicitly time dependent, and add a third field component A0 . It is convenient to introduce the relativistic notation xµ = (x0 , x1 , x2 ) = (ct, x, y), Aµ = (A0 , A) = (A0 , Ax , Ay ),

∂ , ∂xµ Aµ = (A0 , −A), ∂µ =

j µ = (cρ, j). (106)

c is a constant velocity (the speed of light). Then we may add two field equations to obtain the following relativistic form, where is antisymmetric and 012 = 1, µκλ ∂κ Aλ =

2πν µ j . c

(107)

A necessary consistency condition is that the current j µ is conserved, ∂µ j µ = 0. These equations are gauge invariant, so that if Aµ is a solution, then the gauge transformed field eµ = Aµ − ∂µ α A

(108)

e0 = 0, is also a solution. There always exists a “radiation gauge” with A therefore the introduction of A0 does not change the physics. The field equations for Aµ follow from the Lagrangian density L=

~c µκλ Aµ ∂κ Aλ − ~j µ Aµ . 4πν

(109)

The first term is the Chern–Simons term, which exists only when space-time is three-dimensional. L is gauge invariant when the current j µ is conserved, in the sense that it changes by a divergence under the gauge transformation of equation (108), ~c αµκλ ∂κ Aλ − ~αj µ . (110) Le = L − ∂µ 4πν

306

Topological Aspects of Low Dimensional Systems

It is well known that the 2 + 1-dimensional Maxwell theory with an additional Chern–Simons term describes massive photons. If an ordinary Maxwell term is added to the Lagrangian density L above, then the modified theory describes anyons of finite size [262, 263]. When the matter fields that define the current j µ are quantized, either as bosons or fermions, relativistically or non-relativistically, the result is a field theory of anyons [67–69]. We will not discuss these theories any further here. 6

The Feynman path integral for anyons

Laidlaw and DeWitt worked out the Feynman path integral treatment of systems of identical particles [1,49,66], inspired by earlier work of Schulman on the path integral for configuration spaces that are not simply connected [264, 265]. We derive it here from the Schr¨odinger formalism, to show that the two are equivalent. Closely related is the work by Wiegel and by and Inomata and Singh on the entanglement of polymers, and by Gerry and Singh on the path integral treatment of the Aharonov–Bohm effect [266– 269]. See also [211,237] for the path integral treatment of exclusion statistics and parastatistics. What we will derive is the path integral formula for the partition function of the N -particle system, ZN (β) = Tr e

bN −β H

=

∞ X

e−βEN k .

(111)

k=0

We have had little need so far to distinguish explicitly between operators and numbers, but in the present section we will use a “hat” to denote operators, in order to prevent unnecessary confusion. We assume that the particles are confined by an external potential in b N has a discrete eigenvalue such a way that the Hamiltonian operator H spectrum, and the trace is well-defined. At least in principle, ZN as a function of β determines the energy eigenvalues EN 0 ≤ EN 1 ≤ . . . ≤ EN k ≤ . . . . In statistical mechanics β = 1/kB T , where T is the temperature and kB is Boltzmann’s constant. More formally, β may be thought of as imaginary time, since the time evolution operator for a time interval t = b =H bN , −i~β is, with H b

b

e− ~ tH = e−β H . i

(112)

The simplest derivation starts from the “bosonic” description, in which the N -particle wave functions are symmetric complex valued functions on RdN . The Hamiltonian operator is given by equation (99), and any deviation from Bose–Einstein statistics is described by a vector potential a on RdN , as

J. Myrheim: Anyons

307

given in equation (98), with ν = νB = θ/π. The “fermionic” description, where the wave functions are antisymmetric, leads to the same end result, but the derivation is complicated by an extra minus sign associated with every odd permutation of the N particles. Therefore we prefer the bosonic description. 6.1 Eigenstates for position and momentum For distinguishable particles the general position eigenstate is |xi, where x ∈ RdN . We have the orthonormality relation hx|yi = δ(x − y),

(113)

and the completeness relation, also called the resolution of the identity operator, Z ddNx |xihx| . (114) Ib = RdN The bosonic position eigenstates are symmetric under all permutations of particle labels, and with the proper normalization they are 1 X |p(x)i · (115) |xiB = √ N ! p∈SN This definition implies that X 1 X δ(p(x) − q(y)) = δ(r(x) − y), B hx|yiB = N! p,q∈SN

(116)

r∈SN

where r = q −1 p. Note that the permutations p and q here act on RdN as linear operators of determinant ±1, and therefore we may change variables in the Dirac δ-function without introducing an extra Jacobi determinant. It follows that if we restrict both x and y to lie in the true configuration space RdN /SN , then we have the standard orthonormality relation also for the bosonic position eigenstates, B hx|yiB

= δ(x − y).

The identity operator in the space of bosonic states is Z Z 1 dN d x |xi hx| = ddNx |xiB B hx| . IbB = B B N ! RdN RdN/SN

(117)

(118)

When regarded as an operator on the full Hilbert space that includes states of all symmetry classes, IbB is the projection operator onto the subspace of symmetric states. The above definition of |xiB gives that X Z 1 ddNx |p(x)ihq(x)| . (119) IbB = dN (N !)2 R p,q∈SN

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Topological Aspects of Low Dimensional Systems

If we define y = p(x), z = q(x), r = qp−1 and s = pq −1 , then we get two more forms of the above resolution of the bosonic identity operator, Z Z 1 X 1 X dN b d y |yihr(y)| = ddNz |s(z)ihz| . (120) IB = dN dN N! N! R R r∈SN s∈SN The momentum eigenstates for distinguishable particles may be defined as dN Z 2 i 1 ddNx e ~ p·x |xi · (121) |pi = dN 2π~ R They satisfy similar orthonormality and completeness relations as the position eigenstates, Z b ddNp |pihp| . (122) hp|qi = δ(p − q), I= dN R 6.2 The path integral The trace in the definition of the partition function, equation (111), is the trace within the subspace of bosonic states, which we may obtain by inserting the projection operator IbB . Thus [66], Z 1 X b b ddNx hp(x)|e−β H |xi , (123) ZN (β) = Tr e−β H IbB = dN N! p∈SN R b where we have used equation (120) and the general formula Tr(A|ψihχ|) = b hχ|A|ψi. As remarked earlier, the formula would be the same in the fermionic picture, except for a minus sign for every odd permutation, compensated for by the use of ν = νF = νB −1 instead of ν = νB in the statistics vector potential. We now expand as follows, b

b

b

b

e−β H IbB = e− n H Ib e− n H Ib · · · Ib e− n H IbB . β

β

β

(124)

It is simpler to use the identity operator Ib of the full Hilbert space here instead of the identity operator IbB of the bosonic subspace, which would seem more logical. Actually the above formula is equally valid if we replace b and hence the exponential opevery Ib by IbB , because the Hamiltonian H erators in the formula are symmetric under interchange of particle labels, so that they commute with IbB . It is essential to include IbB once, but once is enough. Using the resolution of the identity, equation (114), we get the following formula, Z β b 1 X ddNx ddNu · · · ddNz ddNy hp(x)|e− n H |ui ZN (β) = dN n N! p∈SN R β

b

β

b

β

b

hu|e− n H · · · |zihz|e− n H |yihy|e− n H |xi ·

(125)

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309

The only manifestation of the fact that the particles are identical, is that we integrate over all paths x → y → z → . . . → u → p(x) starting at x and ending at p(x), and we average over all permutations p. These are precisely the closed paths in the configuration space RdN /SN . The desired path integral formula for the partition function is obtained in the limit n → ∞, when a suitable approximation is used for the propagator b

G(y, x; τ ) = hy|e− ~ H |xi · τ

(126)

Here the real variable τ = ~β/n corresponds to the imaginary time interval t = −iτ . The approximation must be sufficiently accurate in the limit τ → 0. By complex conjugation of the definition of G follows the Hermitean symmetry G(y, x; τ )∗ = G(x, y; τ ).

(127)

An approximation which respects this symmetry is the following, τ τ~ 2 τ τ b b exp b·p b exp − b2 a a p G(y, x; τ ) ≈ hy| exp − V − 2~ 4m 2m 2~m τ τ τ~ 2 b·a b exp − Vb − b |xi p a (128) × exp 2m 2~ 4m τ = exp − (V (y) + V (x)) 2~ τ~ 2 2 |a(y)| + |a(x)| − G1 (y, x; τ ), 4m where τ τ τ b·p b exp − b2 Ib exp b·a b |xi· (129) a p p G1 (y, x; τ ) = hy| exp 2m 2~m 2m Here we have inserted the identity operator between two of the exponentials, and when we use equation (122), we get that Z τ dN 2 p d p exp − G2 (p, y; τ )∗ G2 (p, x; τ ), (130) G1 (y, x; τ ) = dN 2~m R with τ b·a b |xi · p G2 (p, x; τ ) = hp| exp 2m

(131)

Without further justification we now introduce the following approximation for G2 , which is formally just a first order approximation for the exponential

310

Topological Aspects of Low Dimensional Systems

function,

τ τ b ·b p a |xi = 1 + p·a(x) hp|xi G2 (p, x; τ ) ≈ hp| Ib + 2m 2m τ i p · a(x) − p · x . (132) ≈ exp 2m ~

It leads to the following approximation for G1 , dN 2π~m 2 τ~ m (a(y) + a(x))2 − (y − x)2 exp G1 (y, x; τ ) ≈ τ 8m 2τ ~ i + (a(y) + a(x)) · (y − x) , 2 (133) where the normalization factor comes from the Gaussian integral 2 ! Z im τ ~ dN (y − x) d p exp − p − (a(y) + a(x)) − 2~m 2 τ RdN dN 2π~m 2 . (134) = τ And finally the desired approximation for G, dN m τ 2π~m 2 2 (y − x) − (V (y) + V (x)) exp − G(y, x; τ ) ≈ τ 2τ ~ 2~ i (135) + (a(y) + a(x)) · (y − x) . 2 We have neglected the following terms in the exponent, τ~ τ~ |a(y)|2 + |a(x)|2 + (a(y) + a(x))2 4m 8m τ~ 2 (a(y) − a(x)) , (136) =− 8m because the Gaussian weight√factor exp −m(y − x)2 /2τ ~ makes the distance |y − x| small, of order τ . In the limit n → ∞ the discrete path x → y → z → . . . → u → p(x) approaches a continuous path x(τ ), with 0 ≤ τ ≤ ~β, while the product of propagators diverges as follows, ~β ~β ~β · · · G z, y; G y, x; G p(x), u; n n n dN n 2πmn 2 − S −iθQ e ~ . (137) → β −

J. Myrheim: Anyons

S is the action in imaginary time, which we define here as ! 2 Z ~β m dx dτ + V (x) , S= 2 dτ 0

311

(138)

and Q is the winding number of the N -particle path, defined by the line integral of the statistics vector potential along the path x(τ ), Z dx · a(x) =

N Z X

dxj · aj (x) = −πνQ = −θQ.

(139)

j=1

We define the path integral measure as D(x(τ )) = lim

n→∞

2πmn β

dN2 n

ddNx ddNu · · · ddNz ddNy ,

(140)

including the divergent normalization factor from the product of propagators. With these definitions, Z 1 X S D(x(τ )) e− ~ −iθQ . (141) ZN (β) = N! C(p) p∈SN

The domain of integration C(p) for a given permutation p ∈ SN consists of all continuous paths x(τ ) with 0 ≤ τ ≤ ~β and with x(~β) = p(x(0)). Let C(p, Q) consist of those paths in C(p) that have the winding number Q, and define R S D(x(τ )) e− ~ C(p,Q) · (142) Pp (Q, β) = R S D(x(τ )) e− ~ C(p) Thus, Pp (Q, β) is the probability of the winding number Q, given the permutation p, and given that the particles are bosons. Define also the Fourier transform of the probability distribution of winding numbers, which is called the probability generating function, Fp (θ, β) =

∞ X

Pp (Q, β) e−iθQ .

(143)

Q=−∞

Below we will usually write simply Pp (Q) and Fp (θ) instead of Pp (Q, β) and Fp (θ, β), but one should remember that these quantities are temperature dependent. These definitions enable us to isolate the dependence of the partition function ZN on the statistics angle θ, Z S 1 X Fp (θ) D(x(τ )) e− ~ . (144) ZN (β) = N! C(p) p∈SN

312

Topological Aspects of Low Dimensional Systems

The fact that the probability Pp (Q) is real implies that Fp (θ)∗ = Fp (−θ). Furthermore, in the definition of Pp (Q), both the action S and the path integral measure D(x(τ )) are invariant under the time reversal transformation e (τ ) = x(~β − τ ). x(τ ) 7→ x

(145)

If the path x(τ ) has the winding number Q, then the time reversed path e (τ ) has the winding number −Q. Hence time reversal invariance implies x that Pp (−Q) = Pp (Q), which in turn implies that the probability generating function is real, Fp (θ)∗ = Fp (−θ) = Fp (θ).

(146)

The fact that Fp (θ) is real, is consistent with our path integral formula for the partition function ZN , equation (144), and with the fact that ZN is real by definition. 6.3 Conjugation classes in SN By definition, a cycle of length L is a cyclic permutation i1 7→ i2 7→ . . . 7→ iL 7→ i1 of the L integers i1 , i2 , . . . , iL , and it is denoted by (i1 i2 . . . iL ). It follows directly from the definition that, for example, (i1 i2 . . . iL ) = (i2 . . . iL i1 ),

(i1 i2 . . . iL )−1 = (iL . . . i2 i1 ).

(147)

A transposition is a cycle (i1 i2 ) of length 2, and the generators of SN introduced in Subsection 2.4 are the transpositions Tj = (j, j + 1). A cycle of length L may be written as a product of transpositions, e.g. in the following way, (i1 i2 . . . iL ) = (i1 i2 )(i2 i3 ) · · · (iL−1 iL ).

(148)

Two cycles commute if they are disjoint. Every permutation p ∈ SN can be factored into a product of disjoint cycles, and the factorization is unique apart from the order of the factors. Let νL be the number of cycles of length L in the factorization of p. Then the sequence of non-negative integers P , . . . , νL , . . . is called a partition of N , because L LνL = N . Let ν1 , ν2P ν = L νL be the number of cycles in the factorization of p, then the sign of p is sgn(p) =

Y p(j) − p(k) Y = (−1)(L−1)νL = (−1)N −ν . j−k

j
(149)

L

Two permutations p, q ∈ SN are conjugate if q = rpr−1 for some r ∈ SN . The conjugation class of p consists of all permutations that are conjugate

J. Myrheim: Anyons

313

to p. The mapping p 7→ rpr−1 preserves the group structure of SN , i.e., it is an automorphism of SN . The conjugate of a cycle is r (i1 i2 . . . iL ) r−1 = (r(i1 ), r(i2 ), . . . , r(iL )).

(150)

It follows that two cycles are conjugate if and only if they have the same length. And in general, two elements of SN are conjugate if and only if they have the same cycle structure, in the sense that they define the same partition of N when factorized into disjoint cycles. Thus the conjugation of N . Let classes in SN are in one to one correspondence with the partitionsP P be the conjugation class in SN corresponding to the partition L LνL = N . Then the number of elements in P is N! · ν (ν L! L L ) L

NP = Q

(151)

Let us now go back to equation (123). The integral there depends on the permutation p, but it is the same for any permutation q = rpr−1 conjugate to p. That is, Z Z b b dN −β H d y hq(y)|e |yi = ddNx hp(x)|e−β H |xi · (152) dN dN R R b

To prove this equality, note that the operator e−β H is permutation invariant, which implies that b

b

hq(y)|e−β H |yi = hr−1 (q(y))|e−β H |r−1 (y)i b

= hp(r−1 (y))|e−β H |r−1 (y)i ·

(153)

The substitution x = r−1 (y) completes the proof. Therefore we need not sum over all permutations p ∈ SN in equation (123), it is enough to pick one arbitrary permutation p from each conjugation class P in SN . The sum reduces to a sum over all conjugation classes, or equivalently, a sum over all partitions of N , Z X 1 b Q ddNx hp(x)|e−β H |xi · (154) ZN (β) = νL ) dN (ν ! L L R L P The path integral formula, equation (144), is modified accordingly, Z X S 1 Q F (θ) D(x(τ )) e− ~ . ZN (β) = νL ) P (ν ! L L C(P) L

(155)

P

The class C(P) of paths consists of all continuous paths inducing one arbitrary, but fixed, permutation p ∈ P.

314

~β 0

Topological Aspects of Low Dimensional Systems

τ 6

r

r

r

r r @ @ r @r

r

xj r-

rH r r HH HH xj r r Hr -

r j=1

r 2

3

1

1

3

2

3

2

xj r-

Fig. 5. Schematic representations of the three classes of closed three-anyon paths x(τ ). The classes 1 + 1 + 1 (left) and 3 (middle) are even, whereas 2 + 1 (right) is odd.

The probability generating function FP (θ) depends implicitly both on β and on the potential V . The values at the boson point θ = 0 and at the fermion point θ = π are known, X PP (Q) = 1, FP (0) = Q

FP (π) =

X

PP (Q) (−1)Q = sgn(P) = (−1)N −ν .

(156)

Q

The fermion value follows because the winding number Q is always even for all even permutations and always odd for all odd permutations. More generally, it follows that FP (θ ± π) = sgn(P) FP (θ).

(157)

Take N = 3 as an example. The 3! = 6 permutations in S3 fall into three conjugation classes, illustrated in Figure 5. The two classes, or partitions, 1 + 1 + 1 and 3 are even, and the class 2 + 1 is odd. 6.4 The non-interacting case The N -boson path integral occurring in equation (155) can be simplified if there is no interaction potential so that the particles are only influenced by an external potential, V (x) =

N X

V1 (xj ).

(158)

j=1

Then the factorization of the permutation p into disjoint cycles implies a similar factorization of the path integral, so that νL X Y 1 1 Z S FP (θ) D(x(τ )) e− ~ , (159) ZN (β) = νL ! L CL P

L

J. Myrheim: Anyons

τ 6

rH r r HH HH r r Hr

3~β 2~β ~β 0

315

rH r r HH H xj r r HHr -

r

r

r

r

r

x1 r -

Fig. 6. A three-particle path x(τ ) inducing a cyclic permutation, and the same path represented as a closed one-particle path over three times the time interval.

where CL is a class of L-particle paths which induce a fixed cyclic permutation of the L particles. But an L-particle path inducing a cyclic permutation is equivalent to a one-particle path over a τ -interval which is L times as long, as illustrated for a three-cycle in Figure 6. Therefore the cyclic L-particle path integral is related to the one-particle partition function, and it follows that X Y 1 Z1 (Lβ) νL FP (θ, β) · (160) ZN (β) = νL ! L P

L

6.5 Duality of Feynman and Schr¨ odinger quantization It is worthwhile observing that the expansion of the partition function given in equation (160) is actually valid much more generally, if we interpret it in a suitable way. The present discussion is partly based on reference [270]. In equation (160) we considered noninteracting anyons in two dimensions, treating them as bosons with no other interactions than a statistics interaction turning them into anyons. But we may consider more generally bosons or fermions in any dimension, or even particles described by wave functions of a more general symmetry class Y , with quite general external potentials and interactions between the particles. We may always write X Y 1 Z1 (Lβ) νL Y (β) = FPY (β) , (161) ZN νL ! L P

L

interpreting the coefficient FPY (β) as a general correction factor describing the effect of all external and internal interactions, including statistics interactions that might transmute the particle identity. For example, bosons

316

Topological Aspects of Low Dimensional Systems

(Y = B) might turn into anyons, as in equation (160), and in particular bosons might turn into fermions. Or fermions (Y = F ) might turn into anyons, and in this connection bosons count as anyons as well. In the general expansion, equation (161), the superscript Y identifies the symmetry class of the wave functions used in computing the N -particle Y (β), and we sum over conjugation classes in the sympartition function ZN metric group SN , because the conjugation classes classify the paths going into the Feynman path integral. Thus the formula relates two approaches that are dual, in a certain sense: either solving the Schr¨ odinger equation for identical particles and classifying the solutions according to the symmetry of the wave functions, or expanding the Feynman path integral as a sum over permutations. A symmetry class is the same as an irreducible representation of the symmetric group SN , which is identified by its Young tableau Y . Consider the Hilbert space H of all wave functions, of arbitrary symmetry. Every permutation p ∈ SN acts as a linear operator pb on H, and so does every conjugation class P ⊂ SN , if we define X b= P pb . (162) p∈P

The projection operator projecting out the subspace HY of the symmetry class Y , is d(Y ) X χ(P; Y )Pb , (163) IbY = N! P

where d(Y ) is the dimension of the irreducible representation Y , and χ(P; Y ) is the character in the representation Y of the permutations belonging to the conjugation class P. The inverse relation is X χ(P; Y ) b = NP IbY . (164) P d(Y ) Y

NP is the number of elements in the conjugation class P, as given in equation (151). Y (β) is the trace of the operator By definition, the partition function ZN b −β H restricted to the subspace HY , e d(Y ) X b bb Y . (165) (β) = Tr e−β H IbY = χ(P; Y) Tr e−β H P ZN N! P

The last equality follows from equation (163). Comparing with equation (161), we see that we may define " νL # Y L d(Y ) bb Y χ(P; Y ) Tr e−β H P . (166) νL ! FP (β) = Z1 (Lβ) N! L

J. Myrheim: Anyons

On the other hand, equation (164) with equation (151) gives that X χ(P; Y ) N! b Y ZN (β) . Tr e−β H Pb = Q νL d(Y ) L (νL ! L )

317

(167)

Y

0

Y (β) Thus we see that if we are able to calculate the partition function ZN 0 odinger equafor every symmetry class Y , for example by solving the Schr¨ tion for wave functions of arbitrary symmetry, then we may calculate every coefficient FPY from the formula ! Y X χ(P; Y 0 ) 1 Y Y0 ZN (β) . (168) FP (β) = d(Y )χ(P; Y ) ν 0) (Z1 (Lβ)) L d(Y 0 L

Y

Note that FPY depends on the symmetry class Y only through the representation dimension d(Y ) and the character χ(P; Y ). In particular, equation (160) refers to the case Y = B, for which d(Y ) = d(B) = 1 and χ(P; Y ) = χ(P; B) = 1. 7

The harmonic oscillator

We can gain some general insight into the properties of anyons, and even learn something about the gas of free anyons, by doing the harmonic oscillator problem. For two anyons it can be solved analytically, and all the energy eigenvalues depend linearly on the statistics angle θ, with a slope of fixed absolute value, sometimes changing sign at the bosonic values θ = 2nπ for integer n. For three anyons, however, only about one third of the energy eigenvalues have the simple linear θ dependence and are known analytically. 7.1 The two-dimensional harmonic oscillator Let us treat the one-particle problem in some detail, in order to have some notation and results for later use. The Hamiltonian for one particle of mass m in a harmonic oscillator potential is H=

1 p2 + mω 2 x2 . 2m 2

(169)

x is the postition, p the canonical momentum, and ω the angular frequency, which characterizes the strength of the potential. H is rotationally symmetric and commutes with the canonical angular momentum L = xpy − ypx . In order to quantize H and L simultaneously, it is convenient to introduce the characteristic length r ~ , (170) λ= mω

318

Topological Aspects of Low Dimensional Systems

the dimensionless complex coordinate z=

x + iy , λ

and the differential operators λ ∂ ∂ ∂ = −i , ∂= ∂z 2 ∂x ∂y

∂∗ =

(171)

∂ λ = ∂z ∗ 2

∂ ∂ +i ∂x ∂y

,

(172)

such that ∂z = ∂ ∗ z ∗ = 1, ∂ ∗ z = ∂z ∗ = 0. In this complex formalism z and its complex conjugate z ∗ are treated as independent variables. We have that L = ~ (z∂ − z ∗ ∂ ∗ ) , |z|2 ∗ · H = ~ω −2 ∂∂ + 2

(173)

The following annihilation and creation operators, z∗ , 2 z b = ∂∗ + , 2

a = ∂ +

z , 2 ∗ z b† = −∂ + , 2

a† = −∂ ∗ +

(174)

satisfy the canonical commutation relations [a, a† ] = [b, b† ] = 1 ,

[a, b] = [a, b† ] = [a† , b] = [a† , b† ] = 0 ,

(175)

and allow us to write L = ~ a† a − b † b ,

H = ~ω a† a + b† b + 1 .

(176)

The non-normalized wave function

|z|2 ψ0 = exp − 2

(177)

is the unique solution of the equations aψ0 = bψ0 = 0, and describes the ground state of the Hamiltonian H. A complete orthonormal set of simultaneous eigenfunctions of L and H are 1 (a† )j (b† )k ψ0 , ψj,k = √ πj!k!

(178)

with j, k = 0, 1, 2, . . . . The state ψj,k has angular momentum `~ = (j − k)~ and energy E = (j + k + 1)~ω.

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319

These one-particle energy levels give the following partition function, with ξ = β~ω, Z1 (β) =

X

e−βEj,k =

j,k

e−ξ 1 · = (1 − e−ξ )2 4 sinh2 2ξ

(179)

The repeated action of the creation operators a† and b† on ψ0 produces wave functions that are polynomials in z and z ∗ , multiplied by ψ0 . It is natural to split off the Gaussian factor ψ0 explicitly and write a general wave function ψ as |z|2 e e · (180) ψ = ψψ0 = ψ exp − 2 e defined such that In consequence, we replace a general operator A by A, e ψ0 . eψ) Aψ = (A

(181)

This gives in particular that e a = ∂, eb = ∂ ∗ ,

e a† = −∂ ∗ + z , eb† = −∂ + z ∗ ,

(182)

and hence, e = ~ e a − eb†eb = ~ (z∂ − z ∗ ∂ ∗ ) = L, L a† e e = ~ω e a + eb†eb + 1 = ~ω (−2 ∂∂ ∗ + z∂ + z ∗ ∂ ∗ + 1) . H a† e

(183)

Thus, one possible approach to the simultaneous eigenvalue problem for L e = L and of and H is to look for polynomials ψe that are eigenfunctions of L j ∗ k e H. The homogeneous polynomial z (z ) , for example, is a solution of the eigenvalue equations z∂(z j (z ∗ )k ) = jz j (z ∗ )k ,

z ∗ ∂ ∗ (z j (z ∗ )k ) = kz j (z ∗ )k .

(184)

e because It is not an eigenfunction of H, ∂∂ ∗ (z j (z ∗ )k ) = jkz j−1 (z ∗ )k−1 ,

(185)

e and L e can be constructed as a but we see that a unique eigenfunction of H linear combination ψej,k = z j (z ∗ )k + α1 z j−1 (z ∗ )k−1 + · · · + αn z j−n (z ∗ )k−n ,

(186)

320

Topological Aspects of Low Dimensional Systems

with n = min(j, k). This is of course the same state as in equation (178). In fact, starting with the ground state ψe0 = 1, we have a† )j (eb† )k ψe0 = (−∂ ∗ + z)j (−∂ + z ∗ )k ψe0 ψej,k = (e = (−∂ ∗ + z)j (z ∗ )k = (−∂ + z ∗ )k z j .

(187)

The lowest order term in equation (186) is either z j−k or (z ∗ )k−j , depending on whether j ≥ k or j ≤ k. Thus the asymptotic form of the wave e 0 as |z| → 0 is |z||`|, where ` = j − k, and `~ is the function ψ = ψψ angular momentum. The quantum number n = min(j, k) describes radial excitations. A standard asymptotic analysis of the eigenvalue equations Hψ = Eψ and Lψ = `~ψ gives the same result. Let r and φ be polar coordinates, that is, x = r cos φ, y = r sin φ, then we have H =−

~2 2m

∂2 1 ∂2 1 ∂ + 2 + 2 ∂r r ∂r r ∂φ2

+

1 mω 2 r2 . 2

(188)

An eigenfunction for L = −i~∂/∂φ has the form ψ(r, φ) = f (r) ei`φ .

(189)

Assuming that f (r) has the asymptotic form f (r) ∝ rµ as r → 0, and inserting into the equation Hψ = Eψ, we get asymptotically as r → 0 the equation µ2 − `2 = 0. The minimum requirement is that ψ must be quadratically integrable, implying the inequality µ > −1, but since there is no reason for ψ to be singular at r = 0, we have to require that µ ≥ 0, and hence µ = |`|. 7.2 Two anyons in a harmonic oscillator potential The two-particle Hamiltonian is H=

1 1 (p1 2 + p2 2 ) + mω 2 (x1 2 + x2 2 ) . 2m 2

(190)

We introduce the anyon statistics by requiring an arbitrary wave function ψ to be multivalued, with ψ(x2 , x1 ) = eiθ ψ(x1 , x2 )

(191)

for an anticlockwise interchange of particle positions. The phase angle θ is defined modulo 2π, and we will assume here that θ = νπ with 0 ≤ ν < 2. Then ν = 0 represents bosons, while ν = 1 represents fermions.

J. Myrheim: Anyons

321

The motion of the two particles can be decomposed into independent motions of the centre of mass position X = (x1 + x2 )/2 and the relative position x = x1 −x2 . The canonically conjugate momenta are P = p1 +p2 and p = (p1 − p2 )/2. This gives the following expression for the Hamiltonian, H=

p2 1 P2 + + mω 2 X2 + mω 2 x2 . 4m m 4

(192)

Thus the centre of mass is represented as a “particle” of mass 2m, whereas the relative coordinate describes a “particle” of the “reduced mass” m/2. The anyonic symmetry condition affects the relative motion only, and takes the following form, still with the anticlockwise convention for changing x into −x, ψ(X, −x) = eiθ ψ(X, x) .

(193)

This condition is singular at x = 0, whenever θ is not an integer multiple of π, and so forces the wave function to behave singularly there, in the sense that it goes to infinity or is not differentiable. Let r and φ be the relative polar coordinates, then the relative motion part of the wave function must have the following asymptotic form as r → 0, ψrel (r, φ) = rµ ei`φ ,

(194)

with `π = θ + 2kπ for some integer k. Like in the one-particle case we get from the energy eigenvalue equation, to leading order in r, that µ2 − `2 = 0. We choose the solution µ = |`|, to make ψ finite in the limit r → 0, even if it should happen that |`| < 1 so that quadratic integrability allows the choice µ = −|`|. We have assumed here that θ = νπ and 0 ≤ ν < 2, hence there are two classes of energy eigenstates: class (I) having µ = ν, ν + 2, ν + 4, . . . , and class (II) having µ = 2 − ν, 4 − ν, 6 − ν, . . . . Let us introduce the complex coordinates z1 and z2 in the same way as before, and define Z = (z1 + z2 )/2 and z = z1 − z2 . Then the Hamiltonian is |z2 |2 ∂2 |z1 |2 ∂2 + −2 + H = = ~ω −2 ∂z1 ∂z1∗ ∂z2 ∂z2∗ 2 2 2 ∂2 |z| ∂2 −4 + |Z|2 + (195) = ~ω − ∂Z ∂Z ∗ ∂z ∂z ∗ 4 and the total angular momentum is ∂ ∂ ∂ ∂ − z1∗ ∗ + z2 − z2∗ ∗ L = ~ z1 ∂z1 ∂z1 ∂z2 ∂z2 ∂ ∂ ∂ ∂ ∗ − Z∗ − z + z · = ~ Z ∂Z ∂Z ∗ ∂z ∂z ∗

(196)

322

Topological Aspects of Low Dimensional Systems

Next we define annihilation and creation operators as follows, Z∗ 1 ∂ +√ , a† = − √ 2 ∂Z 2 1 Z ∂ b† = − √ +√ , 2 ∂Z ∗ 2 √ ∂ z∗ † + √ , c =− 2 ∂z 2 2 √ ∂ z d† = − 2 ∗ + √ · ∂z 2 2

∂ Z +√ , ∗ ∂Z 2 Z∗ ∂ +√ , ∂Z 2 ∂ z + √ , ∂z ∗ 2 2 √ ∂ z∗ + √ , d = 2 ∂z 2 2

1 a = √ 2 1 b = √ 2 √ c = 2

(197)

The non-vanishing commutators among these operators are [a, a† ] = [b, b† ] = [c, c† ] = [d, d† ] = 1 .

(198)

With these definitions we obtain the following form of the total angular momentum and Hamiltonian, L = ~ a† a − b† b + c† c − d† d , (199) H = ~ω a† a + b† b + c† c + d† d + 2 . Two energy eigenstates having the correct symmetry under particle interchange, and belonging to the classes (I) and (II) defined above, are (I)

ψ0 = z ν ψ0 , with

(II)

ψ0

= (z ∗ )2−ν ψ0 ,

(200)

|z1 |2 + |z2 |2 |z|2 2 = exp −|Z| − · ψ0 = exp − 2 4 (I)

(201)

(II)

They have energies E0 = (2 + ν)~ω and E0 = (4 − ν)~ω. We may construct a complete set of energy eigenstates by starting from these “ground states” and acting with the creation operators a† , b† , c† and d† , within certain restrictions. Because c† and d† are antisymmetric under interchange, we have to use either (c† )2 , (d† )2 or c† d† in order to preserve the interchange symmetry of the wave functions. There are two further restrictions, when 0 < ν < 1 or 1 < ν < 2, because the action of either (c† )2 (I) (II) on ψ0 or of (d† )2 on ψ0 produces a singular wave function. However, † † c d is always a “good” operator. Hence the general eigenstates are, with j, k, l, m independent non-negativ integers, ψj,k,l,m = (a† )j (b† )k (c† d† )l (d† )2m ψ0 , (I)

(I)

ψj,k,l,m = (a† )j (b† )k (c† d† )l (c† )2m ψ0 (II)

(II)

.

(202)

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323

Here j, k are quantum numbers of centre of mass excitations, while l, m describe excitations of the relative degrees of freedom. The corresponding energy levels are (I)

Ej,k,l,m = (2 + ν + j + k + 2l + 2m) ~ω , (II)

Ej,k,l,m = (4 − ν + j + k + 2l + 2m) ~ω .

(203)

This gives the two-particle partition function, again with ξ = β~ω, Z2 (β) =

X

e−βEj,k,k,l,m =

j,k,l,m

=

e−(2+ν)ξ + e−(4−ν)ξ (1 − e−ξ )2 (1 − e−2ξ )2

cosh((1 − ν)ξ) · 8 sinh2 2ξ sinh2 ξ

(204)

7.3 More than two anyons Although the complete solution of the N -anyon problem in a harmonic potential can only be obtained numerically when N > 2, it is still possible to find a number of exact energy eigenstates. The N -particle Hamiltonian is a sum of N one-particle contributions, N X 1 1 pj 2 + mω 2 xj 2 . (205) H= 2m 2 j=1 The centre of mass motion can be separated from the relative motion, because the potential is separable by the identity  2 N N N k−1 X X 1  1 XX 2 xj 2 = xj  + (xj − xk ) . (206) N N j=1 j=1 j=1 k=2

The centre of mass energy spectrum is identical to the one-particle spectrum, that is, the energy levels are n~ω with degeneracy n, for n = 1, 2, . . . . Hence the centre of mass motion contributes to the energy but not to the degeneracy of the ground state. The bosonic ground state has all N particles in the lowest one-particle level, hence it is non-degenerate and has energy N ~ω, including the centre of mass contribution. In the fermionic ground state the N one-particle states of lowest energy are filled. Thus there exist “magic numbers” N = n(n + 1)/2 = 1, 3, 6, 10, . . . , with n = 1, 2, . . . , such that the one-particle levels up to and including n~ω are completely filled. The total energy is then √ N 8N + 1 n(n + 1)(2n + 1) 2 ~ω = ~ω. (207) E = (1 + 4 + · · · + n )~ω = 6 3

324

Topological Aspects of Low Dimensional Systems

For these magic numbers the fermionic ground state is non-degenerate, but for other values of N it is degenerate. Thus the degeneracy is 2 for N = 2, 3 for N = 4 and N = 5, 4 for N = 7 and N = 9, 6 for N = 8, and so on. Like in the one-particle case we may introduce complex coordinates zj as well as annihilation and creation operators aj , bj , a†j , b†j , in order to write the Hamiltonian as N N X X 1 (208) a†j aj + b†j bj + 1 , −2∂j ∂j∗ + |zj |2 = ~ω H = ~ω 2 j=1 j=1 and the total angular momentum as L=~

N X

N X a†j aj − b†j bj . zj ∂j − zj∗ ∂j∗ = ~

j=1

(209)

j=1

Assuming that all the operators a†j aj and b†j bj are non-negative, we derive the following inequality relating the energy E and total angular momentum `~, E ≥ (|`| + N )~ω .

(210)

The non-normalized wave function 

 N X 1 |zj |2  ψ0 = exp− 2 j=1

(211)

is annihilated by all the operators aj and bj . Again we may split off the e 0 , at the same Gaussian factor ψ0 and write any wave function ψ as ψ = ψψ e ψ0 . Then e such that Aψ = (A eψ) time as we replace any operator A by A, we have, in the same way as before, that e aj = ∂j ,

e a†j = −∂j∗ + zj ,

ebj = ∂ ∗ , j

eb† = −∂j + z ∗ , j j

(212)

and, e=~ L

N N X X † †e e aj − b j b j = ~ e aj e zj ∂j − zj∗ ∂j∗ = L, j=1

e = ~ω H

j=1

N X aj + eb†jebj + 1 e a†j e j=1

= ~ω

N X j=1

−2∂j ∂j∗ + zj ∂j + zj∗ ∂j∗ + 1 .

(213)

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325

The generalization of the exact two-anyon energy eigenstates of equation (200) are the states (I)

ψ0 = ∆ν ψ0 ,

(II)

ψ0

= (∆∗ )2−ν ψ0 ,

(214)

where ∆ is the lowest degree totally antisymmetric polynomial in the variables z1 , z2 , . . . , zN , ∆=

N k−1 Y Y

(zj − zk ) =

k=2 j=1

Y

(zj − zk ) .

(215)

j
Up to a sign, this is the Vandermonde determinant, 1 1 ... 1 z1 z . . . zN 2 ∆ = ± . .. . . . . ... . . z1 N −1 z2 N −1 . . . zN N −1

.

(216)

∗ , it follows Since ∆ is a function of z1 , z2 , . . . , zN , but not of z1∗ , z2∗ , . . . , zN ∗ ∗ ∗ ∗ that ∆ is a function only of z1 , z2 , . . . , zN . And since ∆ is a homogeneous polynomial, ∆ν and (∆∗ )2−ν are both homogeneous functions, of degrees N (N − 1)ν/2 and N (N − 1)(2 − ν)/2, respectively. The homogeneity means by definition that   N X N (N − 1)  ν ∆ν , zj ∂j  ∆ν = 2 j=1   N X N (N − 1)  (2 − ν) (∆∗ )2−ν . zj∗ ∂j∗  (∆∗ )2−ν = (217) 2 j=1 (I)

(II)

Thus the total angular momentum of the states ψ0 and ψ0 (I)

`0 ~ =

N (N − 1) ν~ , 2

(II)

`0 ~ =

is

N (N − 1) (ν − 2)~ , 2

(218)

respectively, and the total energy is N −1 N −1 (I) (II) ν + 1 N ~ω , E0 = (2 − ν) + 1 N ~ω . (219) E0 = 2 2 In both cases E = (|`| + N )~ω, meaning that these are ground states for those two values of the angular momentum. For ν = 1, in particular, these are fermion states constructed either from the one-particle states z j ψ0 or from the states (z ∗ )j ψ0 , with j = 0, 1, . . . , N − 1. However, the energy

326

Topological Aspects of Low Dimensional Systems

is higher than the fermion ground state energy for every N > 2, and the discrepancy increases with N , since the energy increases like N 2√for large N , whereas the fermionic ground state energy increases like N N . The ground state energy is not exactly known for anyons close to fermions, even in the three-particle case. We may act on these two states by creation operators, taking care to preserve the interchange symmetry and avoid generating singular states. In this way we get energy eigenstates that we may classify as type (I) or (II). Besides the fact that we do not get the fermion states of lowest energy, another way to see that not all energy eigenstates are of either type (I) or (II), is by considering the asymptotic behaviour of the wave functions when two particles approach each other. The states of class (I) will all have the asymptotic form (zj − zk )ν+2m as |zj − zk | → 0, whereas those of class (II) will have the asymptotic form (zj∗ − zk∗ )2−ν+2n , with m and n non-negative integers. But there must exist more general states that somehow mix these possibilities, an example is the three-fermion ground state, ψe = z1 z2∗ + z2 z3∗ + z3 z1∗ − z2 z1∗ − z3 z2∗ − z1 z3∗ .

(220)

Let us look more closely at the states of class (I). They are of the form ∗ ) ∆ν , ψe = f (z1 , . . . , zN , z1∗ , . . . , zN

(221)

where we have split off, as usual, the Gaussian factor ψ0 , and where f ∗ , is a polynomial in the 2N independent variables z1 , . . . , zN , z1∗ , . . . , zN symmetric under interchange of particle labels. The “reduced” Hamiltonian e is given by equation (213). e acting on ψ, operator H, As a starting point for constructing an energy eigenstate, assume that f is a homogeneous polynomial, of degree J in z1 , z2 , . . . , zN and degree ∗ . Then we have that K in z1∗ , z2∗ , . . . , zN e ∆ν ) N (N − 1) H(f ν + N + J + K f = ~ω∆ν 2 −2

N X j=1

∂j∗ ∂j f − 2ν

N k−1 X X ∂j∗ f − ∂k∗ f · zj − zk j=1

(222)

k=2

If the last two terms here do not vanish, they add up to a homogeneous function of degree (J − 1, K − 1), which has to be compensated for by the addition to the polynomial f of a “counterterm”, in fact the same homogeneous function multiplied by some constant coefficient. The counterterm may need a second counterterm, of degree (J − 2, K − 2), and so on, until ∗ are eliminated and the process stops after K the variables z1∗ , z2∗ , . . . , zN steps.

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327

This construction of an energy eigenstate works smoothly and produces a non-singular wave function as long as the counterterms arising are all polynomials. We see from equation (222) that the necessary and sufficient condition is that the polynomial ∂j∗ f − ∂k∗ f must always be divisible by zj − zk . This condition must hold when f is the homogeneous polynomial we start with, or any one of the counterterms that we construct successively. Since we are working with symmetric functions, it is enough to impose the single condition on f that ∂1∗ f − ∂2∗ f must be divisible by z1 − z2 . Equivalently, we must have ∂1∗ f − ∂2∗ f = 0 when z1 = z2 . Remember that we treat z and z ∗ as independent variables, so that z1 = z2 does not imply z1∗ = z2∗ . Explicitly written out the condition is that ∗ )= [∂1∗ f ](z, z, z3, . . . , zN , z1∗ , z2∗ , z3∗ , . . . , zN

∗ ). [∂2∗ f ](z, z, z3, . . . , zN , z1∗ , z2∗ , z3∗ , . . . , zN

(223)

Let us call a symmetric polynomial f “good” if it satisfies this condition. Since the condition is linear, any linear combination of good polynomials is a good polynomial. Almost as easy is it to see that any product of good polynomials is again good. ∗ . Let S be Obviously, f is good if it does not depend on z1∗ , z2∗ , . . . , zN the symmetrization operator, S=

1 X p. N!

(224)

p∈Sn

Then the polynomials cj1 ,j2 , ... ,jN = γj1 ,j2 , ... ,jN S(z1 jN z2 jN −1 +jN · · · zN j1 +j2 +···+jN ) ,

(225)

where j1 , j2 , . . . , jN are non-negative integers and γj1 ,j2 , ... ,jN are (unspecified) normalization factors, form a basis for the vector space of symmetric polynomials in z1 , z2 , . . . , zN . These basis polynomials are homogeneous of degree j1 + 2j2 + · · · + N jN . By definition, the elementary symmetric polynomial ci in z1 , z2 , . . . , zN is the basis polynomial for which ji = 1 and jk = 0 when k 6= i. Explicitly written out, we have for example, c1 = z 1 + z 2 + · · · + z N , c2 = z1 z2 + z1 z3 + · · · + zN −1 zN ,

(226)

c3 = z1 z2 z3 + z1 z2 z4 + · · · + zN −2 zN −1 zN . The corresponding N -anyon energy levels of the harmonic oscillator are therefore N (N − 1) ν + N + j1 + 2j2 + · · · + N jN ~ω . (227) Ej1 ,j2 , ... ,jN = 2

328

Topological Aspects of Low Dimensional Systems

In the system of charged anyons in a magnetic field, discussed in the next section, the same wave functions are even more useful, since they describe the degenerate ground state, the lowest Landau level “LLL”. This is not yet the full story. In fact, any symmetric polynomial which ∗ , is also “good”. It must be of the form is linear in z1∗ , z2∗ , . . . , zN f=

N X

fj (z1 , z2 , . . . , zN ) zj∗ ,

(228)

j=1

with fj (z1 , z2 , . . . , zj , . . . , zN ) = f1 (zj , z2 , . . . , z1 , . . . , zN ), and with f1 (z1 , z2 , . . . , zN ) symmetric in the N − 1 last arguments. The condition for f to be “good”, f1 (z, z, z3 , . . . , zN ) = f2 (z, z, z3 , . . . , zN ) ,

(229)

is seen to hold automatically. It follows that all sums and products of ∗ , are good polynomials. The symmetric polynomials linear in z1∗ , z2∗ , . . . , zN tricky part of enumerating all such good polynomials is to avoid double counting. The enumeration can be done as follows. Let sk =

N X j=1

zj k ,

tk =

N X

zj k zj∗ ,

(230)

j=1

and let G (“G” for “good”) be the set of all polynomials in s1 , s2 , . . . , sN , t0 , t1 , . . . , tN −1 . Then it can be shown that G contains every symmetric polynomial we can generate by taking sums and products of symmet∗ . ric polynomials that are either independent of or linear in z1∗ , z2∗ , . . . , zN Moreover, because the 2N variables s1 , s2 , . . . , sN , t0 , t1 , . . . , tN −1 are independent, the representation of a symmetric polynomial as a polynomial in these particular variables is unique. In order to see that we actually get eigenfunctions of the Hamiltonian, we should start with the basis polynomial f = s1 j1 s2 j2 · · · sN jN t0 k1 t1 k2 · · · tN −1 kN ,

(231)

in which j1 , j2 , . . . , jN , k1 , k2 , . . . , kN are arbitrary non-negative integers, and carry out the construction as outlined above. The construction indeed works, and we get an energy eigenvalue which is (I)

Ej1 , ... ,jN ,k1 , ... ,kN =

N (N − 1) ν + N + j1 + 2j2 + · · · + N jN 2

+ k1 + 2k2 + · · · + N kN ~ω .

(232)

J. Myrheim: Anyons

329

The energy eigenfunctions are † kN ∆ν ψ0 . (233) ψj1 , ... ,jN ,k1 , ... ,kN = (A†1 )j1 · · · (A†N )jN (B0† )k1 · · · (BN −1 ) (I)

when we write A†k =

N X

(a†j )k ,

Bk† =

j=1

N X

(a†j )k b†j .

(234)

j=1

A parallel construction can be carried out for the states of class (II), leading to the energy eigenvalues (II)

Ej1 , ... ,jN ,k1 , ... ,kN =

N (N − 1) (2 − ν) + N + j1 + 2j2 + · · · + N jN 2 ! + k1 + 2k2 + · · · + N kN ~ω. (235)

All of these energy eigenvalues depend linearly on the statistics parameter ν. The partition function obtained by summing over the “linear” energy levels is then L (β) ZN

=

=

e

− N (N2−1) νξ

1 22N −1

+e

cosh

− N (N2−1) (2−ν)ξ

N (N − 1) (1 − ν)ξ 2

e

−N ξ

Y N

N Y

1 −nξ )2 (1 − e n=1

n=1

1

2

sinh

nξ 2

·

(236)

The above arguments left a number of loose ends that we should try to tie up. Consider first the generating function for the elementary symmetric polynomials c1 , . . . , cN , g(w) = 1 +

N X

cj w j =

j=1

N Y

(1 + wzj ) .

(237)

j=1

On the one hand we have that  k ∞ N X c1 2 (−1)k−1 X j cj w = c1 w + c2 − w2 + · · · ln g(w) = k 2 j=1 k=1

=

N X j=1

ln(1 + wzj ) =

∞ X (−1)k−1 k=1

k

sk w k ,

(238)

330

Topological Aspects of Low Dimensional Systems

and this gives every polynomial sk as a polynomial in c1 , . . . , cN . On the other hand we have that ∞ Y (−1)k−1 k sk w exp g(w) = k k=1 s1 2 (239) w2 + · · · , = 1 + s1 w + −s2 + 2 and this gives c1 , . . . , cN as polynomials in s1 , . . . , sN . The fact that the last product of exponentials is a polynomial of degree N gives relations that can be used to express sk for every k > N as a polynomial in s1 , . . . , sN . The values of s1 , s2 , . . . , sN determine the values of c1 , c2 , . . . , cN , which in turn determine the values of z1 , z2 , . . . , zN , in arbitrary order, as the roots of the equation z N + c1 z N −1 + · · · + cN = 0 .

(240)

Next, given z1 , . . . , zN in a definite order, the values of t1 , . . . , tN deter∗ . This shows that the 2N variables s1 , . . . , sN , t0 , mine uniquely z1∗ , . . . , zN . . . , tN −1 are independent, and that every symmetric function of z1 , . . . , ∗ must be functionally dependent on them. However, the zN , z1∗ , . . . , zN functional dependence need not be polynomial, and there do indeed exist ∗ that are non-polynomial symmetric polynomials in z1 , . . . , zN , z1∗ , . . . , zN functions of s1 , . . . , sN , t0 , . . . , tN −1 . As already noted, sk for k > N is always a polynomial in s1 , . . . , sN . Another way to see this is to observe that zj k + c1 zj k−1 + · · · + cN zj k−N = 0

(241)

for every k ≥ N and j = 1, 2, . . . , N , implying the following recursion relations, sk = −c1 sk−1 − · · · − cN sk−N , tk = −c1 tk−1 − · · · − cN tk−N .

(242)

In general, we may write every symmetric polynomial in the 2N variables ∗ as a linear combination of the symmetric polynoz1 , . . . , zN , z1∗ , . . . , zN mials generated by the following generating function, where w00 = 1, ! ∞ N ∞ X Y X l wkl zj k zj∗ g(w10 , w01 , w20 , w11 , . . . ) = j=1

=1+

k=1 l=1 N X j=1

cj w10 j + . . .

(243)

J. Myrheim: Anyons

331

By a similar reasoning as above we find that all such polynomials can in turn be expressed as polynomials in the special symmetric polynomials skl =

N X

zj k zj∗ l ,

(244)

j=1

which include the polynomials sk and tk defined above. Thus in particular, and as we already know, every symmetric polynomial in z, independent of z ∗ , is a polynomial in s1 , s2 , . . . sN . Furthermore, every symmetric polynomial in z and z ∗ , linear in z ∗ , is a polynomial in s1 , . . . , sN , t0 , . . . , tN −1 , linear in t0 , t1 , . . . , tN −1 . One point we have not proved here is that our list of energy levels depending linearly on ν, is complete for states of class (I) when the polynomial part of the wave function is quadratic, cubic or of higher degree in z ∗ . Another point worth noting is that there are three common operators in the two sets of creation operators producing the excited states of class (I) and (II), they are A†1 =

N X j=1

a†j ,

B1† =

N X j=1

b†j ,

B2† =

N X

a†j b†j .

(245)

j=1

Two are linear and one is quadratic in the a† and b† operators. The linear ones produce pure centre of mass excitations, whereas the quadratic operator N X 1 † 1 † † † e † = B † − 1 A† B † = A B − − a b B 2 2 j N 1 1 j=1 j N 1 N 1

(246)

produces excitations of the relative motion, which can be interpreted as radial excitations. The distinguishing property of the two classes of so called “linear” wave functions is their asymptotic behaviour as two particles come together: (zj − zk )ν+2k for class (I) and (zj∗ − zk∗ )2−ν+2k for class (II), with k = 0, 1, 2, ... But all wave functions, even the “non-linear” ones, must show one or possibly both of these two asymptotic behaviours for one given pair of particles. Since both asymptotic behaviours are “good” for the operators A†1 , B1† and B2† , it means that all wave functions belong to their domain of definition. Therefore all energy eigenstates, both “linear” and “non-linear”, e † on a come in so called “towers”, generated by the repeated action of B 2 “bottom state”. Each tower has infinitely many states, with a constant energy spacing of 2~ω.

332

Topological Aspects of Low Dimensional Systems

7.4 The three-anyon problem The three-anyon problem is the simplest example going beyond the solvable two-anyon case. It has received much attention, and yet nobody has succeeded in finding a set of coordinates making it completely solvable by separation of variables. The best choice seems to be the coordinates described below, for which the separation is almost complete. More precisely, an anticlockwise cyclic interchange of the three particles gives a phase factor of e2iθ in the wave function, and this condition is compatible with the separation of variables. The stumbling block is the condition due to the interchange of two particles, which can in general only be satisfied by a superposition of such separated wave functions. We introduce the primitive cube root of unity √ 3 1 2iπ , (247) =− +i η = exp 3 2 2 with the properties that η 2 = −η − 1 = η ∗ = 1/η, and define the dimensionless complex coordinates 1 Z = √ (z1 + z2 + z3 ) , 3λ 1 z1 + ηz2 + η 2 z3 , u = √ 3λ 1 z1 + η 2 z2 + ηz3 . v = √ 3λ

(248)

p Here λ = ~/mω, as before. Z is the centre of mass coordinate, with a slightly unusual normalization, while u and v are relative coordinates. The inverse transformation is λ z1 = √ (Z + u + v) , 3 λ z2 = √ Z + η 2 u + ηv , 3 λ z3 = √ Z + ηu + η 2 v . 3

(249)

This coordinate transformation is a discrete Fourier transformation, and it transforms the cyclic interchange of particle positions, z1 , ze2 , ze3 ) = (z2 , z3 , z1 ) , (z1 , z2 , z3 ) 7→ (e

(250)

into the diagonal form e u (Z, u, v) 7→ (Z, e, ve) = (Z, η 2 u, ηv) .

(251)

J. Myrheim: Anyons

333

The interchange of particles 2 and 3 is just an interchange of u and v. These two permutations generate the whole symmetric group S3 . A similar treatment of permutation symmetry has been used for some time in nuclear physics [271, 272]. Three particles in the plane define a triangle. The ratio s=

z3 − z1 z2 − z1

(252)

is real when the triangle is degenerate so that the particles lie on a straight line. We define the orientation of a non-degenerate triangle as positive or negative depending on whether the imaginary part of s is positive or negative. Thus the orientation is positive when the loop z1 → z2 → z3 → z1 is counterclockwise, and negative when the loop is clockwise. We have that |s + η 2 | |u| = · |v| |s + η|

(253)

Hence |u| = |v| when the particles lie on a line, |u| < |v| when the orientation of the triangle is positive, and |u| > |v| when the orientation is negative. The quantization of the centre of mass motion is trivial, and the interesting part of the problem is the simultaneous diagonalization of the relative Hamiltonian and angular momentum operators, |v|2 ∂2 ∂2 |u|2 + −2 + , Hrel = ~ω −2 ∂u ∂u∗ ∂v ∂v ∗ 2 2 ∂ ∂ ∂ ∂ − u∗ ∗ + v − v∗ ∗ · (254) Lrel = ~ u ∂u ∂u ∂v ∂v The three-particle configuration is completely described by a total scale factor r > 0, a relative scale factor q ≥ 0, and two angles ϕ1 and ϕ2 such that rq e iϕ1 , u= p 1 + q2

r e iϕ2 v=p · 1 + q2

(255)

These are the hyperspherical coordinates of Kilpatrick and Larsen, except that they used the angles ϕ1 ± ϕ2 instead of ϕ1 and ϕ2 [273]. We now have that 2 1 ∂ 3 ∂ 1+q 2 1+q 2 ∂ ∂ 1 ∂2 ∂2 r Hrel =− 3 r − q + + + , ~ω 2r ∂r ∂r 2r2 q ∂q ∂q q 2 ∂ϕ1 2 ∂ϕ2 2 2 ∂ ∂ Lrel = −i + · (256) ~ ∂ϕ1 ∂ϕ2 Assume that the wave function of the relative motion is separable, ψ = ψ(r, q, ϕ1 , ϕ2 ) = f (r) g(q) e i (jϕ1 +kϕ2 ) .

(257)

334

Topological Aspects of Low Dimensional Systems

Then the eigenvalue equation Hrel ψ = Eψ separates into an angular eigenvalue equation, d j2 1 + q2 d q + 2 + k 2 g = Λg , (258) (1 + q 2 ) − q dq dq q with Λ as eigenvalue, and a radial equation, Λ 3 0 r2 E 1 00 f (r) + f (r) . + f (r) = − f (r) − 2 2r 2r2 2 ~ω

(259)

A general wave function can be written as a linear combination of such separated wave functions. As will be seen below, we need linear combinations, where Λ and j + k are constant but j − k varies, in order to satisfy the anyonic boundary conditions. 2 The radial wave function must have the form f (r) = rµ e−r /2 f1 (r2 ), with f1 a polynomial of degree nr = 0, 1, 2, . . . , and in the above radial equation we have to choose Λ = µ(µ + 2) ,

E = (2 + µ + 2nr )~ω .

(260)

We must take µ ≥ 0 if we require the wave function to be finite as r → 0, or µ > −2 if we only require it to be normalizable. Equation (258) has two asymptotic solutions q ±j in the limit q → 0. We exclude the singular solution (for j = 0 the singularity is logarithmic). In fact there is no reason for any singularity at q = 0, where the configuration is an equilateral triangle. The solution g(q) = q |j| (1 + q 2 )κ F (a, b; c; −q 2 )

(261)

is unique up to normalization. The constant κ may be chosen in one of two ways, µ µ or κ=− , (262) κ= +1 2 2 giving two different representations of the same solution. The constants |j| − |k| |j| + |k| +κ, b= +κ, 2 2 define the hypergeometric series a=

F (a, b; c; x) =

c = 1 + |j| ,

∞ X (a)m (b)m xm , (c)m m! m=0

(263)

(264)

where, e.g. (a)0 = 1, (a)n+1 = (a)n (a + n). The convergence radius for this series is 1. A more convergent representation is, e.g., q2 · (265) g(q) = q |j| (1 + q 2 )κ−a F a, c − b; c; 1 + q2

J. Myrheim: Anyons

335

We now need the boundary conditions in order to determine the allowed values of the quantum numbers j, k and Λ = µ(µ + 2) in equation (258). For three identical particles there is a six-fold identification of points in the relative space. We will restrict the wave functions to the region 0 ≤ q ≤ 1, which corresponds to all the positively oriented triangles, but is still a three-fold covering of the true configuration space. The boundary conditions defining the particles to be anyons are of two types, since there are two classes of non-trivial permutations. The first class contains the three-particle cyclic permutations, which leave q invariant. The second class contains the two-particle interchanges, which transform q into 1/q, and so give boundary conditions at q = 1. Consider first a continuous, counterclockwise and cyclic deformation of the configuration, as defined in equation (250), with no extra overall rotation of the triangle. It gives a phase factor e2iθ in the wave function, where θ = νπ is the statistics parameter. We should keep |u| < |v| all the time during the deformation, that is, all the deformed configurations should be positive triangles, since this is the region where we require the wave function to be defined, and since this will ensure that no pair of particles wind around each other separately. Then the phase of v increases continuously from ϕ2 to ϕ2 + (2π/3), whereas the phase of u changes from ϕ1 to ϕ1 − (2π/3) + 2m0π, where m0 is any integer. Note that u = 0 represents a positively oriented equilateral triangle, and by means of small deformations close to u = 0 we may change the phase of u by an arbitrary multiple of 2π. We can not change the phase of v similarly without rotating the whole triangle, or deforming it so that its orientation becomes negative. The condition on the wave function is, therefore, 2π 2π + 2m0 π , ϕ2 + (266) = e 2iθ ψ(r, q, ϕ1 , ϕ2 ) . ψ r, q, ϕ1 − 3 3 That is,

2π 2π 0 + 2m π + k = 2(n0 + ν)π , j − 3 3

(267)

for some integer n0 . Since m0 is an arbitrary integer, j must be an integer. Then k = j + 3(n + ν) ,

(268)

where n = n0 − jm0 is an arbitrary integer, and the eigenvalue of the relative angular momentum Lrel is `~ with ` = j + k = 2j + 3(n + ν) .

(269)

These relations take care of the cyclic permutations of all three particles. What remains is only to take care of one of the three cases where two

336

Topological Aspects of Low Dimensional Systems

particles are interchanged, for example z2 ↔ z3 , or equivalently, u ↔ v. This is the same as q ↔ 1/q and ϕ1 ↔ ϕ2 , if we define angles so that u = v corresponds to ϕ1 = ϕ2 . To be more precise, we consider a continuous interchange, with q = 1 at the beginning and end, and q < 1 during the interchange. The interchange should be anticlockwise, which means that we start with ϕ1 > ϕ2 , and end with ϕ1 < ϕ2 . There is one further restriction, that |ϕ1 − ϕ2 | < (2π/3) when q = 1, meaning that the particle position z1 must not be encircled. Thus, the boundary condition on ψ at q = 1 is ψ(r, 1, ϕ2 , ϕ1 ) = e iθ ψ(r, 1, ϕ1 , ϕ2 ) ,

(270)

for 0 < ϕ1 − ϕ2 < (2π/3). It is a special case of the general condition ψ(r, 1/q, ϕ2 , ϕ1 ) = e iθ ψ(r, q, ϕ1 , ϕ2 ) .

(271)

Since the Schr¨ odinger equation is second order in the q derivative, we need boundary conditions at q = 1 both for the wave function ψ and its normal derivative ψq = ∂ψ/∂q. The derivative condition is easily deduced, ψq (r, 1, ϕ2 , ϕ1 ) = −e iθ ψq (r, 1, ϕ1 , ϕ2 ) .

(272)

The boundary conditions for ψ and ψq can not in general be satisfied by a wave function which is separable in q, ϕ1 and ϕ2 . But we may quantize the relative angular momentum `, and according to equation (269) ` − 3ν = 2j +3n is an integer, either even or odd. Let ν 0 = ν if n = 2m and ν 0 = ν +1 if n = 2m + 1, with m integer. Then ν0 ν0 ` ` , k = +3 m+ · (273) j = −3 m+ 2 2 2 2 Let gm (q) be the function g(q) as given by equation (261). Introducing ϕ = (ϕ1 + ϕ2 )/2 and ξ = 3(ϕ1 − ϕ2 ), and summing over m, including an as yet undetermined coefficient γm for each m, we get the following angular wave function, Ω(q, ϕ1 , ϕ2 ) =

∞ X

γm gm (q) e i (jϕ1 +kϕ2 )

m=−∞

= e i `ϕ

∞ X

γm gm (q) e −i(m+(ν

0

/2))ξ

.

(274)

m=−∞

It is natural to call Ω an anyonic spherical harmonic function, whenever it satisfies the anyonic boundary conditions. The two boundary conditions

J. Myrheim: Anyons

337

that must hold for 0 < ξ < 2π are ∞ X m=−∞ ∞ X

γm gm (1) e imξ =

e i(νπ−ν

0

0 γm gm (1) e imξ = −e i(νπ−ν

m=−∞

ξ)

0

ξ)

∞ X

γm gm (1) e −imξ ,

m=−∞ ∞ X

0 γm gm (1) e −imξ . (275)

m=−∞

0 (1) depend on the three parameters µ, ` and m. Recall that gm (1) and gm For each given `, the parameter µ, which determines the energy E, has to be adjusted so that the above boundary conditions have non-trivial solutions for the coefficients γm . For each ` there will be many solutions, possibly more than one with the same µ, and this procedure should give the complete set of anyonic spherical harmonics. The left hand sides in equation (275) are two functions of ξ with Fourier 0 (1), respectively. They are periodic in ξ components γm gm (1) and γm gm with period 2π, and may be regarded as functions on the interval [0, 2π]. There is a natural scalar product between any two functions φ = φ(ξ) and χ = χ(ξ), with Fourier components φm and χm , Z 2π ∞ X 1 dξ (φ(ξ))∗ χ(ξ) = φ∗m χm . (276) (φ, χ) = 2π 0 m=−∞

Define the linear operator A by [Aφ](ξ) = ei(νπ−ν

0

ξ)

φ(2π − ξ) ,

(277)

for 0 < ξ < 2π. Then A is Hermitean with respect to the natural scalar product, and A2 = I, the identity operator. Note that A is a somewhat 0 singular operator, unless ν is an integer, since the factor ei(νπ−ν ξ) , extended by periodicity outside the interval [0, 2π], is discontinuous at every integer multiple of 2π. This shows up in the slow asymptotic falloff of the matrix elements of A with respect to the basis functions eimξ , Z 2π 0 1 sin(νπ) ·(278) dξ e−imξ ei(νπ−ν ξ) ein(2π−ξ) = Amn = Anm = 2π 0 π(m + n + ν) The bosonic limit ν → 0 is Amn = δm,−n , and the fermionic limit ν → 1 is Amn = −δm,−n−1 . Define operators G and G0 that are diagonal in the Fourier representation, with matrix elements Gmn = gm (1)δmn ,

0 G0mn = gm (1)δmn .

(279)

Then the above boundary conditions may be written as (I − A)Gγ = 0 ,

(I + A)G0 γ = 0 .

(280)

338

Topological Aspects of Low Dimensional Systems

Since A is a real symmetric matrix and A2 = I, the two vectors (I − A)Gγ and (I + A)G0 γ are orthogonal, and we may for example add the two conditions to get one single, equivalent condition, (G + G0 − A(G − G0 ))γ = 0 .

(281)

There exist nontrivial solutions for γ whenever the operator G + G0 − A(G − G0 ) is singular. If we truncate the equation, the condition is that the determinant must vanish. Since the determinant is real, and a small change in ν could move a zero of the determinant, but not remove it, or introduce a new zero (at least not without removing or adding a double zero at once) we conclude, by continuity in ν, that the anyonic solutions are in one to one correspondence with the bosonic and fermionic solutions. To find numerical solutions we must truncate to a finite number M of coefficients. Remarkably enough, this method is capable of giving many energy levels with non-trivial accuracy even if we take M to be very small. This is so when the low Fourier components dominate. On the other hand, the convergence as M → ∞ is sometimes very slow. This is clearly related to the fact that the wave functions for non-integer ν have non-integer power behaviour at ξ = 0, where two particles meet. Hence the approximation by means of a finite Fourier series converges slowly. An empirical rule is that the leading correction term for finite M is of order M −2ν . Using two different M one may therefore extrapolate to M = ∞, and this improves the convergence considerably. Another point to note is that one may take advantage of the supersymmetry in order to get more accurate energy levels. 8

The anyon gas

We will discuss in this section the cluster and virial expansions for the anyon gas, which are by now fairly well understood, although not completely solved. The fact that these expansions exist for anyons, is in itself non-trivial. Three complementary methods for computations, all with their own limitations, are perturbation theory, non-perturbative numerical computation of energy levels, and direct computation of partition functions by the Monte Carlo method. All three methods rely on regularization techniques to obtain the thermodynamic limit from finite systems. Perturbation theory has given important exact information, but will be mentioned only very briefly here. A fourth method is mean field theory, which should be covered in other lectures. The present discussion may seem like an evasion of the most interesting topic, which is the low temperature behaviour of a system of anyons. However, the low temperature problem is a hard nut to crack, and the high

J. Myrheim: Anyons

339

temperature side is one possible direction of approach. Attemts to understand more directly the low temperature behaviour are usually based on mean field theory. 8.1 The cluster and virial expansions The virial expansion for the equation of state of a gas, βP = ρ +

∞ X

An ρn ,

(282)

n=2

is called so because it is related to the virial theorem, see e.g. [274]. Here β = 1/kB T , kB is Boltzmann’s constant, T the temperature, P the pressure, ρ the number density, and An = An (T ) is the n-th virial coefficient. Another representation of the equation of state is the cluster expansion, βP =

∞ X

bn z n ,

n=1

ρ=z

∞ ∂(βP ) X = nbn z n . ∂z n=1

(283)

Any power series with a non-zero radius of convergence defines an analytic function, which in general is well defined in a region in the complex plane at least as large as the circle of convergence of the series. In fact the convergence radius is the smallest distance from the origin to any singularity of the analytic function. In general there need not be any direct relation between the convergence radius of the virial expansion and the region where the equation of state it represents, is physically valid. We will use the cluster and virial expansions here without worrying too much about questions of convergence. Eliminating z in equation (283) gives equation (282), with the following relations between the virial and cluster coefficients, b2 , b21 2b3 4b2 A3 = − 3 + 42 , b1 b1 3b4 18b2 b3 20b32 − , A4 = − 4 + b1 b51 b61

A2 = −

(284)

and so on. Or inversely, b2 = −b21 A2 , b3 b3 = − 1 A3 − 4A22 , 2 b41 A4 − 9A3 A2 + 16A32 , b4 = − 3

(285)

340

Topological Aspects of Low Dimensional Systems

and so on. Note that the coefficient b1 is not determined by the virial coefficients, because the normalization of z is not fixed by the equations (282) and (283). We may define z = eβµ , where µ is the chemical potential. z is usually called fugacity, although strictly speaking it is only proportional to the fugacity, which has the dimension of pressure. If the particles interact by a two-particle potential, then a necessary condition for the existence of the virial expansion in the thermodynamic limit of infinite volume and constant density, is that the potential has sufficiently short range. For example, if it decreases as r−n at large distance r, then the condition is that n > d, the configuration space dimension. However, this criterion does not apply to a vector potential, and the virial expansion exists for non-interacting anyons, even though anyons are two-dimensional and have a statistics interaction which may be represented by a vector potential proportional to 1/r. One may argue that the interaction range is short in the sense that the statistics flux is localized exactly at the particle positions, and the statistics interaction is present only when the particles are close enough to interchange positions or encircle each other. The first clear evidence was the finite and exact result for the second virial coefficient of a gas of free anyons [66, 128]. Perturbation theory gave finite results for the expansion of the higher virial coefficients to first and second order around the boson and fermion points [102,129,130,134–141]. The third virial coefficient was proved to be finite for all θ, and was calculated numerically [122,143–145]. A general proof can be based on the path integral representation for the N -particle partition function, which leads to a path integral representation for the cluster coefficients, valid quite generally for anyons in two dimensions, as well as for bosons and fermions in any dimension, interacting by general scalar and vector potentials. It follows from this representation that the cluster coefficients are finite when the interaction range is “short” in a well defined sense, although it does not follow at the same time that the cluster expansion converges. In particular, the cluster and virial coefficients of the gas of free anyons are finite. What counts is the pointlike nature of the flux more than the 1/r dependence of the vector potential. The range is temperature dependent, however, because the statistics interaction is effective when the particle paths wind around each other, and each path in the path integral represents Brownian motion of a particle in the plane, covering an area inversely proportional to the temperature. 8.2 First and second order perturbative results An important source of exact information about the cluster and virial expansion for anyons is perturbation theory, to first and second order in θ at the boson point θ = 0 and at the fermion point θ = π.

J. Myrheim: Anyons

341

The straightforward way to do perturbation theory is to work, not in the parallel gauge we have used so far most of the time, but in the bosonic gauge, where all wave functions are symmetric, or in the fermionic gauge, where all wave functions are antisymmetric. Then the statistics gauge potential is treated as a perturbation of the bosonic or fermionic Hamiltonian. Since this gauge potential is singular at those points where two particles come together, it is not obvious that perturbation theory should work, especially in the bosonic gauge, where the unperturbed wave functions need not vanish at coincidence points. Furthermore, those wave functions that do not vanish at coincidence points, depend on |θ| rather than θ in their asymptotic behaviour at such points. Thus it might be necessary to treat θ and |θ| as independent expansion parameters. Note that θ is transformed into −θ by either a parity inversion or a time reversal. Equivalently we might say that θ − π is transformed into −(θ − π), since −θ − π and −θ + π represent the same statistics. Thus, if either parity invariance or time reversal invariance hold, implying in particular that there is no external magnetic field to break these invariances, then the energy spectrum and the partition function (but not the energy eigenfunctions) will depend only on |θ|, or equivalently, on |θ − π|. On the other hand, at the fermion point θ = π it must also be possible to treat wavefunctions, energy eigenvalues and the partition function as analytic functions of θ. In fact, the source of the non-analyticity at the boson point θ = 0 is the asymptotic behaviour of some energy eigenfunctions, that |xj − xj |µ as |xj − xj | → 0, with µ = |θ|/π. It is mathematically possible to choose µ = −|θ|/π for small enough |θ|, but that would make the wave functions diverge in the limit |xj − xj | → 0, which is usually considered a physically unacceptable alternative. From this argument we conclude that when we make perturbation expansions around the fermion point θ = π, we have a free choice whether we want to use θ − π or |θ − π| as our expansion parameter. This means that the expansion of the partition function will contain only even powers of θ − π. To be specific, let us consider the transformation from the parallel gauge to the bosonic gauge. It is most easily discussed in terms of dimensionless complex variables zj = (xj + iyj )/λ, where λ is some standard length. Like before, we define ∂j = ∂/∂zj and ∂j∗ = ∂/∂zj∗. Then the kinetic energy operator of particle j is, in the parallel gauge, pj 2 2~2 ∗ =− ∂ ∂j . 2m mλ2 j

(286)

Let ψ be the symmetric wave function in the bosonic gauge, and let Φ be some fixed multivalued function, such that Φψ is the multivalued wave function in the parallel gauge. The gauge independent (covariant) derivatives

342

Topological Aspects of Low Dimensional Systems

are ∂j and ∂j∗ in the parallel gauge, whereas in the bosonic gauge they are ∂j − iaj and ∂j∗ − ia∗j , where by definition ∂j∗ (Φψ) = Φ ((∂j∗ − ia∗j )ψ) .

∂j (Φψ) = Φ ((∂j − iaj )ψ) ,

Thus the “statistics vector potentials” aj and a∗j

aj = i∂j (ln Φ) ,

=

a∗j

(287)

are

i∂j∗ (ln Φ)

.

(288)

The question remaining to be settled is how to choose the gauge transformation factor Φ. One possible choice is the pure phase factor ν ν ν ∆ |∆| ∆ 2 = = , (289) Φ0 = |∆| ∆∗ ∆∗ Q where ∆ = j
a∗0j

=

i∂j∗ (ln Φ0 )

νX 1 = −i , 2 zj∗ − zk∗

(290)

k6=j

give rise to three-body terms in the Hamiltonian, of the type ν2 X X 1 · a∗0j a0j = 4 (zj − zk )(zj∗ − zl∗ )

(291)

k6=j l6=j

A better method is therefore to split off explicitly not only the phase factor Φ0 , but also the factor |∆||ν| . Thus we define |ν|+ν 2

Φ = |∆||ν| Φ0 = ∆

(∆∗ )

|ν|−ν 2

.

(292)

The corresponding vector potentials are aj = i∂j (ln Φ) = i

|ν| + ν X 1 , 2 zj − zk k6=j

|ν| − ν X 1 · a∗j = i∂j∗ (ln Φ) = i 2 zj∗ − zk∗

(293)

k6=j

Note that aj = 0 if ν ≤ 0, and a∗j = 0 if ν ≥ 0. This gives that N X j=1

(∂j∗

−

ia∗j )(∂j

− iaj ) =

N X j=1

∂j∗ ∂j +

|ν| + ν X ∂j∗ − ∂k∗ 2 zj − zk j
|ν| − ν X ∂j − ∂k + , 2 zj∗ − zk∗ j
(294)

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343

and there are no longer any three-body terms present. If we take ν ≥ 0, we have ν = |ν|, and N X

(∂j∗ − ia∗j )(∂j − iaj ) =

j=1

N X

∂j∗ ∂j + |ν|

j=1

X ∂j∗ − ∂k∗ j
zj − zk

·

(295)

If instead ν ≤ 0, we have ν = −|ν|, and the complex conjugate operator, N X

(∂j∗ − ia∗j )(∂j − iaj ) =

j=1

N X j=1

∂j∗ ∂j + |ν|

X ∂j − ∂k · zj∗ − zk∗

(296)

j
Multipyling with −2~2 /(mλ2 ), we have here suitable kinetic energy operators for doing perturbation theory. If parity invariance and/or time reversal invariance hold, then the wave functions and energies can be expanded as power series in |ν|, except that the wave function for ν is the complex conjugate of the corresponding wave function for −ν. The gauge transformation ψ 7→ Φ0 ψ from the bosonic to the parallel gauge is unitary, since Φ0 as defined in equation (289) is a pure phase factor (we do not mind that it is multivalued and hence singular if ν is not an integer). The gauge factor Φ defined in equation (292), on the other hand, is more than a pure phase factor, and hence defines a nonunitary “gauge transformation”. One result of the non-unitarity is that the kinetic energy operator of either equation (295) or equation (296) is not Hermitian in the standard scalar product of the bosonic Hilbert space. This is one of the subtleties involved in the perturbation theoretic treatment of the statistics vector potential [275]. To second order in |ν|, Dasni`eres de Veigy and Ouvry have computed the following result, which does not look particularly encouraging for anybody wanting to attempt third or fourth order computations [138, 276], Λ2 βP = ±

∞ X 1±1 (±z)n (ln(1 ∓ z))2 − |ν| 2 n 2 n=1

1 ± z + 2(1 − (1/24))z (ln(1 ∓ z))2 2(1 ∓ z) X X (±z)s+t+u+v 1 2 +ν s + t + u + v (s + v)2 s,t≥1 u,v≥0 1 + C(s, t, u, v). (s + u)(t + v) + ν2

(297)

The upper signs hold if ν = 0 is defined as Bose statistics, whereas the lower signs hold if one instead defines ν = 0 to mean Fermi statistics.

344

Topological Aspects of Low Dimensional Systems

The coefficient C is defined as (1 − η+ )(η+ − s)(η+ + v) 1 − η+ + (η+ ↔ η− ) , (298) ln C(s, t, u, v) = η+ − η− η+ with η± =

st − uv ±

p (s + u)(s + v)(t + u)(t + v) · s+t+u+v

(299)

The first order term in |ν| vanishes at the fermion point, as it should. The cluster expansion to second order in ν has the convergence radius |z| = 1. 8.3 Regularization by periodic boundary conditions We want to consider free particles in two dimensions, but in order to keep the partition functions finite we need some kind of regularization, by confinement of the particles inside a finite region. We may use a confining potential, such as a harmonic oscillator potential, with soft walls, or a square box, with hard walls, but the fastest convergence to the limit of infinite system size is obtained by using periodic boundary conditions so that there are no edge effects due to the reflecting walls. The periodicity is then used only for normalization, and when we speak about anyons in the path integral formalism, the only restriction is that the starting points of trajectories should be inside the given area. Otherwise the particles propagate freely in the plane and not on the torus. The one-particle partition function is, with periodic boundary conditions in a square box of area A, "

#2 πn2 Λ2 exp − Z1 (β) = A n=−∞ " #2 ∞ X A πn2 A = 2 1+2 exp − 2 · Λ Λ n=1 ∞ X

(300)

The last formula is a Poisson resummation, by Fourier expansion of π(n + x)2 Λ2 exp − , f (x) = A n=−∞ ∞ X

(301)

a periodic function of x [277]. Λ is the thermal de Broglie wave length, r Λ=~

2πβ , m

(302)

J. Myrheim: Anyons

345

with m the particle mass. Below we will take Z1 (β) =

A , Λ2

(303)

using only the leading term in the limit A → ∞. The correction terms for finite A are exponential in A. For comparison, the hard wall boundary conditions would give "

#2 πn2 Λ2 exp − Z1 (β) = A n=1 " #2 ∞ X Λ A 1 πn2 A − √ + = 2 exp − 2 , Λ 2 2 A n=1 Λ ∞ X

(304)

√ with relative correction terms of order 1/ A, which is the ratio between circumference and area of the square box. The formula (303) implies the following scaling relation, valid for one free particle in two dimensions, Z1 (Lβ) =

Z1 (β) · L

(305)

The general relation in dimension d is, with V the d-dimensional volume, √ V Z1 (β) = ( L)d Z1 (Lβ) = d · Λ

(306)

It is convenient to introduce here the following notation. We defined a partition P∞ of N as a sequence of non-negative integers, P = (ν1 , ν2 , . . . ), withS L=1 LνL = N . Let S∞CN denote the set of all partitions of N , and let ∞ C = N =0 CN and C 0 = N =1 CN . In this notation we have that X P∈C

=

∞ X X

=

N =0 P∈CN

∞ ∞ X X ν1 =0 ν2 =0

...

∞ X

...

(307)

νL =0

We will frequently use also another notation, writing for example 311 for the partition 5 = 3 + 1 + 1, which we would otherwise refer to as P = (2, 0, 1, 0, . . . ). The grand canonical partition function is a function of the inverse temperature β and the chemical potential µ, Ξ(β, µ) = 1 +

∞ X N =1

z N ZN (β) ,

(308)

346

Topological Aspects of Low Dimensional Systems

and the relation between quantum mechanics and statistical mechanics is the equation ln Ξ = AβP = A

∞ X

bn z n .

(309)

n=1

The area A enters in the two-dimensional case, but the formula is valid in arbitrary dimension d when A is replaced by the d-dimensional volume. An immediate consequence is that ∞ XY AbL z L exp(Abn z ) = Ξ= νL ! n=1 ∞ Y

νL

n

,

(310)

P∈C L=1

and hence, ZN =

∞ ν X Y (AbL ) L · νL !

(311)

P∈CN L=1

We are more interested in the inverse relation, which follows from the expansion ln Ξ =

∞ X (−1)ν−1

ν

ν=1

=

X

∞ X

!ν z L ZL

L=1 ν−1

(−1)

P∈C 0

∞ Y z L ZL (ν − 1)! νL !

νL ·

(312)

L=1

P∞ Here ν = L=1 νL is the total number of cycles in the partition P. This gives the cluster coefficients in terms of the N -particle partition functions, AbN =

X

(−1)ν−1 (ν − 1)!

P∈CN

∞ Y ZL νL · νL !

(313)

L=1

In particular, Ab1 = Z1 , Ab2 = Z2 −

Z1 2 , 2

(314) 3

Z1 , 3 Z2 2 Z1 4 + Z2 Z1 2 − · Ab4 = Z4 − Z3 Z1 − 2 4 Ab3 = Z3 − Z2 Z1 +

J. Myrheim: Anyons

347

Using these, we obtain the virial coefficients from equation (284), A Z2 1−2 2 , A2 = 2 Z1 A2 Z22 Z3 Z2 1 − 6 2 + 12 4 − 6 3 , (315) A3 = 3 Z1 Z1 Z1 A3 Z22 Z23 Z3 Z3 Z2 Z4 Z2 − 12 4 · 1 − 12 2 + 54 4 − 80 6 − 24 3 + 72 A4 = 4 Z1 Z1 Z1 Z1 Z15 Z1 Clearly many cancellations of leading order terms are needed to produce finite values for the cluster and virial coefficients in the free particle limit A → ∞. Next we use our general expansion of the N -particle partition function as a sum over partitions of N , equation (160), reading explicitly as follows, 1 1 F11 Z1 (β)2 + F2 Z1 (2β) , 2 2 1 1 1 (316) Z3 (β) = F111 Z1 (β)3 + F21 Z1 (2β)Z1 (β) + F3 Z1 (3β) , 6 2 3 1 1 1 F1111 Z1 (β)4 + F211 Z1 (2β)Z1 (β)2 + F22 Z1 (2β)2 Z4 (β) = 24 4 8 1 1 + F31 Z1 (3β)Z1 (β) + F4 Z1 (4β) . 3 4

Z2 (β) =

We get then that AbN

Y 1 Z1 (Lβ) νL = ZN + · · · = + ··· FP νL ! L P∈CN L X Y 1 Z1 (Lβ) νL GP , = Z1 (β) νL ! LZ1 (β) X

P∈CN

!

(317)

L

in terms of a new set of coefficients, GP = (FP + · · · )Z1 ν−1 .

(318)

The “· · · ” in the last formula represents a sum of terms that are products of “F ” coefficients. The “G” coefficients are useful especially because they tend to finite limits in the thermodynamic limit A → ∞, when all interactions have sufficiently short range, as we will prove below. GP is the “connected part” of FP for any partition P. The concept of connectedness will also be made more precise below. We have explicitly that G1 = F1 = 1, GN = FN for N = 2, 3, 4, . . . ,

348

Topological Aspects of Low Dimensional Systems

and G11 = (F11 − 1)Z1 , G111 = (F111 − 3F11 + 2)Z1 2 , G21 = (F21 − F2 )Z1 , G1111 = (F1111 − 4F111 − 3F11 2 + 12F11 − 6)Z1 3 ,

(319)

G211 = (F211 − 2F21 − F2 F11 + 2F2 )Z1 2 , G22 = (F22 − F2 2 )Z1 , G31 = (F31 − F3 )Z1 . So far our formulae are dimension independent. From the specifically twodimensional equations (303) and (305) we now get the equation Λ2 bN =

X P∈CN

GP

Y L

1 , νL ! L2νL

(320)

in which all quantities are finite in the A → ∞ limit. In particular, G2 G11 + , 2 4 G21 G3 G111 + + , Λ2 b3 = 6 4 9 G211 G22 G31 G4 G1111 + + + + · Λ2 b4 = 24 8 32 9 16

Λ2 b2 =

(321)

8.4 Regularization by a harmonic oscillator potential Another useful regularization scheme is by means of an external potential V1 = V1 (x). Assuming that V1 varies slowly enough, we can simply take the pressure, density and fugacity in equation (283) to be functions of the position x. In particular, the local fugacity is z(x) = eβ(µ−V1 (x)) = z0 e−βV1 (x) ,

(322)

with z0 = eβµ independent of position. In this case we should replace equation (309) by the following equation [131], ! Z ∞ X d N z 0 ZN . (323) ln Ξ = d x βP (x) = ln 1 + N =1

To evaluate the integral in equation (323) explicitly, we use the local form of equation (283), with z = z(x) as given by equation (322), and with an external potential of the harmonic oscillator form, V1 (x) =

1 mω 2 |x|2 . 2

(324)

J. Myrheim: Anyons

349

Here m is the particle mass, and ω the angular frequency of the oscillator. We take ω > 0 in order to get a discrete energy spectrum, but in the end we want to take the free particle limit ω → 0. It is convenient to introduce the dimensionless parameter ξ = ~ωβ. Then we have that Z d

d x βP (x) =

Z

∞ X

bn z0n

d

d xe

−nβV1 (x)

n=1

=

∞ X

bn z0n

n=1

Λ √ ξ n

d ·

(325)

Comparison of equation (325) with equation (323) gives the following formulae for the cluster coefficients in terms of the partition functions, in dimension d = 2, Λ2 b1 ξ2 Λ2 b2 2ξ 2 Λ2 b3 3ξ 2 Λ2 b4 4ξ 2

= Z1 , = Z2 −

Z12 , 2

(326)

Z13 , 3 Z2 Z4 = Z4 − Z3 Z1 − 2 + Z2 Z12 − 1 · 2 4 = Z3 − Z2 Z1 +

Comparing with equation (314), we see that we may adopt the mechanical rule of substitution Abn → Λ2 bn /(nξ 2 ). From these equations and equation (284) follow the virial coefficients, 2 Λ Z2 1−2 2 , ξ Z1 4 Λ Z22 Z3 Z2 2 − 10 2 + 16 4 − 6 3 , A3 = ξ Z1 Z1 Z1 6 2 Λ Z Z3 Z2 5 − 42 2 + 138 24 − 160 26 A4 = ξ Z1 Z1 Z1 Z3 Z3 Z2 Z4 −42 3 + 108 − 12 4 · Z1 Z15 Z1 A2 =

(327)

Note that Planck’s constant cancels in the ratio Λ/ξ, but reappears in the partition functions. To obtain the virial coefficients for free particles we must take the limit ω → 0. The existence of the limit is again a far from trivial issue, since the cancellation of the singular factor (Λ/ξ)2(n−1) ∝ ω −2(n−1) in An depends on the vanishing of the factor containing the partition functions to all orders below ω 2(n−1) . That A3 is finite as ω → 0, for the gas of free anyons, was proved in reference [145], by means of the path integral expression for Z3 .

350

Topological Aspects of Low Dimensional Systems

The computation of the free particle GP coefficients by means of the harmonic oscillator regularization is slightly tricky, partly because the substitution rule Abn → Λ2 bn /(nξ 2 ) involves a factor n, and partly because the scaling relation for the harmonic oscillator partition function, equation (179), in the limit ω → 0, Z1 (Lβ) =

Z1 (β) , L2

(328)

differs from the relation Z1 (Lβ) = Z1 (β)/L we assumed when writing equation (320). The correct limit for obtaining the free particle GP is n lim (F + · · · )Z1 ν−1 . νL ω→0 P L L

GP = Q

(329)

8.5 Bosons and fermions For bosons and fermions with no mutual interaction the probability generating functions can be factorized as Y FL νL , (330) FP = L

where FL = 1 for bosons and FL = (−1)L−1 for fermions. The factorization implies, by the equations (160) and (308), that L X Y 1 z L FL Z1 (Lβ) νL Y z FL Z1 (Lβ) = exp · (331) Ξ= νL ! L L P∈C L

L

This is nothing but the standard expression for the logarithm of the grand canonical partition function as a sum over the one-particle energy levels E1k , for an ideal gas bosons or fermions, ln Ξ = ∓

∞ X k=1

∞ L X (±z) Z1 (Lβ) , ln 1 ∓ ze−βE1k = ± L

(332)

L=1

and it gives directly the cluster coefficients, bn =

Fn Z1 (nβ) · nA

(333)

Thus we have GN = FN = (±1)N −1 , and GP = 0 for every partition P containing two or more cycles. In two dimensions we get, using equation (303), and in particular the scaling relation (305), bn =

(±1)n−1 · n2 Λ 2

(334)

J. Myrheim: Anyons

351

Similarly we get in dimension d, using equation (306), bn =

(±1)n+1 √ · n(Λ n)d

(335)

Note that the ideal gas cluster coefficients in any dimension are the same for bosons and fermions, except that the even numbered coefficients b2 , b4 , etc. have the opposite sign. This means that we can formally transform bosons into fermions and vice versa by substituting simultaneously z → −z, P → −P and ρ → −ρ. It follows that the virial coefficients possess the same symmetry as the cluster coefficients, n+1 B An , AF n = (−1)

(336)

with superscripts “B” for boson and “F” for fermion. In particular, the density of the ideal gas of bosons or fermions in dimension d = 2 is ρ=

∞ X

nbn z n = ∓

n=1

1 ln(1 ∓ z) . Λ2

Hence the equation of state in two dimensions takes the form n ∓Λ2 ρ ∞ ∞ 1 − e X X 1 1 ∓Λ2 ρ bn z n = ± 2 = ± Li e , βP = 2 Λ n=1 n2 Λ2 n=1

(337)

(338)

where Li2 is the dilogarithm function [240]. Bose–Einstein condensation occurs in a three-dimensional ideal gas of bosons: above a certain critical density the extra particles condense in the lowest energy level and do not contribute to the pressure. The twodimensional ideal boson gas almost, but not quite, shows the same phenomenon. In fact it has a finite pressure at infinite density, P∞ =

Li2 (0) π2 πm = = · 2 βΛ 6βΛ2 12~2 β 2

(339)

From equation (338) we get that ∞ X n ±Λ2 ρ Bn ∂(βP ) = ±Λ2 ρ ±Λ2 ρ , = ∂ρ e − 1 N =0 n!

(340)

where Bn is the n-th Bernoulli number [240]. Thus, Bn−1 ±Λ2 An = n!

n−1 ·

(341)

352

Topological Aspects of Low Dimensional Systems 2

The function ρ/(e±Λ ρ − 1) is well behaved for all real values of ρ, but has poles in the complex ρ plane at Λ2 ρ = 2nπi, for any non-zero integer n. This means that equation (340) can be integrated to give P as a well defined function of ρ anywhere on the real axis, while the virial expansion converges for Λ2 |ρ| < 2π but diverges for Λ2 |ρ| > 2π. Note that the sum in equation (338) has different convergence properties, it converges for bosons at any non-negative density ρ but for fermions only when Λ2 ρ < ln 2. The relation between the boson or fermion ideal gas virial coefficients in two dimensions, and the Bernoulli numbers, was derived by Sen and by Viefers [134, 147, 278]. In particular, the even numbered virial coefficients A4 , A6 , etc., all except A2 , vanish both for bosons and fermions. Since the odd numbered coefficients are the same for bosons and fermions, the only difference is the sign of A2 . The lowest coefficients are Λ2 Λ4 Λ8 , A3 = , A5 = − , 4 36 3600 A4 = A6 = A8 = . . . = 0. A2 = ∓

A7 =

Λ12 , 211 680 (342)

It may be instructive to rederive the results for bosons and fermions using the harmonic oscillator regularization. The one-particle partition function for the harmonic oscillator in d dimensions is !d ∞ X 1 −ξ (n+ 12 ) e = (343) Z1 (β) = d , ξ n=0 2 sinh 2 with ξ = ~ωβ, as before. We may use equation (333), and translate from one regularization scheme to the other by the rule of substitution √ Abn → (Λ/( nξ))d bn . This gives the same formula as before for the cluster coefficients of free bosons or fermions, bn =

√ d ξ n (±1)n+1 Z1 (nβ) Λ n

→

ω→0

(±1)n+1 √ · n(Λ n)d

(344)

8.6 Two anyons The case of two anyons can be treated most simply by means of the harmonic oscillator regularization, since the energy spectrum is discrete and exactly known. Let us define the periodic sawtooth function α(θ + 2nπ) =

|θ| π

for

|θ| ≤ π,

n = 0, ±1, ±2, . . .

(345)

In terms of it the two-anyon partition function in a harmonic oscillator

J. Myrheim: Anyons

353

potential is Z2 (β) =

cosh((1 − α)ξ) , 2 2 sinh2 ξ 2 sinh ξ2

(346)

where the factor Z1 (β) = 1/(2 sinh(ξ/2))2 is due to the motion of the centre of mass. Note that, in spite of the fact that the function α = α(θ) is nondifferentiable both at the boson point θ = 0 and the fermion point θ = π, the partition function Z2 , as a function of θ, is non-analytic only at the boson point. From equation (346) and equation (327) we get the well-known result for the second virial coefficient of the ideal gas of anyons, in the limit ξ → 0, 1 (1 − α)2 − · (347) A2 = Λ2 4 2 There are two partitions of two, one even, 1 + 1 = 2, and one odd, 2 = 2. According to equation (160) we may write Z2 as a sum over these partitions, Z2 (β) =

1 1 2 F11 (θ) (Z1 (β)) + F2 (θ) Z1 (2β) . 2 2

(348)

By equation (157), F11 (θ + π) = F11 (θ) and F2 (θ + π) = −F2 (θ). The substitution θ → θ + π is essentially the supersymmetry transformation introduced by Sen [132, 133], and it changes α into 1 − α. Hence, splitting the partition function in equation (346) in two parts, one even and one odd under the substitution α → 1 − α, we deduce that cosh α − 12 ξ sinh α − 12 ξ , F2 (θ) = − · (349) F11 (θ) = cosh 2ξ sinh 2ξ From the probability generating functions F11 and F2 we get the probability distributions of the winding number Q, 2ξ tanh ξ2 (Q even), P11 (Q) = 2 ξ + (πQ)2 2ξ coth ξ2 (Q odd). (350) P2 (Q) = 2 ξ + (πQ)2 Note that we have defined the winding number Q as twice the number of windings around the origin, so that the probability distribution of even winding numbers, P11 (Q), is the same distribution as given by Wiegel and by Ouvry [130, 266].

354

Topological Aspects of Low Dimensional Systems

If we do not distinguish between even and odd winding numbers, then we may write Z2 (β) = F (θ)

cosh ξ , 2 ξ 2 2 sinh ξ 2 sinh 2

(351)

where F (θ) is the total probability generating function for even and odd winding numbers, F (θ) =

∞ X

P (Q) e−iθQ =

Q=−∞

cosh((1 − α)ξ) , cosh ξ

(352)

and P (Q) is the probability of the winding number Q, P (Q) =

ξ tanh ξ · + (πQ)2

(353)

ξ2

From the above expressions we also get the G coefficients of free anyons, by equation (329), G11 = 2 lim (F11 − 1)Z1 = α(α − 1) , ω→0

G2 = lim F2 = 1 − 2α . (354) ω→0

8.7 Three anyons The harmonic oscillator regularization is useful also for three anyons, even though the “non-linear” part of the energy spectrum has to be computed numerically. The third virial coefficient A3 has been calculated with high precision by the direct method of computing energy levels numerically and summing to get the partition function [122, 143, 144]. In order to compute the virial coefficient from the spectrum, we should incorporate all the exact knowledge we have. Therefore we subtract the known bosonic partition function Z3B and write the anyonic partition function as Z3 = Z3B + ∆Z3 . Then we split the remainder further into contributions of energy levels that are linear and non-linear functions of the statistics angle θ, writing ∆Z3 = ∆Z3L + ∆Z3NL . The “linear” part is exactly known, Z3L (β) =

cosh(3(1 − α)ξ) , 32 sinh2 2ξ sinh2 ξ sinh2 3ξ 2

(355)

and from Z3L we get ∆Z3L by subtracting the value for α = 0. For the “nonlinear” part the best we can do is to split off the Z1 contribution due to the centre of mass, and the factor due to the tower structure of the spectrum, writing ∆Z3NL =

Z1 ∆Z . 1 − e−2ξ

(356)

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All in all we then get 4 Λ Z22 Z3 Z2 2 − 10 2 + 16 4 − 6 3 A3 = lim ξ→0 ξ Z1 Z1 Z1 ∆Z 1 − 3 lim · = Λ4 ξ→0 ξ 36

(357)

This is an important simplification, since ∆Z has contributions only from “non-linear” states that are “bottom” states of infinite towers. All the θ dependence of the third virial coefficient is seen to come from these states. One may calculate ∆Z numerically by calculating the lowest energy levels, then summing over them and extrapolating to the infinite sum as best one can. The most accurate calculation, including all bottom of tower energy levels up to around 40~ω, gave the result [122] sin2 θ 1 4 + + a sin θ , (358) A3 = Λ2 36 12π 2 with a coefficient a = −(1.652 ± 0.012) × 10−5 = −

1 · (621 ± 5)π 4

(359)

The first and second order perturbative calculations give the exact coefficient of the sin2 θ term, and it is a non-trivial consistency check that the numerical calculation reproduces the exact result with high precision. The coefficient of sin4 θ could in principle be calculated exactly by fourth-order perturbation theory, although this possibility seems rather remote. At this point some general observations might be in order. All observable properties of anyons must be periodic functions of θ with period 2π. Energy eigenvalues and eigenfunctions are analytic functions of θ, except that some are non-analytic at θ = 0, varying like |θ| rather than θ. Hence the partition functions and all thermodynamic quantities derivable from them will be analytic functions of θ, even at the fermion point θ = π, but generally not at the boson point θ = 0. In the absence of an external magnetic field, the theory is both time reversal and parity invariant if each of these transformations is defined so as to include a change in sign of θ. It follows that energy eigenvalues and thermodynamic quantities, as functions of θ, must be symmetric about θ = 0, hence they are functions of the quantity α(θ) defined in equation (345), which is non-analytic in θ at the boson and fermion points. Any even polynomial (or convergent power series) in α is analytic at the boson point, and any even polynomial in 1 − α is analytic at the fermion point. An example is the exact second virial coefficient, which is an even polynomial in 1 − α.

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Topological Aspects of Low Dimensional Systems

The third virial coefficient is analytic at the boson as well as the fermion point, because it is “supersymmetric”, i.e. symmetric under the substitution α → 1 − α [132, 133]. Being analytic everywhere, and periodic in θ with period π, it can not be polynomial in θ, because periodic polynomials do not exist, except constants. But it should be expandable as a rapidly converging power series in sin2 θ, as the above formula indicates. In fact, this is one possible way to represent a Fourier series with the desired properties. The three-anyon partition function may be written as a sum over three partitions, Z3 (β) =

1 1 3 F111 (θ) (Z1 (β)) + F3 (θ) Z1 (3β) 6 3 1 + F21 (θ) Z1 (2β) Z1 (β) . 2

(360)

The first two terms are even and the last term is odd under the supersymmetry transformation of Sen, θ → θ + π [132, 133]. As observed by Sen, the odd part of Z3L , which is Z3L (β, α) − Z3L (β, 1 − α) =

cosh(3(1 − α)ξ) − cosh(3αξ) , 32 sinh2 ξ2 sinh2 ξ sinh2 3ξ 2

(361)

when taken at α = 0, completely accounts for the difference between the bosonic and the fermionic partition functions, which is simply Z1 (2β)Z1 (β). Since the number of states does not vary with θ, it is therefore possible, and indeed true according to Sen, that the odd part of Z3 is identical to the odd part of Z3L also for intermediate values of θ. We compute from this that sinh α − 12 3ξ · (362) F21 (θ) = − sinh 3ξ 2 Note the similarity between the two-particle and three-particle functions F2 (θ) and F21 (θ). From F21 we get the exact G21 coefficient of free anyons, by equation (329), G21 =

3 lim (F21 − F2 )Z1 = 2(1 − 2α)α(α − 1) = 2F2 G11 . 2 ω→0

(363)

8.8 The Monte Carlo method Although A3 has been calculated with very high precision from numerical energy levels, it is useful to discuss here also how it can be calculated by the Monte Carlo method. This method is less precise for three anyons [145], but is so far the only one available for four anyons [146]. The Monte Carlo

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method as such is important because it is a very natural approach to path integrals, and it leads to some general understanding of how they behave. For a Monte Carlo calculation we need a more explicit expression for the propagator, as defined in equation (126). What we need here is only the propagator for the harmonic oscillator in two dimensions, mω ωτ mω exp − tanh |y + x|2 G(y, x; τ ) = 2π~ sinh(ωτ ) 4~ 2 ωτ |y − x|2 , + coth (364) 2 since the many-particle paths we want, may be reinterpreted as consisting of independent cyclic one-particle paths. Consider one cyclic one-particle path from τ = 0 to τ = L~β. In the path integral formula for the partition function Z1 (L~β) every point x(τ ) on the path is integrated out. If we integrate out every point except one, say the point x(0) = x(L~β) = x, then we get instead the propagator G(x, x; L~β). It follows that the probability density for the single point x on the path is Lξ Lξ mω mω G(x, x; L~β) = tanh tanh exp − |x|2 .(365) ρ(x) = Z1 (L~β) π~ 2 ~ 2 This is a two-dimensional normal probability distribution of mean zero and standard deviation s Lξ ~ 1 √ coth . (366) −→ σ0 = ω→0 2mω 2 ω mLβ Thus, loosely speaking, the point x = x(0) = x(L~β) is located inside an area proportional to 1/ω 2 in the limit ω → 0. By a similar reasoning, if we specify three imaginary times τa < τ < τb and the two points x(τa ) = xa and x(τb ) = xb on the path, then the probability density of the position x(τ ) = x on the path is proportional to the product G(xb , x; τb − τ )G(x, xa ; τ − τa ) of two propagators. Again this defines a normal distribution, of mean xτ =

sinh(ω(τb − τ )) xa + sinh(ω(τ − τa )) xb sinh(ω(τb − τa ))

(367)

and standard deviation s s ~ sinh(ω(τb − τ )) sinh(ω(τ − τa )) ~(τb − τ )(τ − τa ) −→ · (368) στ = ω→0 mω sinh(ω(τb − τa )) m(τb − τa ) The fact that στ tends to a finite, non-zero limit when ω → 0, means that the area covered by a single cyclic path tends to a finite limit.

358

Topological Aspects of Low Dimensional Systems

The asymptotic behaviour of σ0 and στ in the limit ω → 0 means that the probability that two cycles overlap, tends to zero as ω 2 . Furthermore, the probability that three cycles overlap simultaneously, tends to zero as ω 4 . Hence the winding number distribution for the three distinct cycles belonging to the partition 1 + 1 + 1 = 3 is determined by the two-particle windings, up to correction terms of order ω 4 . Which means that we may write (4) 3 (369) F111 (θ) = (F11 (θ)) 1 + F111 (θ)ξ 4 + O(ξ 5 ) . (0)

It also follows that the limit F3 (θ) = limω→0 F3 (θ) is finite, so that (0)

F3 (θ) = F3 (θ) + O(ξ).

(370)

These results for the probability generating functions F111 and F3 , together with the exact result for F21 , implies that the third virial coefficient is finite, ! 2 4 7 1 2 (0) 1 1 1 (4) 4 + + − F111 (θ) − F3 (θ) .(371) α− α− A3 = Λ 64 8 2 4 2 9 The supersymmetry of Sen is manifest in this formula for A3 . The equations (366, 367) and (368) are all we need in order to make a Monte Carlo simulation of the three-particle paths. The numerical results (0) [145] suggested the following simple formula for F3 (θ), which has since been proved to all orders in perturbation theory [270], 2 1 9 3 1 (0) (372) = (1 − 3α) 1 − α . α− F3 (θ) = − + 8 2 2 2 (4)

This formula, together with an estimate of the fourth order term F111 (θ), then gave the sin2 θ interpolation between the second order perturbative results for bosons and fermions. 8.9 The path integral representation of the coefficients GP The coefficient GP for a given partition P representing a conjugation class in the symmetric group SN , can be represented as a path integral over all paths inducing one given permutation represented by P, Z S (373) gP . GP Z1 = NP D(x1 (τ ), . . . , xN (τ )) exp − ~ Here xj (τ ) is the path of particle j, as a function of the imaginary time τ , and S is the free particle action in imaginary time, 2 N Z ~β X m dxj (τ ) dτ · (374) S= 2 dτ j=1 0

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This representation is useful for numerical computation by the Monte Carlo method, but can also be used in order to prove that the cluster coefficients are finite. We include the Gaussian factor exp(−S/~) as part of the integration measure, so that it is the integrand gP alone that represents the interaction of the particles, and we include a normalization factor NP so that GP = Z1 ν−1 if gP = 1 identically. Note that NP is then finite (i.e. A independent), since the path integral is proportional to Z1 ν when gP = 1. Note also that this path integral representation is actually very general, and can be applied to any N -particle system with (short range) interactions in any dimension, not just to the N -anyon system considered here. To see what the integrand gP looks like in our case, let us take the partition 2+1+1 of 4 as an example. A closed path in the four-particle configuration space interchanges the positions of two particles, say particles 1 and 2, and takes the remaining two particles back to their starting points. The total winding number Q is the sum of six pairwise winding numbers, Q = Q12 + (Q13 + Q23 ) + (Q14 + Q24 ) + Q34 .

(375)

Note that Q12 is an odd integer and Q34 an even integer (remember that the winding numbers are defined such that a complete revolution corresponds to the winding number 2), whereas Q13 , Q23 , Q14 , Q24 are in general noninteger, because particles 1 and 2 do not return to their starting positions. However, the sums Q(12)3 = Q13 + Q23 and Q(12)4 = Q14 + Q24 are even integers. Hence Q is an odd integer. Let I be any subscript, and introduce the notation eI = 1 + fI = exp(−iθQI ) .

(376)

In order to compute the coefficient G211 Z1 = (F211 −2F21 −F2 F11 +2F2 )Z1 3 we take the integrand to be g211 = e12 e(12)3 e(12)4 e34 − e12 e(12)3 − e12 e(12)4 − e12 e34 + 2e12 = e12 f(12)3 f(12)4 f34 + f(12)3 f(12)4 +f(12)3 f34 + f(12)4 f34 .

(377)

For example, we compute F211 Z1 3 by integrating exp(−iθQ) = e12 e(12)3 e(12)4 e34 ,

(378)

and we compute 2F21 Z1 3 by integrating exp(−iθ(Q12 + Q(12)3 )) + exp(−iθ(Q12 + Q(12)4 )) = e12 e(12)3 + e12 e(12)4 . (379)

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Topological Aspects of Low Dimensional Systems

Equation (377) may be represented diagrammatically as

G211 Z1 =

t t A + tAk t

t t t t A +2 A tAk t tA k t

(380)

The particles are represented as points (filled circles). The two-cycle is represented by e12 in the integrand and by a circle connecting two particles in the corresponding diagram. Each factor fI in the integrand is drawn as a single straight line in the diagram. Note that we should draw labelled graphs to represent the four terms in equation (377). But since the value of a graph is independent of the labelling, it is more natural to draw unlabelled graphs and include instead integer coefficients counting the number of ways each graph can be labelled. Hence the factor 2 in front of the last graph. In a similar way we find the diagrammatic representation t t t t t @ G1111 Z1 = t @t + 6 t t + 12 t t t t t +4 + 12 t t t t

t

t

t

t +3 t

t (381)

The coefficient in front of each diagram is again the number of inequivalent ways of labelling the nodes of the graph. We may also write t

tk t

tm t (382) t We see that only connected diagrams contribute to the cluster coefficients. It follows that the latter are finite in the limit A → ∞. Indeed, any path gives a non-zero contribution to the path integral represented by some diagram only if for every line in the diagram, the corresponding winding number is non-zero. The probability for this to happen for a connected diagram goes to zero as (Λ2 /A)ν−1 when A → ∞, since every L-cycle path gives a Gaussian distribution of points which essentially covers only a finite area, proportional to Λ2 . Here ν is the number of cycles, and ν −1 is the minimum number of links in a connected graph with ν nodes. The factor A−ν+1 cancels exactly the divergence of the factor Z1 ν−1 included in the definition of GP , equation (319). The general meaning of the relations between the F and G coefficients should now be obvious. FP is a sum of both connected and disconnected diagrams, whereas GP is the part of the sum including only the connected diagrams. For example, the relation G22 Z1 =

tk t

,

G31 Z1 =

F211 Z1 3 = G211 Z1 + 2G21 G1 Z1 2 + G2 G11 Z1 2 + G2 G1 G1 Z1 3 ,

(383)

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which follows from (319), is represented as t t A + tAk t

F211 Z1 3 = |

t +

t

t t A +2 tA k t {z G211 Z1 t t

tk t + tk t | {z } | {z } G2 G11 Z1 2 G2 G1 G1 Z1 3

t t t t A +2 A tAk t tAk t } | {z } 2G21 G1 Z1 2 (384)

It is the last term that dominates in the thermodynamic limit, but it is G211 only that contributes to the cluster coefficient. Thus, as usual, the grand partition function is a sum of all diagrams but the thermodynamic potential is a sum of connected diagrams [279]. The Monte Carlo method consists in generating random paths according to the Gaussian distribution of paths valid for bosons. Each four-particle path is closed over the imaginary time interval ~β, in the sense that the final configuration is identical to the initial one, but with the particle positions interchanged by a permutation belonging to the class P ⊂ SN . Consider the partition 2 + 1 + 1 = 4, as in the example above. Then particles 1 and 2 should interchange positions, while particles 3 and 4 should return to their starting points. We take, arbitrarily, the starting point for the path of particle 1 to be at the origin, this is then also the ending point for particle 2. Equivalently, it is the ending point for particle 1 over the imaginary time interval 2~β. The starting point for particle 2, equal to the position of particle 1 after half the imaginary time interval 2~β, can then be generated according to a Gaussian distribution around the origin. The starting and ending point for particle 3 is generated according to a flat distribution inside a square area A centered on the origin. Similarly for particle 4. For each four-particle path generated we count the winding numbers Q12 , Q(12)3 , Q(12)4 , Q34 and increment a histogram n(Q) in the following way. We compute the total winding number Q and add 1 to n(Q), this takes care of the integrand e12 e(12)3 e(12)4 e34 . We subtract 1 from n(Q12 + Q(12)3 ), in order to take care of the integrand −e12 e(12)3 . Similarly, we subtract 1 from n(Q12 + Q(12)4 ) and from n(Q12 + Q34 ), and we add 2 to n(Q12 ). Finally, G211 is the Fourier transform of the histogram n(Q), multiplied by the normalization factor Z1 2 /n, where n is the total number of four-particle paths generated. The net contribution to the histogram vanishes if more than one of the three winding numbers Q(12)3 , Q(12)4 and Q34 is zero, and this is what ensures a finite limit as A → ∞ for the computed G211 .

362

Topological Aspects of Low Dimensional Systems

8.10 Exact and approximate polynomials The first cluster coefficient, with our definition, is b1 = 1/Λ2 . By the harmonic oscillator regularization we found that G11 = α(α − 1) ,

G2 = F2 = 1 − 2α .

(385)

G11 is even and G2 is odd under the substitution α → 1 − α, and together they give the second cluster coefficient of free anyons, Λ2 b2 =

G2 1 G11 1 + = (1 − α)2 − · 2 4 2 4

(386)

We also obtained the exact result G21 = 2F2 G11 .

(387)

The third virial coefficient, A3 = −2

b3 b2 2 + 4 , b1 3 b1 4

(388)

is even under α → 1−α, since the odd part of −2b3 /b1 3 , which is −Λ4 G21 /2, cancels the odd part of 4b2 2 /b1 4 , which is Λ4 G11 G2 . One further result [280], which is exact according to the perturbative calculation of reference [270], is FL =

L−1 Y k=1

Lα 1− k

·

(389)

Various Monte Carlo simulations for single cycles of different lengths are consistent with this formula. Note that only harmonic oscillator energy levels depending linearly on α contribute to the three quantities G2 = F2 , G11 and G21 . The computation of GL = FL for L > 2 by perturbation theory is much more non-trivial, because also states with non-linear α dependence contribute. This is essentially the only exact result known for the non-linear energy levels. This is about as far as one can get with exact results. However, in the diagrammatic expansions shown above, one may argue quite generally that the tree graphs are expected to dominate, because every additional line in a diagram represents another factor of the type fI = exp(−iθQI ) − 1 in the integrand, with QI an even integer. This factor vanishes when QI = 0, which will happen with a certain probability which is definitely non-zero, and even if it does not vanish it will often have an absolute value smaller than 1. Furthermore, one may argue that the path integral represented by a tree graph should approximately factorize in the same way as its integrand.

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These two assumptions, of tree diagram dominance and factorization, lead in a not entirely trivial way to the following polynomial approximation for the general coefficient GP , Y e P = N ν−2 G11 ν−1 (LFL )νL . (390) GP ' G L

There is a factor FL for every cycle of length L, a factor L1 L2 G11 for every single line connecting two different cycles of lengths L1 and L2 (each L-factor counts the number of ways the line can be connected to the cycle), and there is a sum over all ν ν−2 possible ways to connect the cycles into a tree graph. It is perhaps not obvious how this leads to equation (390), but a simple way to understand the connection is by looking at low order examples: consider the case of 3 cycles of lengths L1 , L2 , and L3 . They can be connected to a tree graph in 3 possible ways. This gives a coefficient FL1 FL2 FL3 (L1 L2 G11 L2 L3 G11 + L2 L3 G11 L3 L1 G11 +L3 L1 G11 L1 L2 G11 ) = (L1 + L2 + L3 ) G11 2

3 Y

Li FLi ,

(391) (392)

i=1

which agrees with equation (390) since L1 + L2 + L3 = N . We should point out that equation (390) was first derived empirically as an approximate representation of the Monte Carlo results. Special cases where these polynomial formulae are exact, as already mentioned, are the cyclic coefficients GL = FL , as well as G11 and G21 = 2F2 G11 . In the three-particle case there is one approximate polynomial, e111 = 3G11 2 . G

(393)

The four-particle approximate polynomials are: e 211 = 8F2 G11 2 , e1111 = 16G11 3 , G G e 22 = 4F2 2 G11 , G e 31 = 3F3 G11 . G

(394)

The polynomial approximations for the G coefficients imply the following polynomial approximations for the cluster coefficients, Λ2ebN

N −1 N −1 (−1)N −1 Y 1 Y N (1 − α)2 Ng = = 2 , 1− 1− N2 k N k k=1

(395)

k=1

which imply that the virial coefficients are independent of the statistics, except for the second coefficient, 1 (1 − α)2 1 g − = Λ2 − + · (396) A2 = Λ2 4 2 4 2

364

Topological Aspects of Low Dimensional Systems

The parameter g is defined here by g = 1 − (1 − α)2 .

(397)

One nice property of these polynomials is that they are analytic functions of θ at the fermion point, as the exact cluster coefficients must be. However, they do not give the correct second derivatives at the boson and fermion points, known from perturbation theory, although they do give the correct first derivatives. An alternative way to introduce the same polynomials is to postulate that the second virial coefficient is given by equation (396), while all higher virial coefficients are independent of α. That is, these are just cluster coefficients for two-dimensional exclusion statistics [236], with the statistics parameter given by (397). The corresponding second order diagrams were identified in reference [138]. This correspondence with exclusion statistics is of course only approximate, and it is well known from perturbation theory that the higher virial coefficients of anyons all have a second order variation with θ at the boson and fermion points. Nevertheless it might be interesting to understand better the deeper reasons behind, if any. 8.11 The fourth virial coefficient of anyons Since the third virial coefficient is analytic in θ everywhere, in contrast to the cluster coefficients, which are all non-analytic at the boson point, one may be bold enough to conjecture that all virial coefficients, with the exception of A2 , are analytic functions of θ. If A4 is analytic, then it must have the form " √ 2 1 6 sin θ √ ln 3 + 2 + cos θ A4 = Λ 16π 2 3 # + sin4 θ (c4 + d4 cos θ) + · · · ,

(398)

where the coefficients of the lowest order terms are fixed by perturbation theory at the boson and fermion points. A Monte Carlo calculation of A4 gave indeed a result which was fitted to this form with no more than two parameters [146], c4 = −0.0053 ± 0.0003 ,

d4 = −0.0048 ± 0.0009 .

(399)

This fit is shown in Figure 13. It is rather remarkable how nearly constant the computed A4 is, that is, how close to zero it is for all values of θ. In fact, it is closer to zero than the minimal Fourier series with c4 = d4 = c6 = d6 = . . . = 0. This is one particular example of how well the anyon system realizes approximately Haldane’s

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365

0.1 Re(MC)-polynomial Im(MC) fit 0.08

0.06

0.04

0.02

0

0

0.1

0.2

0.3

0.4

0.5

Fig. 7. G1111 − 16(α(α − 1))3 as a function of α. The imaginary part is plotted to indicate the statistical uncertainty of the real part of the Monte Carlo data. Only the interval 0 ≤ α ≤ 1/2 is plotted, because of the (anti)symmetry about α = 1/2. The curve marked “fit” is given in equation (400). From [146].

so-called exclusion statistics [227], characterized by a continuously variable parameter g, for which only the second virial coefficient depends on g, in two dimensions [236]. The computed G coefficients are plotted in Figures 7 to 11, as functions of α. In each case we subtract the polynomial approximation, which is the main contribution, and plot only the difference, marked by “Re(MC) − polynomial” in the figures. Because of the statistical errors, the Monte Carlo generated curve has also a non-zero imaginary part, marked “Im(MC)”, which is useful because it indicates the statistical errors in the real part. Since the real part is even about α = 1/2 and the imaginary part is odd, or vice versa, depending on whether the partition is even or odd, only the interval 0 ≤ α ≤ 1/2 is plotted in all figures. Figure 7 shows the computed G1111 with the polynomial 16G11 3 = 16(α(α − 1))3 subtracted. The curve marked “fit” is mostly empirical, and

366

Topological Aspects of Low Dimensional Systems

Re(MC)-polynomial Im(MC) fit 0

-0.005

-0.01

-0.015

0

0.1

0.2

0.3

0.4

0.5

Fig. 8. G211 − 8(1 − 2α)(α(α − 1))2 versus α. The curve marked “fit” is given in equation (401). From [146].

is given by fit = −

3 α(α − 1) sin2(απ) . π2

(400)

The figure shows that this is a perfect fit to the Monte Carlo curve, within the statistical uncertainty as indicated by the imaginary part. Figure 8 shows the computed G211 with the polynomial 8F2 G11 2 = 8(1 − 2α)(α(α − 1))2 subtracted. The curve marked “fit” is partly empirical, but with a coefficient which is chosen so as to produce the correct second order derivative at α = 0 [270]. The formula is: fit = −

2 (1 − 2α) sin2(απ) . 3π 2

(401)

Figure 9 shows the computed G22 with 4F2 2 G11 = 4(1 − 2α)2 α(α − 1) subtracted. The “fit” here is √ 2 ln( 3 + 2) sin2(απ) cos2(απ) . (402) fit = √ 3π 2 Figure 10 shows the computed G31 with 3F3 G11 = 3(1 − 3α)(1 − (3/2)α)α

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0.04 Re(MC)-polynomial Im(MC) fit

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.01 0

0.1

0.2

0.3

0.4

0.5

Fig. 9. G22 −4(1−2α)2 α(α−1) versus α. The “fit” here is given in equation (402). From [146].

(α − 1) subtracted. Here we have chosen √ √ 3 ln( 3 + 2) sin2(απ) cos2(απ) . fit = 2 4π

(403)

Figure 11 shows the computed G4 = F4 with the polynomial (1 − 4α) (1 − 2α)(1 − (4/3)α) subtracted. The figure supports the claim that the polynomial is exact. Figure 12 shows the computed cluster coefficient, Λ2 b4 with the polynomial Λ2eb4 of equation (395) subtracted. The parabolas given by the second order perturbation theory at α = 0 and α = 1 are shown. Figure 13 shows the computed virial coefficient, A4 /Λ6 . The parabolas given by the second order perturbation theory at α = 0 and α = 1 are shown. Also plotted are two Fourier series, as given in equation (398). The curve marked “Fourier 1” is a minimal Fourier series having only the two terms required by perturbation theory, i.e. c4 = d4 = . . . = 0. The curve marked “Fourier 2” is a least squares fit with the coefficients c4 = −0.0053 and d4 = −0.0048. The minimal Fourier series is seen to be inconsistent with the Monte Carlo curve, unless there are important systematical errors that dominate over the statistical errors.

368

Topological Aspects of Low Dimensional Systems

Re(MC)-polynomial Im(MC) fit

0

-0.005

-0.01

-0.015 0

0.1

0.2

0.3

0.4

0.5

Fig. 10. G31 − 3(1 − 3α)(1 − (3/2)α)α(α − 1) versus α. The “fit” is given in equation (403). From [146].

8.12 Two polynomial theorems As a mathematical appendix to the present Section we will prove the result that the polynomial approximation (395) for the cluster coefficients is equivalent to a virial expansion which is the same as for the two-dimensional free non-relativistic Bose gas, except that the second virial coefficient is modified according to equation (396). We will also prove that the polynomial approximation (390) for GP implies equation (395). For simplicity we fix the temperature and choose units such that β = Λ = 1. Thus, e.g., the fugacity is z = eµ . We make use of the expansions ρ =

∞ X dP = N bN z N , dµ N =1

∞ X 1 dP dµ 1 dP dµ = = = N AN ρN −2 . dρ ρ dµ dρ ρ dρ

(404)

N =1

We also define ρg (µ) =

X ∞ N −1 ∞ X Ng zN Y (−1)N −1 N g , = zN 1− N N k Ng

N =1

k=1

N =1

(405)

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0.003 Re(MC)-polynomial Im(MC) 0.002

0.001

0

-0.001

-0.002

-0.003 0

0.1

0.2

0.3

0.4

0.5

Fig. 11. F4 − (1 − 4α)(1 − 2α)(1 − (4/3)α) versus α. From [146].

which is the density corresponding to the cluster coefficients ebN of equation (395). For the Bose gas, with g = 0, we have ρ = − ln(1 − z), or equivalently, µ = ln z = ln(1 − e−ρ ). Shifting the second virial coefficient by an amount ∆A2 = g/2 then gives (406) µ = ln 1 − e−ρ + gρ . For every g > 0 and every µ, or for g = 0 and every µ < 0, this equation clearly has a unique solution ρ > 0. We want to prove that the solution is ρ = ρg (µ). For this purpose we rewrite equation (406) as ∞ X z n −ngρ e , ρ = − ln 1 − ze−gρ = n n=1

(407)

and apply the following theorem due to Lagrange (see Vol. 1, pp. 404–405 of [281], or [282]): The equation ρ = f (ρ) has the solution M−1 ∞ X d 1 M f (r) . (408) ρ= M ! dr M=1

r=0

370

Topological Aspects of Low Dimensional Systems

0.005 Re(MC)-polynomial Im(MC) 2nd order 0.004

0.003

0.002

0.001

0

-0.001

-0.002 0

0.2

0.4

0.6

0.8

1

Fig. 12. The fourth cluster coefficient minus the polynomial of equation (395), Λ2 (b4 − e b4 ), as a function of α. Also shown are the parabolas given by the second order perturbation theory at α = 0 and α = 1. From [146].

This gives M−1 ∞ ∞ ∞ X X d 1 X z n1 +···+nM −(n1 +···+nM )gr ··· e ρ = M ! n =1 n1 · · · nM dr n =1 M=1

=

∞ X N =1

1

zN

N X

M

M−1

(−N g)

CN,M ,

r=0

(409)

M=1

where CN,M =

∞ ∞ X 1 X δn1 +···+nM ,N ··· · M ! n =1 n =1 n1 · · · nM 1

(410)

M

What we need to show is that N X M=1

(−N g)M−1 CN,M =

(−1)N −1 Ng

Ng N

.

(411)

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371

Re(MC) Im(MC) Fourier 1 Fourier 2 2nd order

0.006

0.005

0.004

0.003

0.002

0.001

0

-0.001

-0.002 0

0.2

0.4

0.6

0.8

1

Fig. 13. The fourth virial coefficient, A4 /Λ6 , as a function of α. Also plotted are the parabolas given by the second order perturbation theory at α = 0 and α = 1, and two different Fourier series, as given in equation (398). The curve marked “Fourier 1” has c4 = d4 = . . . = 0, whereas “Fourier 2” is the least squares fit with c4 = −0.0053, d4 = −0.0048. From [146], reprinted with permission.

It is straightforward to show that ∞ X

z

N =1

N

N X

M

g CN,M = e

−g ln(1−z)

−1=

M=1

∞ X

N

(−z)

N =1

−g N

,

(412)

and hence, N X

g

M−1

M=1

CN,M

(−1)N = g

−g N

.

(413)

Substituting g → −N g we get equation (411), completing the proof. We next turn to the cluster coefficients X Y FL νL N ν−2 G11 ν−1 , (414) b0N = νL ! LνL P∈CN

L

given by the polynomial approximation in equation (390). We want to prove that b0N = ebN .

372

Topological Aspects of Low Dimensional Systems

0.4 0.3 0.2 0.1

1

2

3

5

4

Fig. 14. d(ln I1 )/dρ2 as a function of ρ. This quantity measures the charge ratio q1 /e, see equation (527). The curves are for N = 2, 3, 4, 5 electrons, the leftmost peak for N = 2 and the rightmost peak for N = 5. The dashed line is 1/3.

We may rewrite the above formula as b0N =

ν N ∞ ∞ X X Y Fnj N ν−2 G11 ν−1 X ··· δn1 +···+nν ,N · ν! nj ν=1 n =1 n =1 j=1 1

(415)

ν

P∞ To evaluate ρ = N =1 N b0N z N we insert equation (415), interchange the summation of N and ν and use the relations N ν−1 z N = (d/dµ)ν−1 z N P∞ order n and n=1 z Fn /n = ρα (µ). We find ν−1 ∞ X d G11 ν−1 ν (ρα (µ)) ρ= ν! dµ ν=1 ν−1 ∞ X d 1 ν = (ρα (µ + G11 r)) ν! dr ν=1

.

(416)

r=0

By the Lagrange theorem, equation (416) is the solution to the equation ρ = ρα (µ + G11 ρ), which, as we saw above, is equivalent to µ + G11 ρ = ln 1 − e−ρ + αρ .

(417)

This is precisely equation (406) with g = α − G11 = 1 − (1 − α)2 , which means that b0N = ebN with ebN as given in equation (395).

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373

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

5

10

15 r

20

25

30

Fig. 15. The quasi-hole charge q1 /e, equation (527), as a function of ρ, the quasihole distance from the origin. The curves are, from left to right, for 20, 50, 75, 100 and 200 electrons. The horizontal line is 1/3. From [286], reprinted with permission.

9

Charged particles in a constant magnetic field

The quasi-particle excitations in the fractional quantum Hall system is so far the best (and maybe the only) experimental realization of anyons. The electrons of the two-dimensional electron gas, as well as the anyon-like quasi-particles, are electrically charged and therefore strongly influenced by the magnetic field. If the field is sufficiently strong, it effectively “freezes out” one degree of freedom, so that in a certain well defined sense the system becomes one-dimensional. The quantization problem for charged particles in a constant magnetic field reduces to the simultaneous quantization of energy and total angular momentum in a harmonic oscillator potential, discussed at length in Section 7 above. In the present context we want to discuss also one more topic, the coherent states, which are interesting because they are maximally localized ground states in the one-particle system. In particular, coherent states of anyons are supposed to be models of localized quasi-particles in the fractional quantum Hall system.

374

Topological Aspects of Low Dimensional Systems

9.1 One particle in a magnetic field The Hamiltonian for one particle of mass m and charge q in a constant magnetic field, in two dimensions, is H=

πx2 + πy2 π2 = , 2m 2m

(418)

where π = p − qA is the kinematical and p = −i~∇ the canonical momentum. The vector potential A = A(x) depends on the gauge. When the magnetic flux density B is constant, a convenient choice is the circular gauge, in which A = (Ax , Ay ) =

B (−y, x) . 2

(419)

In this gauge we have A2 = B 2 x2 /4, and p · A = A · p = BL/2, where L = xpy − ypx is the canonical angular momentum. Hence, H=

1 2 1 1 1 (p − qA)2 = p + mω 2 x2 ∓ ωL . 2m 2m 8 2

(420)

The last sign is − or + depending on whether the product qB is positive or negative, since we define the cyclotron frequency ω to be positive, ω=

|qB| · m

(421)

The commutator [πx , πy ] = i~qB

(422)

is gauge independent, and implies that H is formally just the Hamiltonian of a one-dimensional harmonic oscillator. We may define ∂ |qB| |qB| y = −i~ ± y, 2 ∂x 2 ∂ |qB| |qB| x = −i~ ∓ x, = py ∓ 2 ∂y 2

πx± = px ± πy±

(423)

so that π = π+ if qB > 0 and π = π− if qB < 0. The four operators πx± and πx± are a complete set of observables in the four-dimensional phase space, and they commute, except that [πx+ , πy+ ] = −[πx− , πy− ] = i~|qB| .

(424)

Since only one degree of freedom contributes to the energy, the second degree of freedom contributes only to the degeneracy of the energy levels, which

J. Myrheim: Anyons

375

are called Landau levels. Every level is infinitely degenerate, assuming of course that the system is infinite in extent. Another point of view is that H in the circular gauge is the Hamiltonian of a two-dimensional harmonic oscillator of angular frequency ω/2, plus an angular momentum term, and the large degeneracy is due to cancellations between the two contributions. One usually defines the magnetic length as s r ~ ~ = · (425) λ= |qB| mω In terms of the dimensionless complex coordinate x + iy , z= √ 2λ

(426)

and the corresponding differential operators ∂ = ∂/∂z and ∂ ∗ = ∂/∂z ∗, we have L = ~(z∂ − z ∗ ∂ ∗ ) , |z|2 ~ω ∓ (z∂ − z ∗ ∂ ∗ ) . −2∂∂ ∗ + H = 2 2

(427)

We define annihilation and creation operators, in the same way as before, λ z∗ =√ πy− + iπx− , 2 2~ λ z † ∗ πy− − iπx− , a = −∂ + = √ 2 2~ λ z −πy+ + iπx+ , b = ∂∗ + = √ 2 2~ ∗ λ z = √ −πy+ − iπx+ , b† = −∂ + 2 2~ a =

∂ +

(428)

such that [a, a† ] = [b, b† ] = 1 and [a, b] = . . . = 0. Then the canonical angular momentum is (429) L = ~ a† a − b † b , and the Hamiltonian is ~ω † a a + b † b + 1 ∓ a† a − b † b 2 ~ω b† b + 12 if qB > 0 , = if qB < 0 . ~ω a† a + 12

H=

(430)

376

Topological Aspects of Low Dimensional Systems

For simplicity, we will mostly assume from now on that qB > 0, the case qB < 0 is entirely analoguous. The normalized wave function |z|2 1 (431) χ0 = √ exp − 2 π is annihilated by both operators a and b, and is one of the infinitely many ground states of the one-particle system. When qB > 0, a complete orthonormal set of wave functions in the lowest Landau level are 1 1 |z|2 † n n z exp − , n = 0, 1, 2, . . . (432) χn = √ (a ) χ0 = √ 2 n! πn! The wave function χ0 is distinguished by being maximally localized near the origin. However, the system is completely translation invariant, in fact the operators πx− and πy− are generators of translation that commute with the Hamiltonian, when qB > 0. Therefore we may obtain a wave function of the lowest Landau level which is maximally localized near any arbitrary point z = ζ, simply by translating χ0 . The translated wave function is |z|2 + |ζ|2 1 ∗ √ exp − +ζ z χζ = π 2 ∗ ζ z − ζz ∗ |z − ζ|2 1 exp − · (433) = √ exp π 2 2 In the last expression the first exponential gives the phase and the last exponential the magnitude of the wave function. Thus χζ is complex, except when ζ = 0. It is a coherent state in the sense that it is an eigenstate of the annihilation operator a [64, 283], aχζ = ζ ∗ χζ . For later use, let us define the non-normalized wave function |z|2 + ζ∗z , ψζ = exp − 2 and compute the overlap integral for two such wave functions, Z Z 2 ∗ ∗ ∗ 2 ∗ d z (ψζa (z)) ψζb (z) = d2z e−|z| +ζa z +ζb z = π eζa ζb .

(434)

(435)

(436)

Perhaps the most direct way to obtain this answer is to integrate separately over the real and imaginary parts of z, writing 2 ζa + ζb∗ −|z|2 + ζa z ∗ + ζb∗ z = − Re z − 2 2 −ζa + ζb∗ − Im z − i + ζa ζb∗ . (437) 2

J. Myrheim: Anyons

377

The one-particle Hamiltonian in a constant magnetic field and in an external oscillator potential of angular frequency ω0 is H=

1 1 2 1 (p − qA)2 + mω0 2 x2 = p 2m 2 2m 1 1 + m(ω 2 + 4ω0 2 )x2 ∓ ωL . 8 2

(438)

We introduce the external potential here for the purpose of regularization, thus we want to take the limit ω0 → 0 at the end of our calculations. The oscillator frequency is changed from ω/2 to γω/2, with r 4ω0 2 (439) γ = 1+ 2 · ω As a consequence, if we do not modify the definition of z, we should substi√ tute everywhere γz for z. By quantizing L and H simultaneously, we find the energy eigenvalues 1 1 (440) ~ω1 + k + ~ω2 , Ej,k = j + 2 2 with j, k = 0, 1, 2, . . . , and with 1 p 2 ω + 4ω0 2 + ω , 2 1 p 2 ω + 4ω0 2 − ω . ω2 = 2

ω1 =

(441)

We see that ω1 → ω and ω2 → 0 as ω0 → 0, so that ~ω1 is the modified energy difference between Landau levels, whereas ~ω2 is the energy splitting within one Landau level due to the external potential. We see also that both ω1 → ω0 and ω2 → ω0 in the limit of zero magnetic field, ω → 0. These one-particle energy levels give the partition function Z1 =

X j,k

1

e

−βEj,k

e− 2 β~(ω1 +ω2 ) · = (1 − e−β~ω1 )(1 − e−β~ω2 )

(442)

9.2 Two anyons in a magnetic field The problem of many particles in a magnetic field falls into the class of problems with quadratic Hamiltonian which are exactly solvable for bosons or fermions, but not for anyons, except in a few special cases. Among those special cases is the problem of two anyons in a magnetic field, and this is still solvable if we add a harmonic oscillator interaction potential or an external harmonic oscillator potential, or both.

378

Topological Aspects of Low Dimensional Systems

The two-particle Hamiltonian, including an external potential, is H=

1 1 ((p1 − qA1 )2 + (p2 − qA2 )2 ) + mω0 2 (x1 2 + x2 2 ) . 2m 2

(443)

We introduce the anyon statistics by requiring an arbitrary wave function ψ to be multivalued, with ψ(x2 , x1 ) = eiθ ψ(x1 , x2 )

(444)

for an anticlockwise interchange of particle positions. Thus there is no contribution to the vector potentials from the statistics interaction. We assume that θ = νπ with 0 ≤ ν < 2. The motion of the two particles can again be decomposed into independent motions of the centre of mass position X = (x1 +x2 )/2 and the relative position x = x1 − x2 , with canonically conjugate momenta P = p1 + p2 and p = (p1 − p2 )/2. Let us introduce a similar notation for the vector potentials, writing A=

A1 + A2 , 2

a = A1 − A2 .

(445)

This gives the following expression for the Hamiltonian, H=

q 2 1 1 1 (P − 2qA)2 + p − a + mω0 2 X2 + mω0 2 x2 . 4m m 2 4

(446)

The centre of mass is a “particle” of mass 2m and charge 2q, whereas the relative coordinate describes a “particle” having a “reduced mass” of m/2 and a “reduced charge” of q/2. The ratio of charge to mass is the same for both, so that they have the same cyclotron frequency ω = |qB|/m. We introduce the complex coordinates z1 and z2 by the same definition as before, equation (426), with the same magnetic length λ. Then we define, quite naturally, Z = (z1 +z2 )/2 and z = z1 +z2 . However, we modify slightly the definitions p of annihilation and creation operators, including the scaling factor γ = 1 + (4ω0 2 /ω 2 ), a = b = c = d =

r γ ∗ 1 ∂ √ + Z , 2 2γ ∂Z r ∂ γ 1 √ Z, + ∗ 2 2γ ∂Z r r 2 ∂ γ z∗ + , γ ∂z 2 2 r r 2 ∂ γ z , + γ ∂z ∗ 2 2

r ∂ 1 γ Z, a = −√ + ∗ 2 2γ ∂Z r 1 ∂ γ ∗ b† = − √ + Z , 2 2γ ∂Z r r 2 ∂ γ z c† = − , + ∗ γ ∂z 2 2 r r 2 ∂ γ z∗ d† = − + · γ ∂z 2 2 †

(447)

J. Myrheim: Anyons

379

The non-vanishing commutators among these operators are [a, a† ] = [b, b† ] = [c, c† ] = [d, d† ] = 1 .

(448)

When qB > 0 we obtain the following form of the Hamiltonian, H = ~ω1 b† b + d† d + 1 + ~ω2 a† a + c† c + 1 ,

(449)

with the angular frequencies ω1 and ω2 given by equation (441). Two energy eigenstates having the correct symmetry under particle interchange are |z|2 (I) ν 2 z exp −γ |Z| + , ψ0 = 2 |z|2 (II) · (450) ψ0 = (z ∗ )2−ν exp −γ |Z|2 + 2 (I)

(II)

They have energies E0 = ~ω1 + (1 + ν)~ω2 and E0 = (3 − ν)~ω1 + ~ω2 . A complete set of energy eigenstates are, with j, k, l, m independent nonnegativ integers, ψj,k,l,m = (a† )j (b† )k (c† d† )l (c† )2m ψ0 , (I)

(I)

ψj,k,l,m = (a† )j (b† )k (c† d† )l (d† )2m ψ0 (II)

(II)

,

(451)

and the corresponding energy levels are (I)

Ej,k,l,m = (j + l + 1) ~ω1 + (k + l + 2m + 1 + ν) ~ω2 , (II)

Ej,k,l,m = (j + l + 2m + 3 − ν) ~ω1 + (k + l + 1) ~ω2 .

(452)

This gives the two-particle partition function Z2 =

e−β~(ω1 +ω2 ) − e−β~ω2 )(1 − e−β~(ω1 +ω2 ) ) e−(2−ν)β~ω1 + · 1 − e−2β~ω1

e−β~ω1 )(1

(1 − −νβ~ω2 e 1 − e−2β~ω2

(453)

The above calculation was done under the assumption that qB > 0. Changing the sign of qB is the same as interchanging ω1 and ω2 . The same effect is obtained by substituting 2 − ν for ν, or even simpler by the naive trick of switching the sign of ω. By definition, the lowest Landau level for qB > 0 consists of the energy levels (I)

E0,k,0,m = ~ω1 + (k + 2m + 1 + ν) ~ω2 .

(454)

380

Topological Aspects of Low Dimensional Systems (II)

They lie lower than the lowest energy level of type (II), E0 = (3 − ν)~ω1 + ~ω2 , as long as k and m are small, and ω0 is small so that ω2 << ω1 . At least this is true in the bosonic limit ν → 0+ . In the other bosonic limit, ν → 2− , we see that the Landau levels are no longer clearly separated. For qB < 0, on the other hand, it is in the limit ν → 2− , and not as ν → 0+ , that it has meaning to speak of separated Landau levels. 9.3 The anyon gas in a magnetic field We are now in a position to compute the second virial coefficient of a gas of charged anyons in a magnetic field, neglecting the Coulomb interaction. Using equation (327) we get [102, 284] 2 Λ Z2 1−2 2 A2 = lim ω0 →0 β~ω0 Z1 2 (ν−1)β~ω β~ω 1 e Λ + coth ν −1− · (455) = β~ω sinh(β~ω) 2 2 In the limit of zero magnetic field we recover the well known result, equation (347), 1 (ν − 1)2 − · (456) lim A2 = Λ2 ω→0 4 2 More interesting are the next terms in the expansion around B = 0, q ∂A2 q ν(ν − 1)(ν − 2) ∂A2 , (457) = = −Λ4 ∂B B=0 m ∂ω ω=0 h 6 and

q 2 1 − 2ν 2 (ν − 2)2 ∂ 2 A2 6 · = Λ ∂B 2 B=0 h 24

(458)

The first derivative at B = 0 vanishes for bosons, ν = 0 and ν = 2, and for fermions, ν = 1, but for no other types of anyons. This means that a gas of charged anyons that are not bosons or fermions, should develop a spontaneous magnetization even in the absence of an external magnetic field. The N -anyon problem in a magnetic field can only be solved numerically when N > 2, but it is still possible to treat exactly the limiting case when the field is strong enough, or the temperature low enough, that all energy levels not belonging to the lowest Landau level can be neglected. As already noted, in order to speak meaningfully of separated Landau levels, we have to exclude the bosonic limits ν → 2− when qB > 0, and ν → 0+ when qB < 0.

J. Myrheim: Anyons

381

The N -particle Hamiltonian is N X 1 1 1 2 2 2 2 p + m(ω + 4ω0 )xj ∓ ωLj . H= 2m j 8 2 j=1

(459)

Using our results for the harmonic oscillator we can immediately write down a complete non-orthogonal set of wave functions in the lowest Landau level,   N X γ = s1 j1 s2 j2 · · · sN jN ∆ν exp− |zj |2  , (460) ψjLLL 1 ,j2 , ... ,jN 2 j=1 with j1 , j2 , . . . , jN non-negative integers, sk = (zj − zk ), and the energies, = EjLLL 1 ,j2 , ... ,jN

N ~ω1 + 2

PN

j=1 zj

k

and ∆ =

Q j
j1 + 2j2 + · · · + N jN

! N (N − 1) N + ν ~ω2 . + 2 2

(461)

Hence the N -anyon partition function of the lowest Landau level is e− 2

N

LLL = ZN

(1 −

β~ω1 − N 2 (1+(N −1)ν)β~ω2

e−β~ω2 )(1

e · − e−2β~ω2 ) · · · (1 − e−N β~ω2 )

(462)

Special cases are bosons, ν = 0, and fermions, ν = 1, which can be derived from the grand canonical partition functions ΞB = ΞF =

∞ Y

1 1

− 2 β~(ω1 +ω2 ) e−jβ~ω2 j=0 1 − ze ∞ Y

,

1 1 + ze− 2 β~(ω1 +ω2 ) e−jβ~ω2 .

(463)

j=0

In order to compute the cluster coefficients we use equation (326), substituting ξ = β~ω0 . In the free particle limit ω0 → 0 we then get b n = ρL

n n−1 nν e− 2 β~ω Y , 1− n k

(464)

k=1

where ρL is the surface density of quantum states in the lowest Landau level, ρL =

1 β~ω |qB| = = · Λ2 h 2πλ2

(465)

382

Topological Aspects of Low Dimensional Systems

Note the close resemblance of equation (464) with equation (395), the only significant difference is the factor 1/n in one formula versus 1/n2 in the other. Similarly, we may compute the virial coefficients using equation (327). The general result is An =

ν n − (ν − 1)n · nρL n−1

In particular, the second virial coefficient 1 1 ν− A2 = ρL 2

(466)

(467)

is the same as we get from the exact equation (455) in the strong field limit, or more precisely, when β~ω is large. The difference between (467) and (455) should be a useful measure of the error we make by neglecting the higher Landau levels. An explicit proof that the virial coefficients (466) follow from equation (464), can be given by means of the results from Subsection 8.12. It follows from those results that the equation βP =

∞ X n=1

b n z n = ρL

∞ n−1 X nν (z 0 )n Y , 1− n k n=1

(468)

k=1

1

with z 0 = ze− 2 β~ω , is equivalent to the equation βP βP z 0 = exp ν − exp (ν − 1) · ρL ρL

(469)

Operating on this equation with z 0 (∂/∂z 0 ), and using that z 0 (∂(βP )/∂z 0 ) = ρ, we get that ρ βP βP 0 · (470) − (ν − 1) exp (ν − 1) z = ν exp ν ρL ρL ρL Equating these two formulae for z 0 we obtain the equation of state ρL − (ν − 1)ρ , βP = ρL ln ρL − νρ

(471)

which is equivalent to equation (466). We see that the pressure diverges when the density ρ approaches the maximum value ρL /ν. This result for anyons is a direct generalization of the fact that the maximum density of fermions in one Landau level is ρL . The divergence is clearly unphysical, and would have been avoided if we

J. Myrheim: Anyons

383

had been able to include the higher Landau levels. At some density before the first Landau level is full, the particles should start filling the next level. It was assumed in the above that qB > 0. In the opposite case, qB < 0, the same equation of state holds with the substitution ν → 2 − ν. That is, we have ρL − (1 − ν)ρ · (472) βP = ρL ln ρL − (2 − ν)ρ 10

Interchange phases and geometric phases

In this final Section we will discuss the relation between quantum mechanical phases of somewhat different origins, the interchange phases in systems of identical particles, and the geometric phases, also called Berry phases, associated with cyclic evolution of quantum systems. This relation may be used for studying the statistics of particles or of particle-like excitations in a physical system. 10.1 Introduction to geometric phases The two-level time dependent Hamiltonian cos(2α) sin(2α) e−2iβt , H(t) = ~ω sin(2α) e2iβt − cos(2α)

(473)

with ω, α and β real constants, illustrates well the phenomenon of the geometric phase, or Berry phase [15,57–61]. The instantaneous eigenvectors of H(t), with eigenvalues ±~ω, are cos α sin α e−2iβt (t) = . (474) , χ χ+ (t) = − sin α e2iβt − cos α The exact solution of the time dependent Schr¨ odinger equation i~

dψ = Hψ dt

(475)

is of the form ψ(t) = U (t) ψ(0), with −iβt (cos(ω1 t) − iγ sin(ω1 t)) −iδ e−iβt sin(ω1 t) e , U (t) = e iβt (cos(ω1 t) + iγ sin(ω1 t)) −iδ e iβt sin(ω1 t) (476) and with ω1 = δ=

p ω 2 + β 2 − 2ωβ cos(2α) , ω sin(2α) · ω1

γ=

ω cos(2α) − β , ω1 (477)

384

Topological Aspects of Low Dimensional Systems

In the adiabatic limit when β is very small, but the product βt is not necessarily small, we may put ω1 t ≈ ωt − βt cos(2α) ,

γ ≈ cos(2α) ,

δ ≈ sin(2α) .

(478)

With these approximations the time evolution of the eigenvectors χ± (0) of H(0) is that U (t) χ± (0) ≈ e∓i(ω1 +β)t χ± (t) ≈ e∓iωt e∓iβt(1−cos(2α)) χ± (t) .

(479)

In other words, an eigenvector of H(0) evolves approximately into an eigenvector of H(t). The phase ∓ωt in equation (479) is readily understood as due to the time evolution of states with energies ±~ω. But there is an additional phase, ϑ(t) = ϑ± (t) = ∓βt (1 − cos(2α)) ,

(480)

which can be interpreted as an effect of the geometry of the Hilbert space of spinors to which the eigenvectors χ± (t) belong. In fact, given χ(t) = χ+ (t) or χ− (t), normalized such that |χ|2 = χ† χ = 1, let us ask for the time dependent real phase ϑ(t) such that the curve ψ(t) = eiϑ(t) χ(t) in the Hilbert space has minimal length. The length is, with the time derivative d/dt denoted by a dot, Z Z Z q ˙ 2 + |χ| ˙ ˙ ˙ 2 − iϑ˙ (χ† χ˙ − χ˙ † χ) . (481) dt |ψ| = dt |iϑχ + χ| ˙ = dt (ϑ) To minimize the integral we must minimize the integrand, i.e. choose ϑ(t) such that i † χ χ˙ − χ˙ † χ = − Im χ† χ˙ = iχ† χ˙ . ϑ˙ = 2

(482)

The last equality follows because χ is normalized, so that 1 † 1 d χ χ˙ + χ˙ † χ = χ† χ = 0 . Re χ† χ˙ = 2 2 dt

(483)

If χ = Cχ0 , where χ0 is unnormalized and C is a positive normalization factor such that χ is normalized, then equation (482) takes the form ˙ 0 = −C 2 Im χ†0 χ˙ 0 . (484) ϑ˙ = − Im χ† χ˙ = − Im Cχ†0 C χ˙ 0 + Cχ For the phases ϑ(t) = ϑ± (t) corresponding to χ(t) = χ± (t) we get the two equations † ϑ˙ + = i (χ+ ) χ˙ + = −2β sin2 α = −β (1 − cos(2α)) , † ϑ˙ − = i (χ− ) χ˙ − = β (1 − cos(2α)) ,

(485)

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which are the same as equation (480) if we take ϑ± (0) = 0. The time dependence of H(t) is periodic with period T = π/β, and the eigenvectors χ± (0) of H(0) evolve over one whole period T into U (T ) χ± (0) ≈ e∓iωT e∓iπ(1−cos(2α)) χ± (0) .

(486)

The additional phase over one complete cycle of the variation of H(t) is the Berry phase ϑO ± = ∓π (1 − cos(2α)) .

(487)

The superscript “O” indicates that it has to do with a closed loop. It is independent of the period T , it depends only on the sequence of eigenstates χ± (t) gone through and not on the specific parametrization of the curve. Hence it makes sense to speak of a geometric phase associated with any closed loop in the space of state vectors, irrespective of whether the loop is a physical time evolution due to the adiabatic deformation of a Hamiltonian. In fact, the Berry phase ϑO is unchanged if the eigenvector χ(t) is multiplied by an arbitrary t-dependent phase factor, as long as χ(T ) = χ(0). Thus it depends only on the sequence of physical states gone through. Remember that a physical state is represented in quantum mechanics not by one unique vector in the Hilbert space, but rather by a one-dimensional subspace. That is, two unit vectors in the Hilbert space represent the same physical state if they differ only by a phase factor. 10.2 One particle in a magnetic field As an example of the geometric phase, or generalized Berry phase, let us calculate the phase induced when a charged particle is moved around a loop in a magnetic field. Both the one- and two-particle cases have been discussed by Leinaas [153]. See also [64]. The original derivation of the Berry phase applied to a non-degenerate energy level of a Hamiltonian which was time dependent, although varying slowly. The present example is of a diametrically opposite kind, since the Hamiltonian is time independent and all energy levels are infinitely degenerate. Assume that the localized quantum state χζ , equation (433), is moved once around the circle |ζ| = ρ, in the anticlockwise direction. That is, we parametrize ζ = ρeiα and let the angle α increase from 0 to 2π. Note that, by equationp (426), the dimensionless radius ρ corresponds to a dimensioned √ radius r = x2 + y 2 = 2 λρ. In principle, the circular motion could be induced by a weak central electric field, since a charged particle in crossed electric and magnetic fields drifts perpendicularly to both fields. By direct generalization from equation (482) we define a geometric phase ϑ such that Z Z ∂χζ dϑ = i d2z χ∗ζ = ζ ∗ d2z z|χζ |2 = |ζ|2 = ρ2 , (488) dα ∂α

386

Topological Aspects of Low Dimensional Systems

where we have used that ∂ζ ∂χζ ∂ζ ∗ ∂χζ ∂χζ ∂χζ ∂χζ = + − iζ ∗ ∗ = −iζ ∗ zχζ . = iζ ∗ ∂α ∂α ∂ζ ∂α ∂ζ ∂ζ ∂ζ

(489)

Integrated over α from 0 to 2π this gives the generalized Berry phase, which is independent of the phase convention for the localized states χζ , ϑO = 2πρ2 =

|qB| 2 πr . ~

(490)

Note that we get ϑO > 0 because we assumed that qB > 0. The case qB < 0 corresponds to the complex conjugate wave function, which will give ϑO < 0. Thus we may drop the absolute value sign in equation (490) and write ϑO =

qB 2 Φ πr = 2π , ~ Φ0

(491)

where Φ = Bπr2 is the magnetic flux encircled, Φ0 = h/q is the flux quantum, and both Φ and Φ0 may have either sign. There is an alternative way to compute the same Berry phase, using the non-normalized coherent state wave function ψζ defined in equation (435). By generalization from equation (484), we have that Z 1 ∂ψζ dϑ = − Im d2z ψζ∗ , (492) dα I ∂α where I is the one-particle normalization integral, which by rotation invariance is independent of α, Z 2 (493) I = I(ρ) = d2z |ψζ |2 = π eρ . The point now is that ψζ depends on α only through ζ = ρeiα , and is an analytic function of ζ ∗ , so that 1 ∂I dϑ 1 ∗ ∂I ∗ ∂I = Im iζ +ζ ζ = dα I ∂ζ ∗ 2I ∂ζ ∂ζ ∗ d (494) = ρ2 2 ln I(ρ) = ρ2 . dρ Here we have used that 2ρ2

∂ζ ∂ ∂ζ ∗ ∂ ∂ ∂ ∂ ∂ = ρ + ρ + ζ∗ ∗ · = ρ =ζ 2 ∗ ∂ρ ∂ρ ∂ρ ∂ζ ∂ρ ∂ζ ∂ζ ∂ζ

(495)

We have assumed here that qB > 0. As already mentioned, if qB < 0 instead, we have to use the complex conjugate wave function. Because

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it depends analytically on ζ instead of ζ ∗ , there is a change of sign in equation (494), so that we get d dϑ = −ρ2 2 ln I(ρ) = −ρ2 . dα dρ

(496)

10.3 Two particles in a magnetic field In the case of two bosons or fermions in a magnetic field, in the lowest Landau level, the one-particle coherent states may be used to construct two-particle states where both particles are maximally localized. Simply take the product of the one-particle wave functions localized at ζa and ζb , and symmetrize if the particles are bosons, or antisymmetrize if they are fermions. A continuous variation of the parameters ζa and ζb induces a deformation of the two-particle wave function, and if ζa is changed continuously into ζb , and vice versa, this deformation is a closed loop starting and ending with the same physical state. It is in effect an interchange of the particles, and not unexpectedly, the corresponding Berry phase turns out to be related to the symmetry or antisymmetry of the wave function. For two bosons we define the non-normalized wave function ψζBa ,ζb (z1 , z2 ) = ψζa (z1 ) ψζb (z2 ) + ψζa (z2 ) ψζb (z1 ) ,

(497)

with ψζ as defined in equation (435). For two fermions we define ψζFa ,ζb (z1 , z2 ) =

ψζa (z1 ) ψζb (z2 ) − ψζa (z2 ) ψζb (z1 ) , ζa∗ − ζb∗

(498)

dividing by an extra factor ζa∗ − ζb∗ which serves two purposes. Since the wave functions ψζFa ,ζb and ψζFb ,ζa represent the same physical state, we want them to be completely identical and not just identical up to a sign. Also, it is nice to have ψζFa ,ζb well defined in the limit |ζa − ζb | → 0. The boson and fermion wave functions are both analytic functions of ζa∗ and ζb∗ . To simplify, let us take the one-particle coherent states to be localized symmetrically about the origin, with ζa = −ζb = ζ = ρeiα . Then, since the wave function is analytic in ζ ∗ , the Berry phase is related to the normalization integral in the same way as in the one-particle case above. The boson and fermion normalization integrals are, respectively, Z I B (ρ) =

2

Z I F (ρ) =

2

B d2z1 d2z2 |ψζ,−ζ (z1 , z2 )|2 = 2π (e2ρ + e−2ρ ) , 2

F d2z1 d2z2 |ψζ,−ζ (z1 , z2 )|2 =

2

π (e2ρ − e−2ρ ) · 2ρ2

(499)

388

Topological Aspects of Low Dimensional Systems

Keeping the radius ρ fixed and increasing the angle α from 0 to π corresponds to an anticlockwise interchange. The Berry phase is, for two bosons, ϑB (π) = π

dϑB d = πρ2 2 ln I B (ρ) = 2πρ2 tanh(2ρ2 ) . dα dρ

(500)

And, for two fermions, ϑF (π) = π

dϑF d = πρ2 2 ln I F (ρ) = π (2ρ2 coth(2ρ2 ) − 1) . dα dρ

(501)

The asymptotic limit as ρ → ∞ is 2πρ2 in both cases. This we recognize as the one-particle contribution, due to the displacement of each of the two oneparticle coherent states around a half circle. Subtracting this one-particle contribution, we are left with the genuine two-particle Berry phases, d (ln I2B (ρ) − 2 ln I1 (ρ)) = 2πρ2 (tanh(2ρ2 ) − 1) , dρ2 2 d (ln I2F (ρ) − 2 ln I1 (ρ)) ϑF s = πρ dρ2 = π (2ρ2 (coth(2ρ2 ) − 1) − 1) .

2 ϑB s = πρ

(502)

Here I1 (ρ) is the one-particle and I2 (ρ) the two-particle normalization integral. We will refer to ϑs as the statistics Berry phase. The asymptotic values 0 for bosons and −π for fermions, when ρ → ∞, justify the terminology. Since we define ϑs to depend on the distance ρ, it is not surprising that there is a deviation from the asymptotic values 0 and −π when ρ is so small that the two one-particle coherent states overlap significantly. Note that for ζa = −ζb = ζ, the boson and fermion two-particle coherent states defined in equations (497) and (498) have a common form, apart from constant factors, ψ (ν) =

∞ X k=0

(ζ ∗ )2k z 2k+ν ψ0 , Γ(2k + ν + 1)

(503)

with |z1 |2 + |z2 |2 |z|2 = exp −|Z|2 − , ψ0 = exp − 2 4

(504)

and with Z = (z1 + z2 )/2, z = z1 − z2 . The boson state has ν = 0 and the fermion state ν = 1. Note however that in equation (503) we could take for example ν = 0, 2, 4, . . . and get infinitely many different bosonic two-particle coherent states, with different asymptotic behaviour as z → 0.

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The basis states z 2k+ν ψ0 for fixed ν are orthogonal, and they are normalizable whenever 2k + ν > −1. The normalization integral for ψ (ν) is, again apart from constant factors, I (ν) =

∞ X k=0

(2ρ2 )2k · Γ(2k + ν + 1)

(505)

We could generalize from bosons and fermions to various two-anyon states that localize each of the particles more or less well. The most obvious generalization is simply to allow ν in equation (503) to take any real value, with the only restriction that ν > −1, for normalizability. This can be interpreted as the anyon coordinate eigenstate projected onto the lowest Landau level [285]. Another two-anyon state which has also been proposed as a natural generalization is the coherent state of a particular su(1,1) algebra [64, 285], ψc(ν) =

∞ X

(ζ ∗ )2k q z 2k+ν ψ0 . 1 k k! Γ(k + ν + 2 )Γ(2k + ν + 1) k=0 2

(506)

The normalization integral for this is given by a modified Bessel function Iν−(1/2) , Ic(ν)

∞ X

Iν− 12 (2ρ2 ) (ρ2 )2k = · = ρ2ν−1 k! Γ(k + ν + 12 ) k=0

(507)

The statistics Berry phase = πρ2 ϑ(ν) s

d (ν) (ln I2 (ρ) − 2 ln I1 (ρ)) dρ2

(508)

has the asymptotic value of −νπ for both these two-anyon states, but there (ν) is a difference between them for small ρ. Note that ϑs → 0 for ρ → 0, independent of ν. This does not mean that the bosons, fermions or anyons are not pointlike particles, what it means is that they are not sharply localized. Sharp localization is impossible as long as we admit only states belonging to the lowest Landau level. We may now turn the whole argument around and use the Berry phase to define a distance dependent “anyon parameter” νBerry = −ρ2

d (ν) (ln I2 (ρ) − 2 ln I1 (ρ)) , dρ2

(509)

which is then asymptotically equal to the actual statistics parameter ν at large distances.

390

Topological Aspects of Low Dimensional Systems

In the discussion so far we have assumed that qB > 0. The only difference in the case qB < 0 is that we have to take the complex conjugates of (ν) the wave functions ψ (ν) and ψc , defined in equations (503) and (506), but when we change z into z ∗ , we have to change z ν into (z ∗ )−ν , in order to preserve the meaning of the anyon parameter ν. Thus the anyon states of negative charge are ψ

(ν)

=

∞ X k=0

ζ 2k (z ∗ )2k−ν ψ0 , Γ(2k − ν + 1)

(510)

and ψc(ν) =

∞ X

ζ 2k q z 2k−ν ψ0 . 1 k k! Γ(k − ν + 2 )Γ(2k − ν + 1) k=0 2

(511)

These states are well defined for ν < 1, and are singular as |z| → 0 if 0 < ν < 1. The complex conjugate wave functions depend analytically on ζ instead of ζ ∗ , which implies an opposite sign in the relation between the Berry phase and the normalization integral. Thus, for qB < 0, equation (509) is replaced by νBerry = ρ2

d (ν) (ln I2 (ρ) − 2 ln I1 (ρ)) . dρ2

(512)

10.4 Interchange of two anyons in potential wells The results just derived for two particles in a magnetic field indicate a general relation between the geometric phase, or Berry phase, and the interchange phase in a system of identical particles. The existense of such a relation is not entirely trivial, since the two phases are conceptually rather different. One phase has to do with the geometry, or more precisely the metric, in the Hilbert space of quantum state vectors, the other has to do with the topology of the configuration space. One phase arises when the whole wave function is changed continuously, the other arises when the argument of one single wave function is changed. As another example, we may imagine two identical particles in two dimensions trapped inside two separate deep potential wells, and interchange the particle positions by interchanging the wells [63]. If only one potential well is present at the origin, let ψ0 denote its ground state wave function, of energy E0 . For simplicity we assume that the well is rotationally symmetric, so that ψ0 has angular momentum zero. Let ψa be the wave function ψ0 translated to the position a, that is, ψa (x) = ψ0 (x − a) .

(513)

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391

Table 1. The approximate energies, with the exact and approximate wave functions for two bosons or two fermions in the double well. Normalization constants are ignored.

Then if the two wells are located at a and at b, and if the overlap between the two wave functions ψa and ψb is small, the one-particle ground state is nearly degenerate, since there are two energy eigenstates ψ± ≈ ψa ± ψb .

(514)

The energies are E0 ∓ /2, and the energy splitting is small. The lowest energies and the corresponding wave functions for two particles in the two wells are tabulated in Table 1, for the boson and fermion cases. Note that the single fermionic energy eigenstate has essentially only one particle in each well, but in all three of the bosonic energy eigenstates the probability of finding both particles in the same well is either 50% or 100%, approximately. It follows by interpolation to the anyon case that there is in general no anyonic energy eigenstate with the two particles in separate wells. On the other hand, if the energy splitting is very small, then there surely exists an approximate energy eigenstate with two anyons in separate wells, and the transition probability from this state to other states is small. It is this particular state we are interested in here. Call its normalized wave function χ0 . It is convenient to introduce polar coordinates, (R, Φ) for the centre of mass position and (r, φ) for the relative position, and to work in what we have called the parallel gauge, so that the statistics vector potential vanishes and every wave function ψ satisfies the following periodicity condition, ψ(R, Φ, r, φ + π) = eiθ ψ(R, Φ, r, φ) .

(515)

Assume now that the positions of the two wells are a and b = −a, and that they are interchanged simply by a rotation an angle π about the origin. Define a set of wave functions χα , depending on the real parameter α, such that χα (R, Φ, r, φ) = eiνα χ0 (R, Φ − α, r, φ − α) ,

(516)

392

Topological Aspects of Low Dimensional Systems

with ν = θ/π. The phase factor eiνα is introduced in order that χπ = χ0 . Apart from that, the wave function χα is just χ0 rotated anticlockwise by an angle α, in other words it is the approximate eigenstate for the Hamiltonian where the two wells have been rotated into new positions. The geometric phase ϑ associated with a change in α is determined by the equation Z hLi ∂χα dϑ = i (R dR)dΦ (r dr)dφ χ∗α = −ν + · (517) dα ∂α ~ hLi is the expectation value in the state χα of the angular momentum operator ∂ ∂ + · (518) L = −i~ ∂Φ ∂φ Since we have assumed that the two wells are so far separated that there is negligible overlap of the two ground state wave functions ψa and ψ−a , it can not matter for the expectation value hLi whether the particles are bosons, fermions or anyons. Hence we conclude that hLi = 0 always, as is the case for bosons. Consequently, the geometric phase associated with an interchange by rotation an angle π is the negative of the anyonic statistics angle θ, ϑO = π

dϑ = −νπ = −θ . dα

(519)

10.5 Laughlin’s theory of the fractional quantum Hall effect Arovas et al. used the concept of the geometric phase in order to calculate the charge and statistics of the elementary excitations in Laughlin’s theory of the fractional quantum Hall effect [62,175]. Although the original idea of Laughlin was very simple and elegant, it applied only to the simple fractions 1/3, 1/5, etc., and the hierarchical extensions of the theory needed for other fractions become rather complicated [171, 172]. As our final example, we will discuss the calculation of Arovas, Schrieffer and Wilczek for sufficiently small number of electrons that it can be done either exactly or numerically [285, 286]. Since we want to discuss both particles of positive and of negative charge, we will assume throughout that the magnetic field is positive, B > 0. Then since electrons have negative charge q = −e, we will have qB < 0 for the electrons. The canonical unit of length is the magnetic length for electrons, r ~ , (520) λe = eB

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393

and the one-particle basis states are the complex conjugates of the wave functions χn in equation (432). In √ the state χn the particle is located approximately at a distance |z| = n from the origin, since that is where the probability density |χn |2 is maximal. Hence, if we put an upper limit on n, say n < M , this means that we√ have an approximate description of a system of finite radius r = λe |z| < λe M . If we now distribute N electrons among the M first one-particle states in the lowest Landau level, the filling fraction is νf = N/M . The particles may be electrons, and we may make the (somewhat dubious) assumption that the electron spin is completely polarized in the magnetic field, so that the N -electron system is described by a wave function which is a totally antisymmetric function of the particle positions z1 , z2 , . . . , zN . For a filling fraction νf = 1/µ, where µ is an odd integer, the non-normalized wave function proposed by Laughlin is   N Y X 1 µ µ |zj |2  . zj∗ − zk∗ exp− (521) ψ0 = 2 j=1 j
It is antisymmetric when µ is odd, and is obviously a ground state of a system of non-interacting electrons, built from the one-particle states χ∗0 , χ∗1 , . . . , χ∗M−1 , where M = µ(N − 1) + 1 ≈ µN .

(522)

Due to the factor (zj∗ − zk∗ )µ , the wave function ψ0µ minimizes very well, if not perfectly, the probability of finding two particles j and k close together, so that it is still an approximate ground state of the system when we take into account the Coulomb repulsion between the electrons. The simplest excitation of the system is a vortex at an arbitrary position z = ζ, described by the wave function ψ1 =

N Y

zj∗ − ζ ∗ ψ0µ .

(523)

j=1

The vortex represents a “quasihole”, since it repels the electrons and thereby creates an excess of positive background charge at ζ (we assume that the total electron charge is neutralized by a uniform background density of positive charge). Note that it takes µ vortices at the same position to create a positive excess charge equal in magnitude to the electron charge, thus we expect naively that the charge of the vortex is 1/µ (or rather −1/µ) of the electron charge. We now ask for the geometric phase ϑ1 arising when the vortex is moved around the circle |ζ| = ρ. That is, we parametrize ζ = ρeiα and let α increase

394

Topological Aspects of Low Dimensional Systems

from 0 to 2π. The total phase is Z 2π dϑ1 dϑ1 O = 2π · dα ϑ1 = dα dα 0

(524)

Since ψ1 depends on α only through ζ = ρeiα , and is an analytic function of ζ ∗ , we have, by a similar reasoning as before, that   Z Y N 1 ∂ψ d dϑ1 1 = − Im  d2zj ψ1∗ (525) = ρ2 2 ln I1 (ρ) , dα I1 ∂α dρ j=1 where I1 (ρ) is the normalization integral, Z Y N d2zj |ψ1 |2 I1 (ρ) = j=1

=

N X

ρ

2k

k=0

Z Y N

∗ 2 µ 2 d2zj |cN −k (z1∗ , . . . , zN )| |ψ0 | .

(526)

j=1

Here ck are the elementary symmetric polynomials encountered earlier, see equations (226) and (237). We may try to interpret the phase ϑO 1 as due to the motion of a charge q1 in the magnetic field, and then the vortex charge q1 is related in the following way to the absolute value of the electron charge, |q| = e. According to equation (491), dϑ1 /dα is proportional to the charge q1 , and according to equation (488) we have dϑ/dα = ρ2 for a positive charge e. Hence, 1 dϑ1 d q1 = 2 = 2 ln I1 (ρ) . e ρ dα dρ

(527)

In particular, we conclude immediately that q1 is positive. The normalization integral I1 (ρ) is tabulated in Table 2 for µ = 3 and up to 5 electrons. Based on this table, the quantity d(ln I1 (ρ))/dρ2 is plotted in Figure 14 as a function of ρ. Although N = 5 is a very small number, the plot already suggests that q1 /e = 1/3 for large N . This is confirmed by the results of Monte Carlo integrations with N = 20, 50, 75, 100 and 200, as shown in Figure 15 [286]. Note that the p wave function ψ1 describes µ(N − 1) + 1 (which is 3.6 essentially a system of finite radius ρN ≈ for µ = 3 and N = 5, and 24.5 for N = 200), and that d(ln I1 (ρ))/dρ2 approaches N/ρ2 for ρ > ρN . One should therefore consider only the region ρ < ρN . The wave function for a state with two vortices, at the positions ζ and −ζ, is ψ2 =

N Y j=1

zj∗ − ζ ∗

N Y j=1

N Y zj∗ + ζ ∗ ψ0µ = (zj∗ )2 − (ζ ∗ )2 ψ0µ . j=1

(528)

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Table 2. The one-vortex normalization integral I1 for µ = 3 and for N = 2, 3, 4, 5 particles.

0.5 0.4 0.3 0.2 0.1 1

2

3

4

5

-0.1

Fig. 16. The distance dependent statistics parameter −ρ2 (d/dρ2 )(ln I2 − 2 ln I1 ) as a function of ρ. The curves are for N = 2, 3, 4, 5 electrons, the leftmost peak for N = 2 and the rightmost peak for N = 5. The dashed line is 1/3.

If the two vortices are interchanged by an anticlockwise circular motion an angle π, this again gives rise to a geometric phase Z ϑO 2 =

π

dα 0

dϑ2 dϑ2 =π · dα dα

(529)

By the same derivation as above, we have that d dϑ2 = ρ2 2 ln I2 (ρ) , dα dρ

(530)

396

Topological Aspects of Low Dimensional Systems

where I2 (ρ) is the normalization integral for ψ2 , I2 (ρ) =

Z Y N

d2zj |ψ2 |2

j=1

=

N X k=0

ρ4k

Z Y N

∗ 2 2 µ 2 d2zj |cN −k ((z1∗ )2 , . . . , (zN ) )| |ψ0 | .

(531)

j=1

It is tabulated in Table 3 for µ = 3 and up to 5 particles. Subtracting the one-vortex contribution 2ϑ1 from the two-vortex phase ϑ2 , we would like to interpret the remaining phase as due to the quantum statistics of the vortices. More precisely, we use the definition (509) of the distance dependent anyon parameter, νBerry = −ρ2

d (ln I2 (ρ) − 2 ln I1 (ρ)) . dρ2

(532)

This is plotted as a function of ρ in Figure 16. It clearly depends on the distance between the vortices, at least for small distances. We should in fact expect the vortex statistics to depend on distance, for small distances, since the vortices are not point particles, but have a finite size. More remarkably, the plot indicates that when the vortices are well separated, then there is an approximately constant part of the Berry phase, giving νBerry ≈

1 · 3

(533)

This is again confirmed by Monte Carlo integrations, with N from 20 and up to 200, as shown in Figure 17. In order to compare with the statistics phases of the anyon states given in equations (503) and (506), we have to remember that our present length scale is the electron magnetic length λe , corresponding to the elementary charge e, whereas the vortices that we want to describe as anyons have charge e/µ, with µ = 3 in our numerical example, so that the “vortex mag√ netic length” is λ = µλe . This stretching of the length scale must be compensated for by dividing the dimensionless length in the anyon system √ by µ in every formula we use. The comparison is shown in Figure 18, it indicates that the anyon model with ν = 1/3 is a reasonably good description of the Laughlin quasi-hole states for the filling fraction νf = 1/3. Laughlin also proposed wave functions representing quasi-electron excitations, in which the electron density is increased locally. These quasielectron states were examined in reference [284], and within the approximations used, the results imply that the charge and statistics parameter should have the values −e/µ and +1/µ, respectively, for a filling fraction of 1/µ with µ odd.

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397

Table 3. The two-vortex normalization integral I2 for µ = 3 and for N = 2, 3, 4, 5 particles.

0.4

0.3

0.2

0.1

0 0

5

10

15

20

r 0.1

0.2

Fig. 17. The quasi-hole statistics parameter νBerry , equation (532), as a function of ρ, half the distance between the two quasi-holes. The curves are, from left to right, for 20, 50, 75, 100 and 200 electrons, and the horizontal line is 1/3. From [286].

The proposed wave function for one quasi-electron located at the position ζ is the following polynomial in ζ, ! N Y 0 ∗ (∂i − ζ) (∆∗ )µ , (534) ψ1 = ψ0 i=1

398

Topological Aspects of Low Dimensional Systems

0.5

0.4

0.3

0.2

0.1

0 0

0.5

1

1.5 r

2

2.5

Fig. 18. Comparison of the distance dependent statistics parameter for two Laughlin quasi-holes, and for localized states in the system of two anyons. The lowest lying curve is for 75 electrons, then follows a common curve for the three cases of 20, 50 and 100 electrons, and the third curve is for 200 electrons. Somewhat higher lies the Berry phase curve calculated from the anyon position eigenstate projected onto the lowest Landau level, and even higher the one calculated from the coherent state of the SU (1, 1) algebra. From [286].

with ∆ =

Y

 (zj − zk ) and ψ0 = exp − 21

X

 |zj |2 .

j

j
The normalization integral may be rewritten by partial integration as I10 =

Z d2Nz ψ0 2 |∆|2µ

N Y

(|zk − ζ|2 − 1) .

(535)

k=1

It is again a polynomial in ρ = |ζ|, due to rotational invariance. The difference from the quasi-hole normalization integral is the extra −1 in each factor |zk − ζ|2 − 1. Without this −1, the quasi-electron and quasi-hole

J. Myrheim: Anyons

0

r 10

5

399

15

20

0

-0.1

-0.2

-0.3

-0.4

Fig. 19. The quasi-electron charge q10 /e, equation (536), as a function of ρ, the quasi-electron distance from the origin. The curves are, from left to right, for 20, 50 and 75 electrons. The horizontal line is −1/3. From [286].

integrals would be identical, and the charge of the quasi-electron, d q10 = − 2 ln I1 (ρ) , e dρ

(536)

would be the same as that of the quasi-hole, just with an opposite sign, because the quasi-electron wave function depends on ζ and the quasi-hole wave function on ζ ∗ . This approximation seems hard to justify, nevertheless it may be valid asyptotically for large N , as the results of Monte Carlo integrations shown in Figure 19 seem to indicate. Two quasi-electrons at ζ and −ζ are described, according to Laughlin, by the wave function ! N Y 0 ∗ ∗ (∂i − ζ)(∂i + ζ) (∆∗ )µ , (537) ψ2 = ψ0 i=1

which yields, by partial integration, the normalization integral Z N Y |zk 2 − ζ 2 |2 − 4|zk |2 + 2 . I20 = d2Nz ψ0 2 |∆|2µ k=1

(538)

400

Topological Aspects of Low Dimensional Systems

0.5

0.4

0.3

0.2

0.1

0 0

2

4

6

8

10

12

14

r

Fig. 20. The distance dependent statistics parameter for quasi-electrons, equation (539), versus ρ, half the distance between the quasi-electrons. The lowest lying curve is for 20 electrons. Next are curves for 50 and 75 electrons. The 100 electron curve is cut at ρ = 8 and the 200 electron curve at ρ = 6, to avoid numerical problems. The horizontal line is 1/3. From [286].

Comparing with the quasi-hole integral in equation (531), we see that for two quasi-electrons there are correction terms, like in the case of a single quasi-electron. The distance dependent statistics parameter is now 0 = ρ2 νBerry

d (ln I20 (ρ) − 2 ln I10 (ρ)) . dρ2

(539)

This quantity, as computed by Monte Carlo integration, is plotted in Figure 20. It is found to be positive, although one can hardly justify the conclusion that the asymptotic value is 1/3, as one might want it to be. In Figure 21 the Monte Carlo data are compared with the curves for anyons of negative charge −e/3 and with the anyon parameter ν = 1/3. Note that these two-anyon states are actually singular where the anyon coordinates coincide. It is seen that the small distance behaviour is well represented by the anyon model, but the behaviour at larger distances is not at all well represented. It may be that the calculations were done with

J. Myrheim: Anyons

401

0.5

0.4

0.3

0.2

0.1

0 0

1

2

3 r

4

5

6

-0.1

Fig. 21. The distance dependent statistics parameter for Laughlin quasi-electrons, compared to the two-anyon states. The five curves for 20, 50, 75, 100 and 200 electrons coincide for small ρ. The 200 electron curve overshoots the horizontal line 1/3 at ρ ≈ 3. The curve lying lowest for small ρ represents an anyon eigenstate projected onto the lowest Landau level, whereas the curve going highest for large ρ represents the coherent state of the SU (1, 1) algebra. From [286].

too few electrons, but at least one may conclude that there is a marked difference between the quality of the anyon description for the Laughlin quasi-hole states versus the quasi-electron states. In these calculations the anyon model was compared with the Laughlin wave functions. A more interesting question is perhaps how it compares with experiment, or if not directly with experiment, at least with some less idealized theoretical model. One such model calculation for the quantum Hall system is that of reference [238], which did not rely on any specific trial wave function, but rather on state counting based on numerical simulations for interacting electrons on a sphere. The value of the one-dimensional exclusion statistics parameter [227] was found to be 1/3 in the case of quasi-holes, and 2 − 1/3 for the quasi-electrons, near to the magic filling fraction 1/3. The exclusion statistics parameter is in principle the same parameter as one reads from the Berry phase, although with an opposite sign for

402

Topological Aspects of Low Dimensional Systems

the quasi-electrons, because their charge is negative. Thus the predictions for the anyon parameter, based on the numerical results for the exclusion statistics parameter, would be 1/3 for quasi-holes and −2 + 1/3 for quasi-electrons. The values 1/3 and −2 + 1/3 for the anyon parameter define of course the same particle statistics, but we distinguish between them here in the way we define the correspondence between quasi-particles and anyons. Thus, in the case of quasi-electrons, 1/3 and −2 + 1/3 would represent the same species of anyons, but different anyon states, the 1/3 state singular and the −2 + 1/3 state non-singular. It is interesting to note that the numerical results for the exclusion statistics parameter of realistic quasi-holes and quasi-electrons are easily interpreted in terms of anyons of positive and negative charge, respectively, with non-singular wave functions, and with the same statistics in the two cases. That quasi-holes and quasi-electrons should have the same statistics, is also what one would expect if one regards them as antiparticles of each other. I want to thank my coauthors J.M. Leinaas, A. Kriskoffersen, S. Mashkevich, K. Olaussen and H. Kjønsberg, as well as the publisher, World Scientific, for their permission to reprint some figures. Figure 4 is reprinted from [208]. Figures 7 to 13 are from [146]. Figures 15 and 17 are reprinted from [286].

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COURSE 5

GENERALIZED STATISTICS IN ONE DIMENSION

A.P. POLYCHRONAKOS Institutionen f¨ or Teoretisk Fysik, Box 803, 751 08 Uppsala, Sweden, and Physics Department, University of Ioannina, 45110 Ioannina, Greece

Contents 1 Introduction

417

2 Permutation group approach 418 2.1 Realization of the reduced Hilbert space . . . . . . . . . . . . . . . 418 2.2 Path integral and generalized statistics . . . . . . . . . . . . . . . . 422 2.3 Cluster decomposition and factorizability . . . . . . . . . . . . . . 424 3 One-dimensional systems: Calogero model 427 3.1 The Calogero-Sutherland-Moser model . . . . . . . . . . . . . . . . 428 3.2 Large-N properties of the CSM model and duality . . . . . . . . . 431 4 One-dimensional systems: Matrix model 4.1 Hermitian matrix model . . . . . . . 4.2 The unitary matrix model . . . . . . 4.3 Quantization and spectrum . . . . . 4.4 Reduction to spin-particle systems .

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5 Operator approaches 5.1 Exchange operator formalism . . . . . . . 5.2 Systems with internal degrees of freedom . 5.3 Asymptotic Bethe Ansatz approach . . . . 5.4 The freezing trick and spin models . . . .

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6 Exclusion statistics 6.1 Motivation from the CSM model . . . 6.2 Semiclassics – Heuristics . . . . . . . . 6.3 Exclusion statistical mechanics . . . . 6.4 Exclusion statistics path integral . . . 6.5 Is this the only “exclusion” statistics?

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7 Epilogue

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469

GENERALIZED STATISTICS IN ONE DIMENSION

A.P. Polychronakos

Abstract An exposition of the different definitions and approaches to quantum statistics is given, with emphasis in one-dimensional situations. Permutation statistics, scattering statistics and exclusion statistics are analyzed. The Calogero model, matrix model and spin chain models constitute specific realizations.

1

Introduction

Quantum statistics, as the name suggests, is the way that the quantum mechanical properties of particles influence their statistical mechanics. Two features of quantum mechanics are crucial for the difference between classical and quantum statistics: the indistinguishability of quantum particles and the “finite room” that exists in a given domain of the phase space of a quantum system. The first feature can be incorporated “by hand” into classical statistical mechanics (although there is no underlying principle for doing so) and this will resolve the Gibbs paradox, although by itself it would change little else. The second feature can likewise be incorporated in an ad hoc way, which will assign a unique value to the additive constant of the entropy of a classical system, but again without further physical impact. It is the combination of the two that produces nontrivial results, especially in the domain where quantum phenomena prevail, that is, high densities or low temperatures. The statistics of observed particles seem to be exhausted by the two well-known cases of Bose and Fermi. In this sense, the quest for more exotic possibilities seems to be a bit academic or even quixotic. It is, nevertheless, of interest to explore the possibilities as a way of obtaining a more lucid effective description of the dynamics of (otherwise normal) interacting particles. In this sense, we are trading one kind of intuition for another, hoping to make a profit in the bargain. There are several approaches to defining or implementing quantum statistics. They are roughly classified into “fundamental” (group theoretic, c EDP Sciences, Springer-Verlag 1999

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operator etc.) and “phenomenological” (statistical). While exploring them we must keep in mind that, at the end of the day, we should provide a realization, that is, a system which is (more or less) physical and exhibits these statistics through its dynamics. The less the number of space dimensions, the more easy it is to find such realizations. The concept itself of statistics, actually, depends crucially on the dimensionality of space. Topological considerations allow for generalized abelian statistics in 2+1 dimensions [1–4]. Such particles, called anyons, have found use in the physics of the quantum Hall effect and (probaby) high-temperature superconductivity. There are excellent reviews on anyons [5] and we will not deal with them here. In these lectures the bulk of the material will concern one-dimensional situations, although some considerations in principle applicable to any dimension will also be included. The purpose is to present enough of the concepts and material on the subject to spur the interest of the reader for further study. You are warned that all formulae in these lectures are simply mnemonics of the correct results: mistakes abound! (The written version is only marginally better.) One standard exercise proposed here is to rederive all results, fixing the signs and factors in the process. Hopefully substance survived the sloppyness. So, right to the task. 2

Permutation group approach

As a way of introduction and covering standard ground, we will study the context in which statistics initially arose, namely the permutation group in quantum mechanics [6–14]. The main emphasis will be in connecting to the path integral and pointing the possibilities for generalizations within this approach. The main themes of this lecture are: a) Eliminating the “gauge” multiplicity of states originating from particle permutations. This will, in general, lead to the appearance of internal degrees of freedom that are the residue of “identities” (distinguishability) of particles. b) Implementing the path integral for identical particles. Their indistinguishability calls for including paths that lead to any permutation of the final position. Determining the right weights of these sectors (which are not necessarily phases) will fix the statistics. 2.1 Realization of the reduced Hilbert space We will work in the first-quantized picture, in which the number of particles is fixed but the particles are nevertheless taken to be indistinguishable. This is to be contrasted to the weaker statement that the particles are merely

A.P. Polychronakos: Generalized Statistics in One Dimension

419

identical. To fix the ideas, let us call {xi } the coordinates of these particles, with i = 1 . . . N enumerating the particles, and assume, for now, that there are no internal degrees of freedom. The xi can be in a space of any dimension. The Hilbert space of this system is spanned by the position eigenstates |x1 , . . . xN i ≡ |xi. Let us also define the operators P ∈ SN that permute the quantum numbers of the particles. There are N ! such operators forming the permutation (or symmetric) group of N objects SN . The collection of the above states for a set of distinct xi transforms in the N !-dimensional regular (defining) representation of SN P |xi ≡ |P xi = |xP (1) , . . . xP (N ) i·

(1)

Notice that if any of the coordinates xi coincide the above is not the full defining representation any more. The set of such states, however, is of measure zero (the coordinate space assumed continuous) and thus they can be safely ignored. (We assume that there are no interactions singular at coincidence points that might dynamically make such states of nonzero weight.) If the particles are identical then the hamiltonian treats them all equally and thus we have [H, P ] = 0 (2) for any P . If however, the particles are also indistinguishable, this means that there is no physical way to ever be able to tell the particles from each other. Thus, not only the hamiltonian, but all conceivable perturbations of this hamiltonian must commute with the permutation operators. (These perturbations would correspond to various devices, dies and other tricks that we would attach to the system in our effort to tell the particles from each other.) In short, the permutation operators must commute with all physical observables A: [A, P ] = 0. (3) When such an operator P commutes with all (physical) operators in a system we have a superselection rule: there is no possible transition or interference between states corresponding to different eigenvalues of this operator and thus we can choose a subspace with a fixed eigenvalue and stay with it for ever. (This is also called “picking a theta-vacuum” in particle gauge theories.) In our case, there is a whole set of operators commuting with everything else (the group SN ) and thus the generalized “theta-vacuum” states consist of irreducible representations (irreps) of SN . Further, states that are connected to each other through the action of operators P cannot be distinguished by any physical operator and therefore must be identified as a unique physical state. In other words, the permutation group must be viewed as a discrete gauge group producing unphysical transformations, that is, copies of the same physical system. We must, therefore: • Project on a subspace corresponding to a particular irrep R of SN .

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• Identify elements within this subspace grouping in the same irrep as a unique physical state. The above procedure and choice of R corresponds to a choice of statistics. We will implement this in the coordinate representation |x1 , . . . xN i ≡ |xi. Projecting the Hilbert space to an irrep R of SN amounts to keeping only linear combinations of states within this multiplet that transform in R, that is, X Ca (P )P |xi , a = 1, . . . dR , dR = dim(R) (4) |a; xi = P

where the sum is over all elements of the permutation group and Ca (P ) are appropriately chosen coefficients. If we denote with Rab (P ) the matrix elements of the permutation P in the representations R, X Rab (P −1 )|b, xi· (5) P |a, xi = b

(The appearance of P −1 above is necessary so that successive transformations combine in the right order.) The defining representation decomposes into irreducible components, classified by Young tableaux, each appearing with a certain multiplicity. Should we keep only one irrep out of each multiplicity or the whole multiplet? To decide it, note that if instead of the base state |xi for the construction of the states |a, xi we choose a different permutation Po |xi, then although the new states |a, Po xi constructed through (5) still transform in the irrep R, in general they are not linear combinations of |a, xi but rather span a different copy of R. Since we can continuously move in the configuration space from |xi to Po |xi, we conclude that we must keep all irreps R within each multiplet. (In other words, although for each point in the Hilbert space |xi this multiplet is reducible, the fiber of these representations over the Hilbert space is connected and irreducible.) To realize explicitly the above, we construct the states r dR X Rab (P )P |xi. (6) |ab, xi = N! P

Using the group property of the representation R(P1 )R(P2 ) = R(P1 P2 ), we deduce that under the action of the group SN and under change of base point x the above states transform as: X X Rac (P −1 )|cb, xi, |ab, P xi = Rcb (P −1 )|ac, xi· (7) P |ab, xi = c

c

Thus we see that the first index in these states labels the different elements of a single irrep R, while the second index labels the different equivalent irreps

A.P. Polychronakos: Generalized Statistics in One Dimension

421

in the multiplet. Since both indices take dR values, we recover the standard result that each irrep of SN is embedded in the defining representation a number of times equal to its dimension. Consider now the matrix element hab, x|A|cd, yi, where A is any physical operator, that is, any operator commuting with all elements P of SN . Substituting the definition (6) and using the unitarity of P (P † = P −1 ) and of ∗ (P ) = Rba (P −1 )) we obtain, after a change in summation variable, R (Rab hab, x|A|cd, yi =

dR X Rbe (P 0 )Rea (P −1 )Rcd (P )hx|AP 0 |yi· N! 0

(8)

P,P ,e

Using further the orthogonality (Shur’s) relation (see, e.g. [15]) X

Rab (P )Rcd (P −1 ) =

P

we finally obtain hab, x|A|cd, yi =

X

N! δad δbc dR

δac Rbd (P )hx|A|P yi·

(9)

(10)

P

It is clear from (10) that there is no possible transition between states with different first index. Further, for states with the same index, the amplitude is independent of the index. Thus, the first index a in the state |ab, xi propagates trivially. Since this is the index that corresponds to the different but physically equivalent states within each irrep R, we conclude that the required projection of the Hilbert space to the physical subspace amounts to simply omitting this index from all states. (That is, freeze this index to the same fixed value for all states of the theory; no physical quantity will ever depend on the choice of this value.) On the other hand, the second index, corresponding to different equivalent irreps, does not propagate trivially and must, as argued before, be kept. We are led therefore to the physical states |ba, xi → |a, xi. Let us first choose A = 1. Then (10) provides the overlap between the states X δac Rbd (P )δ(x − P y). (11) hab, x|cd, yi = P

For x in the neighborhood of y it is P = 1 which contributes to the normalization, for which Rbd (1) = δbd and we recover the standard continuous normalization between the states. Choose, now, A = e−iHt , where H is the hamiltonian, and thus find the propagator G(ab, x; cd, y|t) between the states of the system. Projecting to the physical states |ba, xi → |a, xi we obtain the propagator X Rab (P ) G(x, P y; t) (12) GR (a, x; b, y|t) = P

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Topological Aspects of Low Dimensional Systems

where G(x, P y; t) = hx|e−iHt P |yi is the usual many-body propagator. We note that, due to the transformation property (7), the states |a, P xi are linear combinations of states |a, xi. Therefore, projecting down to the physical subspace corresponding to R amounts to trading the original N ! copies of physically equivalent states |P xi for a number dR of global internal degrees of freedom for the system, labeled by the index a. 2.2 Path integral and generalized statistics It is now easy to write down the path integral corresponding to identical particles quantized in the R-irrep of SN . G(x, P y; t) can be expressed as an N -body path integral in the standard way, with particles starting from positions xi and ending in positions P yi = yP (i) . Since all permutations of particle positions are physically equivalent (12) instructs us to sum over all sectors where particles end up in such permuted positions, weighted with the factors Rab (P ) depending on the internal degrees of freedom of the initial and final states. From (6, 11) we can write the completeness relation within the physical subspace Z N X d x |a, xiha, x| (13) IR = N! a and with the use of (13) it is easy to prove that the above path integral is unitary, that is, Z N X d y G(a, x; b, y|t) G(b, y; c, z|t0 ) = G(a, x; c, z|t + t0 ). (14) N! b

We can extend this construction to a more general class of statistics, which includes the cases of the so-called “parabosons” and “parafermions” as special cases. The generalization consists in allowing more than one irrep to appear in the Hilbert space, and keeping more than one state in each irrep. This seems unmotivated and against the spirit of the reduction by the full superselection rule, but it is consistent. It could mean, for instance, that the particles have some hidden internal degrees of freedom accounting for the extra degeneracy, which are invisible to the present hamiltonian but may become dynamically relevant later. (This, then, would augment the space of physical operators in the theory.) The most general situation is that we include CR states from each irrep R. The statistics is fixed by the set of non-negative integers CR . The generalization of the results to this more general situation is straightforward. The internal degree of freedom now takes values A = (R, a, α), where a = 1, . . . dR labels the inequivalent R-irreps and and α = 1, . . . CR labels the states kept within each irrep. (a, α) constitute P the internal degrees of freedom within each irrep. So, overall, A takes R CR dR different

A.P. Polychronakos: Generalized Statistics in One Dimension

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values. The propagator (and corresponding path integral) is obviously X S(P )AB G(x, P y; t) (15) GS (A, x; B, y|t) = P

where S(P )AB = δRA ,RB δαβ (RA )ab (P ).

(16)

The definition of parabosons (parafermions) of order p is that we include once all irreps with up to p rows (columns) in their Young tableaux. We have, therefore, CR = 1 for such irreps and CR = 0 for the rest. We note that the irreps for parafermions are the duals of those for parabosons (the dual of a tableau is the tableau with rows and columns interchanged). In an ˜ are real appropriate basis, the representation matrices of dual irreps R, R and satisfy ˜ ab (P ) = (−1)P Rab (P ) (17) R where (−1)P is the parity of the permutation. We arrive then at the relation between the weights for parabosons and parafermions of order p: SpF (P )AB = (−1)P SpB (P )AB .

(18)

This extends a similar relation for ordinary fermions and bosons [13], for which there are no internal degrees of freedom and SB (P ) = 1. From the path integral we can evaluate the partition function, by simply shifting to the euclidean periodic propagator GE (β) = e−βH and summing over all initial and final states, with the measure implied by (13). Given that X Raa (P ) = trR(P ) = χR (P ) (19) a

we get the expression in terms of the characters of SN Z N X d x ZS (T ) = S(P )hx|GE (β)|P xi N!

(20)

P

where S(P ) =

X

CR χR (P ).

(21)

R

We also note the inversion formula 1 X S(P )χR (P ) CR = N!

(22)

P

which allows to recover CR if we know S(P ) for all permutations. The interpretation in terms of a periodic euclidean path integral is obvious. The characters χR (P ) are a set of integers, and thus the “statistical factors” S(P ) weighing each topological sector of the path integral are

424

Topological Aspects of Low Dimensional Systems

(positive or negative) integers. In the case of parabosons of any order p, however, we note that the statistical weights are positive (or zero) integers. The ones for parafermions can be either positive or negative, as given by SpF (P ) = (−1)P SpB (P ) ,

SpB (P ) ≥ 0.

(23)

We do not have a general formula for SpB (P ) for arbitrary p. For noninteracting particles the partition function can be expressed in terms of the characters of SU (N ) (Shur’s functions) [16, 17]. We note here that the case of distinguishable particles (“infinite statistics” [10, 18, 19]), is also included in this formalism, by accepting all states in each irrep, that is, CR = dR . Since R appears exactly dR times in the defining representation of SN , S(P ) above becomes the trace of P in the defining representation. But all P 6= 1 are off-diagonal in the defining representation, so we get Sinf (P ) = N ! δP,1 , recovering Z (24) Z(T ) = dN xhx|GE (β)|xi for distinguishable particles. 2.3 Cluster decomposition and factorizability Parastatistics particle obey the cluster decomposition principle, in the sense that the density matrix obtained by tracing over a subset of particles which decouple from the system can be constructed as a possible density matrix of the reduced system of remaining particles [8, 12]. From (21), however, we see that the partition function of two dynamically isolated sets of particles N1 and N2 does not factorize into the product of the two partition functions, since the statistical weights S(P ) in general do not factorize into S(P1 )S(P2 ) when P is the product of two commuting elements P1 and P2 . Equivalently, this means that the occupation degeneracy D(p1 , p2 , . . .) of p1 , p2 , . . . particles occupying a set of levels 1 , 2 , . . . does not factorize into the product of individual occupation degeneracies for each level i . There is an effective “coupling” between the particles. This has important physical implications. If the two sets of particles are totally isolated, it does not make sense to evaluate the partition function of the total system, since the statistical distribution can never relax to the one predicted by that partition function. The individual partition functions of the subsystems are the relevant ones. If, however, the two sets are only weakly coupled, then initially each set will distribute according to its reduced partition function, but after some relaxation time (depending on the strength of the coupling between the two sets) they will relax to the joint distribution function, which, we stress, will not even approximately equal the product of the individual ones. Thus, cluster decomposition holds in an absolute sense but fails in a more

A.P. Polychronakos: Generalized Statistics in One Dimension

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realistic sense. In contrast, fermions and bosons respect cluster decomposition in both senses. We summarize by pointing out that the most general statistics of the type examined here is parametrized by any of three possible sets of numbers. The first is, as just stated, the number of states CR accepted for each irrep R of SN . The second is the statistical weights S(P ) appearing in the partition function (euclidean path integral). These weights are invariant under conjugation of P → QP Q−1 . Finally, we could use the degeneracy of a many-body occupancy state, call it D[pi ], as our definition. (This was the starting point of the first known attempt to generalized statistics, the “Gentile statistics” [20].) These three sets contain as many elements as the partitions of N and are all equivalent. What are the restrictions or criteria to be imposed on the above parameters? The first one is unitarity, that is, the existence of a well-defined Hilbert space with positive metric. This requires that CR be non-negative (no negative norm states) integers (no “fractional dimension” states). The other will be what we call “strong cluster decomposition principle” or “factorizability,” that the partition function of isolated systems factorizes. This is a physical criterion, rather than a consistency requirement. To summarize: • Unitarity: CR non-negative integers. Q Q • Factorizability: S(P ) = n∈cycles(P ) S(n) or D[pi ] = i D(pi ). Factorizability, in particular, implies that the grand partition function in the case of noninteracting particles will be obtained by exponentiating the sum of all connected path integrals (P a cyclic permutation of degree n) with weights S(n)/n (1/n is the symmetry factor of this path integral, corresponding to cyclic relabelings of the particles). The grand partition function will factorize into a product of partition functions for each level i . Thus, S(n) are cluster coefficients connected to D(n) in the standard way ! ∞ X S(n) n z D(p)z = exp . n p=0 n=1

∞ X

p

(25)

The above formula, in fact, provides the easiest way to relate D[pi ] and S(P ) in the general case (no strong cluster property) but we will not enter into this. If we assume that S(1) = D(1) = 1, then it is easy to verify that the only solution of the above two criteria is ordinary fermions and bosons. The situation is different, however, when S(1) = D(1) = q > 1 (this would mean, e.g., that the particles come a priori in q different “flavors”). The possibilities are manifold. All these generalized statistics share the following generic features: • The degeneracy of the state where n particles occupy different levels is q n . (Indeed, D(1, 1, . . . 1) = D(1)n = q n .).

426

Topological Aspects of Low Dimensional Systems

• If state A can be obtained from state B by “lumping” together particles that previously occupied different levels, then D(A) ≤ D(B). (E.g., D(3) ≤ D(2, 1) ≤ D(1, 1, 1).) The above possibilities include the obvious special cases of q1 bosonic flavors and q2 fermionic ones (q1 + q2 = q), for which S(n) = q1 − (−1)n q2 , along with many other. As an example, we give the first few degeneracies for many-particle level occupation for all statistics with q = 2: D(1) = 2, D(2) = 4, D(3) = 8 D(1) = 2, D(2) = 3, D(3) = 6, 5, 4(B + B) D(1) = 2, D(2) = 2, D(3) = 4, 3, 2(B + F ), 1, 0 D(1) = 2, D(2) = 1, D(3) = 2, 1, 0(F + F ) D(1) = 2, D(2) = 0, D(3) = 0. The specific choices denoted by B + B, B + F and F + F are the ones corresponding to two bosonic, one bosonic and one fermionic, and two fermionic flavors respectively. The topmost statistics could be termed “superbosons” and the bottom one “superfermions” of order 2. We also remark here that the “(p, q)-statistics” introduced in [12] can be realized as particles with p bosonic and q fermionic flavors, where we identify each multiplet transforming irreducibly under the supergroup SU (p, q) as a unique physical state. Finally, we comment on “Gentile statistics” [20]. The rule is simply that up to p particles can be put in each single-particle level. This corresponds to D(n) = 1 for n ≤ p, and D(n) = 0 otherwise. This has been criticized [14] on the grounds that fixing the allowed occupations for each singleparticle state is not a statement invariant under change of single-particle basis. It should be clear from this lecture that any statistics satisfying the unitarity requirement is consistent and basis-independent. Therefore, Gentile statistics must violate unitarity. Indeed, it is easy to check that all weights CR for such statistics are integers (this is generic for all statistics with integer D(n)), but not necessarily positive. In the specific case of p = 2, e.g., where up to double occupancy of each level is allowed, the degeneracies of each irrep of SN (parametrized, as usual, by the length of Young tableau rows) up to N = 5 are C2 = C21 = C22 = C221 = 1, C111 = C1111 = C2111 = −1, else CR = 0. (26) We see that representations 111, 1111, 2111 correspond to ghost (negative norm) states and their effect is to subtract (rather than add) degrees of freedom. In conclusion, we see that the permutation group approach to generalizing statistics gives results of rather limited interest. The resulting statistics can be though of as particles with internal degrees of freedom, with perhaps some superselection rule that forbids the manifestation of all possible

A.P. Polychronakos: Generalized Statistics in One Dimension

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internal states. To find something more exotic we must consider other approaches. 3

One-dimensional systems: Calogero model

Let us now specialize in one dimension and see in what different angle we can approach the problem of statistics. The peculiarity of a lineal world is that, kinematically, particles cannot exchange positions without “bumping” onto each other. This has tangible physical consequences and makes the notion of statistics in one dimension rather special. Let’s see: A. Since configurations related by particle permutation are “gauge” copies of each other, we could “gauge fix” and restrict ourselves to only one gauge copy. In one dimension the configuration space of N indistinguishable particles breaks into N ! sectors classified by the ordering of the coordinates: x1 < . . . < xN and its permutations. Restricting to one sector we are faced with the problem of boundary conditions on the wavefunction at the boundary of this space, when two or more coordinates become equal, so that we preserve hermiticity. The choice of boundary conditions can be interpreted as a choice of statistics. We mention two possibilities: a) A linear local boundary condition: ψ + λ∂n ψ|boundary = 0

(27)

where ∂n is the normal derivative at the boundary [21]. Clearly λ = 0 (Neumann) corresponds to fermions; we can analytically extend ψ in the other sectors in an antisymmetric way. Similarly, λ = ∞ (Dirichlet) corresponds to bosons. Any other choice would be some intermediate statistics. λ, however, introduces a length scale and is not a very satisfactory statistics definition. At any rate, this system is equivalent to bosons with a deltafunction two-body interaction of strength proportional to λ−1 . b) Fix the analytical behavior of ψ at the boundary as ψ ∼ x` as x → 0.

(28)

This way the probability current would scale as ψ∂n ψ ∼ `x2`−1 and would vanish if ` > 12 or ` = 0. This would define the statistics through the dimensionless parameter `. Notice that this behavior would require a twonear coincidence points. This is the first body potential behaving like `(`−1) x2 glimpse at the inverse-square potential arising in the statistics context. B. We could, alternatively, examine the scattering phase θ between particles. At high relative energies the two-body scattering phase approaches the values θ → 0 for bosons, θ → π for fermions (29) irrespective of the details of their interaction (provided it is not too singular). The corresponding many-body phase is the sum of two-body ones.

428

Topological Aspects of Low Dimensional Systems

So, if we see a system where this phase goes to the value `π we can interpret it as one with generalized statistics of order ` [21]. As we shall see, this definition also leads to the inverse-square potential. There are hints from algebraic considerations also pointing to this type of potential [22–24]. This is enough motivation, at any rate, to examine particles with that type of two-body interaction potentials. This is known as the Calogero model, or, more completely, as the Calogero-SutherlandMoser (CSM) model [25–27]. 3.1 The Calogero-Sutherland-Moser model This is a system of nonrelativistic identical particles on the line with pairwise inverse-square interactions. The basic hamiltonian is H=

N X 1 i=1

2

p2i +

X i<j

g · (xi − xj )2

(30)

This is the “free” (scattering) Calogero system on the line [25]. The particle masses m have been scaled to unity. This is the only scale-free two-body potential that one can have (the potential, quantum mechanically, scales like the kinetic term). An external harmonic oscillator potential can also be added to confine the system without spoiling its features. This is the harmonic Calogero model. Alternatively, one could consider a periodic version of the system. The particles now interact through infinitely many periodic copies of themselves and the two-body potential becomes V (x) =

∞ X

g g = 2 2 (x + 2πn) 2 sin x2 n=−∞

(31)

(we scaled the length of the box to 2π). This is the Sutherland model [26]. There are other versions of this class of models that will not concern us here [28]. Classically the coupling constant g should be positive to ensure particles are not “sucked” into each other. Quantum mechanically the uncertainty principle works in our favor and the minimum allowed value for g is g = − 41 (we put henceforth h ¯ = 1). For later convenience, it is useful to parametrize g in the fashion g = `(` − 1) (32) in which case the minimum value is naturally obtained for ` = 12 . The above system is integrable, which means that there are N integrals of motion in convolution, that is, N functions on phase space with vanishing Poisson brackets: {In , Im } = 0 ,

n, m = 1 . . . N.

(33)

A.P. Polychronakos: Generalized Statistics in One Dimension

429

For the scattering system (no external potential) I1 is the total momentum, I2 is the total energy amd the higher In are higher polynomials in the momenta also involving the two-body potentials. We will not belabor here their form nor prove integrability at this point, since this is beyond our scope and will, at any rate, be dealt with in later lectures. We will only state without proof the qualitative features relevant to our purposes. The key interesting property of the above model, that sets it apart from other merely integrable models, is that, both classically and quantum mechanically, it mimics as closely as possible a system of free particles. Let us first look at its classical behavior. The motion is a scattering event. Asymptotically, at times t = ±∞, the particles are far away, the potentials drop off to zero and motion is free. When they come together, of course, they interact and steer away from their straight paths. Interestingly, however, when they are done interacting, they resume their previous free paths as if nothing happened. Not only are their asymptotic momenta the same as before scattering, but also the asymptotic positions (scattering parameters) are the same. There is no time delay of the particles at the scattering region. The only effect is an overall reshuffling of the particles. Thus, if one cannot tell particles from each other, and if one only looks at scattering properties, the system looks free! This behavior carries over to quantum mechanics. The asymptotic scattering momenta are the same before and after scattering. The fact that there is, further, no time delay translates into the fact that the scattering phase shift is independent of the momenta. Thus it can only be a function of the coupling constant and the total number of particles. It is, actually, a very suggestive function: N (N − 1) `π. (34) 2 Thus the phase is simply `π times the total number of two-body exchanges that would occur in the scattering of free particles. Clearly the case ` = 0 would correspond to free bosons and ` = 1 to free fermions (for these two values the potential vanishes and the system is, indeed, free). For any other value we can interpret this system as free particles with generalized statistics `. A word on the permutation properties of this system is in order. The inverse-square potential is quantum mechanically impenetrable, and thus the “ordinary” statistics of the particles (symmetry of the wavefunction) is irrelevant: if the particles are in one of the N ! ordering sectors they will stay there for ever. The wavefunction could be extended to the other sectors in a symmetric, antisymmetric or any other way, but this is irrelevant for physics. No interference between the sectors will ever take place. All states have a trivial N ! degeneracy. (That is to say all states in all irreps of SN have the same physical properties.) Permutation statistics are therefore θsc =

430

Topological Aspects of Low Dimensional Systems

irrelevant and we can safely talk about the effective statistics as produced by their coupling constant `. The behavior of the wavefunction near coincidence points in as in (28). This system, thus, satisfies both our boundary condition and scattering phase criteria for generalized statistics. Let us also review the properties of the confined systems. In the presence of an external harmonic potential of the form X1 ω 2 x2i (35) V = 2 i the energy spectrum of a system of uncoupled particles would be X N ni ω. E = ω+ 2 i

(36)

The ni are nonnegative integers satisfying n1

≤

. . . ≤ nN

for bosons

(37)

n1

<

. . . < nN

for fermions.

(38)

The actual spectrum of this model is E=

X N (N − 1) N ω+` ω+ ni ω 2 2 i

(39)

with ni being “pseudo-excitation numbers” obeying bosonic selection rules: ni ≤ ni+1 . Defining the “quasi-excitation numbers” n ¯ i = ni + (i − 1)`

(40)

we can check that the expression of the spectrum in terms of the n ¯ i is identical to the free one (36) but with the quantum numbers now obeying the selection rule ¯ i+1 − `. (41) n ¯i ≤ n This is a sort of exclusion principle that requires the particle quantum numbers to be at least a distance ` apart (as contrasted to 1 for fermions and 0 for bosons). Again, a generalized statistics interpretation is manifest [29]. Let us clarify that the above numbers n ¯ i are no more integers. They do, however, increase in integer increments. The rule is that the ground state is determined by the minimal allowed nonnegative values for n ¯ i obeying (41) while the excited states are obtained by all integer increments of these values that still obey (41). The periodic (Sutherland) model has similar properties. Its spectrum is E=

X1 i

2

ki2 + `

X N (N 2 − 1) (kj − ki ) + `2 24 i<j

(42)

A.P. Polychronakos: Generalized Statistics in One Dimension

431

with the “pseudo-momenta” ki being integers satisfying bosonic rules: ki ≤ ki+1 . This looks rather different than the corresponding free expression (for ` = 0). Defining, however, again “quasi-momenta” N +1 (43) pi = ki + ` i − 2 we can check that the expression for the spectrum becomes E=

X1 i

2

p2i

(44)

that is, the free expression. The quasi-momenta satisfy pi ≤ pi+1 − `

(45)

that is, the same selection rule as the n ¯ i before. Again, we observe a generalization of the fermionic and bosonic selection rules corresponding to statistics `. The ground state is the minimal (nearest to zero) numbers satisfying (45) while excitations correspond to integer increments (or decrements) of the pi still satisfying (45). Let us note that the above rule for ` = 1 reproduces the fermionic spectrum of particles with periodic boundary conditions for odd N and anti-periodic ones for even N : in the odd (even) N case the momenta are quantized to (half-) integers. This has a natural interpretation: when we take a particle around the circle it goes over N − 1 other particles. If we require the phase shift of the wavefunction in this process to agree with the minus signs picked up from the N − 1 fermion exchanges we recover the previous rule. We stress that, for free particles, this is not a consistency requirement but rather an aesthetic rule. At any rate, this is what the Sutherland model chooses to do! In conclusion we see that the CSM model can be though of as a system of particles obeying generalized statistics. This manifests in terms of the boundary conditions of the wavefunction, the scattering phases and, most significantly, through a peculiar “level repulsion” of their quantum numbers generalizing the Fermi exclusion principle. 3.2 Large-N properties of the CSM model and duality Let us examine, now, the properties of the CSM model as the number of particles grows large. At zero temperature, a non-interacting fermion system would form a Fermi sea. The corresponding “Fermi surface” in one dimension degenerates to points. For the system in an external harmonic potential there is just one point corresponding to the highest excitation nF = N −1. For the free periodic system we would have two Fermi momenta

432

Topological Aspects of Low Dimensional Systems

at ±pF = ± N2−1 . Excitations over this ground state are, then, conveniently classified in terms of particles (a filled Fermi sea with an isolated particle above or below) and holes (a filled sea with one unoccupied state inside it). Interestingly, the CSM model presents a similar picture. The qualitative features of both the Calogero and the Sutherland model are similar, so we pick the latter as most closely representing a gas of free particles in a box. From (45) above we see that the ground state also forms a “quasi-Fermi sea” (or should we call it a “Luttinger sea”?) with Fermi levels rescaled by a factor `: pF = ` N2−1 . Its minimal excitations are analogous to the ones of a Fermi sea, but not quite: • A particle would be an isolated occupied quasimomentum above or below a completely filled sea of quasimomenta. Particles – are excited in units of 1 (the increments of their quasimomentum); – take up a space ` in quasimomentum (since they cannot be “packed” closer than ` units apart). • A hole would be an isolated empty space inside an otherwise occupied sea. Interestingly, the minimal such excitation is not obtained by removing one particle from the sea, but rather by incrementing all quasimomenta of the sea above the place where we want to create the hole by one unit. Holes – are excited in units of `. Indeed, since the distance of quasimomenta in the sea is `, the possible positions of the hole are at distances ` apart; – take up a unit space in quasimomentum. Indeed, incrementing all quasimomenta above a given place in the sea by two units creates two holes in that place, and so on; by locally reshuffling quasimomenta we can then separate these holes. Note that holes are not antiparticles. Removing a particle for the sea creates a gap of ` spaces and, from above, ` holes. The correspondence is 1 particle ∼ −` holes.

(46)

We already observe a sort of duality between the two types of excitations. This can be summarized as particle ↔ hole ,

`↔

1 , `

p ↔ `p.

(47)

Under the above, the spectrum of excitations of the model remains invariant. This is the simplest manifestation of a coupling-constant duality that goes over to the correlation functions and Green’s functions of the model [30–33]. Obviously, this duality is spoiled by nonperturbative effects, since holes are confined within the sea while there is no “ceiling” for particle excitations. This concludes our brief “physicist’s tour” of the CSM model. We are now going to examine the most useful ways of analyzing this system.

A.P. Polychronakos: Generalized Statistics in One Dimension

4

433

One-dimensional systems: Matrix model

Identical particles can be formulated in terms of their phase space variables modulo permutations. This leads to the symmetric group approach of Section 1. We could try other approaches. For instance, we could formulate them in terms of the eigenvalues of an N × N matrix. There is no a priori ordering of these eigenvalues, so this certainly encodes identical particles. It is clear that the permutation symmetry of the problem has been promoted to the continuous symmetry of unitary conjugations of this matrix, which leaves the eigenvalues intact. This should be viewed as a gauge symmetry and would open the road to yet another definition of statistics. 4.1 Hermitian matrix model With this introductory remark as motivation, let us examine a matrix model that parallels as closely as possible particle mechanics. The kinematical variable is a hermitian N × N matrix M and the lagrangian reads 1 ˙2 M − V (M ) (48) L = tr 2 V (x) is a scalar potential evaluated for the matrix variable M . Clearly the above has a time-translation invariance which leads to the conserved energy 1 ˙2 M + V (M ) . (49) H = tr 2 Moreover, the action is invariant under time-independent unitary conjugations of the matrix M : (50) M → U M U −1 . This nonabelian SU (N ) symmetry leads to the conserved hermitian traceless matrix J = i[M, M˙ ] (51) where [, ] denotes ordinary matrix commutator. These are the “gauge charges” that, when fixed, will determine the “statistics” of the model. But let us first examine the system classically. We are interested in the dynamics of the eigenvalue of M , so we parametrize it as M = U ΛU −1

(52)

where U (t) is the unitary “angular” part of the matrix and Λ(t) = diag{x1 , . . . xN } are the eigenvalues. Clearly the conserved quantity J has to do with invariance under “rotations” of the angular part of M and

434

Topological Aspects of Low Dimensional Systems

thus corresponds to the “angular momentum” of U (t). We define the “gauge potential” (53) A = −U −1 U˙ M˙ , J and the lagrangian L become, in this parametrization, M˙ = U Λ˙ + [Λ, A] U −1 = iU ([Λ, [Λ, A]]) U −1 ≡ U KU −1 1 ˙2 1 Λ + [Λ, A]2 − V (Λ) = tr 2 2

J L

=

N N 1 X 1X x˙ i − (xi − xj )2 Aij Aji . 2 i=1 2 i,j=1

(54) (55) (56) (57)

The matrix elements of A and K are related Kjk = i [Λ, [Λ, A]]jk = i(xj − xk )2 Ajk .

(58)

Finally, solving (58) for Ajk and putting into (57) we obtain L=

X1 i

2

x˙ 2i +

X 1 X Kij Kji − V (xi ). 2 (xi − xj )2 i

(59)

i6=j

The first two terms are kinetic, coming from M˙ 2 , while the last one is potential. Therefore, the hamiltonian H is H=

X1 i

2

p2i +

X 1 X Kij Kji + V (xi ). 2 2 (xi − xj ) i

(60)

i6=j

Note that the eigenvalues are kinematically coupled by an inverse-square type potential with the angular momentum degrees of freedom. The connection of the matrix model to the Calogero model along the lines presented here and below was first established in [34]. Also, the hamiltonian (60) has been proposed independently of the matrix model as an SU (N )generalization of the classical Calogero system [35, 36]. We can now examine special cases: a) The most “gauge invariant” sector is, of course, the one in which the angular momentum charges vanish, that is, J = 0. In that case, (60) for K = 0 becomes the hamiltonian of non-interacting particles in an external potential V (x). This would be the case of “standard” particles. b) For the next simplest case the angular momentum J should be as trivial as possible without vanishing. Only its eigenvalues are really relevant, since we can always perform a time-independent unitary transformation V which would shift U → V U and would rotate J → V JV −1 . The simplest

A.P. Polychronakos: Generalized Statistics in One Dimension

435

choice would be to take the eigenvalues of J to be equal. Unfortunately, this is not possible since the traceless condition would make them vanish. The simplest possible choice is to take all the eigenvalues equal to ` except one, which would cancel the trace by being (1 − N )`. This can be written in terms of an arbitrary column N -vector v as J = `(vv † − 1) ,

v† v = N

(61)

u = U −1 v.

(62)

in which case K becomes K = `(uu† − 1) ,

From (58) we see that Kii = 0 (no sum on i) and thus ui u∗i = 1

(no sum).

(63)

So the coefficient of the inverse-square potential in (60) becomes Kij Kji = `ui u∗j `uj u∗i = `2

(i 6= j).

(64)

Finally, (60) becomes H=

X1 i

2

pi +

X i<j

X `2 + V (xi ). (xi − xj )2 i

(65)

This is the Calogero model! The potential strength g = `2 is related to the conserved charge `. Quantum mechanically, picking this charge will amount to a choice of statistics. We also get, at this stage, an arbitrary external potential V (x). c) More general choices of J amount to more variety in its eigenvalues. Kij Kji now, in general, becomes time-dependent and the dynamics more complicated. We postpone the discussion for the quantum case where it will be shown that this corresponds to Calogero particles having also internal degrees of freedom. This will be a generalization of the discussion of the first section, with irreps of SU (N ) substituting the irreps of SN . Now that we have this new approach we can use matrix technology to demonstrate the integrability of the Calogero model [28, 34]. For V (x) = 0 the matrix motion becomes free and M˙ is conserved. The conjugationinvariant quantities (66) In = tr M˙ n are also conserved and in involution (the matrix elements of M˙ are momenta and have vanishing Poisson brackets). From (54) and (58) we have iKjk −1 ˙ δjk x˙ j − (1 − δjk ) U (67) Mjk = U xj − xk iuj u∗k U. (68) = U −1 δjk x˙ j − (1 − δjk ) xj − xk

436

Topological Aspects of Low Dimensional Systems

When the above expression is inserted in the trace In = tr M˙ n clearly U drops and products of the form ui u∗j uj u∗k . . . u∗i will appear which reduce to powers of `. Therefore, the In reduce to expressions involving only xi , x˙ i and the coupling constant `. These are the conserved integrals of the Calogero model. Starting from the matrix model the actual motion of the Calogero model can be obtained. The solution for M is M = B + Ct

(69)

for arbitrary matrices B, C. The conserved charge becomes J = i[M, M˙ ] = i[B, C] = i`(uu† − 1).

(70)

By unitary transformations we can choose the phases of u such that ui = 1; choices for B, C, then, that satisfy (70) are Bjk = δjk qj ,

Cjk = δjk pj − (1 − δjk )

i` qj − qk

(71)

qi and pi are the initial conditions for xi and x˙ i at time t = 0. Diagonalizing, then, M = B + Ct for the above B, C produces the motion of the system. Another choice is Bjk = δjk aj + (1 − δjk )

i` , Pj − Pk

Cjk = δjk Pj

(72)

Pi and ai are asymptotic momenta and impact parameters. For t → ±∞ the off-diagonal elements of B produce a perturbation of order t−1 to the eigenvalues, so the motion is determined by the diagonal elements ai + Pi t alone. We recover the result that the motion at asymptotic regions is the same as if the particles were free. We conclude by mentioning that the matrix model is also integrable and solvable in the presence of a harmonic oscillator potential V (x) = 12 ω 2 x2 . The non-hermitian matrix Q = M˙ + iωM evolves as Q(t) = eiωt Q(0)

(73)

and the matrix Q† Q is conserved. We leave it as an exercise to derive the conserved integrals and the motion of the corresponding Calogero problem. External potentials with up to quartic dependence on x also lead to integrable, although not so solvable, models [37]. It is an open question to prove this is all there is, or to find yet more integrable potentials. For the purposes of statistics this is a rather academic issue. Finally, we may wonder what restricts us to one dimension. We chose a model with one matrix, and its eigenvalues corresponded to coordinates of

A.P. Polychronakos: Generalized Statistics in One Dimension

437

particles on the line. We could, indeed, start with an appropriate model with many matrices, which would reproduce particle motion in higher dimensions [38]. The integrability and solvability properties of such extended models, however, are much less pleasant. The question of whether they represent a workable extension of statistics remains open. 4.2 The unitary matrix model The hermitian matrix model works well for particles on the line but has trouble representing particles on periodic spaces. The most natural candidate for such models would be a unitary N × N matrix U . Its eigenvalues are phases and naturally live on the circle. We start, therefore, with a lagrangian that represents the invariant kinetic energy on the space of such matrices: 1 (74) L = − tr(U −1 U˙ )2 . 2 A potential could in principle be included but we are interested in the translationally invariant case and will omit it. The treatment is similar as before, and we just summarize the relevant facts. The Lagrangian is, in fact, invariant under separate left- and rightmultiplications of U by time-independent unitary matrices and and there are two corresponding conserved matrix angular momenta L and R: U →VU : U → UW

L =

−1

:

R =

˙ −1 iUU −iU

−1

(75) U˙ .

(76)

The unitary conjugation that preserves the eigenvalues corresponds to W = V and its generator is ˙ U −1 ]. J = L + R = i[U,

(77)

The rest of the discussion is as previously. Parametrizing U = V ΛV −1 with Λ = diag{eixi , . . . eixN }

(78)

the Hamiltonian becomes, after a few steps, H=

X1 i

where, as before,

2

p2i +

1 X Kij Kji x −x 2 4 sin2 i 2 j

(79)

i6=j

K = V −1 JV.

(80)

Choosing J = K = 0 reproduces free particles on the circle, while choosing J = `(uu† − 1) we obtain Kij Kji = `2 and we recover the Sutherland

438

Topological Aspects of Low Dimensional Systems

inverse-sine-square model H=

X1 2

i

x˙ 2i +

1X `2 x −x · 2 2 4 sin i 2 j

(81)

i6=j

This model is integrable and solvable by the same techniques as the hermitian one. The conserved invariant quantities are In = trLn = tr(−R)n = tr(iU −1 U˙ )n

(82)

U = BeiCt

(83)

and the solution is with B a unitary and C a hermitian matrix satisfying BCB −1 − C = J.

(84)

For the Sutherland case with J = `(uu† − 1), ui = 1, B, C become Bjk = δjk eiqj ,

Cjk = δjk pj + (1 − δjk )

i` ei(qj −qk )

−1

(85)

where, clearly, qi and pi are initial positions and momenta. We conclude by mentioning that, upon scaling x → αx, t → α2 t, the Sutherland model goes over to the free Calogero model. This is the “infinite volume” limit. 4.3 Quantization and spectrum We will, now, perform the quantization of this system on a periodic space, using the unitary matrix model. We begin by defining a canonical momentum matrix conjugate to the “coordinate” U Π=

∂L = −U −1 U˙ U −1 . ∂ U˙

(86)

The Poisson brackets are {Ujk , Πlm } = δjm δlk

(87)

Π is somewhat unpleasant, being neither unitary nor hermitian. We prefer to work in terms of the hermitian matrices L and R defined previously ˙ −1 = −iU Π , L = iUU

R = −iU −1 U˙ = iΠU.

(88)

Using (87) we derive the following Poisson brackets: {Ljk , Llm } =

i(Ljm δlk − δjm Llk )

(89)

{Ljk , Rlm } = {Rjk , Rlm } =

0 i(Rjm δlk − δjm Rlk ).

(90) (91)

A.P. Polychronakos: Generalized Statistics in One Dimension

439

The above is nothing but two copies of the U (N ) algebra in disguise. To see this, expand the matrices L and R in the basis of the fundamental generators of SU (N ) T a plus the unit matrix: 2 NX −1

o

L=L +2

La T a

(92)

Ra T a

(93)

a=1

o

R =R +2

2 NX −1

a=1

with Lo , La , Ro , Ra numbers. Then use the SU (N ) commutation relations [T a , T b ] = if abc T c

(94)

as well as the normalization tr(T a T b ) =

1 δab 2

(95)

to show that the expansion coefficients satisfy the Poisson algebra {La , Lb } = {La , Rb } =

f abc Lc 0

(96) (97)

{Ra , Rb } =

f abc Rc

(98)

while Lo , Ro are central. Note that the U (1) charges Lo = −Ro =

1 X 1 tr(−iU −1 U˙ ) = x˙ i N N i

(99)

are essentially the total momentum of the system. We are now ready to perform quantization. In the U -representation, where states are functions of U , Π becomes the matrix derivative Πjk = −iδU , acting as δU tr(U B) = B ,

δU tr(U −1 B) = −U −1 BU −1

(100)

where B is a constant matrix, and similarly on expressions containing more U ’s. L and R, upon proper ordering, are represented as L = −U δU ,

R = δU · U

(101)

where in R it is understood that we first act with the derivative and then right-multiply the result by U . With this ordering, L and R become the proper U (N ) operators acting as Ltr(U B) = −U B,

Ltr(U −1 B) = BU −1

(102)

440

Topological Aspects of Low Dimensional Systems

Rtr(U B) = BU,

Rtr(U −1 B) = −U −1 B.

(103)

It is also useful to express their action on arbitrary functions of U as tr(iL)f (U ) = f ((1 − i)U ) − f (U ) tr(iR)f (U ) = f (U (1 + i)) − f (U )

(104) (105)

where is an arbitrary infinitesimal hermitian matrix, emphasizing their role as generators of left- and right-multiplication on U . Correspondingly, the operators La and Ra satisfy the SU (N ) algebra. Their action can be obtained by taking = εT a with ε an infinitesimal scalar parameter, that is, = f ((1 − iεT a)U ) − f (U )

(106)

iεR f (U ) = f (U (1 + iεT )) − f (U ).

(107)

iεLa f (U ) a

a

The hamiltonian, being classically the kinetic term on the manifold of unitary matrices U (N ), quantum mechanically becomes the laplacian operator on the manifold [39]. Using (92, 93) it is expressed as H=

X X 1 1 1 1 trL2 = (La )2 + N (Lo )2 = (Ra )2 + (Ro )2 = trR2 . (108) 2 2 2 2 a a

It is, therefore, the common quadratic Casimir of the left- and right-SU (N ) algebra plus the square of the U (1) charge, the two parts identifiable as the relative and center-of-mass energy respectively. Quantum mechanical states grouping into irreducible representations of the L and R SU (N ) algebras will, thus, be degenerate multiplets of the hamiltonian. The U (1) (center of mass) part trivially separates: we can boost any state by any desired total momentum N P by multiplying the wavefunction by (det U )P . We will examine only the SU (N ) part from now on. A natural basis of states for the Hilbert space are the matrix elements of the unitary irreducible representations (irreps) of SU (N ). Let R denote such an irrep, R(U ) the matrix that represents U in this irrep and Rαβ (U ) the αβ matrix element of this matrix. Clearly α and β range from 1 to the dimensionality of R, dR . Rαβ (U ) are a complete orthonormal basis of wavefunctions for U , that is Z 0 (U )∗ = δRR0 δαγ δβδ (109) [dU ]Rαβ (U )Rγδ with [dU ] the volume element on the space of SU (N ) matrices as implied by the metric ds2 = −tr(U −1 dU )2 , also called the Haar measure. We will, now, show that each Rαβ (U ) is an eigenstate of the hamiltonian with eigenvalue equal to the quadratic Casimir of R, CR . Qualitatively,

A.P. Polychronakos: Generalized Statistics in One Dimension

441

after the discussion of the last paragraphs, this should be obvious: L and R generate the transformations U → V −1 U and U → U W . R(U ) transforms ¯ under L and in the irrep R under R. Since H is in the conjugate irrep R the common quadratic Casimir of L and R we conclude that all d2R states Rαβ (U ) are energy eigenstates with eigenvalue CR = CR¯ . (If you are confused about L generating U → V −1 U rather than U → V U , think of the difference between active and passive transformations, which is relevant when shifting from classical to quantum: ψ(x − a) shifts the wavefunction by +a. Also, although classical transformations on U compose properly, (110) V1 (V2 U ) = (V1 V2 )U quantum mechanically the operators Vˆ that perform the shift U → V U on the argument of the wavefunction would compose Vˆ1 (Vˆ2 f (U )) = Vˆ1 f (V2 U ) = f (V2 V1 U ) = (Vˆ2 Vˆ1 )f (U ).

(111)

Therefore we need to invert the action of Vˆ to get the right composition law. See also the discussion of irreps of SN in the section about permutation group statistics.) Let us prove the fact HRαβ (U ) = CR Rαβ (U ) more analytically. Since R(U ) is a representation, it obeys the group property X Rαγ Rγβ (V ). (112) Rαβ (U V ) = γ

From (106) we have (1 + iε)La Rαβ (U ) = Rαβ ((1 − iεT a )U ) = Rαγ (1 − iεT a)Rγβ (U )(113) a Rγβ (U ) = Rαβ (U ) − iεRαγ

(114)

where Ra = R(T a ) is the a-th generator of SU (N ) in the R representation. So a Rγβ (U ) (115) La Rαβ (U ) = −Rαγ and X a

(La )2 Rαβ (U ) =

X a

a a Rαγ Rγδ Rδβ (U ) =

X

(Ra )2αδ Rδβ (U ).

(116)

a

P The sum a (Ra )2 appearing above is the quadratic Casimir in the irrep R and is proportional to the identity matrix δαδ . So, finally, HRαβ (U ) = CR Rαβ (U ).

(117)

Incidentally, the spectrum spanned by CR for all R is nothing but the spectrum of N free fermions on the circle with the ground state energy

442

Topological Aspects of Low Dimensional Systems

and the center-of-mass energy subtracted, where the lengths Ri of the rows of the Young tableau of R correspond to the “bosonized” fermion momenta pi = Ri − i + 1

(118)

and where the center-of-mass energy has been subtracted. The condition Ri ≥ Ri+1 for the rows amounts to the fermionic condition pi > pi+1 . The spectrum of the full matrix model, then, is identical to the free fermion one but with different degeneracies. We have, therefore, identified all energy eigenstates of the matrix model. It remains to implement the quantum analog of the choice of angular momentum J, identify the corresponding reduced quantum model, and pick the subspace of states of the full model that belongs to the reduced model. J obeys itself the SU (N ) algebra (it is traceless, no U (1) charge). A choice of value for J amounts to a choice of irrep r for this algebra. States within the same irrep are related by unitary transformations of U and give the same dynamics; they will be discarded as gauge copies, and only the choice of irrep will be relevant. Since J = L + R, we see that states trans¯ R) irreps will transform in the R ¯ × R under forming under (L, R) in the (R, J. So, only irreps r that are contained in the direct product of two mutually conjugate irreps can be obtained for J. This amounts to irreps r with a number of boxes in their Young tableau that is an integer multiple of N . (To get a feeling of this, consider the case N = 2. Then J is an orbital-like realization of the angular momentum through derivatives of U and clearly cannot admit spinor representations.) We must, therefore, project the d2R states in Rαβ (U ) to the subspace of ¯ α; R, β|r, γ) the Clebschstates transforming as r under L + R. Call G(R, Gordan coefficient that projects these states to the γ state of r. Then the relevant states for this model become X ¯ α; R, β|r, γ). Rαβ (U )G(R, (119) ΨR (U ) = α,β

The index γ labeling the states within r, as we argued before, counts the dr gauge copies and does not imply a true degeneracy of states. The degeneracy of the states produced by each R is, then, given by the number of times that ¯ × R or, equivalently, the the irrep r is contained in the direct product R number of times that R is contained in R×r. Calling this integer D(R, r; R), we obtain for the spectrum and degeneracies: ER = CR ,

DR = D(R, r; R).

(120)

In particular, if DR = 0 the corresponding energy level is absent from the spectrum. Concluding, we mention that an approach which also reproduces the spectrum and states of the Sutherland model is two-dimensional Yang-Mills

A.P. Polychronakos: Generalized Statistics in One Dimension

443

theory on the circle [40, 41]. This approach is essentially equivalent to the matrix model above and we will not be concerned with it. 4.4 Reduction to spin-particle systems So we have derived the spectrum, degeneracy and wavefunctions of the matrix model restricted to the sector J = r. Classically these restrictions represented free particles (J = 0), Sutherland particles (J = `(vv † − 1)) or something more general. What are the corresponding quantum systems? To find these, let us reproduce here the expression of the reduced hamiltonian in one of these sectors: X1 1 X Kij Kji p2i + (121) H= x −x − Eo . 2 2 4 sin2 i 2 j i i6=j This expression remains valid quantum mechanically upon a proper definition (ordering) of the operator K. The only residual quantum effect is a constant term Eo that comes from the change of measure from the matrix space to the space of eigenvalues. Let us expand a bit on this without entering too deeply into the calculations. (For details se, e.g. [42].) The Haar measure in terms of the diagonal and angular part of U has the form [dU ] = ∆2 [dV ]

(122)

where [dV ] is the Haar measure of V and ∆ is the Vandermonde determinant ∆=

Y

2 sin

ihj

xi − xj · 2

(123)

To see this, write the “line element” −tr(U −1 dU )2 in terms of V and xi using (78) and obtain −tr(U −1 dU )2 =

X i

dx2i −

X i,j

4 sin2

xi − xj −1 (V dV )ij (V −1 dV )ji . (124) 2

This metric is diagonal in dxi and (V −1 dV )ij . The square root of the determinant of this metric, which gives the measure (volume element) on the space, is clearly ∆2 times the part coming from V which is the standard x −x Haar measure for V . (We get two powers of 4 sin2 i 2 j in the determinant, one from the real and one from the imaginary part of (V −1 dV )ij , so the square root of the determinant has one power of ∆2 .) To bring the kinetic xi -part into a “flat” form (plain second derivatives in xi ) we must multiply the wavefunction with the square root of the relevant measure (compare with the change from cartesian to radial coordinate

444

Topological Aspects of Low Dimensional Systems

in spherical potential problems). The net result is that the wavefunction Ψ in terms of xi and V is the original wavefunction ψ(U ) of the matrix model times the Vandermonde determinant. This, however, also produces an additive constant Eo which comes from the action of the entire xi -kinetic operator on ∆. Noticing that ∆ is nothing but the ground state wavefunction of N free fermions on the circle, we see that Eo is the relevant fermionic ground state energy N (N 2 − 1) · (125) Eo = 24 This is the famous “fermionization” of the eigenvalues produced by the matrix model measure. To determine the proper ordering for K we examine its properties as a generator of transformations. Since U = V ΛV −1 , and J generates U → V 0 U V 0−1 = (V 0 V )Λ(V 0 V )−1 , we see that J generates left-multiplications of the angular part V of U . K = V −1 JV , on the other hand, generates rightmultiplications of V , as can be see from its form or by explicit calculation through its Poisson brackets. As a result, it also obeys the SU (N ) algebra. Its proper quantum definition, then, is such that it satisfies, as an operator, the SU (N ) algebra. It clearly commutes with the diagonal part xi and its momentum pi , since it has no action on it. Its dynamics are fully determined by the hamiltonian (121) and its SU (N ) commutation relations. We can, therefore, in the context of the particle model (121), forget where K came from and consider it as an independent set of dynamical SU (N ) operators. K, however, obeys some constraints. The first is that, as is obvious from K = V −1 JV , K carries the same irrep r as J. The second is subtler: a right-multiplication of V with a diagonal matrix will clearly leave U = V ΛV −1 invariant. Therefore, this change of V has no counterpart on the “physical” degrees of freedom of the model and is a gauge transformation. As a result, we get the “Gauss’ law” that physical states should remain invariant under such transformations. Since K generates right-multiplications of V , and Kii (no sum) generates the diagonal ones, we finally obtain (no sum) Kii = 0 (on physical states).

(126)

˙ (A more pedestrian but less illuminating way to see it is: J = i[U −1 , U], being a commutator, vanishes when sandwiched between the same eigenstate of U . Since K is essentially J in the basis where U is diagonal, its diagonal elements vanish.) Note that the constraint (126) is preserved by the hamiltonian (121). The above fully fixes the reduced model Hilbert space as the product of the N -particle Hilbert space times the dr -dimensional space of K, with the constraint (126) also imposed. The further casting of the model into something with a more direct physical interpretation relies upon a convenient

A.P. Polychronakos: Generalized Statistics in One Dimension

445

realization of K. Any such realization will do: simply break the representation of SU (N ) that it carries into irreps r and read off the spectrum for each r from the results of the previous section. We shall implement K in a construction ` a la Jordan-Wigner. Let ami , a†mi , m = 1, . . . q, i = 1, . . . N be a set of N q independent bosonic oscillators [41]: (127) [ami , a†nj ] = δmn δij . Then

q X

Ka =

ami Tija amj

(128)

m=1

is a realization of the SU (N ) algebra. (Tija are the matrix elements of T a .) The corresponding matrix elements of K are ( ! ) q X 1 X † † amk amk δij · ami amj − (129) Kij = N m=1 k

Correspondingly, the coefficient of the Sutherland potential in (121) is (for i 6= j) X † X † ami ani a†nj amj + ami ami . (130) Kij Kji = m,n

m

We already see that the degrees of freedom of K are redistributed into degrees of freedom for each particle in the above. Specifically, defining ! q 1 X † † a asi δmn (131) Si,mn = ami ani − q s=1 si and comparing with (129) we see that the Si are N independent sets of operators each satisfying the SU (q) algebra. Before expressing Kij Kji in terms of the Si let us see what the constraint (126) implies: Kii =

q X

a†mi ami −

m=1

1 X † amk amk = 0 N

(132)

m,k

P

a†mk amk commutes with all Kij and all Si,mn . It is, therefore, a Casimir and can be chosen as a fixed integer `N equal to the total number operator of the subspace of the oscillator Fock space in which the model lives. The above constraint, then, implies m,k

q X m=1

a†mi ami = `.

(133)

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Topological Aspects of Low Dimensional Systems

(We see why we had to choose the total number operator to be a multiple of N : the operator in (133) above is also a number operator and can have only integer eigenvalues.) Using this in (131) we can express ` a†mi ani = Si,mn + δmn q

(134)

and therefore Kij Kji =

X

Si,mn Sj,nm +

mn

`(` + q) ~i · S ~j + `(` + q) =S q q

(135)

~i = tr(Si Sj ) is the SU (q)-invariant scalar product of the two ~i · S where S SU (q) “vectors.” We finally obtain the hamiltonian as [41] H=

X1 i

2

p2i

~i · S ~j + `(`+q) 1 X 2S q + · x −x 2 4 sin2 i 2 j

(136)

i6=j

So it is a Sutherland-like model but where the particles also carry SU (q) internal degrees of freedom (“spins”) and the potential contains a pairwise antiferromagnetic interaction between the spins. It remains to specify the representation in which the SU (q) spins are and find the irreps contained in this realization of K, therefore obtaining the spectrum. A realization of the form (129) for q = 1 in terms of bosonic oscillators contains all totally symmetric irreps of S(N ) (that is, the ones with a single row in their Young tableau). (129) is essentially the direct product of q such independent realizations, so it contains all direct products of q totally symmetric irreps. This produces all irreps with up to q rows in their Young tableau, perhaps more than once each. The constraint (133), however, implies that the total number of boxes in the Young tableau of these irreps is `N . We recover once more the constraint that we derived ¯ × R. before based on the origin of r as a component of R Similarly, the realization (131) of Si contains all the totally symmetric irreps of SU (q). (133) implies that the number of boxes of these irreps is equal to `, so the spins Si are each in the `-fold symmetric irrep of SU (q). Solving this model amounts to decomposing the tensor product of these N spins into irreducible components of SU (q). Each such component corresponds to a subspace of the Hilbert space with a fixed total spin S. This same irrep, interpreted as an irrep r of SU (N ), will be the corresponding irrep of K, and also of J, and thus will determine the spectrum of this sector through (120). Let us elucidate the above by reproducing the two simplest cases: free particles and (spinless) Sutherland particles, comparing with the classical treatment.

A.P. Polychronakos: Generalized Statistics in One Dimension

447

a) Free particles correspond to J = K = 0. So there is no spin and no potential and we have non-interacting particles. From (120) we see that all DR are one, and thus the spectrum is the free fermion one, as commented before. The matrix model naturally quantizes free particles as fermions. b) Spinless Sutherland particles correspond, classically, to J = `(vv † −1). So J is rank one (ignoring the trace). Quantum mechanically this corresponds to the irrep r of J having only one row and therefore only one independent Casimir. Since q in the realization above corresponds to the number of rows, we must have q = 1. Spins, therefore, are absent. The strength of the potential becomes `(` + 1) where `N is the number of boxes in the one row of r. By standard Young tableau rules we see that the degeneracy DR is one if the row lengths of R satisfy Ri ≥ Ri+1 + `

(137)

else it is zero. The spectrum of this model is, then, the same as the spectrum of free particles but with the selection rule for their momenta pi ≥ pi+1 + ` + 1.

(138)

We recover the “minimum distance” selection rule of the CSM model that led to the interpretation as particles with generalized statistics! Only, in this case, the statistics parameter ` + 1 is a positive integer. We mention here that a Jordan-Wigner realization of K in terms of fermionic oscillators is also useful and leads to particles with spins interacting via ferromagnetic Sutherland-type potentials. The hamiltonian becomes [41] ~i · S ~j + `(`−q) X1 1 X 2S q p2i − (139) H= 2 xi −xj 2 2 4 sin i 2 i6=j where now the spins are in the `-fold antisymmetric irrep of SU (q). We will not elaborate further and leave the details as an exercise to the reader. In conclusion, the matrix model has provided us with the following: 1. An augmentation of the permutation group into the SU (N ) group and a corresponding possibility to define statistics through the irreps of SU (N ). 2. A realization of generalized scalar statistics but with quantized statistics parameter ` + 1 in terms of the CSM model. 3. A realization of generalized “non-abelian statistics” in terms of particles with internal degrees of freedom interacting through a generalized CSM-type potential. 4. A systematic way of solving the above models. What the matrix models has not provided is

448

Topological Aspects of Low Dimensional Systems

1. A realization of generalized statistics for fractional statistics parameter. 2. A realization of spin-CSM systems with the spins in arbitrary (nonsymmetric) representations. 3. A control of the coupling strength of the potential for the spin-CSM ~j terms is fixed to ±2 and also ~i · S models. (Note that the coefficient of S the constant therm `(` + q) is entirely fixed by the spin representation.) There exist generalizations of the above models both at the classical [43,44] and the quantum level [45]. They all share, however, the limitations spelled out above, especially (3). These restrictions are important, in the quest of more general statistics but also from the more practical point of view of solving spin-chain models with spins not in the fundamental representation, as we will shortly explain. For this reason, yet a different approach will be pursued in the next section, namely, the operator approach. 5

Operator approaches

The matrix model connection provided us with a powerful tool that not only allowed us to generalize statistics but also led to the full quantum solution of a set of spin-generalized CSM models. As noted, however, in the conclusion of the preceding lecture, the matrix model fixes the coefficient of the spin-interaction and scalar interaction terms to ±2 and ±`(` ± q) respectively. We cannot choose these coefficients at will. We would like to have an approach that defeats this restriction and leads to spin models with arbitrary coupling strengths. (This is necessary to attack spin-chain systems through the infinite-coupling limit trick to be explained later.) Such an approach should also be able to bypass the excursion to matrix models and deal more directly with these systems in an algebraic way. This will be achieved with the exchange operator formalism [46] (also known as the Dunkl operator formalism [47]). 5.1 Exchange operator formalism Consider the operators Mij that permute the coordinate degrees of freedom of N particles in one dimension which could, in principle, also have internal degrees of freedom (M for metathesis, to avoid confusion with momenta pi ). They satisfy the permutation algebra (symmetric group), in particular Mij

=

[Mij , Mkl ] = Mij Mjk =

Mij−1 = Mij† = Mji 0 Mik Mij

(140) if i, j, k, l distinct if i, j, k distinct.

(141) (142)

A.P. Polychronakos: Generalized Statistics in One Dimension

449

Any operator Ai on the phase space satisfying Mij Ak

= Ak Mij

Mij Ai

= Aj Mij

if i, j, k distinct

(143) (144)

will be called a one-particle operator (even though it may involve the coordinates and momenta of many particles). We construct the following one-particle operators [46]: X X iW (xi − xj )Mij ≡ pi + iWij Mij . (145) πi = pi + j6=i

j6=i

We shall view the πi as generalized momenta. To ensure their hermiticity the prepotential W (x) should satisfy W (−x) = −W (x)∗ .

(146)

We shall construct the corresponding “free” hamiltonian from πi H=

X1 i

2

πi2 .

(147)

In terms of the original pi this hamiltonian will, in general contain linear terms. To ensure that such terms are absent we must further impose W (−x) = −W (x) = real.

(148)

With the above restriction the hamiltonian H and commutation relations of the πi become X Wijk (Mijk − Mjik ) (149) [πi , πj ] = k

H=

X1 i

2

p2i +

X

X Wijk Mijk Wij2 + Wij0 Mij +

i<j

(150)

i<j
where we defined the three-body potential and cyclic permutation Wijk

= Wij Wjk + Wjk Wki + Wki Wij

(151)

Mijk

= Mij Mjk .

(152)

To obtain an interesting and tractable model, Wijk , which appears in the commutator [πi , πj ] and also as a three-body potential, should vanish or at most be a constant. This leads to a functional equation for W (x): W (x)W (y) − W (x + y) [W (x) + W (y)] = const(= Wijk ).

(153)

450

Topological Aspects of Low Dimensional Systems

We present the solutions: a) Wijk = 0 → W (x) = `/x b) Wijk = −`2 < 0 → W (x) = ` cot x c) Wijk = +`2 > 0 → W (x) = ` coth x. Let’s examine each case. a) In this case the πi become [46–48] X i` Mij πi = pi + xij

(154)

j6=i

and satisfy [πi , πj ] = 0.

(155)

The πi commute, so we can consider them as independent momenta. The hamiltonian reads X `(` − Mij ) X1 p2i + · (156) H= 2 x2ij i i<j We obtain a Calogero-like model with exchange interactions. Yet it is nothing but a free model in the commuting momenta πi . Integrability is immediate: the permutation-invariant quantities X πin (157) In = i

obviously commute with each other. If we assume that the particles carry no internal degrees of freedom and are bosons or fermions then Mij = ±1 on physical states. The model becomes the standard Calogero model and we have proved its integrability in one scoop. (You may be left with a question mark: the Hamiltonian and the other integrals In become the standard Calogero ones if Mij = ±1, so these reduced integrals will commute on the bosonic or fermionic subspace; but will they also commute on the full Hilbert space? Prove for yourself that this is indeed the case.) We can also construct harmonic oscillator operators [46, 48]. The commutators between xi and πi are   X Mij  [xi , πi ] = i 1 + ` (158) j6=i

[xi , πj ] =

−i`Mij

(i 6= j).

(159)

Defining ai

=

a†i

=

1 √ (πi − iωxi ) 2 1 √ (πi + iωxi ) 2

(160) (161)

A.P. Polychronakos: Generalized Statistics in One Dimension

we can show

X ω 1+` Mij

[ai , a†i ] =

451

(162)

j6=i

[ai , a†j ] =

−ω`Mij

[ai , aj ] =

[a†i , a†j ]

(i 6= j)

(163)

= 0.

(164)

This is an extended version of the Heisenberg algebra involving the permutation operators. The corresponding oscillator hamiltonian reads H=

X1 i

2

(a†i ai + ai a†i ) =

X1 i

2

p2i +

X1 i

and satisfies

2

ω 2 xi +

X `(` − Mij ) i<j

[H, a†i ] = ωa†i .

[H, ai ] = ωai ,

x2ij

(165)

(166)

This is the harmonic Calogero model with exchange interactions, which becomes again the standard model on bosonic or fermionic subspaces for particles without internal degrees of freedom. Since H=

X i

1 X 1 a†i ai + N ω + `ω Mij 2 2

(167)

i6=j

we see that on bosonic or fermionic spaces the state annihilated by all ai (if it exists) will be the ground state. Solving ai ψ = 0 we obtain for the ground state wavefunction P 2 Y 1 |xij |` e− 2 ω i xi (168) ψB = i<j

ψF

=

Y 1 P 2 sgn(xij )|xij |−` e− 2 ω i xi .

(169)

i<j

For ` > 0 the bosonic state is acceptable, while for ` < 0 the fermionic one is acceptable. In the “wrong” combinations of statistics and sign of ` the ground state is not annihilated by the ai , but it is still annihilated by all permutation-invariant combinations of the ai . From (166) we see that we can find the spectrum of this model for fermions or bosons by acting on the ground state with all possible permutation-symmetric homogeneous polynomials in the a†i . A basis for these is, e.g., X † (ai )n . (170) An = i

So the spectrum is identical to non-interacting fermions or bosons, but with a different ground state energy. For the “right” combinations of `

452

Topological Aspects of Low Dimensional Systems

and statistics, where (169) are the correct ground state wavefunctions, the ground state energy is N (N − 1) N ω+ |`|ω 2 2 which is the correct Calogero result. Finally, the quantities X X † hni = (ai ai )n In =

(171)

Eo =

i

(172)

i

can be shown to commute [46], and therefore this system is also integrable. It is left as an exercise to find the commutation relations of the hi and show that [In , Im ] = 0. b) In the case W (x) = ` cot x we have πi = pi + i cot xij Mij X (Mijk − Mjik ) [πi , πj ] = −`2

(173) (174)

k

so the momenta are now coupled. The hamiltonian becomes   X X `(` − Mij ) X1 N (N − 1) p2 + + − `2  Mijk  . H= 2 i 2 sin2 xij i

i<j

(175)

i<j
We obtain the Sutherland model with exchange interactions plus an extra term. On bosonic or fermionic states this becomes an overall constant and we recover the standard Sutherland model. Again, since H is by construction positive definite, if a state satisfying πi ψ = 0 exists it will be the ground state. We obtain Y | sin xij |` (176) ψB = i<j

ψF

=

Y

sgn(xij )| sin xij |−`

(177)

i<j

which are acceptable for the same combinations of ` and statistics as before. For both cases Mijk = 1 so N (N 2 − 1) (178) 24 is the correct Sutherland model ground state energy. The excited states can again be obtained in a (rather complicated) algebraic way. Finally, the quantities X X 2` Mij = pi + eixi Mij (179) π ˜i = πi + ` eixi − eixj Eo = `2

j6=i

j6=i

A.P. Polychronakos: Generalized Statistics in One Dimension

453

can be shown to have the same commutation relations as the hi defined previously for the harmonic system. Therefore, the integrals constructed from them X π ˜in (180) In = i

commute and the model is integrable. c) For W (x) = ` coth x we have a similar commutation relation and a Hamiltonian   X X `(` − Mij ) X1 N (N − 1) p2i + + + `2  Mijk  . (181) H= 2 2 2 sinh x ij i i<j i<j
This is the inverse-hyperbolic-sine-square model and supports only scattering states. Its integrability can be obtained as for the Sutherland model above, or simply as an “analytic continuation” of that model for imaginary period of space. We will not examine it any further. In conclusion, an exchange-family of models was introduced, solved and related to the standard CSM models in spaces of definite symmetry. It is remarkable that all these proofs work directly, and only, at the quantum domain (there is no classical analog of Mij ). 5.2 Systems with internal degrees of freedom We can easily extend the previous results for particles with internal degrees of freedom. For this, assume that the particles are distinguishable or, equivalently, that they carry a number q of (discrete) internal degrees of freedom (species) that can be used to (partially) distinguish them. Their states are spanned by |x, σi, where σ = 1, . . . q counts internal states. The total permutation operator Tij , then is Tij = Mij σij

(182)

where σij is the operator that permutes the internal states of particles i and j. Let us, then, simply take states that are bosonic or fermionic under total particle exchange: Tij = ±1. On such states Mij = ±σij

(183)

and the Calogero and Sutherland exchange model hamiltonians become [49] Hc =

X1 i

2

p2i +

X1 i

2

ω 2 xi +

X `(` ∓ σij ) i<j

x2ij

(184)

454

Hs =

Topological Aspects of Low Dimensional Systems X1 i

2

p2i +

X `(` ∓ σij ) i<j

sin2 xij

 − `2 

N (N − 1) + 2

X

 σijk  .

(185)

i<j
We get the Calogero and Sutherland models with spin-exchange interactions. From the completeness relation for the fundamental SU (q) generators Ta 2 −1 qX 1 1 a a Tαβ Tγδ = δαδ δγβ − δαβ δγδ (186) 2 2q a=1 we deduce the form of the operators σij ~i · S ~j + 1 σij = 2S q

(187)

where Sia acts as T a on the internal states of particle i. So the spindependent interaction coefficient of the potential in the hamiltonian takes the form [49–52] ~j ∓ ` + 1 . ~i · S (188) ∓` 2S q We have recovered the ferromagnetic and antiferromagnetic spin model of the previous section but with arbitrary coefficient! On the other hand, the spins are necessarily in the fundamental of SU (q). So we have obtained a generalization of the coupling constant with respect to the matrix model but a restriction of the allowed spins. Note that ` here is an arbitrary parameter, while ` in (136) was the size of the symmetric representation of Si . For ` = 1 and spins in the fundamental, the matrix model and exchange-operator model agree. It is interesting to note that we can go from ferromagnetic to antiferromagnetic interactions either by changing the sign of ` or by changing the statistics of the particles. The solution of the above models can be obtained algebraically. For the spin-Sutherland model this is rather complicated and is related to the socalled Yangian symmetry [53–57]. For the spin-Calogero model it is easier ~j ∓ ` − 1 ) ~i · S [58]. Let us concentrate on the model with interaction `(−2S q and define the operators X † X † (ai )n , (Aan )† = (ai )n Sia (189) A†n = i

i

and their hermitian conjugates. These form a complete set for all permutation-symmetric creation and annihilation operators for all species of particles. Yet the commutators among themselves and with H do not involve `. They create, therefore, the same spectrum of excitations over the ground

A.P. Polychronakos: Generalized Statistics in One Dimension

455

state as N noninteracting bosons or fermions with q species. For ` > 0 the ground state is the bosonic one: P 2 Y 1 |xij |` e− 2 ω i xi χs ({σi }) (190) ψB = i<j

. The set of all χs forms the where χs is a totally symmetric state in the σiP N -fold symmetric irrep of the total spin S = i Si . Therefore the ground state is (N + q − 1)!/N !(q − 1)! times degenerate. For ` < 0 the above is not normalizable any more. But we remember that we can obtain the same model by starting from fermions and the opposite coupling −` > 0. The ground state, then, is of a fermionic type ! P 2 X Y Y δα ,α 1 (−1)P δσi ,αi |xij |−` xij i j e 2 ω i xi (191) ψF = P

i

i<j

where P are total particle permutations and αi are a set of fixed values for the indices σi that determine the state. Clearly the ground state will be obtained for the minimal total power of xi appearing above, and that will happen for a maximally different set of values αi . These states form the n-fold antisymmetric irrep of the total spin S, where n = N (modq). The ground state is, thus, q!/n!(q − n)! times degenerate. The above spectra will come handy later. 5.3 Asymptotic Bethe Ansatz approach We already mentioned that there are elaborate algebraic approaches to derive the spectrum of the spin-Sutherland model, based on the Yangian symmetry. We will, instead, take a lower-key approach which reproduces the same spectra and is physically more lucid, although not as rigorous. We will take the ABA route. Consider distinguishable particles of the exchange-Calogero type without external potential, coming in with asymptotic momenta ki and scattering off each other. Before scattering, their positions are in some definite ordering determined by the ordering of their momenta (it is the inverse of that ordering). The key observation is that, after scattering, the particles have simply “gone through” each other with no backscattering [59]. The impenetrable 1/x2 potential has become completely penetrable in the presence of the exchange term! You can prove this fact by examining the asymptotic properties of a simultaneous eigenstate of π1 , . . . πN which is obviously an energy eigenstate: at xi → ±∞ the prepotential terms are vanishing and we simply have eigenstates of the individual pi . Since there are no pieces with the values of pi permuted (coming from backscattering) we have

456

Topological Aspects of Low Dimensional Systems

complete transmission. (To explicitly see how it works, it is instructive to consider the two-body problem, decompose it into symmetric and antisymmetric parts, scatter and recombine the parts after scattering. A relative phase of π between the two parts is what produces the effect.) (Puzzle: what happens with the correspondence principle? With h ¯ back in, the interaction coefficient is `(` − ¯ hMij ). How can a term of order ¯h produce such a dramatic effect, particles going through each other, in the ¯h → 0 limit?) So the only effect of the scattering is a phase shift of the wavefunction which, as we have said, is the sum of two-body phases θsc =

N (N − 1) π`. 2

(192)

This is true on an infinite space. On a periodic space we can still use the above result, together with the requirement for periodicity for the wavefunction, to derive the spectrum. This is the ABA method and is expected to reproduce the correct results in the thermodynamic limit of many particles at constant density [59]. It gives, in fact, the exact answer for the Sutherland model [26], so we can expect it to work also in the present case. For a space of period 2π the result is X π`sgn(ki − kj ) = 2πni . (193) 2πkj + π

The left hand side counts the total phase picked up by a particle going round the space and scattering off to the other particles in the way. ni are arbitrary integers, ensuring periodicity. There are, however, some constraints on the choice of of ni that are imposed by continuity from the ` = 0 case: –if ki ≤ kj then ni ≤ nj ; –if ni = nj there is a unique solution, that is, ki < kj and ki > kj represent the same state. These rules are important to avoid overcounting and to discard spurious solutions. With these, the spectrum obtained is the same as the one derived with more rigorous methods. For the ordering n1 ≤ . . . nN the solution for ki is N +1 ) (194) ki = ni + `(i − 2 and similarly for other orderings. We see that the ABA momenta ki are the same as the quasimomenta that we have previously defined. The bottom line is that the spectrum and degeneracies are the same as those of distinguishable particles obeying generalized selection rules for their momentum. Still, what fixes the degeneracy of states is the different ways that we can distribute the particles to the quantum numbers ni , rather than ki (see, especially, the second rule above). A state of N particles with

A.P. Polychronakos: Generalized Statistics in One Dimension

457

the same ni , for instance, is nondegenerate although they, seemingly, have different ki which would imply a permutation degeneracy. For particles with spin the construction above, in combination with the trick of the previous subsection of starting with fermions or bosons, produces a spectrum with degeneracies the same as those of free particles (the ni are “free” quantum numbers). As argued before, for ferromagnetic interactions we must choose bosons and combine their spins accordingly, while for antiferromagnetic interactions we must choose fermions. To spell it out, this means the following: 1. Choose a set of quantum numbers ni . The ordering is immaterial, since we have identical particles, so you can choose n1 ≤ . . . nN . 2. Place your particles on these quantum numbers and put their spins in the appropriate state. For the ferromagnetic case treat them as bosons: the total spin of particles with the same ni transforms in the symmetric tensor product of their spins. For the antiferromagnetic case treat them as fermions: the total spin of of particles with the same ni transforms in the antisymmetric tensor product of their spins; clearly up to q can have the same ni in this case. 3. Calculate the energy of this state in terms of the ABA momenta (194): P E = i ki2 . It should be obvious that similar rules applied to the spin-Calogero system reproduce the spectrum derived in the last subsection. This method can be used to calculate both the statistical mechanics (large N ) of these systems and the few-body spectra. 5.4 The freezing trick and spin models Now that we have a tractable way of solving spin-CSM systems with arbitrary strength of interaction we can introduce the freezing trick [64] and deal with spin chain models. Consider, first, the previous ferromagnetic or antiferromagnetic spinSutherland model. Take the limit ` → ∞. The potential between the particles goes to infinity, so for any finite-energy state the particles will be nearly “frozen” to their classical equilibrium positions. In fact, even the excitation energies around that configuration will go to infinity: the ground state energy scales like `2 , while the excitations scale like N `n + n2 with n some excitation parameter. So, to leading order in ` the spectrum becomes linear and of order `. These excitations correspond, essentially, to phonon modes of small oscillations around the equilibrium positions of particles. The “stiffness” of oscillations is, of course, proportional to the strength of the potential `2 and the spectrum is proportional to the frequency, of order `. The quantum fluctuations of the particle positions in any state will scale √ like the inverse square root of the oscillator frequency, that is, like 1/ `.

458

Topological Aspects of Low Dimensional Systems

But, in the hamiltonian, the piece coupling the spins to the kinematical degrees of freedom is proportional to 1/ sin2 xij . In the large-` limit, thus, this term becomes a constant equal to its classical equilibrium value; so, in that limit, spin and kinematical degrees of freedom decouple. (Note that the spin part is also of order `.) The hamiltonian becomes H = HS + `Hspin

(195)

with HS the spinless Sutherland hamiltonian and Hspin the spin part Hspin = ∓

X 2S ~i · S ~j x ¯ 4 sin2 2ij i<j

(196)

where the classical equilibrium positions x ¯j are equidistant points on the circle: 2πj · (197) x ¯j = N The hamiltonian (196) above describes a spin chain consisting of a regular periodic lattice of spins in the fundamental of SU (q) coupled through mutual ferro- or antiferromagnetic interactions of strength inversely proportional to their chord distance. It is the well known SU (q) Haldane-Shastry (HS) model [60,61]. According to the above, its spectrum can be found by taking the full spectrum of the corresponding spin-Sutherland model in the large-` limit, “modding out” the spectrum of the spinless model and rescaling by a factor 1/`. Each state will inherit the spin representation of its “parent” spin-Sutherland state. So, both the energy and the total spin of the states of the HS model can be determined this way. Commuting integrals of this model [62] can also be obtained this way [63]. At the level of the partition function at some temperature T we have Zspin (T ) = lim

`→∞

Z(`T ) · ZS (`T )

(198)

From this, the thermodynamics of the spin chain model can be extracted [59]. We will not give the details of this construction here. We urge anyone interested to solve this way a few-site (two or three) spin chain, see how it works and deduce the “construction rules” for the spectrum of a general spin chain. Let us simply state that the many degeneracies of the spectrum of the HS model (larger that the total spin SU (q) symmetry would imply), which is algebraically explained by the existence of the Yangian symmetry, can, in this approach, be explained in terms of the degeneracies of free particles. (The degeneracies are not identical, due to the modding procedure, but related.)

A.P. Polychronakos: Generalized Statistics in One Dimension

459

For the spin-Calogero model a similar limit can be taken, scaling also the external oscillator frequency as ω → `ω to keep the system bound. The classical equilibrium positions of this model are at the roots of the N -th Hermite polynomial. We obtain, therefore, a non-regular lattice of spins interacting with a strength inversely proportional to the square of their distance [64]. The spectrum of this model can be found quite easily with the above method. Again, we refer to the literature for details [58, 65]. In the continuum limit (N → ∞) the antiferromagnetic version of both the above models become c = 1 conformal field theories, the HS containing both chiral sectors while the inhomogeneous harmonic one containing just one sector. Other models exist and can be solved in this spirit: hierarchical (manycoupling) models [66], supersymmetric models [67], “twisted” models [68] etc. All, however, work only for the fundamental representation of some internal group. The big, important open problem is to crack a particle system with a higher representation for the spins and arbitrary coupling strength. If this is done, through the freezing trick we will be able to solve a spin chain with spins in a higher representation. This is interesting since we could then see if the antiferromagnetic system for integer SU (2) spins develops a mass gap, according to the Haldane conjecture [69]. 6

Exclusion statistics

So far we approached the problem of statistics in a “fundamental” way, trying to give a reasonable definition and presenting systems that realized this definition. In this last section we shall give a “phenomenological” approach, based on the state-counting properties of a system whose dynamics may remain, otherwise, undetermined. It is based on a notion already familiar from the CSM model, the notion of “state repulsion” or “exclusion”. This will lead to Haldane’s “exclusion statistics”. 6.1 Motivation from the CSM model Although Haldane derived his definition from the properties of the HS spin chain, we will use the CSM model instead. Consider the “principal” quantum number of this model, that is, the one in terms of which the energy eigenvalues are a sum of independent terms: quasiexcitation numbers ni for the Calogero and quasimomenta pi for the Sutherland model. They obey an “enhanced” exclusion principle where nearby values can be no closer that ` units. It seems as if each of them occupies ` places in the single-particle Hilbert space, instead of one. So, let us define the dimensionality d(N ) of the Hilbert space of states available to an additional particle given that there are already N particles in the system. (Some high-energy cutoff is

460

Topological Aspects of Low Dimensional Systems

needed, of course, to make this finite.) The “exclusion statistics parameter” g is, then defined as [70] ∆d · (199) g=− ∆N If this parameter is independent of N , or becomes a constant for high enough N , then we say that the system obeys exclusion statistics of order g. Clearly g = 0, 1 corresponds to bosons and fermions, respectively. From the discussion of the CSM model we would conclude that it obeys exclusion statistics with g = `. Note that, in principle, this definition applies to systems of arbitrary dimensionality. The fact that only one-dimensional exclusion statistics systems have been as yet identified presumably points to something essentially one-dimensional in this definition. Note also that g can be fractional: d(N ) could be either ill-defined (as, in fact, in the CSM model) so that it can assume “approximate” fractional values, or it could depend on N in a way that g becomes constant only for large enough ∆N , like, e.g., d(N ) = [N/2] with [ ] the integer part, giving g = 1/2. 6.2 Semiclassics – Heuristics Before examining the consequences of (199) let us make some heuristic semiclassical arguments about phase space volume and exclusion to give more substance to it. Please view the following discussion simply as additional motivation -do not take it seriously! Semiclassically, the number of states of a system is given by the volume of its phase space in units of h for each pair of canonical variables. Let us consider for simplicity the minimal space (q, p). It could correspond, for instance, to the relative coordinate and momentum of two particles on the line. The standard lagrangian would look like L = pq˙ − H(p, q).

(200)

If we want the presence of the one particle to “knock out” g states out of the Hilbert space, we should include some term in the classical action that, effectively, reduces the volume (in this case, area) of phase space by gh. This area is given as I Z dpdq =

A= D

pdq

(201)

∂D

where V is a domain in phase space and ∂D its boundary. If the domain does not include the point q = p = 0 the area should be the standard one, since we are talking about a region of phase space where the particles are apart. If, on the other hand, q = p = 0 is included, the particles are together and the area should diminish by gh = 2πg (taking h ¯ = 1).

A.P. Polychronakos: Generalized Statistics in One Dimension

461

This reminds us of the Aharonov-Bohm effect. There must be a “Dirac string” piercing the point (0, 0) in phase space giving this extra contribution when circled. So we must add to the action a term Z Z pdq − qdp λq = −g d atan (202) §g = −λg p2 + λ2 q 2 p where λ is any positive constant. This amounts to adding to the Lagrangian the extra term pq˙ − q p˙ · (203) Lg = −λg 2 p + λ2 q 2 This extra contribution is a total time derivative (a topological term) so it will not change the equations of motion. It is expected, though, that it will change the quantum mechanical states as described above. Let us consider the above in the specific example of two particles in an external harmonic oscillator potential. After separating the center of mass phase space, the lagrangian for the relative part becomes Lg = pq˙ − ωg

pq˙ − q p˙ 1 − ω 2 (p2 + q 2 ). p2 + ω 2 q 2 2

(204)

For later convenience we chose λ = ω in the g-term. The effect of this term is to shift the Poisson brackets between p and q. We will follow the simple approach to canonical quantization of defining new “polar” variables ρ, θ as √ (205) p + iωq = ωρeiθ . In terms of these the lagrangian becomes 1 1 2 ρ − g θ˙ − ωρ2 L= 2 2

(206)

so the canonical momentum of θ is πθ =

1 2 ρ −g 2

(207)

and the hamiltonian becomes H = ω(πθ + g).

(208)

Quantum mechanically the operator θ is not quite well defined. In the absence of g, it would correspond to the “phase” of the annihilation operator which cannot be a well-defined hermitian operator: eiθ should be unitary, yet it decreases the eigenvalue of ρ2 = a† a by one unit, which is impossible for the ground state. Nevertheless, we will proceed qualitatively and see what we get in this case.

462

Topological Aspects of Low Dimensional Systems

θ being a phase, its momentum πθ should be quantized to integer values. Since the particles are identical, the change of relative coordinates p → −p, q → −q should also be a symmetry. This corresponds to θ → θ+π. Choosing the wavefunction to transform as ± itself under this shift, which amounts to quantizing the particles as bosons or fermions, further restricts the values of πθ to even or odd integers. Finally, since ρ2 is a positive definite operator, we have 1 2 ρ = πθ + g ≥ 0 (209) 2 (clearly the problems with the definition of θ are hidden in the above constraint). Choosing the even eigenvalues for πθ (bosonic case), and for g not greater that 2, the spectrum of the relative Hamiltonian becomes E = ω(2n + g)

(210)

for n a nonnegative integer. This is the excitation spectrum of the Calogero model! To see this, add the center of mass oscillator energy Ecm = ωm for m a nonnegative integer and define n1 = n ,

n2 = m + n + g.

(211)

Then the full spectrum becomes E = ω(n1 + n2 ) with n1 ≤ n2 − g

(212)

that is, the excitations of the Calogero model in terms of the quasiexcitation numbers n1,2 satisfying the “least distance `” constraint for ` = g. This will serve as enough motivation that the idea of phase space exclusion should be related to eigenvalue repulsion a` la CSM model. The questions about the ground state energy, g > 2 etc. that are left hanging are simply set aside – this model is not treated seriously here. If anyone is interested, of course, they are welcome to polish it! 6.3 Exclusion statistical mechanics We will implement exclusion statistics in terms of the possible quantum numbers of N particles placed in K single-particle states. Arranging the K states in a linear fashion and implementing the “least distance g” constraint for the particles, for integer g we get [70, 71] D(K, N ) =

[K − (g − 1)(N − 1)]! N ![K − g(N − 1) − 1]!

(213)

possible combinations for allowed values for the particle quantum numbers (to prove it is a simple combinatorial matter). Extrapolated to arbitrary

A.P. Polychronakos: Generalized Statistics in One Dimension

463

(fractional) g, this defines the state multiplicity of N particles placed in K states. Clearly g = 0, 1 reproduces the Bose and Fermi result. This can be used to derive the statistical mechanical properties of a grand ensemble of such particles. Considering all K states to be at nearby energy and maximizing the Gibbs factor D(K, N )eβ(µ−)N = maximum

(214)

in terms of N for fixed temperature T = 1/kβ and chemical potential µ we obtain for the distribution n = N/K for large N, K [29, 71, 72] n=

w = eβ(µ−) . (1 − w)g

w , 1 + (g − 1)w

(215)

The expression for n cannot be found analytically except for a few values of g. It is obvious from the above that fractional statistics particles are not “independent”, in the sense that one cannot derive their statistical mechanics by considering each single-particle state as an independent system to be filled by particles. (Operator constructions that incorporate some of the features of exclusion statistics do exist [73] but they further require specific interactions, else they produce Gentile statistics.) There is no microscopic formulation: we need to start from K states and take the limit of large K. Further, it is not clear how we could implement exclusion statistics for an interacting system, where we cannot use single-particle states as a convenient basis. In the following we will sketch how the above difficulties can be overcome [74]. The starting point will be the grand partition function for particles in K states of like energy Z(K, z) =

∞ X

D(K, N )z N

(216)

N =0

where z = exp β(µ − ) is the fugacity. Z(K, z) should be extensive, which means that, for large K it should become the K-th power of a function of z ∞

X X 1 1 ln Z(K, x) = lim ln D(K, N )z N = ln Pn z n . lim K→∞ K K→∞ K n

(217)

N =0

In the above we introduced the quantities Pn which play a role analogous to the allowed states of occupation of a single level in the cases of fermions and bosons: Pn = 1 for bosons, while P0,1 = 1, Pn>1 = 0 for fermions. We can call them “a priori probabilities of occupation” [74] or “fractional dimensionality of states” [75] according to taste. That Pn exist should be guaranteed from the proper extensive behavior of the exclusion statistical mechanics. The problem is to calculate them.

464

Topological Aspects of Low Dimensional Systems

For this, we make a technical trick: to derive the multiplicity of states D(K, N ) above we placed the K states on a line. Let’s place them on a circle instead, and implement the “minimum distance g” rule there. That shouldn’t influence the statistical behavior at large K. This modifies the combinatorics into K[K − (g − 1)N − 1]! ˜ · D(K, N) = N !(K − gN )!

(218)

˜ We can check that D(K, N ) leads to the same statistical distribution n ˜ (215) as D(K, N ). We now notice that Z(K, z) defined in terms of D(K, N) becomes an exact K-th power for all K. We can, then, calculate Pn from ˜ n). The result is D(1, n Y gn . (219) 1− Pn = m m=2 Unless g = 0, 1, Pn are fractional and always become negative for some values of n [76]. So their interpretation as probabilities or space dimensions must be taken with a grain of salt. They are, at any rate, useful tools for describing these systems. The single-level grand partition function Z(z) can be shown to satisfy Z g − Z g−1 = z

(220)

from which the corresponding relation (215) for n = z∂z ln Z follows. Remember that the CSM system enjoyed a particle-hole couplinginverting duality. Since exclusion statistics are directly extracted from the properties of these systems, we anticipate a similar duality here [46, 76]. Indeed, we can show that the grand partition function satisfies 1 1 + = 1. −g −1 Z(g, z ) Z(g , z)

(221)

Note that, interestingly, a similar relation holds for Gentile statistics of maximum occupancy p. The grand partition function obviously is Z(p, z) = 1 + z + z 2 . . . + z p =

z p+1 − 1 · z−1

(222)

Identifying the maximum occupancy p of Gentile statistics states with the parameter 1/g, the above expression satisfies (221) although Z(p, z) is not a priori defined for fractional p. From the distribution function n(g, z) (215) we can derive in the standard way the low-temperature Sommerfeld expansion for the energy Z ∞ X Cn (g)T n+1 En (223) E(T ) = dρ()n(g, ) = E(0) + n=0

A.P. Polychronakos: Generalized Statistics in One Dimension

where ρ() is the density of states and µ is fixed through Z N = dρ()n(g, )

465

(224)

En are T - and g-independent energy integrals. The coefficients Cn essentially determine the low-temperature heat capacity of the system. The are calculated as [77] C0

=

C1

=

C2

=

0 (third law of thermodynamics) 2 π (same as in conformal field theory) 6 2ζ(3)(1 − g) etc.

(225) (226) (227)

where ζ(x) is the Riemann zeta function. In general, Cn is a polynomial of degree n − 1 in g. The grand potential G = −kT ln Z can be expressed in terms of cluster coefficients wn : ∞ ∞ X X wn n z . Pn z n = (228) ln n n=0 n=1 We find for wn : wn =

n−1 Y m=1

1−

gn m

(229)

which is remarkably similar to Pn (except for the range of m). Similar results were obtained in [78] for the case of anyons in a strong magnetic field. This is not surprising: the lowest Landau level becomes, essentially, a two-dimensional phase space, so this is yet another realization of fractional statistics in an effectively one-dimensional space. Note that the connection between Pn and wn is the same as the one between “disconnected” and “connected” diagrams in field theory. This, in view also of the discussion of the first section, will lead us to a path-integral representation for the partition function of exclusion statistics particles in an arbitrary external potential. 6.4 Exclusion statistics path integral We start from the usual euclidean path integral with periodic time β for N particles with action the sum of N one-particle actions; we further sum over all particle numbers N with chemical potential weights eβµN /N ! to obtain the grand partition function. Since the particles are identical we include a symmetry factor of 1/N !, but we must also sum over paths where particles have exchanged final positions. Thus the path integral for each N decomposes into sectors labeled by the elements of the permutation group

466

Topological Aspects of Low Dimensional Systems

P erm(N ). (See the discussion in Sect. 1.) Such permuted sectors will be summed with appropriate extra weighting factors, to be determined in the sequel. By the usual argument, the Gibbs grand potential (the logarithm of the grand partition function) will be given by the sum of all connected path integrals. It is obvious that these are the ones where the final positions of the particles are a cyclic permutation of the original ones, since these are the only elements of SN that cannot be written as a product of commuting elements. Clearly all diagrams corresponding to different cyclic permutations are equal. There are (N − 1)! such permutations. So, overall, this diagram will carry a factor of ((N − 1)!/N ! = 1/N . This is nothing but the “symmetry factor”, familiar from Feynman diagrams, corresponding to cyclic relabelings of particle coordinates (compare, already, with the factors 1/n included in (228)). Cyclic permutation diagrams being connected, they only contain one “thread” of particle worldline. They really correspond to one particle wrapping N times around euclidean time and thus will reproduce the singleparticle partition function with temperature parameter N β. So, in the grand partition function they will contribute the terms proportional to e−N β eβµN = z N (the first factor is the single-particle Boltzmann factor for temperature parameter N β and the second is the chemical potential factor). But these are the terms appearing in (228) for each energy level! The extra weighting factors that must be included for these diagrams in order to reproduce exclusion statistics are, then, identified to the cluster coefficients wN . For g = 0, 1 we have wN = 1 and wN = (−)N −1 , respectively, which is, indeed, the correct factor implied by χR (P ) as discussed in Section 1, where R is the trivial (bosons) or the antisymmetric (fermions) irrep of SN and P a cyclic permutation. In conclusion, if we weight these configurations with the extra factors wN we will reproduce the grand potential of a distribution of g-ons on the energy levels of the one-body problem, that is [74] Ω(β, µ) =

∞ X N =1

eµN

wN N

Z

e−

PN n=1

SE [xn (tn )]

N Y

Dxn (tn )

(230)

n=1

where SE is the one-particle euclidean action and the paths obey the boundary conditions (231) xn (β) = xn+1 (0), xN (β) = x0 (0) (x can be in arbitrary dimensions.) By exponentiating, the full N -body partition function will be the path integral over all disconnected components, with appropriate symmetry factors and a factor of wn for each connected n-particle component.

A.P. Polychronakos: Generalized Statistics in One Dimension

467

More directly, we can omit the symmetry factors, sum explicitly over all equivalent permutations in each sector and divide by N !: Z N ∞ PN X Y eµN X WP Dxn (tn ) e− n=1 SE [xn (tn )] Z(β, µ) = N! n=1 N =1

(232)

P

where the paths obey the boundary condition {x1 (β), . . . xN (β)} = P {x1 (0), . . . xN (0)}.

(233)

The weighting factor WP for each permutation depends only on the conjugacy class of the permutation and is calculated through wn as WP = wn1 · · · wnk

(234)

where n1 , . . . nk are the cycles of P . It is clear that the above path integral is not unitary, since the weights wn are not phases and they do not provide true representations of the permutation group (unlike the g = 0, 1 cases). This does not matter: it is simply used as a tool to derive the statistical mechanics of exclusion statistics. We are not going to calculate any propagators or other processes with it. Once we have the path integral (232) we can easily extend the notion of exclusion statistics to interacting particles: we simply replace the action P S [x ] by the full interacting N -particle action, thus circumventing all E n n difficulties with combinatorial formulae. In the interacting case one has to work with the full grand partition function, rather than the grand potential, since topologically disconnected diagrams are still dynamically connected through the interactions and do not factorize. In conclusion, we have a path-integral way of defining exclusion statistics for interacting systems through appropriate symmetry factors included in each permutation sector (analogous to S(P ) of Sect. 1), calculated via (228). There is, yet, no application of this procedure to an interesting interacting situation. 6.5 Is this the only “exclusion” statistics? After giving so much motivation for it, the question seems almost offensive! But let us examine it for the sake of ensuring that we have the full picture. Exclusion statistics in terms of “repulsion rules” for the quantum numbers is, certainly, as we defined it. But we started from (199) which defines statistics in terms of the Hilbert space dimension of an additional particle in the system. Consider, then, a chunk of K levels in the CSM model (K quasimomentum values in the Sutherland model or K quasiexcitation levels

468

Topological Aspects of Low Dimensional Systems

for the harmonic Calogero model). Place N particles there, and see what space is left for an additional particle. If the N particles are packed “densely” (like, e.g., in the Fermi sea-like ground state) they certainly take up a piece of approximately `N spaces. So the levels left for an extra particle are d = K − `N and we get g = −∆d/∆N = ` as expected. What if, however, the N particles are “sparse” in K; that is, if the distances between them are all bigger than, say, 2`? Then each particle makes unavailable ` states either way around it, so, overall, the available space for the extra particle has diminished by (2` − 1)N . In this situation we would get g = 2` − 1. Clearly this definition implies different statistics for different situations (dense or dilute). Can we define another statistics that matches closer the Hilbert space definition? Clearly any choice of combinatorial formula for D(K, N ) that has the right extensive properties is an alternative definition of some statistics. Alternatively, each choice of Pn or wn amounts to some statistics. Let’s make the simplest choice: wn = (−α)n−1

(235)

which, in the path integral, corresponds to one factor of −α for each unavoidable particle crossing. This leads to the statistical distribution n ¯ n ¯=

1 e(−µ)β + α

(236)

which was analyzed in [79] as the simplest imaginable generalization of the Fermi and Bose distribution. The combinatorial formula for D(K, N ) for the above α-statistics is D = αN

(K K(K − α)(K − 2α) · · · (K − (N − 1)α) α )! · = K N! N !( α − N )!

(237)

This can be thought as a realization of the exclusion statistics Hilbert space idea: the first particle put in the system has K states to choose, the next has K − α due to the presence of the previous one an so on, and dividing by N ! avoids overcounting. Fermions and bosons correspond to α = 1 and α = −1 respectively. α = 0 corresponds to Boltzmann “classical” statistics. This should also be clear from the path integral: no configurations where particles have exchanged positions are allowed, since their weighting factor is zero, but factors of 1/N ! are still included. The corresponding single-level Pn are n−1 Y 1 − mα · (238) Pn = 1+m m=1

A.P. Polychronakos: Generalized Statistics in One Dimension

469

For α = 1/p with p integer (a fraction of a fermion), the above “probabilities” are all positive for n up to p and vanish beyond that. For α < 0 all probabilities are positive and nonzero. Thus, the above system has a bosonic (α < 0) and a fermionic (α > 0) sector, with Boltzmann statistics as the separator. It is a plausible alternative definition of exclusion statistics, and has many appealing features, not shared by the standard exclusion statistics, such as positive probabilities, a maximum single-level occupancy in accordance with the fraction of a fermion that α represents, and analytic expressions for all thermodynamic quantities. This should demonstrate that the route to alternative definitions, with nice features too, is open. The real issue is whether a dynamical system realizes these other statistics, and, at this point, this is not clear. 7

Epilogue

The topic of statistics will not cease fascinating at least a few physicists working in various fields. It challenges our understanding of the fundamentals of quantum mechanics and matter, expands the mathematical tools of the trade, promises new results in specific systems and is fun to think about. It is anyone’s guess whether considerations coming purely out of statistics will produce a (major or minor) breakthrough in any physics problem. A safer prediction, however, is that the idea of generalized statistics will re-emerge in different contexts as we strive to understand new physical systems and develop useful frames of mind. After all, statistics has barely been touched for objects such as strings and membranes, which are increasingly the entities of theoretical choice for the fundamental constituents of the universe. The future promises to be fun! References [1] Leinaas J.M. and Myrheim J., Nuovo Cimento 37B (1977) 1. [2] Goldin G.A., Menikoff R. and Sharp D.H., J. Math. Phys. 21 (1980) 650; 22 (1981) 1664; Phys. Rev. D 28 (1983) 830. [3] Wilczek F., Phys. Rev. Lett. 48 (1982) 1114; 49 (1982) 957. [4] Wu Y.S., Phys. Rev. Lett. 52 (1984) 2103. [5] Myrheim J., Anyons (Notes for the Course on Geometric Phases, ICTP, Trieste 6-17 Sept. 1993) and lecture notes in in this Les Houches Lectures. [6] Green H.S., Phys. Rev. 90 (1953) 270. [7] Steinmann O., Nuovo Cimento 44 (1966) A755. [8] Landshoff P.V. and Stapp H.P., Ann. Phys. 45 (1967) 72. [9] Ohnuki Y. and Kamefuchi S., Phys. Rev. 170 (1968) 1279; Ann. Phys. 51 (1969) 337. [10] Doplicher S., Haag R. and Roberts J., Comm. Math. Phys. 23 (1971) 199; 35 (1974) 49.

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Topological Aspects of Low Dimensional Systems

[11] Messiah A.M. and Greenberg L.O.W., Phys. Rev. B 136 (1964) 248; 138 (1965) 1155. [12] Hartle J.B. and Taylor J.R., Phys. Rev. 178 (1969) 2043; Stolt R.H. and Taylor J.R., Phys. Rev. D 1 (1970) 2226; Nucl. Phys. B 19 (1970) 1; Hartle J.B., Stolt R.H. and Taylor J.C., Phys. Rev. D 2 (1970) 1759. [13] Laidlaw M.G.G. and DeWitt C.M., Phys. Rev. D 3 (1971) 1375. [14] Greenberg O.W., et al., hep-ph/9306225. [15] Hammermesh M., Group Theory (Addison-Wesley Eds., 1962). [16] Suranyi P., Phys. Rev. Lett. 65 (1990) 2329. [17] Chaturvedi S., Phys. Rev. E 54 (1996) 1378. [18] Govorkov A.B., Theor. Math. Phys. 54 (1983) 234. [19] Greenberg O.W., Phys. Rev. Lett. 64 (1990) 705. [20] Gentile G., Nuovo Cimento 17 (1940) 493. [21] Polychronakos A.P., Nucl. Phys. B 324 (1989) 597. [22] Leinaas J.M. and Myrheim J., Phys. Rev. B 37 (1988) 9286. [23] Isakov S. and Leinaas J.M., Nucl. Phys. B 463 (1996) 194. [24] Isakov S., Leinaas J.M., Myrheim J., Polychronakos A.P. and Vernhagen R., Phys. Lett. B 430 (1988) 151. [25] Calogero F., J. Math. Phys. 10 (1969) 2191 and 2197; 12 (1971) 419. [26] Sutherland B., Phys. Rev. A 4 (1971) 2019; Phys. Rev. A 5 (1972) 1372; Phys. Rev. Lett. 34 (1975) 1083. [27] Moser J., Adv. Math. 16 (1975) 1. [28] Olshanetskii M.A. and Perelomov A.M., Phys. Rep. 71 (1981) 314; 94 (1983) 6. [29] Isakov S.B., Mod. Phys. Lett. A B8 (1994) 319 and Int. Jour. Mod. Phys. A 9 (1994) 2563. [30] Gaudin M., Saclay preprint SPht-92-158. [31] Minahan J.A. and Polychronakos A.P., Phys. Rev. B 50 (1994) 4236. [32] Lesage F., Pasquier V. and Serban D., Nucl. Phys. B 435 (1995) 585. [33] Ha Z.N.C., Phys. Rev. Lett. 73 (1994) 1574. Erratum, ibid. 74 (1995) 620. [34] Kazhdan D., Kostant B. and Sternberg S., Comm. Pure Appl. Math. 31 (1978) 481. [35] Gibbons J. and Hermsen T., Physica D 11 (1984) 337. [36] Wojciechowski S., Phys. Lett. A 111 (1985) 101. [37] Polychronakos A.P., Phys. Lett. B 277 (1992) 102. [38] Polychronakos A.P., Phys. Lett. B 408 (1997) 117. [39] A physicist’s exposition of relevant material is in Y. Nambu, “From SU (3) to Gravity,” edited by E. Gotsman and G. Tauber; Phys. Rev. D 26 (1992) 2875, and references therein. [40] Gorsky A. and Nekrasov N., Nucl. Phys. B 414 (1994) 213. [41] Minahan J.A. and Polychronakos A.P., Phys. Lett. B 326 (1994) 288. [42] Mehta H., Random Matrices (Academic Press, New York, 1991). [43] Blom J. and Langmann E., Phys. Lett. B 429 (1988) 336. [44] Polychronakos A.P., UUITP-8-98, hep-th/9810211, to appear in Nucl. Phys. B. [45] Polychronakos A.P., UUITP-7-98, hep-th/9806189, to appear in Nucl. Phys. B. [46] Polychronakos A.P., Phys. Rev. Lett. 69 (1992) 703. [47] Dunkl C.F., Trans. Amer. Math. Soc. 311 (1989) 167. [48] Brink L., Hansson T.H. and Vassiliev M., Phys. Lett. B 286 (1992) 109. [49] Minahan J.A. and Polychronakos A.P., Phys. Lett. B 302 (1993) 265.

A.P. Polychronakos: Generalized Statistics in One Dimension

[50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79]

471

Ha Z.N.C. and Haldane F.D.M., Phys. Rev. B 46 (1992) 9359. Kawakami N., Phys. Rev. B 46 (1992) 1005, 3191. Hikami K. and Wadati M., Phys. Lett. A 173 (1993) 263. Haldane F.D.M., Ha Z.N.C., Talstra J.C., Bernard D. and Pasquier V., Phys. Rev. Lett. 69 (1992) 2021. Bernard D., Gaudin M., Haldane F.D.M. and Pasquier V., J. Phys. A 26 (1993) 5219. Cherednik I., RIMS-1144 (1997) and references therein. Lapointe L. and Vinet L., Comm. Math. Phys. 178 (1996) 425. Avan J. and Jevicki A., Nucl. Phys. B 469 (1996) 287 and B 486 (1997) 650. Polychronakos A.P., Nucl. Phys. B 419 (1994) 553. Sutherland B. and Shastry B.S., Phys. Rev. Lett. 70 (1993) 4029. Haldane F.D.M., Phys. Rev. Lett. 60 (1988) 635 and 66 (1991) 1529. Shastry B.S., Phys. Rev. Lett. 60 (1988) 639 and 69 (1992) 164. Inozemtsev V.I., J. Stat. Phys. 59 (1989) 1143. Fowler M. and Minahan J.A., Phys. Rev. Lett. 70 (1993) 2325. Polychronakos A.P., Phys. Rev. Lett. 70 (1993) 2329. Frahm H., J. Phys. A 26 (1993) L473. Kawakami N. and Kuramoto Y., YITP-K-1065 (1994). Kato Y. and Kuramoto Y., cond-mat/9409031. Fukui T. and Kawakami N., Phys. Rev. Lett. 76 (1996) 4242; and Nucl. Phys. B 483 (1997) 663. Haldane F.D.M., Phys. Lett. 93A (1983) 464 and Phys. Rev. Lett. 50 (1983) 1153. Haldane F.D.M., Phys. Rev. Lett. 67 (1991) 937. Y.-Wu S., Phys. Rev. Lett. 73 (1994) 922. Rajakopal A.K., Phys. Rev. Lett. 74 (1995) 1048. Karabali D. and Nair V.P., Nucl. Phys. B 438 (1995) 551. Polychronakos A.P., Phys. Lett. B 365 (1996) 202. Ilinskaia A.V., Ilinski K.N. and Gunn J.M.F., Nucl. Phys. B 458 (1996) 562. Nayak C. and Wilczek F., Phys. Rev. Lett. 73 (1994) 2740. Arovas D., Isakov S., Myrheim J. and Polychronakos A.P., Phys. Lett. A 212 (1996) 299. de A.Veigy D. and Ouvry S., Phys. Rev. Lett. 72 (1994) 600 and Mod. Phys. Lett. A 10 (1995) 1. Acharya R. and Narayana Swamy P., J. Phys. A 27 (1994) 7247.

COURSE 6

LECTURES ON NON PERTURBATIVE FIELD THEORY AND QUANTUM IMPURITY PROBLEMS

H. SALEUR Department of Physics, University of Southern California, Los-Angeles, CA 90089-0484, U.S.A.

Contents 1 Some notions of conformal field theory 1.1 The free boson via path integrals . . . . . . 1.2 Normal ordering and OPE . . . . . . . . . . 1.3 The stress energy tensor . . . . . . . . . . . 1.4 Conformal in(co)variance . . . . . . . . . . 1.5 Some remarks on Ward identities in QFT . 1.6 The Virasoro algebra: Intuitive introduction 1.7 Cylinders . . . . . . . . . . . . . . . . . . . 1.8 The free boson via Hamiltonians . . . . . . 1.9 Modular invariance . . . . . . . . . . . . . .

. . . . . . . . .

2 Conformal invariance analysis of quantum impurity 2.1 Boundary conformal field theory . . . . . . . 2.2 Partition functions and boundary states . . . 2.3 Boundary entropy . . . . . . . . . . . . . . . 3 The 3.1 3.2 3.3 3.4

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

483 483 485 488 490 493 494 497 500 502

fixed points 503 . . . . . . . . . . . . 503 . . . . . . . . . . . . 506 . . . . . . . . . . . . 509

boundary sine-Gordon model: General results The model and the flow . . . . . . . . . . . . . . . . . . . Perturbation near the UV fixed point . . . . . . . . . . . . Perturbation near the IR fixed point . . . . . . . . . . . . An alternative to the instanton expansion: The conformal invariance analysis . . . . . . . . . . . . . . . . . . . . . .

512 . . . . . 512 . . . . . 513 . . . . . 515 . . . . . 518

4 Search for integrability: Classical analysis

520

5 Quantum integrability 524 5.1 Conformal perturbation theory . . . . . . . . . . . . . . . . . . . . 524 5.2 S-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 5.3 Back to the boundary sine-Gordon model . . . . . . . . . . . . . . 531 6 The thermodynamic Bethe-ansatz: The gas of particles with “Yang-Baxter statistics” 6.1 Zamolodchikov Fateev algebra . . . . . . . . . . . . . . . . . 6.2 The TBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A standard computation: The central charge . . . . . . . . 6.4 Thermodynamics of the flow between N and D fixed points

. . . .

532 532 534 536 538

7 Using the TBA to compute static transport properties 7.1 Tunneling in the FQHE . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Conductance without impurity . . . . . . . . . . . . . . . . . . . . 7.3 Conductance with impurity . . . . . . . . . . . . . . . . . . . . . .

541 541 542 543

. . . .

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LECTURES ON NON PERTURBATIVE FIELD THEORY AND QUANTUM IMPURITY PROBLEMS

H. Saleur

Abstract These lectures provide a simple introduction to non perturbative methods of field theory in 1 + 1 dimensions, and their application to the study of strongly correlated condensed matter problems – in particular quantum impurity problems. The level is moderately advanced, and takes the student all the way to the most recent progress in the field: many exercises and additional references are provided. In the first part, I give a sketchy introduction to conformal field theory. I then explain how boundary conformal invariance can be used to classify and study low energy, strong coupling fixed points in quantum impurity problems. In the second part, I discuss quantum integrability from the point of view of perturbed conformal field theory, with a special emphasis on the recent ideas of massless scattering. I then explain how these ideas allow the computation of (experimentally measurable) transport properties in cross-over regimes. The case of edge states tunneling in the fractional quantum Hall effect is used throughout the lectures as an example of application.

Introduction Quantum impurity problems have been for many years, and increasingly so recently, a favorite subject of investigations, for theorists and experimentalists alike. There are many good reasons for that. First, these problems often represent the simplest setting in which some qualitatively fascinating physical properties can be observed and probed. For instance, the Kondo model (for a comprehensive review1 , consult [1]) provides a clear cut example of asymptotic freedom. The basic experimental fact is that normal metals with dilute impurities exhibit an unusual 1 These being only lecture notes, I have tried to refer to papers that were pedagogically inclined, if at all possible, rather than to original works.

c EDP Sciences, Springer-Verlag 1999

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Topological Aspects of Low Dimensional Systems

minimum in the temperature dependence of the electrical resistivity. The ultimate explanation is that the interaction of the electrons with the impurity spins produces an increase of the resistivity as T is lowered, counteracting the usual decrease in resistivity arising from interactions with lattice phonons. Low temperature means low energy, or large distance: we thus have a problem where interactions increase at large distance, a characteristic of asymptotic freedom2 . As another example I would like to mention the recent experiments [2,3] about point contact tunneling in the fractional quantum Hall effect (for a review on this active topic, see [4]) at filling fraction ν = 13 . Measurments of the shot noise reveal a behaviour I 2 ∝ 3e IB in the limit of weak backscattering, where few quasiparticles tunnel, and do so independently. If one compares this formula with the standard Schottky formula for Fermi liquids, one sees that this noise has to be due to the tunneling of fractional charges e∗ = 3e : although the existence of these (Laughlin quasiparticles) had been conjectured for a long time, the noise is the first direct evidence of their existence3 . As a final example, let me recall that the basic archetype of dissipative quantum mechanics (for a review, look at [5, 6]), the two state problem coupled to a bath of oscillators with Ohmic dissipation, is described by another quantum impurity problem: the anisotropic Kondo model. Crucial fundamental issues are at stake here, as well as a large array of applications in chemistry and biology. The second reason of our fascination for quantum impurity problems is that they are, to a large extent, manageable by analytic methods. This has led to incredibly fruitful progress in the past. For instance, the renormalization group was, to a large extent, borne out of the efforts of Kondo, Anderson and Wilson to understand the low temperature behaviour of the Kondo model (see [1]). Also, the works of Andrei [7], Wiegmann [8] and others showed that the Bethe ansatz could be used to analyze situations of experimental relevance: this spurred a new interest in quantum integrable models, an area which, together with its various off-springs like quantum groups, knot theory and others, has become one of the most lively in mathematical physics. Finally, it is fair to say that quantum impurity problems are not only of fundamental interest: they are at the center of the most challenging problems of today’s condensed matter, like Kondo lattices, heavy fermions, and, maybe, high Tc superconductors. In these lectures, I will concentrate on what is usually considered the most important about quantum impurity problems: their properties as strongly interacting systems. There is no doubt that strongly correlated

2 The analogy with QCD can be made more complete, including the logarithmic dependences encountered in both problems, and the “dimensional transmutation”. 3 The conductance itself is not a measure of the charge of the carriers.

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electrons are of the highest interest. On the practical side, besides high Tc superconductivity, let me mention the remarkable recent developments in manufacturing and understanding small systems like quantum wires, carbon nanotubes and the like, where, because of the reduced dimensionality, Fermi liquid theory is not applicable, and the interactions have to be taken into account non perturbatively. On a more fundamental level, strongly interacting systems exhibit rather counter intuitive properties, the most spectacular being probably spin charge separation. It is a challenge for the theorist to understand these properties, and quantum impurity problems provide one of the best theoretical and experimental laboratory to do so. Maybe it is time now to define what I mean by a quantum impurity problem. The general class of systems I have in mind have the following features: (i) There are extended gapless (critical) quantum mechanical degrees of freedom, which live in an infinite spatial volume, the “bulk” (ii) These interact with an impurity, localized at one point in position space. This impurity may carry quantum mechanical degrees of freedom. To have an example in mind, consider the Kondo problem: (i) The extended degrees of freedom are those of the bulk metal. The presence of a Fermi surface means that the metal sits at a RG fixed point (see e.g. [9]). Physically this is easily understood since the system of electrons has (“particle-hole”) excitations of arbitrarily low energy about its Fermi-sea ground state, providing the critical degrees of freedom in the bulk of the metal (ii) The impurity spin, located at one point in space (say the origin), is a dynamical quantum mechanical degree of freedom (the dynamical process is the spin-flip). In the Kondo model we also see another feature of quantum impurity problems: they are generically one-dimensional. The problem of an impurity in a Luttinger liquid to be discussed below is inherently one-dimensional, but the Kondo model needs to be reduced to one dimension. Since the impurity spin sits only at one point in space, it is only the s-wave wavefunctions of the metal electrons that can interact with the spin. Second-quantizing this s-wave theory, we get a (non-interacting) quantum field theory of onedimensional Fermions, defined on half-infinite (radial) position space (a halfinfinite line), which interacts with the quantum spin at the end of the line. Considering a path-integral representation of the 1D theory of Fermions, we have a (1 + 1)-dimensional Lagrangian field theory, one dimension from the (half-infinite) radial space coordinate, and another dimension from the (say, euclidean) time coordinate. All interactions take place at one point in space, the end of the line, where the impurity is located. In the (1 + 1) dimensional space-time picture, the impurity sits at the “boundary” of space-time, which can be viewed as the upper-complex plane, the “boundary” being the real axis (these points of view are schematically illustrated in Figs. 1 and 2).

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Topological Aspects of Low Dimensional Systems

This will be the general picture that we use in the study of all quantum impurity problems.

Fig. 1. We will think of quantum impurity problems in various ways. The figure on the left represents a 1 + 1 quantum point of view where the bulk right and left degrees of freedom are confined to a half line, with the impurity at the origin. Altrnatively, because the theory is massless in the bulk, one can unfold this picture to get only right degrees of freedom on the full line, as indicated on the right.

Fig. 2. Alternatively again, one can go to imaginary time and obtain a 2D statistical mechanics point of view, with a theory defined in a half plane, that often we will chose to be the upper complex plane. In this figure, the arrows are supposed to “represent” the bulk degrees of freedom; later, we will see that they can be associated with integrable quasiparticles.

In the Kondo problem we see one way how a quantum impurity problem can be realized experimentally: a bulk system (here the 3D metal) contains a finite but small concentration x of quantum impurities. In the limit of very dilute impurities (x < 1) the impurities do not interact with each other (to lowest order in x), and the single-impurity theory may be used to describe the physics of the bulk material in the presence of dilute impurities. Actually, experiments performed at very low concentrations are known to be in good agreement with the single-impurity theory for the ordinary (onechannel) Kondo model. A quite different realization of quantum impurity models occurs in the context of point contacts. These are basically electronic devices: two leads (capable of transporting electrical current) are attached to a single quantum impurity. Each one of the leads is connected to a battery, so that electrical current is driven through the quantum impurity. One can then measure experimentally the electrical current I, flowing through the quantum impurity, as a function of the applied driving voltage V (from the battery) [10]. The I(V ) curve, the differential (non-linear) conductance

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Gdiff = ∂I(V )/∂V , or the temperature dependence of the linear response conductance Glr = limV →0 I(V )/V are examples of experimental probes characteristic of the quantum impurity. Notice that since the I(V ) curve as well as the differential conductance are non-equilibrium properties, the

V/2

A A

-V/2

Fig. 3. A schematic experimental set up to study point contact tunneling in the fractional quantum Hall effect (the magnetic field points towards the reader). Details are provided in the text.

point-contact realizations of quantum impurity problems are theoretically more challenging than other realizations, in that more than equilibrium statistical mechanics is involved to achieve a theoretical understanding of these quantities. Ideally, the most interesting point contact situation would involve onedimensional leads, where electrons are described by the Luttinger model, the simplest non-fermi-liquid metals [11]. It consists of left- and right-moving gapless excitations at the two fermi points in an interacting 1-dimensional electron gas. In the past, this model had been difficult to realize experimentally however. This is simply because in a one-dimensional conductor (such as a quasi-one-dimensional quantum wire so thin that the transverse modes are frozen out at low temperature), random impurities occur in the fabrication. These impurities lead to localization due to backscattering processes between the excitations at the two fermi points. In other words, the random impurities generate a mass gap for the fermions. Fortunately, there is another possiblity: the edge excitations at the boundary of samples prepared in a fractional quantum Hall state should be extremely clean realizations of the Luttinger non-fermi liquids, as was observed by Wen [12]. In contrast to quantum wires, these are stable systems because for 1/ν an odd integer, the excitations only move in one direction on a given edge. Since the right and left edges are far apart from each other, backscattering processes due to random impurities in the bulk cannot localize those extended edge states. Moreover, the Luttinger interaction parameter is universally related to the filling fraction ν of the quantum Hall state in the bulk sample by a topological argument based on the underlying Chern-Simons theory, and does therefore not renormalize. The edge

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Topological Aspects of Low Dimensional Systems

states should thus provide an extremely clean experimental realization of the Luttinger model. We now describe an experimental set-up in more details [13, 14] (see Fig. 3). A fractional quantum Hall state with filling fraction ν = 1/3 is prepared in the bulk of a quantum Hall bar which is long in the x-direction and short in the y-direction. This means that the bulk quantum Hall state is prepared in a Hall insulator state (longitudinal conductivity σxx = 0), and that the (bulk) Hall resistivity is on the ν = 1/3 plateau where σxy = (1/3)e2 /h. This is achieved by adjusting the applied magnetic field, perpendicular to the plane of the bar. Since the plateau is broad, the applied magnetic field can be varied over a significant range without affecting the filling of ν = 1/3. Next, a gate voltage Vg is applied perpendicular to the long side of the bar, i.e. in the y direction at x = 0. This has the effect of bringing the right and left moving edges close to each other near x = 0, forming a point contact. Away from the contact there is no backscattering (i.e. no tunneling of charge carriers) because the edges are widely separated, but now charge carriers can hop from one edge to the other at the point contact. The left-moving (upper) edge of the Hall bar can now be connected to battery on the right such that the charge carriers are injected into the leftmoving lead of the Hall bar with an equilibrium thermal distribution at chemical potential µL . Similarly, the right-moving carriers (propagating in the lower edge) are injected from the left, with a thermal distribution at chemical potential µR . The difference of chemical potentials of the injected charge carriers is the driving voltage V = µR − µL . If V > 0, there are more carriers injected from the left than from the right, and a “source-drain” current flows from the left to the right, along the x-direction of the Hall bar. In the absence of the point contact, the driving voltage places the right and left edges at different potentials (in the y-direction, perpendicular to the current flow), implying that the ratio of source-drain current to the driving voltage V is the Hall conductance G = νe2 /h (both in linear response and at finite driving voltage V ). When the point-contact interaction is included, at finite driving voltage, more of the right moving carriers injected from the left are backscattered than those injected from the right, resulting in a loss of charge carriers from the source-drain current. In this case we write the total source-drain current as I(V ) = I0 (V ) + IB (V ), where IB (V ) is the (negative) backscattering current, quantifying the loss of current due to backscattering at the point contact. It is this backscattering current that I ultimately want to show how to compute. Let me write up some formulas as a preamble. I will not have time in these lectures to discuss bosonization or edge states in the fractional quantum Hall effect: I will thus simply claim that, in its bosonized form,

H. Saleur: Lectures on Non Perturbative Field Theory

the problem is described by the Hamiltonian Z β 1 ∞ dx Π2 + (∂x φ)2 + λ cos √ (φL − φR )(0), H= 2 −∞ 2

481

(1)

2

β = ν. Here, the free boson part describes the massless edge states where 8π [12], and the cosine term describes the effect of the gate voltage, with λ ∝ Vg . In general of course, the backscattering term induced by this gate voltage should be represented by a complicated interaction; but we keep only the most relevant term (the only one for ν = 13 ), which is all that matters in the scaling regime (see below) we will be interested in. The interaction is a relevant term, that is, in a renormalization group transformation, one has, b being the rescaling factor4

dλ = (1 − ν)λ + O(λ3 ). db

(2)

This means that at large gate voltage, or, equivalently, at small temperature (since then, elementary excitations have low energies, so the barrier appears big to them), the point contact will essentially split the system in half, and no current will flow through5 . The questions the theorist wants to answer are: how do we study the vicinity of the weak-backscattering limit? How do we find out more precisely what the strong back-scattering limit looks like? How about its vicinity? Finally, can we be more ambitious and compute say the current at any temperature, voltage and gate voltage? For this latter question, let me stress that we are interested in the universal, or scaling, regime, which is the only case where things will not depend in an complicated way on the microscopic details of the gate and other experimental parameters. In practice, what the experimentalist will do is first sweep through values of the gate voltage, the conductance signal showing a number of resonance peaks, which sharpen as the temperature is lowered. These resonance peaks occur for particular values Vg = Vg∗ of the gate voltage, due to tunneling through localized states in the vicinity of the point contact. Ideally, on resonance, the source-drain conductance is equal to the Hall conductance without point contact, i.e. Gresonance = νe2 /h. This value is independent of temperature, on resonance. Now, measuring for instance the linear response conductance as a function of the gate voltage near the resonance, i.e. as a function of δVg ≡ Vg − Vg∗ , at a number of different temperatures T , one gets resonance curves, one for each temperature. These 4 In particle physics language, dλ = −β(λ), so our relevant operator corresponds to a db negative beta-function, i.e. an asymptotically free theory. 5 This feature is actually remarkable. What it means is that, for one dimensional electrons with short distance repulsive interactions, an arbitrarily small impurity leads to no transmittance at T = 0 [10]: compare with the effects of barriers on non-interacting electrons you studied in first year quantum mechanics.

482

Topological Aspects of Low Dimensional Systems

peak at δVg = 0. Finally, these conductance curves should collapse, in the limit of very small T and δVg , onto a single universal curve when plotted as a function of δVg /T 1−ν . This is what the field theorist wants to compute. To proceed, it is useful to formulate the problem as a boundary problem. For this, a few manipulations are needed. We decompose φ = φL + φR and set6 : 1 ϕe (x + t) = √ [φL (x, t) + φR (−x, t)] 2 1 ◦ ϕ (x + t) = √ [φL (x, t) − φR (−x, t)] . 2

(3)

Observe that these two fields are left movers. We now fold the system by setting, for x < 0: φeL = ϕe (x + t), φoL

o

= ϕ (x + t),

φeR = ϕe (−x + t), φoR = −ϕo (−x + t),

(4)

e,o and introduce new fields φe,o = φe,o L + φR , both defined on the half infinite o line x < 0. The odd field φ simply obeys Dirichlet boundary conditions at the origin φ◦ (0) = 0, and decouples from the problem. The field φe , which we call rather Φ in the following, has a non trivial dynamics Z 1 0 βΦ(0) e · (5) dx Π2 + (∂x Φ)2 + λ cos H≡H = 2 −∞ 2

The aspects we have to understand are, by increasing order of complexity: the fixed points, their vicinity, and what is in between. This is the order I will follow in these lectures. Part I: Conformal field theory and fixed points The first difficulty one encounters in that field is how to describe the low energy fixed points. This may sound rather simple in the tunneling problem, but in other cases, for instance in a tunneling problem for electrons with spin, the matter is quite involved. The reason for this is, that fixed points are not necessarily described in terms of nice linear boundary conditions for the bulk degrees of freedom. It does seem to be true however, that even if the quantum impurity has internal degrees of freedom, interaction 6 More details are given in Part I. A common objection to the following manipulations is that they are good only for free fields, but not when there is a boundary interaction. This, in fact, depends on what one means by “fields” – the safest attitude is to imagine one does perturbation theory in λ. Then, all the quantities are evaluated within the free theory, on which one can legitimately do all the foldings, left right decompositions, etc.

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and renormalization effects do turn the dynamical quantum impurity into a boundary condition on the extended bulk degrees of freedom, at large distances, low energy or low temperature. At low temperatures the system may be in the strong coupling regime (for instance, this is where Kondo’s result diverges). The boundary condition is thus a way to think about the strongly interacting system. Nozi`eres’ physical picture of screening [15] illustrates how this works for the simplest case, the one-channel Kondo model: the antiferromagnetic interaction of the impurity spin with the spin of the conduction electrons, which has renormalized to large values at low temperatures, causes complete screening of the impurity spin. A modified boundary condition on the electrons that are not involved in the screening, is left. This mechanism, however, appears to be much more general, and seems to apply to all quantum impurity problems. The boundary conditions generated in this process may be highly nontrivial (see e.g. [17, 18]). However, since the bulk is massless (critical), the induced boundary condition is scale invariant asymptotically at large distances and low temperatures. Actually, it is, in most cases, conformally invariant. Quantum impurity problems are thus intimately related with scaleinvariant boundary conditions: these are RG fixed points, and, like in bulk 1 + 1 quantum field theories, (recall that the bulk is always critical in the type of systems that we are considering here), conformal symmetry is the best way to describe them. Now, conformal invariance is a long story. All I can do is provide, in the next sections, what I believe is the minimal set of ideas necessary to understand what is going on, and tackle without fear the literature on the subject. In several instances, I will have to discard entire discussions of key issues, substituting them with some intuitive comments, and only providing the final result. Additional bits and pieces are then provided in the text in small characters, together with specific references, to help the reader bridge the gaps. Good reviews on this subject are the Les Houches Lectures of 1988 [19], the article by Cardy [20], the lectures by Polchinski [21], and the textbook [22]. The relevant chapters in [23] can also be quite useful. In the following, I will intimately mix path integral and Hamiltonian points of view. The two are of course equivalent, but each has its own advantages. 1

Some notions of conformal field theory

1.1 The free boson via path integrals We consider the free bosonic theory, with action Z h i 1 2 2 dx1 dx2 (∂1 Φ) + (∂2 Φ) . S= 2

(6)

484

Topological Aspects of Low Dimensional Systems

To start, let us discuss briefly the issue of correlators and regularization. To keep in the spirit of condensed matter, we initially define the Gaussian model on a discrete periodic square lattice of constant a by setting Slatt =

1X 2 [Φ(rj ) − Φ(rk )] , 2

(7)

hjki

where the sum is taken over all pairs of nearest neighbours. Introduce the lattice Green function Glatt (r) =

0 1 X eik.r 2πn 2 2a L n ,n 4 − 2 cos L1 a − 2 cos 2πn L 1 2

(8)

L , and the where the sum is restricted to the first Brillouin zone |ni | ≤ 2a 7 prime means the zero mode is excluded . One has then (where the points r, r0 belong to the lattice)

hΦ(r)Φ(r0 )ilatt = Glatt (r − r0 )

(9)

while Glatt satisfies the discrete equation (where ∆latt is the discrete Laplacian) (10) ∆latt Glatt = −δlatt (r). 1 a ln L , L a, The important points here are the behaviours Glatt (0) ≈ − 2π 1 r and Glatt (r) ≈ − 2π ln L for a r L. We recall now Wick’s theorem, according to which the average of any quantity can be obtained as a sum of all pairwise contractions. It follows that E D

2 1 iβ1 Φ(r) iβ2 Φ(r 0 ) 2 e = exp − (β1 +β2 ) Φ (r) latt e 2 latt

× exp −β1 β2 hΦ(r)Φ(r0 )ilatt − Φ2 (r) latt . (11)

To define a continuum limit for this model, we look at distances large compared to the lattice spacing but small compared to L, where the right hand side of the previous expression simplifies into D

0

eiβ1 Φ(r) eiβ2 Φ(r )

E latt

≈

a (β1 +β2 )2 /4π |r − r0 | β1 β2 /2π L

a

.

(12)

The well known observation follows that the correlator vanishes unless charge neutrality is satisfied, that is β1 + β2 = 0. We then have, where 7 The zero mode divergence simply occurs because the action is invariant under the symmetry Φ → Φ + cst.

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r, r0 are now arbitrary points in the continuum, E D 0 eiβΦ(r) e−iβΦ(r ) ≈

a |r − r0 |

β 2 /2π ·

(13)

In the following we will sometimes, but not always, set a = 1. 1.2 Normal ordering and OPE We now introduce complex coordinates z = x1 + ix2 , z¯ = x1 − ix2 . We have 1 2 ¯ ∂ = 12 (∂1 − i∂ 2 , and we define the delta R 2 ),2 ∂ 2= 2 (∂1 + i∂2 ), d2 z = 2dx11 dx function by d zδ (z, z¯) = 1, so δ (z, z¯) = 2 δ 2 (x1 , x2 ). The action reads now Z ¯ (14) S = d2 z∂Φ∂Φ, and the Laplacian ¯ ∆ = 2∂ ∂¯ + 2∂∂. Of crucial importance is the result8 1 ∂ = 2πδ 2 (z, z¯) z¯

(15)

(16)

from which it follows that9 ∂ ∂¯ ln |z|2 = ∂ ∂¯ ln z¯ = 2πδ 2 (z, z¯).

(17)

It is customary to write the basic correlator as10 hΦ(z, z¯)Φ(z 0 , z¯0 )i = −

1 ln |z − z 0 |2 . 4π

(18)

Note that, in this expression, we have completely discarded the L dependence that occurs in the lattice system. A reason for doing so is that Φ is not a “good” field anyway, and that we will usually consider rather derivatives of Φ, for which this ambiguity does not matter. The L dependence is however crucial for exponentials of the field Φ. When we discard it, we have to remember that, at the end of the day, only correlators which have vanishing charge are non zero. 8 A physical way to prove this is to observe that, in practice, 1 has to be regulated z with a short distance cut-off which does introduce a z dependence, as for instance in H(z z ¯−a2 ) , H the Heavyside function. z 9 Note that the two derivatives cannot

be interchanged on singular functions, that is why ∆ ln r = 2πδ2 (x1 , x2 ), and not twice as much. 10 Of course the notation is somewhat redundant, since the value of z determines x 1 and x2 ; but in what follows, we will reserve the notation f (z) for analytic functions.

486

Topological Aspects of Low Dimensional Systems

Now, still using Wick’s theorem it follows that ∂ ∂¯ hΦ(z, z¯)Φ(z 0 , z¯0 ) . . .i = −

1 2 δ (z − z 0 , z¯ − z¯0 ) . . . , 2

(19)

where the dots stand for any other insertions in the path integrals, that involve no field either at z or z 0 . A relation that holds in this sense is simply rewritten 1 ¯ ∂ ∂Φ(z, z¯)Φ(z 0 , z¯0 ) = − δ 2 (z − z 0 , z¯ − z¯0 ). 2

(20)

This is the first example of equations we will write quite often between “operators” in the theory – the word operator here occurs naturally when one splits open the path integral to obtain a Hamiltonian description, see later. Recall that the equations of motion for the field Φ read, on the other hand ¯ ∂ ∂Φ(z, z¯) = 0. (21) It follows that the product Φ(z, z¯)Φ(z 0 , z¯0 ) obeys the equation of motion except at coincident points. In the sequel, we shall constantly use the concept of normal ordering. We define the normal ordering of the product of two bosonic fields by11 : Φ(z, z¯)Φ(z 0 , z¯0 ) : ≡ Φ(z, z¯)Φ(z 0 , z¯0 ) +

1 ln |z − z 0 |2 . 4π

(22)

This definition is such that the normal ordered product of fields now does satisfy the equation of motion even at coincident points ∂ ∂¯ : Φ(z, z¯)Φ(z 0 , z¯0 ) : = 0.

(23)

As a result of this, the normal product is (locally) the sum of an analytic and antianalytic function, and can be expanded in powers of z. Thus, for instance Φ(z, z¯)Φ(0, 0) = − +

1 ln |z|2 + : Φ2 (0, 0) : +z : ∂ΦΦ(0, 0) : 4π ¯ z¯ : ∂ΦΦ(0, 0) : + . . .

(24)

This is the first example of an operator product expansion (OPE). Like equation (20), its precise meaning is that it holds once inserted inside a correlation function. OPEs in conformal field theories are not asymptotic, 11 If one wishes to keep the a and L factors, the normal ordering formula reads:

1 Φ(z, z¯)Φ(z 0 , z¯0 ) : ≡ Φ(z, z¯)Φ(z 0 , z¯0 )+ 4π ln (|z − z 0 |/a)2 . One has then : Φ2 : = ln L/a.

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but rather convergent expansions; their radius of convergence is given by the distance to the nearest other operator in the correlation functions of interest. The right hand side of (24) involves products of fields at coincident points, which turn out to be well defined in this theory. Notice that : Φ2 (0, 0) : in (24) could be defined equally well by point splitting, as will be discussed later. For a product of more than two bosonic operators, the definition of normal order can be extended iteratively : Φ(z, z¯)Φ(z1 , z¯1 ) . . . Φ(zn , z¯n ) : = Φ(z, z¯) : Φ(z1 , z¯1 ) . . . Φ(zn , z¯n ) : 1 ln |z − z1 |2 : Φ(z2 , z¯2 ) . . . Φ(zn , z¯n ) : + permutations (25) + 4π such that the classical equations of motion are still satisfied (a quick way to understand normal order, as is clear from the previous formula, is that quantities inside double dots are not contracted with one another when one computes correlators). Of crucial importance is the normal ordered exponential: eiβΦ :. It is a good exercise to recover the OPE : eiβ1 Φ(z,¯z ) : : eiβ2 Φ(0,0) : = |z|

β1 β2 2π

: eiβ1 Φ(z,¯z )+iβ2 Φ(0,0) :

(26)

−1 : eiβΦ(0,0) : + : ∂Φ(z, z¯)eiβΦ(0,0) : 4πz

(27)

and ∂Φ(z, z¯) : eiβΦ(0,0) : =

The quantities inside the normal ordering symbols can now be expanded in powers of z, z¯ like an ordinary function. Exercise: Show that the “quantum Pythagoras” theorem holds: 2 2 √ √ ¯ = −4π∂Φ∂Φ. : cos 2 πΦ : + : sin 2 πΦ : finite

Notice that in (27), we could have treated ∂Φ as an analytic function: a perfectly legitimate thing to do when one computes correlators. Hidden here is, in fact, the very convenient decomposition of the field Φ itself into the sum of an analytic and antianalytic component (one has to be very ¯ z ); first, because such careful however when one writes Φ(z, z¯) = φ(z) + φ(¯ a decomposition does not hold for the general fields summed over in the path integral, and second, because the field Φ does not obey the equations of motion at coincident points). Pushing that line of thought a bit further however, one has 1 ln(z − z 0 ) 4π

1 ¯ z )φ¯0 (¯ ln(¯ z − z¯0 ) z0) = − φ(¯ 4π hφ(z)φ(z 0 )i = −

(28)

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Topological Aspects of Low Dimensional Systems

this up to phases due to the branch cuts. We will also use in the following the dual field ¯ ˜ = φ − φ. Φ (29) The exponentials are scalar operators, i.e. they are invariant under rotations. More general operators also have a spin, so their two point function reads 1 (30) hO(z, z¯)O(0, 0)i = 2h 2h¯ · z z¯ ¯ = 0. In The simplest example is provided by O = ∂Φ which has h = 1, h ¯ general, single valuedness of physical correlators requires h − h to be an ¯ are called usually right and left conformal integer. The numbers h and h ¯ while its spin is s = h − h. ¯ dimensions. The dimension of O is d = h + h, 1.3 The stress energy tensor The stress energy tensor is defined in the classical theory as follows. Consider a coordinate transformation xµ → xµ + µ (that is, changing the arguments of the fields in the action from xµ → xµ + µ ). The variation of the action reads, to lowest order12, Z 1 (31) ∂µ ν Tµν dx1 dx2 . δS = − 2π Elementary calculation shows that Tµν = −2π∂µ Φ∂ν Φ + πδµν ∂ρ Φ∂ρ Φ.

(32)

The stress energy tensor in the quantum theory is defined through Ward identities: the end result is the same formula as (32), but where products of fields are normal ordered. It enjoys some very important properties: the symmetry T12 = T21 as a result of rotational invariance, and the tracelessness T11 + T22 = 0 as a result of scale invariance. In addition, the stress energy tensor is always conserved, ie the operator equation ∂µ Tµν = 0 holds. This can be checked explicitely for the free boson using (32) and the classical equations of motion, which we recall are satisfied by normal ordered products13 . In general, one introduces complex components

12 The

Tzz

=

Tz¯z¯

=

1 (T11 − T22 − 2iT12 ) 4 1 (T11 − T22 + 2iT12 ) 4

factor of 2π is peculiar to the conformal field theory literature. generally, that T is classically conserved follows simply from the fact that the action is stationary as a consequence of the classical equations of motion, so δS must vanish for arbitrary . 13 More

H. Saleur: Lectures on Non Perturbative Field Theory

Tzz¯ = Tz¯z

=

1 (T11 + T22 ) 4

489

(33)

which, in the free boson case, read simply 2

Tzz

=

Tz¯z¯

=

−2π : (∂φ) : 2 −2π : ∂¯φ¯ :

Tzz¯

=

0.

(34) (35) (36) (37)

Using the equations of motion, Tzz is analytic (again, in the special sense that, when inserted in correlation functions, the dependence is analytic away from the arguments of the other operators) and will simply be denoted T (z) in what follows; similarly, Tz¯z¯ is antianalytic. It is easy to chek that these properties extend to other models: a remarkable consequence of locality (so there is a stress energy tensor to begin with) and masslessness is thus the existence of a conserved current, a field of dimensions (2, 0). Currents provide a powerful tool to classify the fields of the theory, as we will see shortly. Plugging back our results into (31), one checks that the variation of the action δS vanishes exactly for a conformal transformation z → z + (z). This is the celebrated conformal invariance, which we will discuss in more details below. The short distance expansion of T with itself reads T (z)T (0) =

2 1 1 + 2 T (0) + ∂T (0) + analytic. 4 2z z z

(38)

The coefficient of the 1/2z 4 term is uniquely determined once the normalization of T has been chosen. For other massless relativistic field theories, this coefficient takes the value c/2 where c is a number known as the central charge. For a free boson we see that c = 1. For n independent free bosons, c = n. For a free Majorana fermion c = 12 . The relative normalization of the two other factors is fixed by the requirement that T (z)T (0) = T (0)T (z). Exercise: show this. The fact that T is analytic except when its argument coincides with the one of some other field inside a correlator has an interesting consequence for the trace ¯ (z)T (0)i is now a derivative of the delta of the stress energy tensor. Indeed, ∂hT function! Using the conservation equation ∂µ Tµν , which reads in complex compo¯ + 1 ∂(T11 + T22 ) = 0, if follows that the two point function of the trace nents, ∂T 4 of the stress energy tensor is non zero: h[T11 + T2 2](x1 , x2 )[T11 + T22 ](0, 0)i =

πc 2 ∆δ (x1 , x2 ). 3

(39)

This is a simple example of an anomaly, a quantity which is zero classically, but non zero quantum mechanically. It is a bit dangerous to give too much meaning to

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Topological Aspects of Low Dimensional Systems

(39) however, since the trace is not really an independent object – it is much safer to remember again that classical equations of motion hold except at coincident points.

Remarkably, the stress energy tensor was introduced in a statistical mechanics long ago by Kadanoff and Ceva [24]. These authors were interested in the way Ising correlators change under shear and scaling transformations. They recognized that, in a critical theory, rescaling in the x1 and x2 directions was equivalent to changing the horizontal and vertical couplings, and thus that the effect of shear and scaling could be taken into account by introducing an operator “conjugated” to these changes in the correlators, just like, say a change in temperature could be taken into account by x2

x2

x1

x1

w=x+iy

Fig. 4. Some of the geometries used in the text.

introducing the total energy in the correlators. It is thus possible to physically identify T , and to wonder how its continuum limit behaves, and how the various algebraic properties we are going to derive emerge. 1.4 Conformal in(co)variance To fix ideas, let us now consider the free bosonic theory on a cylinder of circumference T1 and length L. Introducing the complex coordinate w = x + iy such that the imaginary axis is parallel to the cylinder’s length (see Fig. 4), the two point function of the field Φ in that geometry is easily found to be 2 1 1 0 0 0 ln sin πT (w − w ) . ,w ¯ )icylinder = − (40) hΦ(w, w)Φ(w ¯ 4π πT From this, it follows that

E D 0 0 ¯ : : e−iβΦ(w ,w¯ ) : : eiβΦ(w,w)

= cylinder

πT | sin πT (w − w0 )|

β 2 /2π .

(41)

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Let us focus on the holomorphic part D :e

iβφ(w)

::e

−iβφ(w 0 )

E :

= cylinder

πT sin πT (w − w0 )

β 2 /4π .

This can be shown to be, equivalently, h 0 h E D dz dz iβφ(w) −iβφ(w 0 ) ::e : = :e dw w0 cylinder D E 0 × : eiβΦ(z) : : e−iβΦ(z ) :

(42)

,(43) plane

where we used the mapping z = e−2iπT w . The latter formula expresses the covariance of the two point function under the conformal transformation. Another example of such covariance is provided by the derivative of the field 0 dz dz 0 (44) h∂z φ(z)∂z0 φ(z 0 )i , h∂w φ(w)∂w0 φ(w )i = dw dw0 where we suppressed mention of the geometry, which is implicit in the variables used. Relations like (43,44) are well expected, since the Gaussian action is, in fact, conformal invariant; this follows, as discussed above, from the properties of the stress energy tensor, and thus is expected to generalize to other local massless field theories. More directly, this invariance is easily established for the free boson, since upon changing the argument of the field from z → w in the action, S is invariant, the Jacobian cancelling the term coming from the partial derivatives. Of course, one has to be quite careful in using the conformal invariance of the action, since the correlators are not invariant – i.e., one has for instance, 0 ,w ¯0 )i = 6 hΦ(z, z¯)Φ(z 0 , z¯0 )i, while the naive change of variables hΦ(w, w)Φ(w ¯ in the action would suggest the propagators to map straightforwardly, and thus the equality to hold. The reason for this discrepancy comes from the cut-off, which is also modified in a conformal transformation. We, on the other hand, wish to use the same regularization whatever the geometry, i.e., for instance, use a square lattice of constant a to regularize both the problem in the plane and on the cylinder; hence, there is an “anomaly”. Fields obeying the general covariance relation (and a similar one for the antiholomorphic part) h 0 h dz dz 0 hO(z)O(z 0 )i (45) hO(w)O(w )i = dw dw0 are called primary fields. The field φ itself is not primary, though, in a way, it satisfies the equivalent of the previous relation with h = 0, since dz dz 0 1 0 0 ln · (46) hφ(w)φ(w )i = hφ(z)φ(z )i + 8π dw dw0

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Topological Aspects of Low Dimensional Systems

Fields which are not primary exhibit in general more complicated covariance relations. An example is provided by the second derivative of φ, which we leave to the reader to work out. A more interesting example is furnished by the stress energy tensor. Though we have defined it so far by normal ordering, it is clear that an equally good definition is obtained by point splitting, i.e.14 1 T (z) = −2π lim ∂φ(z + d/2)∂φ(z − d/2) + · (47) d→0 4πd2 We have thus, from the change of variables T (z) = ×

−2π lim [w0 (z + d/2)w0 (z − d/2)∂w φ d→0 1 (w(z + d/2))∂φ(w(z − d/2)) − · 4πd2

To define the stress energy tensor on the cylinder, we use the same definition (47), but with z replaced by w15 . Therefore T (z) = [w0 (z)] T (w) + 2

1 {w, z}, 12

(48)

where the added term is 1 1 w0 (z + d/2)w0 (z − d/2) 1 {w, z} = lim − 2· d→0 2 [w(z + d/2) − w(z − d/2)]2 12 2d This is known under the name of Schwartzian derivative, and reads {w, z} =

2w000 w0 − 3w04 · 2w02

(49)

It enjoys nice properties under the composition of successive conformal transformations, that we leave to the reader to investigate. An important property following from (48) is that T acquires a finite expectation value on the cylinder, while it did not have one in the plane hT icylinder = −

c 2 (2πT ) . 24

(50)

Exercise: show this by using the propagators on the plane and the cylinder, together with appropriate definitions of normal ordering. 14 Here, it does not matter how and by what amount the two points are split of course, provided they both tend to z at the end. 15 That is, normal ordering is always defined by subtracting the short distance, geometry independent, divergences.

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The OPE of the stress energy tensor with any field of the theory has the general form T (z)O(0, 0) = . . . +

1 h O(0, 0) + ∂O(0, 0) + . . . z2 z

(51)

The two terms explicitely written follow from the fact that O has dimension h and the use of the Ward identity (57). It can be shown that 1/z 2 is the highest singularity if O is primary. Exercise: check this for the free boson by considering various examples. 1.5 Some remarks on Ward identities in QFT Suppose in general that there is a transformation of the field Φ0 (x1 , x2 ) = Φ(x1 , x2 ) +δΦ(x1 , x2 ) that leaves the product of the path integral measure and the Boltzmann weight e−S invariant. Examples of such transformations are provided for instance by translations or rotations in ordinary isotropic homogeneous physical systems. Consider then a transformation Φ0 (x1 , x2 ) = Φ(x1 , x2 )+ρ(x1 , x2 )δΦ(x1 , x2 ). For general ρ, this is not a symmetry of the problem anymore. On the other hand, we can always change variables in the functional integral and reevaluate any correlator in terms of the new field Φ0 . This means that we have the identity Z Z 0 0 = [dΦ0 ]e−S − [dΦ]e−S (52) where in S 0 one maybe had to add up terms coming from the change of variables in the path integral. On the other hand, one can expand the right hand side of this equation to first order in the change of fields assumed small. Since for ρ a constant the product of the measure and the weight would be invariant, this means that the right hand side of (52) must depend on the gradient of ρ only, ie one has Z Z i rhs = [dΦ]e−S jµ ∂µ ρdx1 dx2 . (53) 2π The quantity jµ is called a Noether current. Since it comes from local manipulations, it must be a local quantity. Now, that (53) vanishes is something that must hold for any reasonably smooth function ρ. Let us choose ρ to be equal to unity inside a disk of radius R1 , and to vanish on and outside of R2 , while it is arbitrary in between. Integrating (53) by parts, we get an integral of jµ on the circle R1 , together with an integral on the annulus between R1 and R2 of ∂µ jµ . Since the functions ρ is quite arbitrary there, it follows that the current has to be conserved, that is ∂µ jµ = 0. (54) This result would still hold of course with fields inserted far from R1 and R2 , so (54) truly holds as an operator equation, in the sense explained above. As an application, consider a translation xµ → xµ + µ , where µ is small and constant: we obtain a current which, in the classical case, coincides with ijµ = Tµν ν . The foregoing procedure is a generalization to the quantum theory, and the conservation equation follows from Noether’s theorem.

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Topological Aspects of Low Dimensional Systems

Now consider some field O inside the circles, say right at the origin. Under the transformation Φ → Φ0 , this field becomes O0 = O + δO (the change is expanded to first order as before, but of course O0 might as well depend on the derivatives of ρ in general). We now have, since all we are doing is changing variables in the path integral Z Z 0=

0

[dΦ0 ]O0 e−S . . . −

[dΦ]O e−S . . .

Expand this to first order. This time, the integration by part gives Z Z 1 1 −iδO = (dx2 j1 − dx1 j2 )O(0, 0) = (dzj − d¯ z¯ j)O(0, 0). 2π 2iπ

(55)

(56)

As an application, consider a transformation z → z + α(z). where α is small. The Noether current associated with it is given by j = iα(z)T (z) and ¯ j = iα(¯ ¯ z )T¯(¯ z ). For a transformation that is conformal inside a contour C, and differentiably connected to a (necessary non conformal) transformation vanishing at large distances, one finds from (56) the key result Z Z 1 1 δO = α(z)T (z)O(0)dz − α ¯ (¯ z )T¯ (¯ z )O(0)d¯ z. (57) 2iπ C 2iπ C Exercise: derive from this (51) and the fact that there are no singularities stronger than 1/z 2 for a primary operator. Notice finally that for the free boson, the expression of the stress energy tensor is almost the classical one, up to normal ordering, and it appears as if the integration measure essentially plays no role in the construction of the Ward identities. That one can forget about the behaviour of the measure in conformal transformations is justified a posteriori, by the fact that the quantum currents are indeed conserved. The measure would play a more subtle role for theories defined on curved two-dimensional manifolds.

1.6 The Virasoro algebra: Intuitive introduction As noticed before, the main consequence of conformal invariance is the existence of a conserved current, the stress energy tensor T . In general, one sets ∞ X Ln (58) T (z) = z n+2 n=−∞ that is, plugging this expansion into the OPE provides a definition of what the field Ln O actually is: for instance L0 O = hO, L−1 O = ∂O, L2 T = 1, etc. In general, one does not expect fields with negative dimensions to appear, or at least not fields with arbitrarily large negative dimensions (weird things can occur in non unitary theories adapted to disordered systems in particular though). This means that for every field, Ln O must vanish for n positive large enough. Of particular interest are the primary fields, for which the highest singularity is 1/z 2, i.e. they satisfy Ln O = 0, n > 0.

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495

C z = C ' - C '' z C C''

O

C C'

Fig. 5. The contour manipulations that lead to the definition of a commutator in radial quantization.

For the moment, we can contend ourselves with the intuitively reasonable notion that the Ln are “operators” acting on the space of fields of the theory – here, exponentials multiplied by normal ordered polynomials in derivatives of the field Φ - so the Ln are not unlike differential operators acting on functions (in fact, the (L−1 )n are just that). It is then tempting to ask oneself what the algebra of these operators is, that is how do Ln (Lm O) compare with Lm (Ln O)? This is easily done by using contour integration together with the short distance expansions. The commutator [Ln , Lm ]O can be computed as follows. We have Z Ln Lm O =

dz n+1 z T (z) 2iπ

C0

Z C

dw m+1 w T (w)O(0) 2iπ

(59)

where the contours encircle the origin and C is inside C 0 (see Fig. 5). Indeed, imagine writing the OPE of the integrand. First we expand T (w)O(0) to extract the field Lm O, on which the action of Ln is then obtained by the second integration. That the countour C is inside C 0 is natural from the point of view of “radial quantization” which, as we will see later, gives a precise operatorial definition to the Ln ’s. It is also necessary if one wishes to use the OPE in the order we just said for convergence reasons. The product Lm Ln O is computed in the same fashion with this time with a contour C 00 inside C: this forces one to expand first the product T (z)O, resulting in the opposite order for the operators. Comparing the two, and forgetting the operator O itself, we see that Z [Ln , Lm ] = C

dz 2iπ

Z Cz

dw n+1 m+1 z w T (z)T (w) 2iπ

(60)

496

Topological Aspects of Low Dimensional Systems

and a computation using the OPE of T with itself gives [Ln , Lm ] = (n − m)Ln+m +

c 3 (n − n)δn+m . 12

(61)

This is the celebrated Virasoro algebra, the Ln being called Virasoro generators. It is an infinite dimensional Lie algebra. Exercise: compute the action of the Virasoro generators of say derivatives of the field φ, and directly check the Virasoro commutation relations. A very important use of this algebra is to provide one with a natural structure to organize and recognize the fields in a theory. Of course, one does not quite need this powerful tool for the free boson, whose fields are easily built “by hand”, but for more complex theories, this is really very useful. Given a lattice model with microscopic variables, arbitrary combinations of neighbour variables can be built, whose scaling limit may or may not give rise to new scaling fields: which are truly new, which are nothing but “Ln ”’s (descendents) of others? What happens is that a theory has a certain number of primary fields, which is very often finite (eg, three for the Ising model), and all the other fields are just descendents of these ones. The whole set of fields is thus organized into products of representations of the left and right Virasoro algebras, for which the primary fields are heighest weight states. This can be expressed by the compact form X V irh ⊗ V irh¯ . (62) H= ¯ h,h

The situation is quite similar to the case of angular momentum in ordinary quantum mechanics, where the space of say the possible electronic states of some atom can be organized in terms of representations of the angular momentum algebra. Of course, here we have an algebra with an infinite number of generators, instead of three for angular momentum in three space dimensions. Qualitatively, there is an infinite number of Virasoro generators because there are an infinite number of elementary conformal transformations, one for each power of z: z n . As in the theory of angular momentum, unitarity contrains the quantum numbers, that is the values of conformal weights for a given central charge. This in turn gives rise to strong constraints for multipoint correlation functions; this is beyond the scope of these lectures, but not by far. In particular, correlations at strong coupling fixed points in the Kondo model can be computed just by using that technique [18]. The reason why we focused on the algebra of the Ln ’s is because of the special role of the stress energy tensor in conformal transformations. Of course, we could define other operators and other algebras associated with any field that has an integer dimension (so the contour integrals can be closed in the complex plane. Generalizations also occur for fields with

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497

non integer, rational dimensions, and a cut plane, but this is more complicated). A natural candidate in condensed matter is provided by currents, for instance ∂φ. Set therefore 1 X α ¯n 1 X αn , i∂¯φ¯ = √ · i∂φ = √ n+1 4π n z 4π n z¯n+1 From h∂φ(z)∂φ(w)i =

1 1 , 4π (z − w)2

(63)

(64)

it follows that [αn , αm ] = nδm+n .

(65)

Here we recognize the oscillator algebra standard in the quantization of the free boson, and maybe it is time to discuss more what the mode expansion has to do with Hamiltonian quantization. 1.7 Cylinders I shall mostly discuss what happens in the case of the cylinder. The key idea here is to remember that path integrals are scalar products of states. If we insert a field O at y = −∞ on the cylinder, this corresponds to having prepared the system in a state (an “in” state) |Oi, to which is associated, in the field representation a wave function ΨO [ΦS 1 ], result of a partial path integration Z [dΦΩ ]O(−∞) e−SΩ

ΨO [ΦS 1 ] =

(66)

ΦS 1 fixed

where the integral is taken over all configurations of the field in the bottom part of the cylinder Ω = (−∞, 0] × S 1 , the values at the boundary S 1 being held to ΦS 1 16 . Similarly if we insert a field O0 at ∞, this corresponds to projecting the system on an “out” state |O0 i, to which is associated a wave fucntion ΨO0 (ΦS 1 ). The scalar product of these two states is then Z (67) hO|O0 i = [dΦS 1 ]Ψ∗O0 (ΦS 1 )ΨO (ΦS 1 ), and this is essentially the correlation function of the two fields O, O0 . Of course, by translation invariance on the cylinder, this does not depend on the particular place where we have cut open the path integral. To make things concrete, let us discuss an example we will use explicitly later, with O = O0 = I, the identity operator – i.e., nothing is actually inserted at ±∞. In this case, (67) is just the partition function Z of the 16 I

am not being too careful here about what happens at −∞.

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Topological Aspects of Low Dimensional Systems

problem. To find the wave functions, let us split open the path integral at y = 0, and let us Fourier decompose X Φn eiωn x , (68) ΦS 1 (x) = n

where ωn = 2πnT . Introduce then the solution of the Laplace equation ∆Φ0 = 0 subject to the constraint Φ0 (x, y = 0) = Φ(x). One finds easily X Φn e−|ωn y| eiωn x . (69) Φ0 (x, y) = n

We can now split the field in the path integral into Φ = Φ0 + Φ0 where Φ0 vanishes at y = 0. Because of this, together with the fact that Φ0 solves Laplace equation, integration by parts shows that the path integral factorizes into the partition fucntions of two half cylinders with Dirichlet boundary conditions (we will get back to these later; the point here is that they are independent of ΦS 1 ), and an interesting term ! Z 1X 2 2 |ωn ||Φn | · (70) Z = ZD [dΦS 1 ] exp − T n Comparing with (67), it follows that, up to a phase " 2 # Z Φ(x) − Φ(x0 ) πT 2 ΨI [ΦS 1 ] ∝ exp − 2 S 1 sin πT (x − x0 )

(71)

where we Fourier transformed back the integrand in (70). We notice here as a side remark that the issue of finding wave functions is more than formal: the computation above was carried out for instance by people interested in finding the wave function of the Thirring model in terms of the original fermions [25]. In this point of view, we have a Hilbert space made up of states in one to one correspondence with the various fields of the theory. For operators other than the identity, one has to be a little bit careful. While the in state is always obtained by inserting O at −∞, the out state is obtained actually 2h −2iπT w¯ 2h¯ O at +∞. e by inserting e−2iπT w The same analysis can be carried out in the plane in the framework of “radial quantization”, where time is ln |z|. This is why the expansions we used earlier were called OPE. To be correct however, they do have a meaning as OPE’s only when the operators are radially ordered, since recall that, to an Euclidian Green function computed with a path integral, there corresponds a time ordered Green function in the quantum field theory. Of course, other Hamiltonian descriptions (for instance, the standard one where

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imaginary time runs say along x2 ) could be obtained by splitting open the path integrals differently. The remarkable thing now is, that the Hamiltonian on the cylinder has a very simple spectrum. Indeed, first observe that for primary operators, using the formula (43), the two point function on the cylinder decreases at large distance along the cylinder as ¯ h)(y − y 0 ) exp −2iπT (h − h)(x − x0 ) . hO(w)O(w0 )i ≈ exp −2πT (h + ¯ (72) For non primary operators, the same can be shown to hold, the more complicated terms in the covariance formula decreasing more quickly. On the other hand, suppose we want to describe the theory on the cylinder in a Hamiltonian formalism with imaginary time along the y axis. As is well known, the rate of decay of correlation functions is given by the gaps of the Hamiltonian, and the decay (72) indicates that there is an ¯ over the eigenstate of the Hamiltonian whose eigenenergy is 2πT (h + h) ground state. More generally, the computation of any physical property of the theory boils down to evaluating correlators, which all obey (72): therefore, the whole space of the quantum field theory must be organized in states associated with the various observables, such that their eigenenergy ¯ over the ground state! This is exactly what we expected from is 2πT (h + h) the Hamiltonian formalism described before, with one additional piece of information: the spectrum of H. In addition, it is important to stress that the stress tensor acquires a non vanishing expectation value on the cylinder, due to the schwartzian derivative (50). As a result, the Hamiltonian on the cylinder reads ¯0 − c (73) H = 2πT L0 + L 12 and the momentum ¯0 P = 2πT L0 − L

(74)

¯ ¯ 0 eigenvalues h. where L0 has eigenvalues h, L We can of course define the whole set of Virasoro generators on the cylinder by ! X c 2 −2iπnT w · (75) Ln e − T (w) = (2πT ) 12 n Now we have a precise meaning to give the Ln as operators, and their commutator can be computed, of course giving rise to the same Virasoro algebra derived more intuitively before. Notice the identity 1 H= 2π

Z

1/T

(T + T¯)dx. 0

(76)

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Topological Aspects of Low Dimensional Systems

This is independent of x, a result of analyticity. It should be clear that the whole conformal invariance analysis could be written within the Hamiltonian formulation. For instance, the OPE of T with itself corresponds to the commutator c 1 [T (x), T (x0 )] = δ(x − x0 )T 0 (x) − 2δ 0 (x − x0 )T (x) + δ 000 (x − x0 ), (77) 2iπ 6 and the OPE of T with a primary field i [T (x), O(x0 )] = δ(x − x0 )∂x O − δ 0 (x − x0 )hO. 2π

(78)

1.8 The free boson via Hamiltonians We now discuss the Hamiltonian formalism more specifically for the free boson. The Lagrangian is Z h i 1 1/T 2 2 dx (∂t Φ) − (∂x Φ) (79) L= 2 0 from which the momentum follows Π = ∂t Φ

(80)

and the standard Hamiltonian 1 H= 2

Z

1/T

h i 2 Π2 + (∂x Φ)

(81)

0

with the canonical equal time commutation relations [Φ(x, t), Π(x0 , t)] = iδ(x − x0 ).

(82)

The field is periodic in the space direction, that is Φ(x, t) ≡ Φ(x + 1/T, t). We chose to compactifiy the field on a circle of radius r, that is we identify Φ ≡ Φ + 2πr. The mode expansion of the field reads then (see e.g. [26] for many more details on this) i X 1 2iπT n(x−t) −α ¯ −n e2iπT n(x+t) , αn e Φ(x, t) = xˆ + T pˆt + 2πT rwx + √ n 4π n6=0

(83) R is the boson zero mode, while p ˆ = where x ˆ = T Φ(x, t)dx = Φ 0 R Π(x, t)dx = Π0 is the total momentum. w in an integer (the winding number); pˆ is quantized such that rpˆ = k is also an integer. The commutation relations of the operators are [ˆ x, pˆ] = i and αn , α ¯ m ] = nδn+m , [αn , α ¯ m ] = 0. [αn , αm ] = [¯

(84)

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501

¯ n are related with the usual creation and annihilation The operators αn , α operators of the free boson harmonic oscillators by √ √ (85) αn = −i nan , n > 0, αn = i −na†−n , n < 0 and

√ √ a−n , n > 0, α ¯ n = i −n¯ a†n , n < 0. α ¯ n = −i n¯

(86)

Note that if we go to euclidian space time, replacing t by −iy and then use the conformal coordinates z = e2πT (y−ix) (z = e−2iπT w , z¯ = e2πT (y+ix) ), we obtain the expansion pˆ wr pˆ wr + − ln z − i ln z¯ Φ(z, z¯) = Φ0 − i 4π 2 4π 2 i X1 αn z −n + α ¯ n z¯−n . (87) + √ n 4π n6=0

When the winding number is non zero, the field is not periodic around the origin; rather, a “vortex” is inserted there. When w = 0, we can set ¯ 0 = √pˆ4π , one recovers the expansion (63) for i∂φ. In general, we will α0 = α set √ √ pˆ pˆ ¯ 0 = √ − wr π. (88) α0 = √ + wr π, α 4π 4π The Hamiltonian (81) reads, before regularization   2 X 1 pˆ + (α−n αn + α ¯ −n α ¯ n ) . H = 2πT π(wr)2 + 4π 2

(89)

n6=0

A question that arises now is the relation between the normal ordering defined in the field theory and the normal ordering in the usual sense of ordering free bosonic operators in quadratic expressions: : αn αm := αinf(n,m) αsup(n,m) . The two might differ by a constant; in the present case actually, they coincide provided one uses zeta P regularization. Indeed, by ordering H, we encounter a divergent term n, which we can regularize by (for results on the zeta function, see [27]) ∞ X 1

n = ζ(−1) = −

1 · 12

(90)

With this prescription, the Hamiltonian with the vacuum energy divergence subtracted reads as it should (73), with the Virasoro generators Ln =

X 1X ¯n = 1 αn−m αm , L α ¯ n−m α ¯m 2 m 2 m

(91)

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Topological Aspects of Low Dimensional Systems

together with L0 =

∞ ∞ X 1 2 X ¯0 = 1 α α0 + ¯ 20 + α−n αn , L α ¯ −n α ¯n. 2 2 n=1 n=1

(92)

¯ n are annihilation operators for n > 0 and creation opThe modes αn and α erators for n < 0. The whole space of fields is thus obtained starting from highest weight states |w, ki which are annihilated by the annihilation operators and are eigenstates of the zero modes, and applying creation operators to them. Schematically, one has H=

X

Heisw,k ⊗ Heisw,k .

(93)

w,k

Of course, the |w, ki states are primary, and thus highest weight of the Virasoro algebra. Accordingly, one could as well build the whole space of fields by acting on them with L0n s. This would be more complicated; for instance, the field ∂φ, which is simply the result of α−1 |0, 0i, is not obtained from the action of L−1 on that state at all. This means in general that more primary fields are necessary than the |w, ki in the Virasoro description. 1.9 Modular invariance A convenient way of encoding the field content of the theory is to write the torus partition function, that is, the partition function when one imposes periodic boundary conditions in the imaginary time direction, too. One has, using (73) i h ¯0 − c · (94) Z = T r exp −2πT L L0 + L 12 Using the mode decomposition, one finds easily Z=

X 1 ¯ q hwk q¯hwk η(q)¯ η (¯ q)

(95)

w,k

where q = e−2πT L = q¯ (the notation allows consideration of more complicated parallelograms), 1

η(q) = q 24

∞ Y

(1 − q n )

(96)

n=1

and

hwk = 2π

k wr + 4πr 2

2 ¯ wk = 2π , h

k wr − 4πr 2

2 ·

(97)

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x2

R

x1

Fig. 6. The geometry for boundary conformal field theory.

An important property of this partition function is that it is modular invariant. What this means is, suppose one considers quantization of the free boson with time in the x instead of y direction. The radius being the same, this will lead to the same expression as (95) but with L and 1/T exchanged, that is X ¯ 1 h h (q 0 ) wk (¯ q 0 ) wk · (98) Z= η(q 0 )¯ η (¯ q0 ) w,k

where q 0 = e−2π/T L . The expressions (95) and (98) do turn out to be equal thanks to some elliptic functions identities (see next section). They ought to be, of course, since they represent the same physical object from two different points of view. For more sophisticated theories, the partition function cannot be computed a priori, but it is possible to determine it by imposing that it does not depend on the description, ie is modular invariant. See [22] and references therein for more details. 2

Conformal invariance analysis of quantum impurity fixed points

2.1 Boundary conformal field theory An excellent reference for this part is the original work of Cardy [28]. Consider now a field theory defined only on the half plane x2 > 0 (Fig. 6) – it might be for instance the continuum limit of a 2D statistical mechanics model which is at its critical point in the bulk, that is T = Tc , the usual critical temperature of the system. Various situations could occur at the

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Topological Aspects of Low Dimensional Systems

boundary depending on whether the coupling there is enhanced, or whether some quantum boundary degrees of freedom have been added. Consider, to fix ideas, the simplest case where the statistical mechanics model would have the same couplings in the bulk and the boundary (the so called “ordinary transition”). Intuitively, one expects the system to still be invariant under global rotations, dilations and translations that preserve the boundary, and that this invariance should be promoted to a local one, ie conformal invariance in the presence of the boundary. Physical fields are now characterized both by a bulk and a boundary anomalous dimension. If both fields are taken deep inside the system, they behave as in the bulk case. On the other hand, if they are near the boundary, one has, for example, hO(x1 , x2 )O(x01 , x02 )i ≈

1 , |x1 − x01 |2ds

|x1 − x01 | >> x2 , x02

(99)

i.e. the large distance behaviour of the correlators parallel to the surface is determined by the boundary dimension. We quote also the formula hO(x1 , x2 )O(x01 , x02 )i ≈

1 Rd+ds

(cos θ)

ds −d

.

(100)

A condition of boundary conformal invariance is that T12 = T21 = 0 when x2 = 0, which means physically that there is no flux of energy through the boundary. As a result, the left and right components of the stress tensor are not independent anymore, but T = T¯ for Imz = 0; this is expected, since the theory is invariant only under the transformations that preserve this boundary, that is satisfy w = w ¯ for Im z = 0. As a result however, one can define formally the stress tensor in the region Imz < 0 by setting T (z) = T¯ (z), Imz < 0.

(101)

Instead of having a half plane with left and right movers, we can thus equivalently describe the problem with only right movers on the full plane. For instance, the two point correlation function in the half plane is related with the four point correlation function in the full plane. Also radial quantization corresponds to propagating outwards from the origin in the upper half plane, with Hamiltonian (see Fig. 7) Z T (z)dz + cc. (102) C

Using the continuation (101), this becomes a closed contour integral of T only: thus, the Hilbert space of the theory with boundary is described by a sum of representations of a single Virasoro algebra this time: X V irh . (103) H= h

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x2

C

x1 C

Fig. 7. Geometry of the contours for the boundary case.

L The natural mapping in this problem is w = − iπ ln z, which maps the half 17 plane onto a strip of width L with the same boundary conditions on both sides18 . The Hamiltonian now reads c π L0 − · (104) H= L 24

Note that there are, roughly, two factors of two differing from the periodic Hamiltonian: the prefactor has a π instead of 2π, and there is a single Virasoro generator in the bracket. The space onto which the periodic Hamiltonian (73) acts is uniquely defined by the (bulk) theory one is dealing with, say the Ising model - as we discussed, this specification amounts to giving the various representations of V ir ⊗ V ir defining the model. For (104), the space depends on the boundary conditions; it is specified by a set of representations of a single Virasoro algebra. By a careful study of the two point function in the plane, together with the conformal transformation (where the jacobians still involve the bulk dimension), one can show that the gaps of H are given by the corresponding surface dimensions. 17 Notice that here I have put L in the mapping, instead of T . In the periodic case, L we could have used the mapping w = − 2iπ ln z to produce a similar result. This is all equivalent, but I prefer the present choice, where 1/T is always the periodic direction in the problem. 18 A strip with different boundary conditions on either side would correspond to a half plane with different boundary conditions x < 0 and x > 0, with a “boundary conditions changing operator” inserted right at x = 0.

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Topological Aspects of Low Dimensional Systems y

y

a

a

open

closed

L

b

b x

x

Fig. 8. The open and closed channel geometries when boundaries are present.

It is important to stress again that the same physical obervable will be associated with different representations of the Virasoro algebra in the bulk and boundary cases. For instance, the spin in the Ising model coresponds, in the bulk, to V ir1/16 ⊗ V ir 1/16 , while with free boundary conditions, it corresponds to V ir1/2 (with fixed boundary conditions, the spin is the same as the identiy operator). 2.2 Partition functions and boundary states To classify boundary conditions, it is extremely useful to deal with partition functions a bit. We consider thus a cylinder with a periodic direction of length 1/T and a non periodic one of length L: on either side, boundary conditions of type a, b have been imposed. We can describe the situation in two ways (see Fig. 8): either imaginary time runs in the direction parallel to the boundary (“open channel”), in which case we can write the partition function as (105) Z = T r e−Hab /T where Hab is the Hamiltonian (104) with boundary conditions a and b, or imaginary time can run in the direction perpendicular to the boundary (“closed channel”), in which case

(106) Z = Ba |e−LH |Bb where |Ba i , |Bb i are boundary states, and H is the periodic Hamiltonian (73). Observe that the the boundary states are not normalized: they are entirely determined, including their norm, by the condition that (106) gives the right partition function. To make things more concrete, fixed boundary conditions in the Ising model for instance Q are represented, in the micro= scopic Hilbert space, by the state |Bi fixed i |+i, while for free boundary Q conditions one has |Bifree = i (|+i + |−i).

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Here the boundary states are states in the Hilbert space of the bulk theory, ie in V ir ⊗ V ir. Conformal invariance at the boundary requires ¯ −n |Bi = 0. (107) Ln − L A solution to this equation is provided by so called Ishibashi states [30] X |h, ni ⊗ h, n (108) |hi = n

where |h, ni denotes an orthonormal basis of the representation V irh , and h, n the corresponding basis of V irh . In the case of the free boson, a boundary state will satisfy (107) if it satisfies a stronger constraint ¯−m ) |Bi = 0. (αn ± α

(109)

This in fact corresponds to Neumann and Dirichlet boundary conditions, for which T12 ∝ ∂1 Φ∂2 Φ = 0. The negative sign in (109) is solved by # " ∞ X α−n α ¯ −n |0, ki · (110) |Bi ∝ exp − n n=1 Therefore, we can build boundary states by # " ∞ X α−n α X ¯ −n |0, ki · ck exp − n n=1

(111)

k

The question of interest is to determine the coefficients ck . A quick way to proceed19 is to recognize here a Dirichlet state: indeed, suppose we act with Φ(x, t = 0) on the boundary state. Because of the condition (109), the oscillator part just does not contribute; what does contribute is only the x ˆ ∂ . Therefore, we have part, which acts as x ˆ = i ∂p |BD (Φ0 )i = ND

∞ X k=−∞

# ∞ X α−n α ¯−n |0, ki · exp − n n=1 "

e

−ikΦ0 /r

(112)

The last question, which is actually of key importance for what follows, is the determination of the overall factor N : in other words, what is the overall normalization of boundary states? This is where the consideration of partition functions is useful. 19 This topic goes back to the early days of open string theory. A nice recent paper on the subject is [31], where the following computations are carried out in many more details.

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Topological Aspects of Low Dimensional Systems

To answer this, we observe that, if we compute the partition function with height Φ0 on both sides, the identity representation should appear once and only once. On the other hand, the partition function is easily computed in the closed channel from the boundary states: one finds, for more general pair of values at the boundary ∞

1 X k2 /8πr2 ki(Φ0 −Φ00 )/r q˜ e (113) Z = BD (Φ0 )|e−LH |BD (Φ00 ) = ND2 η(˜ q) k=−∞

where q˜ = e−4πT L . We now perform a modular transformation to reexpress this partition function in terms of the other parameter q = e−π/LT . One has (the proof of this is a bit intricate. See e.g. [29], Chap. 3). 1 η(q) η(˜ q) = √ 2T L

(114)

and, by using Poisson resummation formula for the infinite sum, X n

1 X π exp −πan2 + bn = √ exp − a a k

one finds ZDD =

2 b k+ 2iπ

2 √ 0 1 N 2 X 2π π2r D q (Φ0 −Φ0 +2πnr) . η(q) n

(115)

(116)

This expression has a simple interpretation: one sums over all the sectors where the difference of heights between the two sides of the cylinder is Φ0 − Φ00 + 2πrn. For each such sector, the partition function is the product of a basic partition function corresponding to heights equal (without the 2πr identification) on both sides, times the exponential of a classical action. Φ −Φ0 +2πnr y, whose The latter is easily obtained: the classical field is Φ = 0 0L classical action is 1 0 2 (Φ0 − Φ0 + 2πnr) · exp − 2LT Consider now (116). We know that the partition function must write as a sum of characters (that is, T rV irh q L0 −c/24 , as follows from (103) and (104)) of the Virasoro algebra with integer coefficients; even though I will not spend time discussing what the characters at c = 1 are (q h /η for generic h), it is easy to see that this implies that the prefactor in (116) has to be an integer. Since we do not expect the normalization of the boundary states to change discontinuously with Φ0 , this integer is actually a constant, whatever Φ0 , Φ00 . We can in particular choose Φ0 = Φ00 , for which the identity representation

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V irh=0 appears in the spectrum; of course it should appear only once, and therefore 1 (117) ND = p √ · 2r π The other condition corresponds to Neumann boundary conditions, or, ˜ =Φ ˜ 0 . One equivalently, Dirichlet boundary conditions on the dual field Φ finds the boundary state # "∞ ∞ E X X α−n α ¯ ˜ −n ˜ 0 ) = NN |w, 0i · (118) e−2iπrwΦ0 exp BN (Φ n w=−∞ n=1 The Neumann Neumann partition function reads then ZNN =

2 1 1 X 2π ˜ ˜0 q (Φ0 −Φ 0 +n/r) , η(q) n

(119)

q √ 1 2r π. (120) 2 The Neumann Dirichlet partition function is actually independent of the ˜ 0 , since then the field cannot wind in any direction. values of Φ0 , Φ Exercise: show the following 1 X 14 (n−1/2)2 q . (121) ZND = 2η(q) n

and one has

NN =

The consideration of boundary states is extremely powerful to find out and study boundary fixed points. A general strategy is, knowing the Virasoro algebra symmetry of the model at hand, to try to find out combinations of Ishibashi states that are acceptable boundary states. Solving this problem involves rather complicated constraints. For instance, if one has several possible candidates |Bi i, the partition function with boundary conditions ij can easily be evaluated in the closed channel; after modular transformation to the open channel, it should expand as a sum of characters of the Virasoro algebra with integer coefficients. Another constraint is that the identity representation should appear at most once in all open channel partition functions. Clearly, this becomes a rather technical subject; more details can be found in the paper of Cardy [32]. Questions like the completeness of boundary states (i.e. whether all the boundary fixed points of a given bulk problem are known) are still open in most cases. 2.3 Boundary entropy Let us now suppose that we have a one dimensional quantum field theory defined on a segment of length L, with some boundary conditions at x = 0

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Topological Aspects of Low Dimensional Systems

and x = L. As is well known, the partition function at temperature T of this theory will be given by the same expression as the partition function of the two dimensional systems considered previously; notice however that I have changed conventions calling now x (resp. y) what was y (resp. x) previously (see Fig. 9)20 . y=it

a

b

x

Fig. 9. The geometry for defining boundary entropy.

In terms of the parameter q, this partition function is expressed as a h sum of terms qη with integer coefficients: the spectrum of Hab is discrete, and its ground state has integer degeneracy, nothing very exciting. In the limit L → ∞, the spectrum becomes gapless however, and one has to be more careful about the concept of degeneracy. If we take this limit, the free energy of the quantum field theory behaves as F = −Lf + fa + fb

(122)

where f is a free energy per unit length, fa , fb are boundary contributions. These contributions will involve, as T → 0, a boundary energy that is non universal, but also a boundary entropy. It is easy to see what this entropy will be by using a modular transformation. The same partition function q˜h expresses then as a sum of η(˜ q ) with some non integer coefficients that come from the Poisson resummation formula (in general, from the modular S matrix). In the large L limit, q˜ → 0. From the fact that F = −T ln Z

(123)

we see that, as T → 0, f = O(T 2 ) (the ground state energy of Hab is set to zero in this approach; for the exact dependence of f on T 2 see Sect. 6), while fa and fb are of the form fa = −T ln ga , fb = −T ln gb

(124)

20 This is to match as much as possible with the literature; in any case, there is no perfect notation that would be convenient all the way through.

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where [33]: ga = hBa |0i , gb = h0|Bb i ·

(125)

A one dimensional massless quantum field theory defined on a line with boundary conditions (or boundary degrees of freedom as we will see next) therefore has a non trivial zero temperature boundary entropy, or ground state degeneracy. Exercise: Show that the precise meaning of this degeneracy is related with the behaviour of the density of states r cn ga gb c 1/4 exp 2π (126) D(n) ≈ 2 6n3 6 where we parametrized the excitation energies of Hab by en = nπ L , n, L large (when computing the partition function and its logarithm, do not forget to integrate the fluctuations around the saddle point!). As we have seen in the previous subsection, some boundary conditions have a degeneracy g < 1, i.e. a negative boundary entropy. This is a bit shocking, but of course we should remember, first, that g is more a prefactor in an asymptotic formula for degeneracies (126) than a true ground state degeneracy (at L = ∞, there is no gap), and second, that we are dealing with quantum field theories and that this is only a finite, properly regularized “entropy”. The same remark applies, somehow, to g being non integer. However, it is perfectly possible to have non integer degeneracies for semiclassical systems involving kinks [34]. Intermezzo: Perturbation near the fixed points A scale-invariant boundary condition is a RG fixed point (recall that the bulk is always critical in the type of systems that we are considering here). As with any RG fixed point, there is a set of relevant/marginal/irrelevant boundary operators (and couplings) associated with each scale-invariant boundary condition. These operators have support only at the boundary, i.e. at one point in position space (at the position of the impurity). If no relevant boundary operators are allowed, then the scale invariant boundary condition represents a stable fixed point (the zero temperature fixed point, describing the Kondo model at strong coupling, is an example; so is the Dirichlet fixed point in the tunneling problem, to which we will get back soon). Irrelevant boundary operators give perturbatively calculable corrections to physical properties evaluated at the RG fixed point. Many important physical features of the Kondo model are actually due to the effect of the leading (dominant) irrelevant boundary operator [18]. Adding a relevant boundary operator to the Hamiltonian describing a particular scale-invariant boundary condition, destroys that boundary condition, and causes crossover to a new, scale-invariant boundary condition at

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Topological Aspects of Low Dimensional Systems

large distances and low temperatures (in the infrared). In other words, we have a (boundary) RG flow, describing the crossover from the initial scaleinvariant boundary condition (in the ultraviolet, i.e. at short distances or high temperatures) to a new scale invariant boundary condition (in the infrared, i.e. at large distances and low temperature). Note that at every stage of this flow, the bulk remains always critical and unchanged; the only action is at the boundary. An interesting observation concerning general boundary RG flows was made in [33]: the zerotemperature boundary entropies (s = ln g in the previous section) generally obey (decrease of boundary entropy). sUV > sIR , This may be viewed as a boundary analogue of the well known c-theorem of bulk conformal field theory [35]. (Note, however, that the universal numbers sUV and sIR do not seem to be obviously related to a dynamical quantity, in contrast with the central charge, which is related to the stress tensor of CFT). A well known example is the one-channel Kondo model. Initially, at weak coupling (at high temperature, in the ultraviolet), we have a quantum mechanical spin decoupled from the electron degrees of freedom of the metal. An isolated (s = 1/2) spin has a zero-temperature entropy of sUV = ln 2. At strong coupling (at low temperature, in the infrared), this impurity spin is completely screened by the conduction electrons. This means that no dynamical degrees of freedom are left, and thus we have sIR = 0. 3

The boundary sine-Gordon model: General results

3.1 The model and the flow We consider now the model we had decided to tackle in the introduction Z Z Z h i β 1 0 2 2 (127) dx dy (∂x Φ) + (∂y Φ) + λ dy cos Φ(0, y). S= 2 −∞ 2 This model is called the boundary sine-Gordon model since it has a sineGordon type interaction, but at the boundary. In more general terms than those of the edge states tunneling, the physics of this model is rather clear. The limits λ = 0 and λ = ∞ are fixed points, corresponding to conformal invariant boundary conditions, respectively of Neumann and Dirichlet types. Away from these limits, the model is not scale invariant because of the boundary interaction. In the vicinity of λ = 0, the RG equation is dλ = (1 − g) λ + O(λ3 ), db 2

(128)

where we have set g = ν = β8π . It is natural to expect that λ flows all the way from 0 to ∞ under renormalization. Equivalently, the boundary conditions

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look like Neumann at very high energy (UV) but like Dirichlet at low energy (IR) – the dimension of the physical coupling is [λ] = Lg−1 , so the typical 1 energy scale for the cross over between UV and IR behaviours is TB ∝ λ g−1 . Equivalently also, the field Φ feels Neumann boundary conditions close to the boundary, but feels Dirichlet boundary conditions instead far from it, with a cross over distance 1/TB . Notice that the boundary entropies of the UV and IR fixed points are different. To compute them, we can use the results of the previous section after having identified the radius of the boson. In the shift Φ → Φ + 2πr, the interaction cos β2 Φ must be unchanged, which requires r= It follows that

gN =

β2 4π

2 · β

−1/4

Notice the ratio gN = gD

(129)

, gD =

β2 8π

β2 16π

1/4 ·

(130)

−1/2 ·

(131)

For the case of a relevant perturbation we are considering here, this ratio is larger than one: the boundary entropy is greater in the UV than it is in the IR. This is in agreement with the intuitive idea that degrees of freedom disappear under the renormalization group, leading to a loss of information. There is a well known conjecture stating that for any allowed flow in a unitary system (that is, roughly, a system with real, local Hamiltonian), gUV > gIR . For the case of irrelevant perturbation, one finds gN < gD , so according to this the flow should not be possible, which is indeed the case: since the operator is irrelevant, it does not generate any flow, and one should observe N boundary conditions both at small and large distance. 3.2 Perturbation near the UV fixed point The first question we will be interested in is the calculation of the boundary free energy at any temperature T and coupling λ. This can be represented by a Coulomb gas expansion as follows. First, by using a conformal mapping, one finds the two point function of the free boson with Neumann boundary conditions on the half cylinder sin πT (y − y 0 ) 0 · (132) hΦ(y)Φ(y )i = −g ln πT Exercise: derive this, by first computing the two point function on the half plane.

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Topological Aspects of Low Dimensional Systems

We can then evaluate the ratio of partition functions with and without boundary interaction as follows Z 1/T ∞ X 1 Z(λ) 2n = 1+ λ dy1 . . . dy2n Z(λ = 0) (2n)! 0 n=0 β β × cos Φ(y1 ) . . . cos Φ(y2n ) · (133) 2 2 Of course, only electrically neutral configurations with n positive and n negative charges contribute. After some rescaling, one finds ∞ X Z(λ) ˜ 2n I2n , =1+ (λ) Z(λ = 0) n=1

(134)

where the dimensionless coupling is ˜ = λ (2πT )g , λ 2T

(135)

and the integrals are I2n

1 = (n!)2

Z

Z

2π

du1 . . . 0

0

2π

Q u0 −u0 j i<j 4 sin ui −u sin i 2 j 2 0 du2n Q ui −u0j i,j 2 sin 2

2g ·

(136)

This is the partition function of a classical Coulomb gas in two space dimensions, with the charges moving on a circle of unit radius (see Fig. 10). The integrands have small distance behaviours 1/u2g . It follows that there is no short distance divergence, and the integrals are all finite for g < 12 (there are never large distance divergences here since we have a temperature). When g > 12 , the integrals have divergences. In the sequel, I will always regularize integrals dimensionally, not by introducing a cut-off. To explain what this means, consider the case n = 1, which can be done by elementary computations I2 =

Γ(1 − 2g) · Γ2 (1 − g)

(137)

This can then be continued beyond g = 12 simply by using the known continuation of Γ to negative arguments. How to do this in the case of arbitrary n is a bit more tricky. A way to do it relies on the remarkable fact that the integrals I2n can be expressed in an almost closed form by appealing to techniques of Jack polynomials [36, 37]. I will only give the result here 2 n XY Γ[mi + g(n − i + 1)] 1 (138) I2n = [Γ(g)]2n m i=1 Γ[mi + g(n − i) + 1]

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d

u2 u1

Fig. 10. Charges of a two dimensional Coulomb gas that move on a circle 1 d = 2 sin u2 −u . 2

where the sum is over all sets (Young tableaux) m = (m1 , . . . , mn ) with integers mi obeying m1 ≥ m2 . . . ≥ mn ≥ 0. This expression can be used to compute the I2n numerically to high values of n, or, more fundamentally, to perform the analytic continuation in g. I will not discuss this further, and get back now to the physics of this model. For g < 12 at least, the perturbative expansion is well defined, giving a series in λ with positive coefficients. This series will presumably have a finite radius of convergence – although one does not expect the appearance of a singularity on the positive real axis (this would correspond to the existence of a phase transition on the one dimensional boundary). Beyond this radius, some other technique has to be used to understand quantitatively what happens. It is possible to argue what the leading behaviour of the partition function at large λ should be. Indeed, Z actually depends only on the ratio of the two energy scales TB and T , so large λ is like small temperature. But small temperature corresponds, going back to an euclidean description, to a cylinder of large diameter. In this limit, the partition function per unit length of the boundary should have a well defined, “thermodynamic” limit, so Z should go as Z ∝ exp TTB . This means our perturbative series has to ˜ 1/1−g ). go as exp(cst λ 3.3 Perturbation near the IR fixed point A natural idea to find out what happens beyond the radius of convergence is to think of the problem from a “dual” point of view, ie around the

516

Topological Aspects of Low Dimensional Systems

λ = ∞ infra red fixed point. The first question one may ask is along which irrelevant operator this fixed point is approached. There are several, equally interesting ways to answer this question. The first one starts by considering the model where the bulk degrees of freedom have been integrated out, leading to the action (at zero temperature, which makes the formulas a bit simpler) S=

1 2π

Z

dydy 0

Φ(y) − Φ(y 0 ) y − y0

2

Z +λ

dy cos

m β Φ(y) + 2 2

Z dy (∂y Φ)2

(139) where we have added an irrelevant mass term to make some integrals finite. We can find kinks interpolating between adjacent vacua and satisfying the equations of motion [38] m∂y2 Φ = −

β λβ sin Φ. 2 2

A simple solution of this equation is indeed " 8 2π −1 + tan exp Φ ≡ fins (y) = β β

(140)

β 2

r

λ y m

!# ·

(141)

The energy of this kink is infinite, but can be made finite by subtracting a constant term from the action, replacing the cos β2 Φ by cos β2 Φ − 1. If we then consider a configuration of the field Φ made of a superposition of far apart instantons and anti-instantons, X (142) Φ= j fins (y − yj ) the kinetic term of the action can be conveniently evaluated by Fourier transform Z dω (143) Skin = |Φ(ω)|2 |ω| 2π At large distances, one finds Skin ≈

16π X j k ln |yj − yk |. β2

(144)

j
This is in exact correspondence with the Coulomb gas expansion discussed β2 → 8π previously, but with the exchange 8π β 2 . It follows that the IR action reads, at leading order Z Z Z 2 2 4π ˜ 1 0 ˜ ˜ y), (145) dx dy ∂x Φ − ∂y Φ + λd dy cos Φ(0, S≈ 2 −∞ β

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˜ is the dual of the boson Φ. It follows that the IR where we recall that Φ 1 fixed point is approached along an operator of dimension h = 8π β 2 = g . One 1

also checks that λd ∝ λ− g . It is important to stress now that, while the flow away from the UV fixed point is fully specified by a single perturbing term, the situation is very different for the approach towards the IR fixed point. Of course, one is free if one wishes to perturb the D boundary conditions by a single irrelevant operator as represented in (145), though of course one has to be especially careful in defining the theory because of the strong short distance divergences in the integrals. The point is, that there is only one particular way of approaching the IR fixed point that corresponds to the trajectory originating at our UV fixed point. This means that the large λ behaviour of the series we are interested in would be computable from the knowledge of an action of the form Z Z 2 2 1 0 ˜ ˜ dx dy ∂x Φ + ∂y Φ S = 2 −∞ Z X hk −1 4π ˜ Φ(0, y) + λ g−1 Ok , (146) + λd dy cos β k

where Ok belong to a very large class of operators allowed by symmetry: n ˜ ˜ 2 , and many others. Since all Φ, (∂ Φ) there are for instance all the cos 2β these operators come with appropriately scaled powers of the coupling constant, they all give contributions to physical properties that depend on our ˜ and no operator can be discarded (let me stress single scaling variable λ, that an expansion such as (146) does not make much sense until one specifies the regularization procedure employed). Here of course the reader should ask: but why didn’t we add that collection of operators near the UV fixed point as well? The point is that we had control of what we wanted to do near the UV fixed point, and only a maniac would want to use such an irrealistically finely tuned combination of operators to perturb a fixed point. However we have no control about the way the IR fixed point is approached: this is entirely determined by the dynamics of the quantum field theory, and it turns out to be quite complicated. It is important in particular to realize that, starting from (146) and trying to go against the renormalization group flow, there is, most probably, only one choice of IR perturbation that would get back to our UV fixed point. It seems very hard to push the instanton expansion beyond the first non trivial order to try to get at (146), or get the expansion of physical quantities for large λ: this has for a long time made IR perturbation theory impossible to carry out beyond the first trivial order. Remarkably however (146) can be entirely determined using ideas of integrability. In a scheme where everything is dimensionally regularized,

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˜ the only vertex operator that is present near the IR fixed point is cos 8π β Φ: none of the other harmonics actually appear! There are also very strong constraints on the other operators. In any case, the non-perturbative region of large λ is very hard to access quantitatively using perturbation of the IR fixed point. Fortunately, the problem can be tackled by using ideas of integrability, a topic to which we will turn soon. 3.4 An alternative to the instanton expansion: The conformal invariance analysis Clearly, the instanton expansion, if physically appealing, is a bit laborious, especially when one considers how little information it finally provides. The conformal invariance analysis gives an alternative way, usually more reliable, to know which operators are present near the IR fixed point. Indeed, this information is encoded in the partition function ZDD (116) for Φ0 = Φ00 : the 2 modes in the sum correspond to operators with dimension h = 8πn β 2 , the cosines identified previously, while the other terms obtained by expanding the eta function correspond to powers of derivatives of the field. Part II: Integrability and the complete flow The constraint of integrability has been used with much success to study crossover scaling in bulk 2D theories exactly. However, it is often objected that integrable models are not so relevant for experimentally observable physics for at least two reasons: (a) In order to achieve integrability, extensive fine-tuning of parameters is often required. Therefore, it is often believed that exact predictions made by studying an exactly integrable model might often not be generic and therefore difficult to observe experimentally. For example integrable spin chains with spin greater than 1/2 are gapless, while the generic even-spin spin chain and those observed have gaps in the spectrum. (b) A very important set of experimentally accessible observables are transport properties. Amongst those is the conductance which is usually computed from (equilibrium) Green’s functions using the Kubo-formula. It is usually very difficult or impossible to compute exact Green’s functions, even when the system is (Bethe-Ansatz) integrable. Therefore, before the progress made in the last couple of years, integrability was largely restricted to the computation of thermodynamic quantities, excluding transport properties at finite temperature. In fact, the situation is quite different for integrable quantum impurity problems. For these problems, exact transport properties can be computed (even

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out of equilibrium) and integrability can answer directly experimentally important questions. In particular: (i) In order to achieve integrability in quantum impurity problems, one often needs to adjust very few parameters – sometimes none! For instance, both the Kondo effect and the point contacts in fractional quantum Hall Effect devices, provide an experimental realization of an integrable system without any fine-tuning. Similarly, while integrable higher spin quantum spin chains are non generic, the higher spin Kondo problem, is integrable. (ii) Exact transport properties (at non-zero temperature) can be computed from integrability [39]. The linear response conductance for the quantum Hall point contact, for instance, agrees quantitatively with recent experiments by Milliken/Webb/Umbach [13]: this is the quantity we will discuss in what follows. Many other properties can also be computed, and I will discuss them briefly at the end. The method to obtain those exact results is a bit unconventional: it relies crucially on a judicious choice of basis of the Hilbert space of the system. We use a basis that is natural from the point of view of integrability. It is simply the basis in which all the infinite number of conservation laws (that exist since we have an integrable system) are diagonal. This basis turns out to have a “Fock-space” like structure, i.e. it is spanned by “quasiparticles”. It is in this basis that the quantum impurity interaction becomes tractable. In order to compute transport properties, we use a kinetic equation for those quasiparticles of the Bethe-Ansatz. This is non-trivial, since we are really describing a fully interacting system, where a single-particle (Fermi-liquid) concept such as a kinetic equation seems out of place, at first sight. The single particle kinetic equation would fail to produce exact results in interacting systems due to the existence of particle production processes in the single-particle basis. However, the quasiparticle basis dictated to us by integrability is precisely characterized by the absence of quasiparticle production (and “factorized scattering”). This particular and special feature of an integrable theory, allows us to use a kinetic equation to compute transport properties exactly from integrability. A historical note is necessary here. The point of view I will use is different from the original works on integrable quantum impurity problems. In the latter works, the authors started from a “bare” theory, and proved by hand that there were simple eigenstates obtained by making a two body scattering ansatz, the Bethe ansatz (there won’t be much about the Bethe ansatz per se in these lectures). They then proceeded to build the “physical” theory by filling up the ground state, and studying excitations above it. Nowadays, it has become customary to start directly with the physical theory, and prove its integrability using a very different and powerful tool that I will introduce below, perturbed conformal theory. The spectrum of excitations and the

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S matrix are then deduced (some would say guessed) by using symmetry arguments; in particular by analyzing non local currents (for this aspect, see e.g. [40]). The approach gives rise somehow naturally to the computation of transport properties, in particular by making physical sense of massless scattering, and that has definitely been a progress. Another key advantage is that, for a given “physical” theory, there are many possible “bare” choices, that is many different possible regularizations. Usually only one of them is integrable, and not always the obvious one; for instance, Zamolodchikov showed in his pioneering work that the Ising model at Tc with a magnetic field is an integrable quantum field theory, but it is well known that its standard regularized square lattice version is not integrable (in the case we are interested in, the boundary sine-Gordon model, I am actually not aware of any simple integrable bare Hamiltonian). All this is not to diminish the beauty and astonishing insight of the pioneering works about quantum impurity problems [7, 8], nor the huge body of work on the Bethe ansatz and Yang Baxter equation that it is impossible to even start to acknowledge here. 4

Search for integrability: Classical analysis

As emphasized in the first sections, one of the main uses of conformal invariance is to provide a convenient basis to the Hilbert space of observables in terms of representations of the infinite dimensional symmetry. For the free boson, the basis furnished by irreducible representations of the Virasoro algebra is just one of many choices: the basis furnished instead by representations of the Heisenberg algebra is also possible, and sometimes more convenient. When one wishes to study the problem with a boundary interaction, the question arises, of which basis will be the most convenient to work. It turns out it is still a third choice, provided by a “massless scattering theory”. To understand what this means, it is good to first consider the classical case. In the classical limit, one can scale the parameter β off the action. Going to real time, one obtains a classical scalar field Φ(x, t) satisfying the KleinGordon equation in the bulk x ∈ [−∞, 0): ∂t2 Φ − ∂x2 Φ = 0 together with the boundary conditions (where λ is also rescaled): 1 Φ . ∂x Φ|x=0 = λ sin 2 x=0

(147)

(148)

Now, the point is that the “most natural” basis of solutions for the bulk problem, that is plane waves, behaves badly with respect to the boundary

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interaction: if a plane wave is sent towards the boundary, what bounces back is a complicated superposition. Is it possible to find a better basis made of wave packets that will bounce nicely? To find such a basis, we make a detour through a more complicated problem which everybody knows is integrable, the massive sine-Gordon model. That is, we want to think of our Klein-Gordon problem as the Λ → 0 limit of the sine-Gordon equation: ∂t2 Φ − ∂x2 Φ = −Λ sin(Φ).

(149)

It turns out that this massive model, in the presence of the boundary interaction, is still integrable. We will discuss this point in more details below, and start instead by considering how things look like in the massless limit. There are two types of finite-energy solutions of the classical sine-Gordon equation: solitons, which are time-independent and topologically nontrivial, and breathers, which are time-dependent and topologically trivial. Intuitively, a breather can be thought of as a bound state of a kink and an antikink oscillating in and out (i.e. breathing). Here, we will discuss only the solitons; the analysis for the breathers follows analogously. A major triumph of the theory of non-linear partial differential equations was the construction of explicit solutions of (149) for any number of moving solitons. The solitons’ energies and momenta are conveniently expressed in terms of rapidities αj , defined by Ej = M cosh αj and Pj = M sinh αj , M = Λ1/2 . The velocity of each is thus given by tanh αj (positive for a right-moving soliton). We have set the speed of “light” to be 1. Consider now a two-soliton solution of (149) on (−∞, ∞). This solution is usually expressed as: Im(τ ) (150) Φ(x, t) = 4 arg(τ ) ≡ 4 arctan Re(τ ) where the τ -function solution is given by: 2 −E (x−a)−E (x−b)+P t+P t 2 2 1 2 e 1 τ = 1− 1 2 tanh α1 −α 2 −E (x−a)+P t 1 + i 1 e 1 + 2 e−E2 (x−b)+P2 t .

(151)

The constants a and b represent the initial positions of the two solitons, and j = +1 if the j th soliton is a kink, while j = −1 if it is an anti-kink. What happens if we try to take the massless limit of this solution? For a wavepacket to have finite energy in the massless limit m → 0, the rapidity |α| must go to infinity. We thus define α ≡ A + θ, and let A → ∞ such that the parameter m ≡ 12 M eA remains finite. The energy and momentum of a right-moving “massless” soliton then reads E = P = meθ .

(152)

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For a left mover, α ≡ −A + θ, and its energy and momentum read E = −P = me−θ .

(153)

Suppose that both of these solitons are right-moving. Then the massless limit yields: h i τ = 1 − 1 2 e−∆ e−E1 (η−a)−E2 (η−b) + i 1 e−E1 (η−a) + 2 e−E2 (η−b) (154) where η = (x − t) and ∆ ≡ − log (tanh(θ1 − θ2 ))2 . This leads to an a priori strangely complicated solution of the Klein-Gordon equation. Observe that: h io n arg 1 − 1 2 e−E1 (η−a)−E2 (η−b) + i 1 e−E1 (η−a) + 2 e−E2 (η−b) i h i h = arg 1 + i1 e−E1 (η−a) + arg 1 + i2 e−E2 (η−b) h i h i (155) = tan−1 1 e−E1 (η−a) + tan−1 2 e−E2 (η−b) . This is easily checked to be the sum of two one-soliton solutions; the factor ∆ thus measures the extent to which the two-soliton solution is not a superposition of one-soliton solutions. More precisely, consider the limit a → ∞, η → ∞ so that E1 (η − a) is finite. This corresponds to moving the first kink off to x = +∞ and following it. The τ function collapses to the one-kink form τ = 1+i1 e−E1 (η−a) . Moving this soliton through the second one corresponds to taking it to x = −∞, or taking the limit a → −∞, η → −∞ (with E1 (η − a) finite). Discarding an overall multiplicative factor (which is irrelevant in the computation of φ = 4 arg(τ )), we see that in this limit, τ = 1 + i1 e−E1 (η−a)−∆ . Thus these preferred Klein-Gordon wave packets exhibit non-trivial monodromy. The foregoing time delay ∆ is precisely the classical form of a massless scattering matrix. One obtains the same ∆ for two left-moving solitons. For a left-moving and a right-moving soliton colliding one easily sees that the massless limit 2 of tanh 12 (α1 − α2 ) is unity. The solution collapses to the superposition of a left-moving wave packet and a right-moving wave packet exactly as in (155), with no time delay. This is the classical manifestation of the fact that the left-right quantum scattering matrix SLR elements are at most rapidity-independent phase shifts. Consider now the Klein-Gordon equation on [−∞, 0] with the boundary condition (148). Here is a direct way of seeing the integrability of the Klein-Gordon (and indeed, sine-Gordon) equation with boundary conditions

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(148). The idea is to show that the method of images can be used on (−∞, ∞) even in the non-linear system, so as to replicate the boundary conditions (148) on [−∞, 0]. The scattering of a kink, or anti-kink, from the boundary can be described by a three-soliton solution on (−∞, ∞). These three solitons consist of the incoming soliton, its mirror image with equal but opposite velocity, and a stationary soliton at the origin, to adjust the boundary conditions (see [42] for more details). If one takes the infinite rapidity limit of this three-soliton solution then the stationary soliton simply collapses to an overall shift of φ by a constant, while the mirror images (since they are moving in opposite directions) reduce to a superposition of two wave packets. One thus obtains: Φ = Φ0 + 4 arg 1 + i1 e−E(ξ−a) + 4 arg 1 + i2 e−E(η−b)

(156)

where ξ = x + t, η = x − t. By direct computation one finds that this solution satifies (148) with: eE(a+b) = −1 2

(2E + λ) · (2E − λ)

(157)

The constant ∆B ≡ −E(a + b) represents the delay of the reflected pulse. If one defines the classical boundary scale θB via g = 2me−θB , then this delay may be written as

1 ∆B = log −1 2 tanh (θ − θB ) · 2

(158)

Note that the sign 2 is to be chosen so as to make the argument of the logarithm real. This determines whether the reflection of a kink will be a kink or an anti-kink. Thus we see that θB is the scale at which behavior crosses over from the region of the Neumann critical point (where the classical boundary scattering is completely off-diagonal) to the Dirichlet boundary critical point (where classical boundary scattering is diagonal). The conclusion of this section is clear: there are classical wave packets that scatter very nicely at the boundary. The price to pay is that they are considerably more complex than plane waves, and “weakly interacting” – that is, they scatter non trivially through one another. The natural attitude after having established such results classically is to see whether they are preserved quantum mechanically. One can for instance establish integrability order by order in a loop expansion (which here amounts to an expansion in powers of β). Here I want to show a more direct way to proceed, that generalizes to all sorts of theories.

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Topological Aspects of Low Dimensional Systems

Quantum integrability

5.1 Conformal perturbation theory As in the previous section, we start by considering theories with a bulk interaction. All what follows is based on the very insightful work of Zamolodchikov [41]. Consider therefore the usual sine-Gordon model with action21 Z Z i h 1 2 2 S= (159) dxdy (∂x Φ) + (∂y Φ) + Λ dxdy cos βΦ. 2 As in the classical case, quantum integrability is established by proving the existence of non trivial integrals of motion. Rather than doing perturbation around β = 0, what one can do instead is perturbation around the conformal limit. This requires first the understanding that a conformal theory is integrable22 , a rather straightforward property. Indeed, let us try to build a set of conserved quantities for a quantum field theory. We consider Euclidian space with imaginary time in the y-direction: a quantity will thus be conserved if its integral along two horizontal contours at different values of y gives the same result. Using complex coordinates, this will occur if we have a pair of quantities, say Tn and Θn ¯ n = ∂Θn . Right at the conformal point, by analyticity, T , all such that ∂T its derivatives and (regularized) powers, do provide conserved quantities. To clarify this a little, let us consider the classical case. Going to imaginary time and complex coordinates, the equation of motion is ¯ = ∂ ∂Φ

Λ sin Φ. 4

(160)

Exercise: Show that the first pairs leading to conserved quantities are 2

T2

=

(∂Φ) ,

T4

=

∂2Φ

2

Θ2 = − −

Λ cos Φ 2

1 Λ 4 2 (∂Φ) , Θ4 = (∂Φ) cos Φ, 4 4

(161)

¯ 2n = ∂Θ2n holds. Go back to real time, and find out that is, the relation ∂T which quantity is, indeed, conserved by time evolution. 2

is safer to assume here that β < 12 , so no counter term is necessary to define 8π the perturbed action. Only a finite number of such counter terms would be required in general anyway, because of the “super-renormalizability” common to most perturbed conformal field theories. 22 At least partly integrable – this subtlety does not seem to matter for most problem occuring in condensed matter. 21 It

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Away from the conformal point, what will happen is that there will sometimes be a deformation of (some of) these quantities that is still conserved. To see that, let us start by looking at the stress energy tensor, and ¯ becomes with a perturbation. To make sense of this question, see what ∂T we have to insert T inside a correlator, as usual. The difference with the conformal case is that now the action reads, quite generally Z (162) S = Scft + Λ dxdyO so we have to expand the Boltzmann weight in powers of Λ. This gives an infinity of terms, each of which has now a Boltzmann weight with a pure Scft , so the results right at the conformal point can be used for them. But then it seems that T being analytic at the conformal point, nothing will make it non analytic away from it! That is not true because of what happens at coincident points. The integral of the perturbing field will affect only the behaviour near z, so we can use the OPE of T with the perturbation T (z)O(z 0 , z¯0 ) =

hO(z 0 , z¯0 ) ∂O(z 0 , z¯0 ) + + ... (z − z 0 )2 z − z0

(163)

Now, using the identity (16) it follows that ¯ = π(1 − h)Λ∂O. ∂T

(164)

Hence, for T2 = T , Θ2 = −πΛ(1 − h)O, and we have a conserved quantity - to first order in Λ that is. Before wondering about higher orders, let us stress what ensured the existence of a conserved quantity: the fact that the residue of the simple pole of the OPE of T with the perturbation was a total derivative. One can then try to see whether there are other quantities for which a similar thing holds. Let us consider therefore the combination 2 2 T4 = 4π 2 : (∂φ) : +A : ∂ 2 φ :. After laborious computation, one finds that the residue of the simple pole, in the sine-Gordon case of interest, is β3 Aiβ 3 +i : ∂ 3 φeiβφ : −3β 2 : ∂φ∂ 2 φeiβφ : −4iπβ : (∂φ) eiβφ : 2π 8π which is a total derivative when (notice this would still hold with β subsituted with 8π β ) 8π β2 − 2 · A = 2π 3 − 8π β The same sort of argument can be built to show that a conserved quantity can be obtained for every even n, which is the integral of a local field of dimension n.

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We now have to discuss what happens beyond first order. Suppose we carry out the computation to order n; a priori, we expect the result to be something like ¯ 4 = ∂Θ4 + O(Λn ) (165) ∂T where Θ4 is of order one in Λ. The left hand side has dimensions (4, 1), 2 2 while Λn has dimensions (n(1 − β8π ), n(1 − β8π )). This means that a local 2

2

field with dimensions (4 − n + n β8π , 1 − n + n β8π ) has to appear multiplying the Λn term. Since β 2 < 8π, only a finite number of cases allow a positive set of dimensions, and for each of these except n = 1, and for β 2 generic, one checks that there is no field with these dimensions. Hence, the conservation at lowest order extends, generically, to conservation at arbitrary order23. This proves quantum integrability, perturbatively that is. Finally, these conserved quantities also turn out to be in involution, ie they define mutually commuting operators. The proof is based on the Jacobi identity, which implies that, if two conserved quantities do not commute, then their commutator is also a conserved quantity – a little more thinking then establishes this is not possible. I also would like to remark that in the early literature about quantum integrable models, there is the implicit “suspicion” that quantization might destroy integrability, i.e. reduce the classical symmetry of the theory. It is important to realize that such a thing is not always true. In fact, in two dimensions at least, the quantum theory often has more symmetry than the classical theory. 5.2 S-matrices The next step in the analysis of a massive integrable quantum field theory requires going to a scattering description [43]. Let us assume quite generally that we have massive particles distinguished by some label a, with mass Ma . We write their two momentum pµ in terms of a rapidity variable α: E = M cosh α, P = M sinh α. We now consider scattering processes. There are the “in-states”, corresponding physically to a bunch of particles arranged on the x-axis by decreasing order of rapidities, which we describe formally by a ket |α1 , . . . , αN iain1 , ... , an . At large time, they give rise to “out-states”, fora0 , ... , a0

1 n0 , with an it a priori different set mally described by |α01 , . . . , α0n0 iout of particles arranged on the x-axis by increasing order of, a priori different, rapidities (see Fig. 11). Both the in and the out states are a complete set of states in a local quantum field theory, and they are connected by the S-matrix.

23 Non generic cases could be more complicated; consider the example of h = 1 for instance, where now arbitrary orders are allowed in the right hand side of (165)!

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p2

p3 p4

p4

p3 p1 p2

Fig. 11. Scattering of quasiparticles in an integrable, 1 + 1 quantum field theory.

Now, the existence of infinitely many conserved quantities has very drastic consequences on the scattering of these particles. Indeed, the conserved quantities have to act simply on the multiparticles states - they are in fact proportional to the sums of odd powers of the momenutm. As a result, it follows that, in the scattering process: (i) The number of particles is conserved; in fact, the number of particles of the same mass is conserved. (ii) The final set of two-momenta coincides with the initial set of twomomenta. From this in turn, it follows that the S matrix factorizes into a product of 2-body scattering processes. Toµ see this, consider for instance conjugating the S matrix by the operator eip1 xµ . Since the S matrix conserves momenta, it actually does commute with this operator, so we don’t change anything. On the other hand, the operator has a non trivial physical action: it changes the space time coordinates of particle 1. If we chose p1 appropriately, we can arrange for particle 1 to scatter with the other particles only after they are very highly separated, so the scattering of this particle is a succession of two particle scatterings. By proceeding inductively, one deduces that indeed, the S matrix factorizes. Moreover, for the whole thing to be consistent, the scattering must be “associative”; that is, the scattering of three particles can be decomposed into three pairwise scatterings, with a result independent of which particular decomposition is used. This is illustrated graphically in Figure 12. This constraint, the so called Yang Baxter equation, is the pillar of the algebraic approach to integrable quantum field theories (for a review see for instance [44]).

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Topological Aspects of Low Dimensional Systems

Fig. 12. Factorization of the scattering. a' 2

a' 1

a1

a2

1

2

a0 ,a0

Fig. 13. The matrix element Sa11a22 corresponds to the process illustrated here.

Notice that the YB equation is trivial if all the particles have different masses. It becomes more interesting in the case where several particles have the same mass, but differ by some other quantum number, e.g. the charge. In general, we define the S matrix elements by the relation (Fig. 13) in

a0 ,a0

out

|α1 , α2 ia1 ,a2 = Sa11a22 |α1 , α2 ia0 ,a0 . 1

2

(166)

Relativistic invariance contrains the S matrix to depend on the difference of rapidities α1 − α2 . In the case of the sine-Gordon model, perturbation in β, considerations of (quantum affine) symmetry, minimality and consistency assumptions, lead to the following results. The spectrum is made up of the kink and 1 antikink of mass M ∝ Λ 2−2g , together with breathers. For n − 1 < 1 g ≤ n, there are n − 2 such breather states. Their masses are Mk = 2M sin [kπg/2(1 − g)].

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+

+

+

+ a

+

-

_

+

_

+

_

+

b

529

c

Fig. 14. The three types of possible processes involving kink and antikink.

The kink S-matrix is closely related to the matrix of Boltzmann weights in the 6-vertex model. There are three key amplitudes a(α)

= sin[γ(π + iα)]Z(α)

b(α) c(α)

= − sin(iγα)Z(α) = sin(γπ)Z(α),

(167)

where γ = g1 − 1. The element a(α1 − α2 ) describes the process |α1 α2 i++ → |α1 α2 i++ , as well as |α1 α2 i−− → |α1 α2 i−− , b describes +− → +−, c describes the non-diagonal process +− → −+ (see Fig. 14), and there is a symmetry under interchange of kink to antikink (corresponding to Φ → −Φ). Exercise: Show that (167) gives rise, indeed, to a solution of the Yang Baxter equation. Hint: proceed graphically as sketched in Figure 15 – to represent the matrix multiplications of YB, simply draw all the physical processes that connect a given pair of initial and final states, and add up their amplitudes. The function Z(α) is a normalization factor, which can be written as Z ∞ 2αγy sinh[(γ − 1)y] dy −1 exp i sin · Z(α) = sin[γ(π + iα)] π sinh y cosh(γy) −∞ 2y The breather-kink and breather-breather S matrices are well known; we do not write them down here (see below for some examples, and in the appendix). These S matrices of course are not only characterized by the fact that they must solve the Yang-Baxter equation: there are several other physical requirements, like unitarity and crossing symmetry. In the formula above, these translate into the relations a(α) = b(iπ − α) c(α) = c(iπ − α) form which unitarity S(α)S(−α) = 1 follows.

(168)

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Topological Aspects of Low Dimensional Systems 1

2

3 _

+

+

+

+

1

_

2

3

Fig. 15. The three types of possible processes involving kink and antikink.

In addition, one must have a “closed bootstrap”: for instance, breathers appear as poles in the kink antikink scattering, and their S matrix can be computed using this fact (for a review see [45, 46]). When 1/g is an integer, the bulk scattering is diagonal (c vanishes) and a = ±b. Therefore, the only allowed processes are transmissions: particles go through one another without exchanging quantum numbers. This is the simplest case, to which we will restrict in what follows. It is reasonable to think of the S matrix, together with the mass spectrum, as the quantum equivalent of the knowledge of the τ function in the classical case24 . We can then take the massless limit exactly as we did before, by letting the mass parameter M → 0, and at the same time boosting the rapidities. One obtains then a collection of left and right moving kinks, antikinks and breathers. The LL and RR scattering matrices take exactly the same expressions as in the massive case in terms of the new rapidities θ, while the LR S matrices go to constants, which can in most cases just be forgotten. This provides an alternative description of the free boson in terms of “massless scattering”. We will see in a little while how quantities of the conformal field theory can be recovered, if one wishes, within that description. 24 There

are more accurate formulations of this statement.

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It is fair to stress here that the RR or LL scattering are hard to make sense of in the context of a true, physical scattering process. If particles have the label R say, this means they are moving at the speed of light in the right direction, so, for instance, all the lines in the Figure 11 which illustrates the Yang Baxter equation become parallel! The point is that the scattering has to be interpreted in the massless limit as a set of commutation relations for creation operators, an idea which we will discuss below. The idea of massless scattering appeared a bit weird at the beginning [47, 48, 50], but its predictive power and favorable comparison with experiments gained it respectability quickly. A more rigorous approach along the lines of lattice regularizations is proposed in [49]. 5.3 Back to the boundary sine-Gordon model For the moment, we finish following the logic of the classical analysis in the quantum case. First, one can prove that the quantum sine-Gordon model with a bulk and boundary interaction Z Z h 1 0 dx dy (∂x Φ)2 + (∂y Φ)2 S = 2 −∞ Z i β (169) + Λ cos βΦ(x, y) + λ dy cos Φ(0, y) 2 allows the existence of conserved quantities as well [51]. The proof proceeds in the same spirit as for the bulk case, and we are not going to reproduce it here, though it is an excellent exercise for the diligent reader. Exercise: Show that, in the classical theory, the pair T4 , Θ2 still leads to a conserved quantity provided there exists still another local quantiy θ3 such that d ¯ 2 − T¯4 − Θ2 θ3 . = (170) T4 + Θ x=0 dy Show then, that for the sine-Gordon model with a potential V [Φ(x = 0, y)] at the boundary, only the choice V ∝ cos 12 (Φ + Φ0 ) allows the existence of the integral of motion. Integrability means that the particles also have to scatter nicely at the boundary, i.e. scatter one by one, without particle production, in a way that is compatible with the bulk scattering. This latter condition can simply be expressed graphically as shown in Figure 16, and corresponds to the “boundary Yang-Baxter” equation. The problem of determining the reflection matrix is purely technical. It does have quite a simple answer in the massless limit. Introduce R+− (θ) = R−+ (θ) =

−i exp(γθ) exp [iχg (θ)] 1 − i exp(γθ)

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Topological Aspects of Low Dimensional Systems

Fig. 16. The boundary Yang-Baxter equation.

R++ (θ) = R−− (θ) =

1 exp [iχg (θ)] 1 − i exp(γθ)

where χg is a phase that will disappear at the end of the computations, γ = g1 − 1. The reflection matrix for coupling λ is then given by R(θ − θB ) where TB = meθB ∝ λ1/(1−g) (the exact correspondence depends on the regularization scheme; it is given in [36] in the case of dimensional regularization) is the equivalent of the Kondo temperature. As θ → ∞, R++ → 0 (the scattering is completely off-diagonal) corresponding to Neumann boundary conditions, while as θ → −∞, R+− → 0 (the scattering is completely diagonal) corresponding to Dirichlet boundary conditions. Notice also the unitarity condition |R+− |2 + |R++ |2 = 1. 6

The thermodynamic Bethe-ansatz: The gas of particles with “Yang-Baxter statistics”

The thermodynamic Bethe ansatz was probably written first in [52] in the context of the XXZ model. Its use in quantum field theory, in particular to compute the central charge and study RG flows, was pioneered by a beautiful series of papers of Zamolodchikov; see for instance [53]. A useful and pedagogical review on many of these topics can be found in [55]. 6.1 Zamolodchikov Fateev algebra It is convenient to think of the particles in terms of creation and annihilation operators. For this, let us introduce, still denoting the type of the particles

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by a label a, operators Za (θ) and Za† (θ) satisfying the relations Za1 (θ1 )Za2 (θ2 ) Za†1 (θ1 )Za†2 (θ2 )

= Sa1 a2 (θ1 − θ2 )Za2 (θ2 )Za1 (θ1 ) = Sa1 a2 (θ1 − θ2 )Za†2 (θ2 )Za†1 (θ1 )

Za1 (θ1 )Za†2 (θ2 )

= Sa1 a2 (θ1 − θ2 )Za†2 (θ2 )Za1 (θ1 ) + 2πδa1 a2 δ(θ1 − θ2 ).

(171)

Here, we restricted to the case of diagonal scattering. Note that the compatibility between the first two relations uses unitarity in the form S † (θ) = S −1 (θ) = S(−θ). The space of states is generated by the kets |θ1 , . . . , θn ia1 , ..., an = Za†1 (θ1 ) . . . Za†n (θn ) |0i ,

(172)

where |0i denotes the physical vacuum. Similarly, the dual space is generated by the bras hθn , . . . , θ1 | = h0| Zan (θn ) . . . Za1 (θ1 ).

(173)

The metric is, from (171), induced by a1

hθ1 |θ2 ia2 = 2πδa1 a2 δ(θ1 − θ2 ).

(174)

If for instance θ1 > θ2 , then the in and out states are, respectively in

|θ1 , θ2 ia1 a2 = |θ1 , θ2 ia1 a2 out

|θ1 , θ2 ia1 a2 = |θ2 , θ1 ia1 a2 .

(175)

When the rapidity sets are not ordered, one obtains states which are neither in nor out; or course they are related to either of these by products of S matrix elements. To make things more concrete, let us discuss briefly wave functions in coordinate representation, restricting for simplicity to two particles. To satisfy the relations (171), it is easy to see that the wave function must have a singularity at coincident coordinates, and be of the form Z dx1 dx2 ei(P1 x1 +P2 x2 ) |x1 , x2 ia1 a2 |θ1 θ2 i > a1 a2 ∝ x1 <x2 Z − Sa1 a2 (θ1 − θ2 ) dx1 dx2 ei(P1 x1 +P2 x2 ) |x1 , x2 ia1 a2(176) x1 >x2

where we assumed that the particles are fermions, S(0) = −1. Equivalently, one has Z h dθ1 dθ2 ei(P1 x1 +P2 x2 ) |x1 x2 ia1 a2 ∝ θ1 >θ2 i + Sa1 a2 (θ1 − θ2 )ei(P1 x2 +P2 x1 ) |θ1 , θ2 iin (177) a1 a2 where we see the appearance of the well known Bethe wave function [55].

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Topological Aspects of Low Dimensional Systems

6.2 The TBA The next step is to get a handle on the massless scattering description. The latter turns out to be quite convenient to discuss thermodynamic properties, and this is what we shall start with. As a simple example we consider a hypothetical theory made up of a single type of massless particle, say right-moving, with energy and momentum parametrized as in (152). The scattering is described by a single S-matrix element SRR . Quantizing a gas of such particles on a circle of length L requires the momentum of the ith particle to obey (we have set ~ = 1) Y SRR (θi − θj ) = 1. (178) exp im eθi L j6=i

One can think of this intuitively as bringing the particle around the world through the other particles; one obtains a product of two-particle S-matrix elements because the scattering is factorizable. A bit more rigorously, one can deduce this from the wave function as in (176) Going to the L → ∞ limit, we introduce the density of rapidities indeed occupied by particles ρ(θ) and the density of holes ρ˜. A hole is a state which is allowed by the quantization condition (178) but which is not occupied, so that the density of possible rapidities is ρ(θ) + ρh (θ). Taking the derivative of the log of (178) yields Z ∞ K(θ − θ0 )ρ(θ0 )dθ0 , (179) 2π[ρ(θ) + ρh (θ)] = mLeθ + −∞

where

1 d ln S(θ). i dθ To determine which fraction of the levels is occupied we do the thermodynamics, following the pioneering work of Yang and Yang. The energy is Z ∞ ρ(θ)meθ dθ, E= K(θ) =

−∞

and the entropy is Z ∞ (ρ + ρh ) ln(ρ + ρh ) − ρ ln(ρ) − ρh ln(ρh ) dθ. S= −∞

Exercise: derive this relation by using Stirling’s formula Γ(z) ≈ √ 1 z z− 2 e−z 2π. The free energy F = (E − T S) is found by minimizing it with respect to ρ. The variations of E and S are Z ∞ δρmeθ dθ δE = −∞

H. Saleur: Lectures on Non Perturbative Field Theory Z δS

∞

=

−∞

535

(δρ + δρh ) ln(ρ + ρh ) − δρ ln(ρ) − δρh ln(ρh ) dθ

It is convenient to parametrize ρ(θ) = exp − ρh (θ) T giving

Z

∞

δS =

(180)

h i δρ ln 1 + e/T + δρh ln 1 + e−/T dθ.

−∞

Using (179) allows us to find ρ˜ in terms of ρ. Denoting convolution by ?, this gives 2π(δρ + δρh ) = K ? δρ so Z ∞ K + ? ln 1 + e−/T δρdθ. δS = 2π −∞ T Hence the extremum of F occurs for meθ = + T

K ? ln 1 + e−/T 2π

and one has then, expressing ρh from (179) and using (181) Z ∞ 2 m eθ ln 1 + e−/T dθ. F = −LT 2πT −∞

(181)

(182)

It is a simple exercise to show that this formula, together with (181), generalizes to a theory with several species of particles, provided the scattering is β2 = g = 1t , t an integer, to which we diagonal. This corresponds to the case 8π restrict in what follows. In that case, recall that we have a kink and antikink kπ , of mass parameter m, and breathers of mass parameter mk = 2m sin 2(t−1) with k = 1, . . . , t − 2. We will also allow for different chemical potentials µk for the various particles. Defining now ’s through µj − j ρj (θ) = exp (183) T ρhj (θ) the equivalent of (179) is now 2π[ρj (θ) +

ρhj (θ)]

θ

= mj Le +

XZ k

∞

−∞

Kjk (θ − θ0 )ρk (θ0 )dθ0 ,

(184)

and the equivalent of (181) mj eθ = j + T

X Kjk k

2π

µk −k ? ln 1 + e T .

(185)

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Topological Aspects of Low Dimensional Systems

The equivalent of (182) is, in turn: G = E −T S −

X k

For the case g =

X mk Z ∞ µj −k µk Nk = −LT eθ ln 1 + e T dθ. (186) 2πT −∞ 2

k

1 3

for instance, one has 2 cosh θ

Kbb

= 2K++ = 2K+− = −

Kb+

√ cosh θ · = K+b = −2 2 cosh 2θ

(187)

It has become common in the literature to reformulate the TBA in a convenient form by using simple diagrams. It is a laborious but straightforward exercise to demonstrate, using the kernels given in the appendix, that (185) is equivalent to the following simple system25 X k −µk s ? ln 1 + e T j = T Njk . (188) 2π k

t−1 , Njk = 1 if the nodes j and k are neighbours on the Here, s(θ) = cosh(t−1)θ following diagram, 0 otherwise

. +

1 2 s t−3

—— – – – – – – – —— /

t−2

− Exercise: establish this for the case g = 13 . The equations (188) have to be supplemented by the boundary conditions (189) j ≈ mj e θ , θ 1 6.3 A standard computation: The central charge The thermodynamics of a chiral theory like the one we just studied is not so exciting; this is because, after all, the 1 + 1 theory is conformal invariant, so the results at different temperatures are essentially equivalent. This can easily be seen 25 The case where g is not of the simple form 1/integer can also be handled of course. It is technically more difficult because the scattering is non diagonal, so an additional Bethe ansatz is necessary to diagonalize the scattering to start with, before the periodicity of the wave function can be imposed [54].

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on the TBA equations: a change in T , or the mass scale m, can be fully absorbed by a boost of the particles, i.e. a shift of rapidities, exactly like for the changes in mass scale encountered before. As a result, we see that the integrals in (182) are independent of the temperature, so F ∝ LT 2 . That this is so, and the coefficent of proportionality, are directly related with considerations from the beginning of these lectures. Indeed, we have F = −T ln Z, where Z is the partition function of the one dimensional quantum field theory at temperature T . In Euclidean formalism, this corresponds to a theory on a torus with finite size in time direction R = 1/T . By modular invariance, identical results should be obtained if one quantizes the theory with R as the space coordinate. For large L, Z = e−E(R)L , where E(R) is the ground-state (Casimir) energy with space a circle of length R. Thus F = LE(R)/R. Conformal invariance requires πc that at a fixed point this Casimir energy is E(R) = − 6R , where c is the central 2 charge. Going back to the thermal point of view, F = − LπcT and the specific 6 26 LπcT heat is C = 3 . It is possible to analytically find this central charge from (181). This is a bit technical, but worth studying, since it is a crucial a posteriori test of the whole thing. We take the derivative of (181) with respect to θ and solve for eθ . Substituting this in (182), we have Z TL d F = − dθ ln(1 + e−/T ) 2π dθ Z dθ0 1 −(θ)/T 0 d − )K(θ − θ ) 0 ln(1 + e 2π dθ 1 + e(θ0 )/T Z TL − meθ d 1 = − ln(1 + e−/T ) + dθ 2π dθ T 1 + e(θ)/T Z TL d /T −/T = −F − , (190) dθ ln(1 + e )+ 2π dθ 1 + e/T where we use (181) again to get to the second line. We can replace the integral over θ with one over , giving an ordinary integral Z /T TL ∞ F =− · d ln(1 + e−/T ) + 4π (−∞) 1 + e(θ)/T A change of variables gives T 2L F =− L 2π

1 1 + x0

,

(191)

where L(x) is the Rogers dilogarithm function Z 1 x ln(1 − y) ln y L(x) = − + dy, 2 0 y 1−y 26 With massive particles or with nontrivial left-right massless scattering, F does depend on M/T , giving a running central charge.

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Topological Aspects of Low Dimensional Systems

and x0 ≡ exp[(−∞)/T ] is obtained from (181) as I 1 1 = 1+ , x0 x0 R 1 with I = 2π K. For example, when the S matrix is a constant, K = 0, x0 = 1 and F =−

LT 2 π , 24

(192)

(193)

2

where we used L(1/2) = π12 . Here we find cL = 14 . In a left-right-symmetric quantum field theory, the right sector makes the same contribution, giving the total central charge c = 12 required for free fermions. Similar computations can be carried out for more complicated theories, leading to beautiful expressions of central charges in terms of sums of dilogarithms (see e.g. [56]). In the case of interest, one finds of course c = 1.

6.4 Thermodynamics of the flow between N and D fixed points We now wish to do the thermodynamics in the presence of the boundary, to obtain the boundary free energy, and the associated flow of boundary entropies. To start, it is better to map the problem onto a line of length 2L (−L < x < L) by considering the left movers to be right movers with x > 0. Thus we have only R movers scattering among themselves and off the boundary, which can now be thought of as an impurity (a particle with rapidity θB ). The reflection matrix becomes a transmission matrix, with appropriate relabellings, for instance R+− → T++ , etc. (This trick is the same than what we did for boundary conformal field theory, and can only be used in the massless limit.) For simplicity, we put periodic boundary conditions on the system; these do not change the boundary effects at x = 0. Recall we consider only the case γ = g1 − 1 a positive integer, where the bulk scattering is diagonal. The impurity scattering still is not, but we √ can redefine our states to be |1, 2i ≡ (|+i ± |−i)/ 2 so that the impurity scattering is now diagonal27 : T11 (θ) = R++ + R+−

=

T22 (θ) = R++ − R+−

=

exp [iχg (θ)] γθ iπ − tanh exp [iχg (θ)] . 2 4

(194)

We can now write the Bethe equations. These differ from the bulk ones only by the presence of the additional impurity scattering X 1 κj (θ − θB ) Kjk ? ρk (θ) + (195) 2π(ρj (θ) + ρhj (θ)) = mj eθ + 2L k

27 If γ is even, this actually makes the bulk scattering completely off-diagonal (e.g. |11i scatters to |22i), but the TBA equations turn out the same.

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where Kjk (θ)

=

κj (θ − θB ) =

1 d ln Sjk (θ) i dθ 1 d ln Rj (θ − θB ). i dθ

(196)

The effect of the boundary is seen in the last piece of (195) proportional to 1/L. The minimization equations are independent of the boundary terms, since these do not appear directly in E or S, and they disappear when one takes a variation of (195). Thus equations (185) still hold. Boundary terms do enter the free energy or the grand potential however when one rewrites it in terms of the ’s. One finds Z t dθ X κj (θ − ln(T /TB )) ln(1 + e−j (θ) ). (197) F = Fbulk − T 2π j=1 2 As discussed before, Fbulk = − πc 6 T L in a massless bulk theory, where c is the central charge of the conformal field theory, c = 1 here. The second term in (197) is the boundary free energy. Although the equations (185) for (θ) cannot be solved explicitly for all temperatures, the free energy is easy to evaluate as T → 0 and T → ∞, as we will show next. Moreover, one can extract the analytic values of critical exponents by looking at the form of the expansions around these fixed points. Also, they are straightforward to solve numerically for any T . Several notes of caution are necessary. At the order we are working, the formula for the entropy is not quite correct, because there are 1/L corrections to the Stirling formula used in its derivation. Also, at this order, the logarithm of the partition function is not E −T S: it depends not only on the saddle point value of the sum over all states, but also on fluctuations. Their net effect is that we cannot compute the g factors from F alone. However, both of these corrections are subleading contributions to the bulk free energy, and do not depend on the boundary conditions. Therefore we can still compute differences of g factors from F ; the corrections are independent of the boundary scale θB and cancel out of the difference. We can evaluate the impurity free energy explicitly in several limits. In the IR limit T /TB → 0 the integral is dominated by θ → ∞ where the source terms in (185) become very big. Hence r (∞) = ∞ and the impurity free energy vanishes in this limit. In the UV limit T /TB → ∞ the integrals are dominated by the region where −θ is large so that the source terms disappear in (185) and the r go to constants. These are found by using the alternative form (188), which reads here, denoting xj = ej /T Y xj = (1 + xk )Njk /2 . (198) k

540

Topological Aspects of Low Dimensional Systems

One finds xn ≡ en (−∞)/T = (n + 1)2 − 1;

x1,2 = γ

(199)

Therefore we obtain ln

gN = gD =

Pt−2 n=1

−Fimp T UV

−

−Fimp T IR

I (n) ln(1 + 1/xn ) + (I (+) + I (−) ) ln(1 + 1/x± )

where

Z I (r) ≡

(200)

dθ κr (θ) = κ ˜ (0). 2π

From results given in the appendix, one finds I (n) = n/2 and I (+) + I (−) = γ/2, and thus gN ln gD

=

γ γ + 1 X n (n + 1)2 ln + ln 2 γ 2 n(n + 2) n=1

=

1 1 ln(γ + 1) = ln t. 2 2

γ−1

(201)

This is in agreement with the ratio calculated from conformal field theory. We can also find the dimension of the perturbing operators. From the equations one deduces the following expansions for T /TB large: Yr (θ) = er (θ) =

X

Yr(j) e−2jγθ/(γ+1) .

j

As a result it is straightforward to see that near λ = 0, F can be expanded in powers of (TB /T )2γ/(γ+1). On the other hand we expect F to be an analytic function of λ2 . Hence λ ∝ meθB

γ/(γ+1)

.

(202)

This agrees with the conformal result that the perturbing operator cos[βΦ(0)/2] has boundary dimension d = 1/(γ + 1) = β 2 /8π. In the IR limit of T /TB small, one can expand out the kernels κr in powers of exp(θb − θ). This leads to the fact that the irrelevant operator which perturbs the Dirichlet boundary conditions has dimension d = 2. This is the energy-momentum tensor. (Recall that there is another irrelevant operator in the spectrum with dimension d = γ + 1, which for 0 ≤ γ < 1 is the appropriate perturbing operator.)

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7

541

Using the TBA to compute static transport properties

Let us pause for a moment to compare the gas of Yang-Baxter interacting quasi particles to say free fermions. Within the TBA, the interactions have been fully encoded into non trivial pseudo energies j (θ): that is, at temperature T , the filling fractions of the various species are not independent, but correlated via the coupled integral equations discussed previously. This has some striking consequences. For instance, we see from (199) that the filling fraction of kinks or antikinks at rapidity −∞ (i.e. at vanishing bare energy) is f = 1t . Except for t = 2 (which is a free fermion theory) there is no symmetry between particles and holes. It is important to realize that the interactions would have other effects, in general, for other questions asked. For instance, in the case of free fermions, the total density n = ρ + ρh , ρ = nf , the fluctuations also depend on the j through the well known 2

formula (∆ρ) = nf (1 − f ). Such a formula does not hold in the present case: the fluctuations of the various species are correlated - their computation plays an important role in the DC noise at non vanishing temperature and voltage, see [57]. Similarly, physical operators have complicated matrix elements in the multiparticle basis; the current for instance is able to create any neutral configuration of quasiparticles by acting on the vacuum. There is thus a somewhat deceptive simplicity in what we have done so far. However, for the DC conductance, it turns out that the knowledge of the distribution functions is all that is necessary, so for that particular aspect, our quasiparticles are not so far from free ones. 7.1 Tunneling in the FQHE At this stage, it is useful to recall the tunneling problem of the introduction: we had L and R moving electrons that were backscattered by the impurity. Certainly if a R moving particle bounces back on the gate voltage to become a left one, the charge QR + QL is conserved. Now QL + QR is essentially the charge of the even field in the manipulations discussed in the introduction, which we found has no dynamic indeed. Now when a R mover bounces back as a L mover, there is a change in the non conserved charge Q ≡ QR − QL ; this one is proportional to the charge of the odd field, which has a non trivial dynamics. Now, following carefully the formulas for bosonization, β R ∂x Φ = one finds the simple result that a right moving kink, for which 2π 1, also has physical charge Q = 1, and similarly for antikinks and left moving particles. Therefore, the non conservation of the physical charge due to backscattering is the same as the non conservation of charge in the boundary sine-Gordon model. More precisely, when a kink comes in and bounces back as an antikink, as happens most of the time near the UV fixed point (Neumann boundary conditions), the charge Q is conserved in the

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Topological Aspects of Low Dimensional Systems

original problem. On the other hand, when a kink bounces back as a kink, as happens near the IR fixed point (Dirichlet boundary conditions), the charge in the original problem is not conserved; rather, ∆Q = −2. Let me stress here that the kink in the boundary sine-Gordon theory however would look horribly complicated in the original problem, because the changes of variables we have performed are non local. Only the conserved charge is easy to follow28 . 7.2 Conductance without impurity In the absence of impurity, that is with Neumann boundary conditions in the original boundary problem, charge is straightforwardly transported. A right moving kink or antikink just goes through. Of course, if there are as many particles of each specie, no current is transported overall. If however, a voltage V is applied, kink and antikink are at a different chemical potential, µ = ± V2 - this follows since the U (1) charge in the boundary sine-Gordon model is nothing but the physical charge Q. The current that flows through the system is thus Z ∞ (ρ+ − ρ− )(θ)dθ. (203) I= −∞

We can use our TBA to evaluate this expression quickly. First, we introduce the filling fractions 1 · (204) f± = V ( ± ∓ 2 )/T 1+e Second, we observe that the very convenient identity nj = ρj + ρhj = holds, and that, moreover, + = − ≡ .

1 dj 2π dθ

Exercise: Prove these two statements by staring at the TBA equations. It thus follows that Z ∞ d 1 (205) (f+ − f− ) dθ I= 2π −∞ dθ and thus T I= 2π

1 + e−V /2T e−/T d dθ ln · dθ 1 + eV /2T e−/T −∞

Z

∞

(206)

The current is thus entirely determined by the values of at ±∞, exactly like for the central charge. As before, (∞) = ∞, but the value of (−∞) now does depend on the voltage. One finds in fact, solving again (188) but 28 Charge

is like current here, where we have set the Fermi velocity equal to one.

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with a voltage,

2

sinh(t − 1)V /2tT sinh V /2tT (207) (observe one recovers the result (199) as V → 0) from which an elementary computation shows the simple result (recall g = 1t ) en (−∞)/T =

sinh(n + 1)V /2tT sinh V /2tT

− 1, e± (−∞)/T =

I=

gV · 2π

(208)

The bizarre factor of 2π occurs here because we have set e = ~ = 1 (recall 2 that in physical units, I = g eh V ). Exercise: Prove the last two formulas. This is what one expected, and of course there are quicker ways to derive this result. The point however, is that the same computation carries over without much additional difficulty to the case where the impurity is present. 7.3 Conductance with impurity In the general case, we will write the source drain current as I = I0 + IB where I0 = gV 2π is the current in the absence of backscattering, and IB is the backscattered current. In the original problem, IB is for instance the rate at which the charge of the right moving edge is depleted. Of course, ∂t QL = −∂t QR in each hopping event, so IB = ∂t Q 2 . In the steady state this rate is constant. When for instance V is positive, there are more kinks than antikinks injected with a thermal distribution into the system from their respective infinite reservoirs; it is assumed that these reservoirs are so big that the backscattering does not change their properties. We now derive an analytic expression for this backscattering current using a kinetic rate equation for quasiparticles of the Bethe ansatz. It is possible to compute the rate of change of ∆Q/2 in the basis of the Bethe ansatz quasiparticles, since each scattering event of a kink (antikink) into an antikink (kink) changes the physical charge ∆Q/2 by −1 (by (+1)). This kinetic equation is of course very familiar. However, in general there would be no reason why it should be applicable to an interacting system. But it is exact in the case that we are considering, even though the system is interacting. The reason for this lies in the constraints of integrability: as discussed above, in the very special quasiparticle basis of the Bethe ansatz, these quasiparticles scatter off of the point contact independently (“one-byone”), and all quasiparticle production processes are absent29 . 29 Some

more detailed justifications are available; see [58] and references therein.

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Topological Aspects of Low Dimensional Systems

This allows us to express the rate of change in ∆Q, in terms of the transition probability |T+− |2 (recall that in the unfolded point of view T+− = R++ ) and the number of kinks and antikinks (carriers of charge ∆Q = ±1) in the rapidity range between θ and θ + dθ n± (θ)f± (θ)dθ, where n± is the density of states and f± are the filling fractions. The number of kinks of rapidity θ that scatter into antikinks per unit time is |T+− (θ)|2 ρ+− dθ

(209)

where ρ+− is the probability that the initial kink state is filled and the final antikink state is empty (in all these quantities, there is also a V dependence, which we keep implicit here). For a system of free fermions we would have ρ+− = f+ (1 − f− ) but in our interacting system we only have ρ+− = f+ − f+− where f+− is the probability that both, the kink and the antikink states are filled. For the number of antikinks of rapidity θ that scatter into kinks per unit time one finds a formula similar to (209), with ρ+− → ρ−+ . In the final rate equation, only the difference between these two probabilities ρ+− − ρ−+ = f+ − f− appears. Notice that the unknown f+− = f−+ has cancelled out (it can in fact be determined by techniques more elaborate than the TBA [57]). Therefore, the backscattering current is Z (210) IB (V ) = − dθ n(θ)|T+− (θ − θB )|2 [f+ (θ) − f− (θ)]. All ingredients in this formula are exactly known: the scattering matrix has a simple analytic form and the occupation factors and densities of state are obtained exactly from the thermodynamic Bethe ansatz (TBA). Notice that this equation is valid for any value of the driving voltage V . It thus automatically describes non-equilibrium transport. By the same manipulations as before, it then follows that Z ∞ 1 2 d dθ (f+ − f− ) |T++ | I = 2π −∞ dθ Z ∞ 1 + e−V /2T e−/T 1 d T ln dθ = · (211) 2π −∞ 1 + e−2(t−1)(θ−θB ) dθ 1 + eV /2T e−/T

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Conductance 1

0.1

G h/e^2

0.01

exact curve Monte Carlo experimental data

0.001

0.0001 0.01

0.1

1

10

X=.74313(T_B/T)^(2/3)

Fig. 17. Comparison of the field theoretic result with MC simulations and experimental data for g = 13 .

Of special interest is the linear conductance, which we obtain by taking a derivative at V = 0; this gives, after reinserting the factor 2π, Z 1 1 (t − 1) ∞ · (212) dθ G= 2 /T 2 1 + e cosh [(t − 1)(θ − θB )] −∞ The resulting curve is shown in Figure 17, together with experimental results [13] and the results of Monte Carlo simulations [14], for g = 13 . The agreement with the simulations is clearly very good (the is one and only one fitting parameter – the horizontal scale – , accounting for the unknown, non universal ratio of the experimental gate voltage (a “bare” quantity) to the parameter λ in our renormalized field theory). As far as the experimental data go, it is also very satisfactory, except in the strong backscattering regime. Recall however that the field theoretic prediction holds true only in the scaling limit: the experimental data are still quite scattered for low values of G, indicating that this limit is not reached yet -

546

Topological Aspects of Low Dimensional Systems

actually the “noise” is of the same order of magnitude as the discrepancy from the theoretical curve, as reasonably expected. Exercise: The problem had been solved previously [10] in the simplest case of g = 12 , where one can refermionize the Hamiltonian for the boson φe . Look at this solution, and compare with what we have just done: what is the meaning of kink and antikink, what is the bulk scattering, the boundary scattering? Conclusions: Further reading and open problems I am now leaving you at the beginning of a very exciting domain. Let me suggest some further reading and open problems. These lectures have stopped short of really tackling the problem of boundary fixed points classification. Equipped with what you learned here, you should not have much difficulty reading the paper of Cardy on boundary states [32]. This paves the way to questions that are still open. For instance, the problem we have studied at length generalizes, for tunneling in quantum wires where the spin of the electrons has to be taken into account, to a “double sine-Gordon problem”, involving two bosons. Surprisingly it has been shown [10] that new non trivial fixed points do exist in that case, besides the obvious Neumann and Dirichlet possibilities. With a few exceptions [64, 65], nobody knows how these fixed points precisely look like! As I will mention again below, what we have discussed is also very close to the Kondo problem. You can learn more about fixed points and conformal invariance by reading the papers of Affleck and Ludwig on the multichannel Kondo problem [16]. There, you will also discover an aspect that I have neglected by lack of space: how multipoint correlators can be evaluated at fixed points by further using conformal invariance [18]. The integrable approach can also be pushed further to allow the computation of AC properties, together with space and time dependent Green functions, in the cross-over regime. The idea here still relies on massless scattering; but now, one has to evaluate matrix elements of physical operators, and these are usually pretty complicated. Moreover, an infinity of these matrix elements are a priori needed: for instance, the current operator is able to create any neutral configurations of quasiparticles out of the vacuum! It turns out however that, first, the matrix elements can be determined by algebraic techniques [59,60] (the latter reference is recomended as a first reading; the first is a bit hard to read), and second, in many cases, only a few of these matrix elements are required to obtain controlled accuracy all the way from the UV to the IR fixed point. Using that technique, for instance the current current correlator itself can be evaluated, at least at T = V = 0 [61] (there does remain a non trivial dependence upon space,

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547

time, and the coupling λ). Determining correlators with a finite temperature or voltage is still more difficult; some progress in that direction has been made [62, 63], but a lot remains to be done. In another direction, for those of you who are more formally oriented, it should be clear that what I just described is the tip of an iceberg of beautiful mathematical structures: see [36, 66] and the series [67]. Let me just mention here that the Kondo problem, which would be described by Z 1 0 e dx Π2 + (∂x Φ)2 + λ S + e−iβΦ(0)/2 + S − eiβΦ(0)/2 , H≡H = 2 −∞ (213) is just around the corner: it actually does have deep relations with the boundary sine-Gordon model, and with the subject of quantum monodromy operators. Especially exciting results have actually appeared recently, concerning an exact duality between the UV and IR regimes of the problem [68, 69], and exhibiting tantalizing relations with the recent breakthroughs in 4D SUSY gauge theories [70]. It is also fair to stress that the methods developed within the context of quantum impurity problems can be generalized to different systems of physical interest in 1 + 1 dimensions: an example is the amazing recent mapping of the two-ladder problem onto an SO(8) Gross Neveu model [71]. It is very likely indeed that more such problems are awaiting us in the near future. Finally, the traditional question is, can any of this be generalized to more than 1 + 1? Well, the recent excitments in string theory are centered around somewhat similar ideas in 3 + 1, where, roughly, integrability is replaced by supersymmetry, an incredibly powerful tool. As for 2+1, I don’t quite think it’s over yet. The material in these notes is partly based on collaborations with I. Affleck, P. Fendley, A. Leclair, F. Lesage, A. Ludwig, M. Oshikawa, P. Simonetti, S. Skorik and N. Warner; I also greatly benefitted from questions by the school students, the organizers, and my fellows co-lecturers - many thanks to all of them. The work was supported by the USC, the DOE, the NSF (through the NYI program), and the David and Lucile Packard Foundation. Many friends from “Le plateau” have disappeared since I left France, and their memory weighs very much on my mind. I am especially thinking of Heinz Schulz as I correct these proofs it is to him that I would like to dedicate these notes.

Appendix: Kernels We use the following convention for Fourier transform: Z ∞ dθ i2γθy/π ˜ e f (θ), f (y) = 2π −∞

548

Topological Aspects of Low Dimensional Systems

where γ = t − 1. The bulk kernels Kjk are well known; they can be written in the form (± stand for kink and antikink) ˜ jk K

=

˜ j,± K

=

˜ ±,± K

=

cosh y cosh(γ − j)y sinh ky cosh γy sinh y cosh y sinh jy − cosh γy sinh y ˜ ±,∓ = − sinh(γ − 1)y K 2 cosh γy sinh y δjk − 2

j, k = 1 . . . γ − 1; j ≥ k

(214) (215)

with Kjk = Kkj . The boundary kernels are κ ˜j

=

κ ˜−

=

κ ˜+

=

sinh jy 2 sinh y cosh γy 1 sinh(γ − 1)y + 2 sinh 2y cosh γy 2 cosh y sinh(γ − 1)y · 2 sinh 2y cosh γy

(216) (217)

Finally, s˜ =

1 2 cosh y .

References [1] Hewson A.C., The Kondo Problem to Heavy Fermions (Cambridge University Press, 1997). [2] Saminadayar L., Glattli D.C., Jin Y. and Etienne B., cond-mat/9706307, Phys. Rev. Lett. 75 (1997) 2526. [3] de-Picciotto R., Reznikov M., Heiblum M., Umansky V., Bunin G. and Mahalu D., Nature 389 (1997) 162. [4] Perspectives in Quantum Hall Effects, edited by S. Das Sarma and A. Pinczuk, (Wiley, 1997). [5] Leggett A.J., Chakravary S., Dorsey A.T., Fisher M.P.A., Garg A. and Zwerger W., Rev. Mod. Phys. 59 (1987) 1. [6] Weiss U., Dissipative Quantum Mechanics (World Scientifc, Singapore, 1998). [7] Andrei N., Furuya K. and Lowenstein J., Rev. Mod. Phys. 55 (1983) 331. [8] Wiegmann P.B. and Tsvelick A.M., JETP Lett. 38 (1983) 591. [9] Shankar R., Rev. Mod. Phys. 66 (1994) 129. [10] Kane C.L. and Fisher M.P.A., Phys. Rev. B 46 (1992) 15233; B 46 (1992) 7268. [11] Haldane F.D.M., J. Phys. C 14 (1981) 2585. [12] Wen X.G., Phys. Rev. B 41 (1990) 12838; Phys. Rev. B 43 (1991) 11025. [13] Milliken F.P., Umbach C.P. and Webb R.A., Solid State Comm. 97 (1996) 309. [14] Moon K., Yi H., Kane C.L., Girvin S.M. and Fisher M.P.A., Phys. Rev. Lett. 71 (1993) 4381.

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[15] Nozi` eres P., J. Low Temp. Phys. 17 (1974) 31; Nozi` eres P. and Blandin A., J. Phys. France 41 (1980) 193. [16] For reviews, see I. Affleck, in Correlation Effects in Low-Dimensional Electron Systems; edited by A. Okiji and N. Kawakami (Springer-Verlag, Berlin, 1994), condmat/9311054; Ludwig A.W.W., Int. J. Mod. Phys. B 8 (1994) 347; in Proceedings of the ICTP Summer School on Low Dimensional Quantum Field Theories for Condensed Matter Physicists, Trieste (Italy), Sept. 1992 edited by S. Lundqvist, G. Morandi and Lu Yu (World Scientific, New Jersey, 1995); Physica B 199 & 200 (1994) 406 (Proceedings of the International Conference on Strongly Correlated Electron Systems, San Diego, 1993). [17] Affleck I. and Ludwig A.W.W., Nucl. Phys. B 352 (1991) 849; ibid B 360 (1991) 641. [18] Ludwig A.W.W. and Affleck I., Nucl. Phys. B 428 (1994) 545. [19] Les Houches, session XLIX, 1988, Fields, Strings and Critical Phenomena, edited by E. Bre´zin and J. Zinn-Justin (Elsevier, New York, 1989). [20] Cardy J.L., Conformal Invariance, in Phase Transitions, edited by C. Domb and J.L. Lebowitz, Vol. 11 (Academic Press, New York, 1987). [21] Polchinski J., in Proceedings of the 1994 Les Houches Summer School, hep-th/9411028. [22] Di Francesco P., Mathieu P. and Se´ ne´ chal D., Conformal Field Theory (Springer, New York, 1997). [23] Tsvelik A.M., Quantum Field Theory in Condensed Matter Physics (Cambridge University Press, Cambridge, 1995). [24] Kadanoff P. and Ceva H., Phys. Rev. B 11 (1971) 3918. [25] Stone M., Bosonization (World Scientific, Singapore, 1994). [26] Green M., Schwarz J. and Witten E., Superstring Theory (Cambridge University Press, 1987). [27] Whittaker E.T., Watson G.N., A Course of Modern Analysis (Cambridge University Press, 1990). [28] Cardy J.L., Nucl. Phys. B 240 (1984) 512. [29] Apostol T., Modular Functions and Dirichlet Series in Number Theory (Springer, New York, 1990). [30] Ishibashi N., Mod. Phys. Lett. A 4 (1989) 251. [31] Affleck I. and Oshikawa M., Nucl. Phys. B 495 (1997) 533. [32] Cardy J., Nucl. Phys. B 324 (1989) 581. [33] Affleck I. and Ludwig A.W.W., Phys. Rev. Lett. 67 (1991) 161. [34] Fendley P., Phys. Rev. Lett. 71, (1993) 2485. [35] Zamolodchikov A.B., JETP Lett. 43 (1986) 730. [36] Fendley P., Lesage F. and Saleur H., J. Stat. Phys. 79 (1995) 799, hep-th/9409176. [37] Macdonald I.G., Symmetric Functions and Hall Polynomials (Clarendon Press, 1979); Stanley R.P., Adv. in Math. 77 (1989) 76. [38] Schmid A., Phys. Rev. Lett. 51 (1983) 1506. [39] Fendley P., Ludwig A.W.W., Saleur H., Phys. Rev. Lett. 74 (1995) 3005, cond-mat/9408068. [40] Bernard D., Leclair A., Comm. Math. Phys. 142 (1991) 99. [41] Zamolodchikov A.B., Adv. Stud. Pure Math 19 (1989) 1. [42] Saleur H., Skorik S., Warner N.P., Nucl. Phys. B 441 (1995) 412. [43] Zamolodchikov A.B. and AL Zamolodchikov B., Ann. Phys (N.Y.) 120 (1979) 253L.

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[44] Jimbo M., Yang-Baxter Equation in Integrable Systems, Adv. in Math. Phys. 10, World Scientific (Singapore). [45] Mussardo G., Phys. Rep. 218 (1992) 215. [46] Dorey P., Exact S matrices in Two Dimensional Quantum Field Theory (Cambridge University Press, 1996). [47] Faddeev L.D., Takhtajan L.A., Phys. Lett. 85A (1981) 375. [48] Fendley P., Saleur H., Massless Integrable Quantum Field Theories and Massless Scattering in 1+1 Dimensions, Proceedings of the Trieste Summer School on High Energy Physics and Cosmology (World Scientific Singapore, 1993). [49] Reshetikhin N.Yu. and Saleur H., Nucl. Phys. B 419 (1994) 507. [50] Zamolodchikov A.B., Al. Zamolodchikov B., Nucl. Phys. B 379 (1992) 602. [51] Ghoshal S. and Zamolodchikov A.B., Int. J. Mod. Phys. A 9 (1994) 3841, hep-th/9306002. [52] Yang C.N. and Yang CP C., J. Math. Phys. 10 (1969) 1115. [53] Zamolodchikov Al B., Nucl. Phys. B 342 (1991) 695. [54] Fendley P. and Intriligator K., Nucl. Phys. B 372 (1992) 533 [55] Korepin V.E., Bogoliubov N.M. and Izergin A.G., Quantum Inverse Scattering Method and Correlation Functions (Cambridge Univ. Press, Cambridge, 1993). [56] Kirillov A.N. and Reshetikhin N.Yu., J. Phys. A 20 (1987) 1565, 1587. [57] Fendley P. and Saleur H., Phys. Rev. B 54 (1996) 10845. [58] Lesage F. and Saleur H., Duality and IR Perturbation Theory in Quantum Impurity Problems, cond-mat/9812045 [59] Smirnov F.A., Form Factors in Completely Integrable Models of Quantum Field Theory (World Scientific, Singapore) and references therein. [60] Cardy J., Mussardo G., Nucl. Phys. B 410 (1993) 451; Delfino G., Mussardo G., Simonetti P., Phys. Rev. D 51 (1995) 6620. [61] Lesage F., Saleur H., Skorik S., Nucl. Phys. B 474 (1996) 602. [62] Leclair A., Lesage F., Sachdev S. and Saleur H., Nucl. Phys. B 483 (1996) 579. [63] Lesage F. and Saleur H., Nucl. Phys. B 493 (1997) 613. [64] Yi H. and Kane C., Quantum Brownian Motion in a Periodic Potential and the Multi Channel Kondo Problem, cond-mat/9602099. [65] Affleck I., Oshikawa M. and Saleur H., Boundary Critical Phenomena in the Three State Potts Model, cond-mat/9804117. [66] Fendley P., Lesage F. and Saleur H., J. Stat. Phys. 85 (1996) 211, cond-mat/9510055. [67] Bazhanov V., Lukyanov S. and Zamolodchikov A.B., Comm. Math. Phys. 177 (1996) 381, hep-th/9412229; Comm. Math. Phys. 190 (1997) 247, hep-th/9604044; Nucl. Phys. B 489 (1997) 487, hep-th/9607099. [68] Fendley P., Duality Without Supersymmetry, hep-th/9804108. [69] Fendley P., Saleur H., Self-duality in Quantum Impurity Problems, cond-mat/9804173, Phys. Rev. Lett. to appear. [70] Seiberg N. and Witten E., Nucl. Phys. B 426 (1994) 19, hep-th/9407087; Nucl. Phys. B 431 (1994) 484, hep-th/9408099. [71] Lin H., Balents L. and Fisher M.P., Exact SO(8) Symmetry in the Weakly Interacting Two-leg Ladder, cond-mat/9801285.

SEMINAR 1

QUANTUM PARTITION NOISE AND THE DETECTION OF FRACTIONALLY CHARGED LAUGHLIN QUASIPARTICLES

D.C. GLATTLI ´ Service de Physique de l’Etat Condens´e, CEA Saclay, 91191 Gif-sur-Yvette, France

Contents 1 Introduction

553

2 Partition noise in quantum conductors 2.1 Quantum partition noise . . . . . . . . . . . . . . . . . . . . . . 2.2 Partition noise and quantum statistics . . . . . . . . . . . . . . 2.3 Quantum conductors reach the partition noise limit . . . . . . . 2.4 Experimental evidences of quantum partition noise in quantum conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 558

3 Partition noise in the quantum hall regime and determination of the fractional charge 3.1 Edge states in the integer quantum Hall eﬀect regime . . . 3.2 Tunneling between IQHE edge channels and partition noise 3.3 Edge channels in the fractional regime . . . . . . . . . . . . 3.4 Noise predictions in the fractional regime . . . . . . . . . . 3.5 Measurement of the fractional charge using noise . . . . . . 3.6 Beyond the Poissonian noise of fractional charges . . . . . .

. . . . . .

. . . . . .

. . . . . .

554 . . 554 . . 555 . . 557

. . . . . .

562 562 563 564 567 569 570

QUANTUM PARTITION NOISE AND THE DETECTION OF FRACTIONALLY CHARGED LAUGHLIN QUASIPARTICLES

D.C. Glattli

Abstract Quantum partition noise is a recent ﬁeld, both theoretical and experimental, of the physics of quantum conductors. We review here some basic important properties of the electron shot noise. We discuss how its sensitivity to both quantum statistics and to carrier charge manifests in ordinary conductors or can be used to detect the topological excitations of the Fractional Quantum Hall Eﬀect.

1

Introduction

Many fundamental phenomena lead us to think that the quantum of charge is indivisible. Indeed, above a typical length scale of a few Fermi, all the free elementary charged particles carry a well deﬁned charge quantum e. At a macroscopic level, charge conservation requires that the total charge of an isolated system be also a multiple of e. This robust property, established by Millikan in 1911 [1], has been even extended to non isolated systems. When the electrons of a conductor are delocalized by tunneling into a macroscopic circuit, single electron tunneling eﬀects show that e is still the quantum unit which governs the transport properties [2]. At a more microscopic level, it was less evident that, for conductors made of a large number of interacting electrons, the elementary excitations above the ground state carrying the current, the quasiparticles, should necessarily be associated with the quantum of charge e or its multiples, although this is what was observed in all ordinary conductors. However during the last ﬁfteen years it has been predicted that low-dimensional systems may exhibit quasiparticles with fractional charge. These systems – for example one-dimensional commensurate conductors [3], 2D electrons in high magnetic ﬁelds [4], and the theoretical one-dimensional integrable models such as the Calogero-Sutherland model [5] – share the common property that the number of single particle quantum states involved to built the ground state c EDP Sciences, Springer-Verlag 1999

554

Topological Aspects of Low Dimensional Systems

is larger than the number of particles and the ratio can be fractional. An elementary excitation above the ground state is built by simply emptying a quantum state thus leaving a hole with unbalanced fractional charge. 2D electrons in perpendicular magnetic ﬁeld were the best candidate to show this eﬀect. While the electron number N is ﬁxed by construction, the number of quantum states NΦ can be varied at will by the magnetic ﬁeld, one quantum state for one ﬂux quantum φ0 . A measure of the ratio N/NΦ , also called ﬁlling factor ν, is the Hall conductance whose classical expression can be re-written as (e2 /h)N/NΦ , where e2 /h is the conductance quantum. When by varying the magnetic ﬁeld the ratio approaches a fraction p/(2k+1) one observes the Fractional Quantum Hall Eﬀect (FQHE) [6] signaled by p e2 /h. A gap occurs resulting from the a quantized Hall conductance 2k+1 interactions and the Fermi statistics. An elementary excitation above the gap is for example a hole in the collective ground state wavefunction with the size of a ﬂux quantum, i.e. an empty quantum state, with fractional charge −e/m. In addition theses excitations are believed to obey fractional statistics. A physical quantity which can be sensitive to both charge and statistics is the current noise. The current ﬂuctuations due to the discreteness of carriers, called shot noise, provides a unique tool to measure the carrier charge. The noise is a very fundamental quantity which brings information diﬀerent from the conductance. While photon noise has been studied for years in quantum optics [7] this is only recently that is has been considered in quantum conductors [8–14]. One can show that the noise not only measures the carrier charge but is in general sensitive to the quantum statistics of the carriers. We will review here some of the recent theoretical and experimental progress made in the understanding of the shot noise. In the ﬁrst part we will discuss the sensitivity of noise to quantum statistics and show how Fermion noise, in ordinary conductors, reaches the limit of binomial partition noise. All these properties established for Fermionic quasiparticles will be useful, in the second part, to understand the noise properties in the special case mentioned above where interacting electrons lead to fractionally charged quasiparticles. We will review the recent theoretical results in this regime and the recent experimental work where the use of current noise has given the ﬁrst direct observation of the Laughlin quasiparticle charge. Finally we will mention possible extensions to use noise as a probe of fractional statistics. 2

Partition noise in quantum conductors

2.1 Quantum partition noise Consider the gedanken experiment shown in Figure 1. A well deﬁned number NI of particles (photons for instance) initially occupy a single quantum

D.C. Glattli: Quantum Partition Noise and the Detection

j k i

i

j k t

i

(> k

j k r

k i

k t

555

e V

r

i

(> s e m i- tr a n s p a r e n t m ir r o r

o n e c h a n n e l q u a n tu m

c o n d u c to r

Fig. 1. A semi transparent mirror or a tunnel barrier are examples of scatters responsible for partition noise.

state |kI representing a wave incident on a transparent mirror. The ﬁnal states correspond to two transmitted |kT and reﬂected |kR outgoing states. The NI particles, ﬁrst in a superposition of the outgoing states, interact with the macroscopic phototubes and, when decoherence occurs, NT and NR = NI − NT particles are recorded in each detector. After many repetitions of the experiment, the average numbers of transmitted and reﬂected particles deﬁne the transmission T = NT /NI and reﬂection R = NT /NI = 1 − T coeﬃcients. The random partitioning of incident particles into outgoing states gives rise to ﬂuctuations of the particle number ∆NT,R = NT,R − NT,R around the average value. Using the binomial NT NT T (1 − T )NI −NT to ﬁnd NT transmitted partiprobability P (NT ) = CN I cles, one obtains: 2 (1) ∆NT,R = T (1 − T )NI = − ∆NT ∆NR · This result expresses the quantum partition noise. As we will see below, it is observable in quantum conductors. The one dimensional conductor with a tunnel barrier of transparency T, schematically shown in Figure 1, is equivalent to a mirror from the partition noise point of view. Electrons incident from the left in the energy range eV are either transmitted or reﬂected. Understanding the weak and strong transmission limit of this fundamental noise will be useful in the second part of this contribution to understand the noise of fractional excitations. 2.2 Partition noise and quantum statistics To perform the previous experiment is not straightforward as one needs to prepare a single incident state occupied by a given number NI of particles. In general, ordinary sources emitting particles (light sources or electron reservoirs for instance) are described by many quantum states [15].

556

Topological Aspects of Low Dimensional Systems

The number of particles in each incident quantum state |kI is no longer well deﬁned and can be represented by a probability distribution, for instance the thermal distribution. The previous results have to be averaged over all possible NI according to a statistical weight [16]. To calculate the new values of the ﬂuctuations in the outputs we only need to know NI and NI2 (here bars correspond to averaging over the NI while brackets correspond to the previous averaging over the binomial distribution). Using this scheme, one ﬁnds NT = T NI and NT2 = T 2 NI2 +T (1−T )NI , from which we get: ∆NT2 = T 2 ∆NI2 + T (1 − T )NI .

(2)

We see that the particle noise is the sum of the incident particle noise ∆NI2 , reduced by the square of the probability to cross the scatter, and of the partition noise in the absence of ﬂuctuations in the incident channel. Similarly the cross correlation gives NT NR = T (1 − T )(NI2 − NI ) and: ∆NT ∆NR = T (1 − T ) ∆NI2 − NI .

(3)

Results (2) and (3) are valid for any distribution of the NI . For zero ﬂuctuations of the incident particle beam, (1) is recovered. For 2 Poissonian ﬂuctuations, ∆NI = NI , the output ﬂuctuations are also 2 Poissonian, ∆NT,R = NT,R , and are uncorrelated: ∆NT ∆NR = 0. But the main interesting point here is the eﬀect of the quantum statistics. We show below that electrons in quantum conductors behave very diﬀerently than photons in quantum optics [14,17]. For Bosons (like photons) the particle ﬂuctuations ∆NI2 = NI (1+NI ) [18] are larger than the Poissonian noise. From (3) we see that the statistics gives a striking positive correlation in contrast with the negative correlation of the partition noise. The large noise of the incident particles results from the bunching eﬀect of photons because the statistical interaction forces the particles to try to condense in the same states. This noise makes partition noise diﬃcult to observe. For Fermions a very diﬀerent result is found: the Fermi-Dirac statistics makes the incident particle noise sub-Poissonian ∆NI2 = NI (1 − NI ) due to Pauli exclusion. From (3) we see that the outputs are always anticorrelated. Partition noise is easier to detect as we will see in the next section. Using Bose-Einstein (sign + ) and Fermi-Dirac (sign − ) distributions, the explicit calculation of the ﬂuctuations and correlations gives [14,16,17]: ∆NT2 = T NI (1 ± T NI ) = NT (1 ± NT ) 2

∆NT ∆NR = ±T (1 − T )NI .

(4) (5)

D.C. Glattli: Quantum Partition Noise and the Detection

557

2.3 Quantum conductors reach the partition noise limit We will now focus on the Fermi-Dirac statistics corresponding to the case of quasiparticles in ordinary conductors. First let’s consider a single mode quantum conductor with a barrier characterized by a transmission T1 as in Figure 1. Upon an electrochemical potential diﬀerence ∆µ = eV between the left and right reservoirs a current I = (2e2 /h)T1 V ﬂows through the conductor. In the zero temperature limit, the current is only due to states |kI in the energy range eV emitted by the left reservoir and elastically transmitted to the right in states |kT with probability T1 . Because of Fermi-Dirac statistics, the incident states are occupied by one and only one electron (NI = 1 for each state |kI ) and the incident ﬂow of electrons is noiseless. The ﬂuctuations in the population NT of the transmitted states, which in turn gives the current noise, can only come from the partition 2 noise. Using (4) and NT = 1, one ﬁnds: ∆NT,R = NT (1 − T1 ). This is the binomial partition result expressed in the form of a Poissonian law with a reduction factor (1 − T1 ). We are now ready to calculate the current noise. For Poisson’s statistics, it is given by the well known Schottky formula ∆I 2 = 2e I ∆f where ∆f is the frequency bandwidth [19]. The expression is valid for frequencies I/e (d.c. limit). For energy independent transmissions in the energy range eV, it is now straightforward to write the spectral density SI = ∆I 2 /∆f of the current noise in a single mode quantum conductor [10]: SI = 2eI(1 − T1 ).

(6)

This is the fundamental result for the Fermions noise. The noise of a quantum conductor is sub-Poissonian and, at zero temperature, reaches the partition noise limit. In particular for unit transmission the current noise vanishes as the noiseless incident ﬂow of electrons is fully transmitted. Observation of such remarkable eﬀect, as shown below, is a direct proof of the long range temporal ordering of the electrons by the Fermi statistics. While Poissonian shot noise is the hallmark of electron granularity, the absence of noise expresses that indistinguishable Fermions completely loose all granular features. The results can be generalized to more realistic situations. For energy eV dependent transmission: SI = 2e(2e/h) 0 dεT1 ()(1 − T1 ()) with I = eV (2e/h) 0 dεT1 (). For a general quantum conductor, with many modes, transmission Tn , the spectral noise density is: Tn (1 − Tn ) Tn (1 − Tn ) = 2eI n (7) SI = 2e(2e/h)V n Tn n with I = (2e2 /h)V n Tn , and energy independent transmissions are used for clarity. Eﬀect of a ﬁnite temperature θ can also be included. There is a

558

Topological Aspects of Low Dimensional Systems

continuous transition from thermal noise, or Johnson-Nyquist noise, where SI = 4GkB θ, to shot noise. In short, thermal noise can be viewed as the emission noise of the reservoirs because the number NI of particles occupying the states emitted by the left and right reservoirs is no longer 1 or 0 but ﬂuctuates according to the Fermi distribution [11]. The results derived in section II-B could be used when eV kB T , however, when eV becomes comparable or lower than kB θ there is an important correction not included. As the energy range to be considered now extends a few kB θ beyond eV, there is a non negligible probability that particles simultaneously emitted from right and left reservoirs at the same energy end up in the same outgoing state. This situation is forbidden by the Pauli principle and a special counting of these events has to be done. A proper account of this eﬀect has been done using various approaches. Reference [14] uses a wavepacket approach and counting arguments, reference [16] uses a multinomial distribution approach and in references [12, 17] a second quantiﬁcation approach similar to that used in quantum optics has been chosen. All approaches give the following result (again written here for energy independent transmissions for clarity): eV Tn2 + 2e(2e/h)V Tn (1 − Tn ) coth . (8) SI = 4kB θ(2e2 /h) 2kB θ n n 2.4 Experimental evidences of quantum partition noise in quantum conductors The last ﬁfteen years achievements in molecular beam epitaxy techniques have provided experimentalists with nearly ideal 2D metals. They are made from a 2D electron gas conﬁned at the interface between two semiconductors (GaAs and Ga(Al)As) as described in the previous lectures. Electrons can move ballistically over distance as long as 10 µm. At low temperature θ 1 Kelvin the phase coherence is limited by electron-electron collisions and the coherence length is larger than few µm. In addition achievable low densities, typically nS 1015 m−2 , give large Fermi wavelengths λF 75 nm comparable to present lithography resolutions. Also as the 2D electron gas is not far from the surface sample 100 nm and the density is low, a metallic gate evaporated on the surface can deplete easily the electrons with a few hundred mVolt negative voltage. Combining these techniques, it is possible to realize Quantum Point Contacts (QPC) which are constrictions in the 2D electron gas whose size can be continuously varied below and above λF by changing the potential of gates patterned on the surface sample, as shown in Figure 2. The QPC can be view as a short waveguide connecting two parts of the 2D electron gas (the reservoirs). The control of the waveguide width allows for selection of the number of transmitted electronic modes and for

D.C. Glattli: Quantum Partition Noise and the Detection

559

Fig. 2. Upper left: vertical structure of a GaAs/Ga(Al)As structure used to deﬁne a 2D electron system; lower left: a negative potential on gates evaporated on the surface can be used to realize a QPC in the 2D electron system; on the right is shown the QPC conductance versus gate for the ﬁrst two modes (black circles represent the transmission values used for the noise measurements displayed in Fig. 3).

Fig. 3. left part: noise versus voltage bias for T1 = 1/6, 1/4, 1/2, and 3/4; the solid lines are comparison with theory (no adjustable parameters) and for clarity each curves are horizontally shifted by 100 mk; right part: noise measured at ﬁnite temperature showing the transition from thermal to shot noise (solid lines are comparison with theory).

560

Topological Aspects of Low Dimensional Systems

the accurate control of their transmission Tn . The sum of the transmission coeﬃcients can be determined by measuring the conductance via the Landauer formula G = (2e2 /h) n Tn [20]. In particular a conductance plateau at G = p(2e2 /h) is found each time p modes are fully transmitted [21, 22]. By using a QPC as an artiﬁcial tunable scatter and by combining both current noise and conductance measurements, the fundamental prediction of noise suppression due to the Fermi-Dirac statistics has been successfully tested and the electron partition noise formula at zero and ﬁnite temperature have been checked with high accuracy [23, 24]. Noise measurements are diﬃcult because of the weak signal to detect. Indeed, the Fermi energy, which gives the energy scale separating consecutive electronic modes, is small for 2D electrons in GaAs/Ga(Al)As heterojunctions, typically 40 Kelvin. The electro-chemical potential diﬀerence eV must be smaller than a few Kelvin and, for conductances lower than the quantum of conductance the associated currents are less than a√few nA. The Poissonian noise for a current of 1 nA corresponds to 18 fA/ Hz and a higher resolution is needed to observe the quantum noise suppression. Early attempts have been made which [25, 26] showed indication of noise suppression but the ﬁrst clear observation of suppressed shot noise can be found in reference [23]. A quantitative test of the partition noise, including thermal eﬀects, and the observation of noise suppression as high as 90% can be found in reference [24]. Here we show in Figure 3, left part, the data taken for four values of the transmission T1 of the ﬁrst mode at negligible temperature. For convenience, the noise is plotted in units of equivalent noise temperature T ∗ = SI /4GkB and the voltage bias is also plotted in temperature units. This allows to plot all curves in the same graph. We see a linear variation of noise with bias as expected. The slope of the variation decreases with T1 . In this units the slope is the partition noise reduction factor 1 − T1 , as T ∗ = (1 − T1 )eV/2kB . There are no adjustable parameters and the agreement with predictions is excellent. In the right part, we have displayed the continuous transition from thermal noise to partition shot noise. Again quantitative agreement is very good. A plot of the noise reduction factor SI /2eI in Figure 4 shows that the partition noise limit is actually reached for the ﬁrst two modes. At nearly unit transmission T1 1, T2 0 and T1 = 1, T2 1 the noise suppression is larger that 90% showing that Fermi statistics generates fundamentally noiseless electrons. Other evidence of the fact that electrons reach the partition noise limit have been found in diﬀusive samples. As the probability distribution of the transmission eingenvalues F ({Tn }) is known for a diﬀusive system, the fac1 tor n Tn (1−Tn) in equation (7) has to be replaced by 0 T (1−T )F (T )dT . The result is a noise suppression factor equal to 1/3 plus small weak localization and universal ﬂuctuations corrections [27]. The ﬁrst term of the noise reduction factor, the 1/3 term, is expected to survive in the regime

D.C. Glattli: Quantum Partition Noise and the Detection

561

Fig. 4. Noise reduction factor SI /2eI measured for the ﬁrst two modes T1 + T2 ≤ 2. The dashed curve show comparison with theory assuming ideal QPC. For T1 + T2 = 1 and 2 the noise suppression is larger than 90%.

where coherence is lost (but Pauli exclusion still holds) [28, 29]. In this regime, the 1/3 suppression has indeed been observed in conductors [30,31]. The eﬀect of the Fermi-Dirac statistics has also been tested using multiterminal conductors [32] following a suggestion made in references [17] and [14]. Reference [17], followed by more recent work [33], emphasizes the role of exchange in multiterminal conductors where correlation between the current ﬂuctuations of diﬀerent contacts are measured. Partition noise is also aﬀected by a radiofrequency electrical ﬁeld, frequency f , and shows singularities at eV = nhf . This comes from photon assisted mixing at energy ε and ε ± nhf of the electron population of right and left moving outgoing states [31, 34]. The partition noise in quantum conductors is now so well established that it can even be used as a sensitive tool to determine the Tn in atomic point contacts complementary to that obtained with conductance [35]. This is a whole ﬁeld of mesoscopic physics now and a fair review goes beyond the scope of this contribution. How interaction aﬀect partition noise? Above we have considered non interacting electrons. In real ordinary 3D or 2D metals, which are Fermi liquid, bare electrons do not participate directly to the conduction but are to be replaced by the Landau quasiparticles deﬁned close to the Fermi surface. For time scales lowers than an energy dependent decay time, which gives an upper bound to the coherence length, the quasiparticles behave like fermions carrying charge e. This is why the above shot noise predictions are experimentally observed. In the special case of strong electron correlations built in normal metals by the proximity of a superconducting

562

Topological Aspects of Low Dimensional Systems

conductor, noise and conductance can still be treated in a similar frame with little complications for energies smaller than the superconducting gap. Here again, because of Fermi statistics, electrons and holes emitted by the normal region toward the superconductor are noiseless. The only noise is the binomial two-quasiparticle partition noise. Are two normal quasiparticles Andre’ev reﬂected as a Cooper pair or not? In the case of a single 2 mode, if |she | denotes the Andre’ev reﬂection probability, the current be2 tween normal metal and the superconductor is proportional to |she | while 2 the noise is reduced from the Poissonian value by (1 − |she | ), in perfect analogy with (1 − T1 ) reduction of pure normal conductors [36–38]. Can simple partition noise models be found to describe other correlated electron systems such as the FQHE or one-dimensional conductors? In these systems electron interactions have a stronger eﬀect and the Fermi liquid description is lost and must be replaced by a Luttinger liquid description. In the second part of this contribution we discuss recent theoretical ﬁndings going into this direction [39–41, 43–45]. This is a hot and topical question. From the experimental side only the Poissonian shot noise regime, which recently allowed determination of the Laughlin fractional charge [46, 47], seems to be well understood. 3

Partition noise in the quantum hall regime and determination of the fractional charge

3.1 Edge states in the integer quantum Hall eﬀect regime In the simplest picture of the Integer Quantum Hall regime [48] where interactions can be neglected as a ﬁrst approximation, valid at ﬁlling factor ≥ 2, the fundamental current noise associated with electron conduction is not fundamentally diﬀerent than the partition noise discussed previously in ballistic samples. Here we are not discussing the regime of macroscopic samples where the conduction occurs in the bulk of the sample via hopping through localized states and where edge eﬀects can be neglected. Instead we are going to consider clean narrow samples where the conduction occurs only on the edges [49]. By the use of the QPC technique, a narrow constriction can bring opposite edges close together. The overlap between wavefunctions can induce an electron transfer from one edge to the others [50]. The current noise can be described by the same model as in the previous section for low ﬁeld quantum conductors. In order to simply extract the physics in high magnetic ﬁeld, it is useful to recast the canonical pairs of electron coordinates [x, px ] and [y, py ] into a new set of conjugate pairs of coordinates [ξ, η] = [vy /ωc , −vx /ωc] = −i/eB and [X, Y ] = i/eB, where (x, y) = (X + ξ, Y + η). The new coordinates describe respectively the (fast) cyclotron motion relative to the (slow)

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563

cyclotron orbit center motion. The Hamiltonian of electrons in magnetic ﬁeld H = (p + eA)2 /2m becomes H = 12 mωc2 (ξ 2 + η 2 ) and gives quantized cyclotron orbits and Landau levels Ep = (p − 12 )ωc . As the system is 2D and the energy depends on a single quantum number, there is a “missing” quantum number. This means that the Landau levels are highly degenerate. The degeneracy is equal to the number of way to put the center of cyclotron orbits in the plane. It is equal to the number of ﬂux quanta Φ0 = h/e in the plane. Indeed the orbit center coordinates (X, Y ) do not commute and the 2D electron plane is analogous to the phase space [P, Q] of a one-dimensional system. In the latter a quantum state occupies an area h, in the former a quantum state occupies the area of a quantum ﬂux h/eB. This gives the number of states available in the plane. The Quantum Hall Eﬀect oﬀers the unique fascinating possibility to inject electrons in the four corners of the phase space and to be able to tune the “Planck’s constant” h/eB by changing the magnetic ﬁeld. In presence of an external potential V (X, Y ) not strong enough to mix Landau levels, the dynamic of the cyclotron orbit center within a Landau level corresponds to a drift along the equipotential lines. The motion given by X˙ = (1/eB)∂V /∂Y , Y˙ = −(1/eB)∂V /∂X expresses the compensation of the electric ﬁeld by the Lorentz force to keep 12 m |v|2 constant. As a result, the conﬁning potential at the edge of the sample gives persistent currents running along the boundary. The potential bends the Landau levels and at the crossing with the Fermi level the gap to create a hole vanishes. The lines of gapless excitations, one per Landau level, gives rise to one-dimensional chiral conductors called Edge Channels, see Figure 5. This is where conduction takes place in clean narrow samples. 3.2 Tunneling between IQHE edge channels and partition noise Using the analogy between the phase space (P, Q) of a one-dimensional conductor and the plane (X, Y ) for the motion of cyclotron orbit centers of electron condensed within the ﬁrst orbital Landau level, it is easy to ﬁnd the noise. Upon applying a potential diﬀerence between contacts, electrons can be injected from the upper left edge within an energy range eV above the Fermi energy of electrons coming from the lower right edge. A Quantum Point Contact can be used to create a barrier coupling the lower and upper right channel, see Figure 6. Electrons from the upper left can be either transmitted (transmission T ) into the upper right channel or reﬂected (R = 1 − T ) into the lower left channel. The incident electrons at the upper left and lower right channels are in equilibrium and, at low tem2 perature, the Fermi-statistics makes the incident current I0 = eh V noiseless. The outgoing electrons at upper right and lower left edges give rise 2 2 to non-equilibrium forward I = T eh V and backward IB = R eh V currents.

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Topological Aspects of Low Dimensional Systems

B z^

(N ,O )

. F

( e d g e c h a n n e l )

X , Y

E

h w

n = 3 c

n = 2

V

n = 1

D

E

C o n f.

Fig. 5. The bending of Landau levels by the conﬁnement potential leads to edge channels. They can be view as 1D chiral conductors on the edges.

The electron partition generates current 2 ﬂuctuations which are anticorre = T (1 − T )2eI0 = 2eI(1 − T ). lated with − ∆I∆IB = ∆I 2 = ∆IB The result is identical to the case of a one mode quantum conductor except that here one can physically separate incoming and outgoing electrons. Generalization to ﬁnite temperature and to many edge channels gives the same noise formula as equation (8) but where the Tn represent the transmission of each edge channel (note 2e2 /h has to be replaced by e2 /h as spin degeneracy is lifted). There are two interesting limits which suggest a duality between electrons and holes and which will be useful to consider later in the fractional regime. For strong backscattering T 1, or I I0 , the noise SI 2eI is due to Poissonian transmission of electrons with current I. For weak backscattering R 1, or IB I0 , the noise SI 2eIB is due to Poissonian reﬂection of electrons or alternatively Poissonian transmission of holes with current IB as schematically shown in Figure 6. 3.3 Edge channels in the fractional regime The fractional Quantum Hall eﬀect is fundamentally an intra Landau level physics [51]. The Coulomb correlation between cyclotron orbit center combined with the Fermi statistics of the electrons favors the formation of new quantum liquids at simple fractional ﬁlling factors ν = p/(2k + 1). The ground state of a fractional quantum Hall liquid is incompressible.

D.C. Glattli: Quantum Partition Noise and the Detection S T R O N G

B A R R IE R

i

t

: T < < 1

E

W E A K B A R R IE R

e V

r

> > 1

e V F

Y n

: T

565

I

Y X n

IB

X

Fig. 6. A QPC can be used to create a potential reﬂecting edge states. The weak and strong barrier are two limiting cases giving rise to Poissonian noise of transmitted holes SI = 2eIB or electrons SI = 2eI respectively.

To change locally the electron density or the local ﬁlling factor quasi-electron 1 or quasi-hole excitations with gap ∆ ∼ e2 /(/eB) 2 are needed. The most striking prediction about the excitations is that they should carry fractional charge ±e/(2k + 1) [4] and obey fractional statistics [52, 53]. Here we will focus on the physics of edge states in this regime where the shot noise can be used to measure the charge of the fractional quasiparticles. As in the integer regime, the gap vanishes when the ground state energy level, bent by the external potential, crosses the Fermi energy. A new chiral one-dimensional conductor forms, called fractional edge channel. Its properties are determined by the bulk. It is believed that it is not a onedimensional Fermi liquid but that it is similar to a Luttinger liquid [54], i.e. like one dimensional electron systems with short range interactions characterized by a parameter g [55]. For the simplest and the more robust fractional states with ν = 1/(2k + 1) on have g = ν. For higher fractions in the bulk such as ν = k/(2k + 1), theoretical models based on fundamental symmetries of the wavefunctions propose that k fractional edge channels are formed [56], but the understanding is not yet complete and, in experiments, the smooth decrease of the electron density on the edges may leads to more edges states [57, 58]. However these outer edge states in most cases should not participate in the backscattering mechanism that we are going to discuss below. Here we will focus on the most understood ν = 1/3 Fractional Quantum Hall state. A simpliﬁed view, which nevertheless contains most of the fractional edge state physics, is to consider the ν = 1/(2k + 1) quantum Hall liquid as an incompressible strip of width 2D along the x ˆ axis with shape deformations y+/− (X, T ) of the upper and lower edge [54]. The electron density is n(X, Y, t) = ns {Θ(Y − y+ − D) − Θ(Y + y− + D)} where ns = νeB/h.

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Topological Aspects of Low Dimensional Systems

˙ The conservation of the density ∂n/∂t + X∂n/∂X + Y˙ ∂n/∂Y = 0 and 1 ∂V /∂Y and of Y˙ = 0 leads to chiral propagation the use of X˙ = eB ∂y

∂y

+/− +/− of the shape deformation at each edges: ± vD ∂X = 0, where ∂t 1 ∂V vD = ± eB ∂Y |±D is the drift velocity. The energy for the upper edge 2 ∂V deformation is ER = dX 12 ns y+ ∂y . By deﬁning the charge variation inteX φR grated on the upper edge as π = −∞ ns y+ dX, the following Lagrangian ∂φR R ∂φR dXdt ∂φ LR = − πν ∂X ( ∂t + vD ∂X ) leads to the previous propagation equation for y+ , to the Hamiltonian HR = ER and to the conjugate momen R tum πR = − ν ∂φ ∂X , with [πR (X), φR (X )] = iδ(X −X ). Similar deﬁnitions for the lower edge give the total Hamiltonian for the two decoupled modes:

vD H= ν π

dX

∂φR ∂X

2 +

∂φL ∂X

2 ·

(9)

This bosonization of the edge modes is the starting point of the Luttinger liquid model for the edge states. The physics of the bosonic modes would not be interesting unless one have to consider the transfer of bare electrons or Laughlin quasiparticles when a scatter couples one edge to an other. The non trivial physics arise from the fact that adding an electron on one edge involves an inﬁnite number of bosonic modes as the electron creφ(X) ation operator takes the form Ψ+ e (X) ∼ exp(−i ν ). This is the source of non-linear transport. Another remarkable consequence of this simple phenomenological model is that the requirement that Ψ+ describes Fermions implies ν = 1/(2k + 1), a Laughlin fraction. Finally, in this model the Laughlin quasiparticle creation operator becomes Ψ+ q (X) ∼ exp(−iφ(X)). Non linear transport is the hallmark of the fractional quantum Hall edge conduction because of Luttinger liquid properties. The tunneling conductance G() depends on the energy = eV or kB θ to which charge is transferred. At energy lower than a characteristic coupling energy εB , strong 1 backscattering occurs with G() ∼ (/εB )2( ν −1) , while for B one have a weak backscattering regime where the conductance asymptotically reaches 2 the Hall conductance: ν eh − G() ∼ (B /ε)2(1−ν) [54]. Clear evidence of Luttinger liquid have been found in tunneling experiments. For tunneling from a metal to the edge, a good quantitative agreement with theory has been found for ν close to 1/3 [59]. For inter-edge tunneling quantitative agreement is less convincing but at least the features found are qualitatively those predicted [60]. Understanding the transport is important if we want to have a complete understanding of the partition noise in this regime. Fortunately the weak and strong backscattering limits can lead to very simple Poissonian noise predictions independent on the exact dynamics of the edge states.

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567

3.4 Noise predictions in the fractional regime Determination of the Laughlin quasiparticle charge using shot noise has been suggested by experimentalists rather early [51]. The Poissonian shot noise of the current is a direct consequence of charge granularity. Indeed, the general Schottky formula tells that SI = 2qI where q is the carrier charge. The simultaneous measure of the average current and its ﬂuctuations gives a simple direct measurement of q, free of geometrical parameters. In addition, this is an non-equilibrium experiments, a necessary condition to probes the quasiparticles, that is the excitations above the ground state carrying current. A theoretical proposition taking into account the speciﬁc Luttinger liquid dynamics was given in reference [39]. A single localized scatter, for example a QPC or an impurity, couples two fractional edges and induces tunneling. The results, obtained in the weak and strong backscattering regime, are respectively: SI 2eI coth(eV/2kB θ) SI 2(νe)IB coth((νe)V /2kB θ)

I I0 =

1 e2 V 3 h

IB = I0 − I I0 .

(10) (11)

Here, the thermal noise contribution is not included. For low conductance, the Poissonian noise of electrons, charge e current I, is found: only electrons can tunnel. Fractional excitations, being a collective electron phenomena, are not expected in a nearly insulating region. For high conductance, or weak eﬀective barrier, the Poissonian noise of fractional charges νe, current IB , is found: fractional excitations may be expected inside the weakly perturbed fractional region. In this limit, the noise thus provides the way to measure νe. A fractional νe charge is also found in the coth function. The cross-over from thermal noise to shot noise occurs at a voltage ν −1 larger than the one found in the case of pure electron tunneling. Indeed, the thermal cross-over corresponds to electro-chemical potential diﬀerence ∆µ = νe comparable to kB θ. Observation of a larger voltage for thermal cross-over in noise experiments, see below, has been an important conﬁrmation of equation (11). However, this is not a measure of the fractional quasiparticle charge. This is a measure of the average charge per quantum state at equilibrium, like the conductance νe2 /h. Only the shot noise SI 2(νe)IB really measures the quasiparticle charge. The zero temperature limit of expressions (10) and (11) have been also derived in [40] using Luttinger liquid approach. An exact solution for a single δ scatter has been obtained, also at zero temperature. The result predicts both the non-linear conductance versus bias and the shot noise [41]. The tunneling problem is shown to be integrable using results of the conformal ﬁeld theory. Separating the ﬁelds φR,L into even and odd ﬁelds φe,o = √12 (φR (X, t) ± φL (−X, t)) the even modes are conserved and do not

568

Topological Aspects of Low Dimensional Systems

contribute to conductance and noise while the odd modes do. For negative X, φe describes a unit noiseless current of charges incident on the scatter and for positive X it describes a noisy (smaller) current of outgoing charges. Once folded into a semi-inﬁnite line the model corresponds to the integrable boundary Sine-Gordon model. Using a special basis of kink and √ anti-kink solitons, charge ± 2e, and noting that the scatter reduces the transmitted current by transforming incoming kinks into outgoing antikinks, a Landauer type approach, similar to the wavepacket approach of reference [14] for fermions, gives the conductance and the noise. As for the Fermion noise discussed previously, one have a binomial partition noise: 2 is a kink transform into an antikink or not? If |S+− (α − αB )| denotes the probability to transform a kink of energy parametrized by α into an antikink, αB is related to the scatter strength, and ρ(α) is the density of states of incoming kinks, one have:

IB (V ) = −evD SI (V ) = 2e2 vD

αMax (V )

−∞

αMax (V ) −∞

2

dαρ(α) |S+− (α − αB )| 2

(12) 2

dαρ(α) |S+− (α−αB )| (1− |S+− (α−αB )| ).(13)

Exact expression and technical mathematical details can be found in references [41] and in this book in a previous lecture by Saleur. The special simple form of |S+− (α − αB )| leads to a relation between current and noise v dI v B where SI = 1−v (V dV − I) = 1−v (IB − V dI dV ). From it, using the weak and strong backscattering limits of the Luttinger theory, we can easily check that SI → 2(νeIB ) and 2eI respectively in agreement with the zero temperature limit of (10-11). Finite temperature predictions can also be found in reference [42]. Finally, we would like to mention the theoretical work of reference [43] on the noise between an ordinary metal and a ν = 1/3 edge. Here no e/3 tunneling can be expected as fractional charges cannot exist in the metal. Surprisingly the weak (high bias) scattering limit gives an apparent noise of charge e/2. Asymptotically the conductance G → 12 e2 /h and the current is noiseless. This means that one half of the electrons incoming from the metal side and transmitted in the ν = 1/3 are strongly correlated. If they were uncorrelated a one-half transmission would correspond to maximum partition noise. Thus the outgoing states correspond to a regular ﬂow of one half fractionally ﬁlled quantum states. For slightly less conductance some of these one half ﬁlled states are empty leading to Poissonian noise of charge 1/2. This is a remarkable proximity eﬀect. We don’t know yet if this could be experimentally realizable.

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569

3.5 Measurement of the fractional charge using noise A major diﬃculty of an experimental implementation of the shot noise measurements in the FQH eﬀect is that the extremely low shot noise has to be extracted from the background of relatively large ampliﬁers noise. Shot noise levels are extremely small both due to the smaller charge and the small available current. The latter is restricted by the fact that the FQH eﬀect breaks down when the applied voltage is larger than the excitation gap. This excitation gap, in turn, depends crucially on the quality of the material in which the 2DEG resides. The state of the art technology currently yields samples with an excitation gap of the of the order of a few 100 µ eV, leading to shot noise levels in the 10−29 A2 /Hz range. Recently two groups [46, 47], succeeded in doing these diﬃcult measurements nonetheless. A QPC was used by both groups in order to realize a local and controllable coupling between two ν = 1/3 fractional edges to partially reﬂect the incoming current. Both experiments where designed to have a best sensitivity for the weak coupling limit where Poissonian noise of the e/3 Laughlin quasiparticles were expected. In one experiment a cross correlation technique [46] detects at low frequency the anticorrelated noise of the transmitted current I and the reﬂected current IB , i.e. SI,IB = ∆I∆IB /∆f −2(e/3)IB . The magnetic ﬁeld corresponded to a ﬁlling factor 2/3 in the bulk of the sample and a small region of ﬁlling factor 1/3 was created close to the QPC using the depletion eﬀect of the gates. The size of the 1/3 region was estimated about 150 φ0 . The advantage of doing this is that the coupling between edges occurs on a shorter scale and the controllable QPC potential is larger than the potential ﬂuctuations inherent of sample fabrication. In the two samples measured, the combination of QPC and random potential lead to two dominant paths for backscattering. The coherent interference between paths gives rise to nearly perfect resonant tunneling peaks in the conductance. Careful measurements of the conductance resonance showed that tunnelling was coherent. This was an important check for the quasiparticle charge measurement because this ruled out the possibility of noise suppression due to multiple uncorrelated steps, similar to the 1/3 noise reduction factor in zero ﬁeld diﬀusive conductors. Also the resonant conductance showed non-linear dependence on bias voltage consistent with Luttinger liquid model provided the ﬁlling factor of the bulk is used. The other group [47] used a high frequency technique in order to increase the signal bandwidth and measured the autocorrelation of the transmitted current. Here the magnetic ﬁeld corresponded to a ﬁlling factor 1/3 throughout the sample. They found few non-linearities in the conductance, in contrast with the Luttinger liquid predictions, and this allowed them to deﬁne a bias voltage independent transmission. In the Poissonian limit IB I0 , both groups arrived at the same conclusion (see Fig. 7) that near ﬁlling factor 1/3, shot noise is threefold

570

Topological Aspects of Low Dimensional Systems

Fig. 7. Experimental Poissonian noise of the fractionally charged excitations of the FQHE, from reference [46] (left) and reference [47] (right). Authors of the last reference used a Fermion noise reduction factor to analyze their data.

suppressed – giving a the ﬁrst direct evidence that the current can be carried by quasiparticle with a fraction of e and that Laughlin conjecture was correct. In addition, the data showed a cross-over from thermal noise to shot noise when the applied voltage satisﬁes the inequality eV/3 > 2kθ (rather than eV > 2kθ), indicating that the potential energy of the quasiparticles is threefold smaller as well as predicted in equation (11). More recent measurements close to ν = 2/5 give indications that the e/5 quasiparticles are the relevant excitations in this regime [61]. This last result is analyzed in a model of non-interacting composite Fermions where Luttinger eﬀects are neglected [62]. 3.6 Beyond the Poissonian noise of fractional charges In the ﬁrst part of the paper we have discussed the diﬀerence between the noise of Bosons and Fermions and shown that SI = 2eIB (1 − R). Is there a similar relation for quasiparticles for which one believe they obey fractional statistics? Beyond the Poissonian noise, i.e. for IB /I0 no longer small, we may expect a reduced noise due to the correlation between quasiparticles: SI = 2(e/3)(1 − R(IB /I0 )) < 2(e/3)IB . This is indeed what implicitly tells equation (13) and the reduction factor 1 − R(IB /I0 ), now a non trivial function of IB /I0 , can be obtain from the explicit solution given in reference [41]. Experiments indeed show a reduced Poissonian noise at ﬁnite IB /I0 . In reference [47] where transport seems linear, a good agreement is even found using the Fermionic (1 − R) reduction factor, while in reference [46], where nonlinearities compatible with Luttinger properties are found, the noise seems systematically close but above the Fermion noise. More accurate experiments are needed to understand this non-Poissonian

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571

regime. It would be also very interesting to physically understand in a uniﬁed frame the noise of Bosons, Fermions and of quasiparticles obeying fractional statistics. In the excluson approach of fractional statistics as deﬁned by Haldane [53, 63], the symmetry is broken between electrons and hole. For a statistics with β = 3 corresponding to ν = 1/3, the electrons fractionally ﬁll the quantum states while the holes correspond to empty quantum states as Laughlin excitations do. There is a duality where electrons obey a super-Pauli principle by excluding β quantum states while holes obey fractional statistics 1/β, intermediate between Fermions and Bosons. Noting that the average of the density of electron ne and of holes nh per quantum states satisﬁes nh + βne = 1 and using the thermal distribution for exclusons [63], one can get the thermal ﬂuctuations ∆n2e = ne nh (ne + nh ). From this we can see that the noise of holes is above the Fermion noise: ∆n2h = nh (1 − nh )(1 + (β − 1)nh ) in qualitative agreement with the idea that ﬂuctuations should be between that of Fermions and that of Bosons. If we consider the backscattered current as a forward current of transmitted holes IB = (e/3)(eV/h)nh , and deﬁne the noise power as SI = 2(e/3)(eV/h)∆n2h , the correct limits for the noise at weak (nh → 0) and strong (nh → 1) backscattering are found. Alternatively, one can use a dual electron representation where the forward current is I = (eV/h)ne and the noise power is SI = 2e(eV/h)∆n2e with the limits ne → 1/3 or 0 and ﬁnd the same result. However there is no existing justiﬁcation for this procedure, which assumes that the relation between ﬂuctuations and average values is the same for partition and thermal noise. Note that this would be correct in the case of Fermions and Bosons. Also the composite Fermion model of reference [62] gives a similar result. The exclusonic thermal ﬂuctuations however could correctly be plug in equation (2) when considering the partition noise of a single reservoir. In the general case where two reservoirs emit counterpropagating particles, a complete treatment of both fractional statistics and partition noise is needed, but still awaiting. Recent theoretical progress in this direction have been made [44, 45]. Full understanding may give a hope that partition noise experiments in the FQHE regime would allow experimental observation of fractional statistics. The author would like to acknowledge the contribution of his close collaborators having actively participated in the noise experiments in Saclay: A. Kumar, L. Saminadayar, Y. Jin and B. Etienne, and of P. Roche for critical reading. Invaluable discussions with theoreticians H. Saleur, Th. Martin, S. Ouvry, V. Pasquier, S. Isakov, I. Saﬁ, N. Sandler; and many others are also acknowledged.

References [1] Millikan R.A., Chicago: Univ. of Chicago Press (1917). [2] Fulton T.A. and Dolan G.J., Phys. Rev. Lett. 59 (1987) 109.

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Topological Aspects of Low Dimensional Systems

[3] Su W.P. and Schrieﬀer J.R., Phys. Rev. Lett. 46 (1981) 738. [4] Laughlin R.B., Phys. Rev. Lett. 50 (1982) 1395-98. [5] Calogero F., J. Math. Phys. 10 (1969) 2191; Sutherland B., J. Math. Phys. 12 (1971) 246; see for example Serban D., Lesage F. and Pasquier V., Nucl. Phys. B 466 (1996) 499 and refernces therein. [6] Tsui D.C., St¨ ormer H.L. and Gossard A.C., Phys. Rev. Lett. 48 (1982) 1559. [7] See for example “Quantum Fluctuations”, edited by Reynaud S., Giacobino E. and Zinn-Justin J.J., Les Houches Nato ASI Session LXIII (North Holland, 1995); see also Loudon R., “The Quantum Theory of Light”, Clarendon, Oxford (1973). [8] For a review, see Th. Martin, in Coulomb and Interference Eﬀects in Small Electronic Structures, Glattli D.C., edited by Sanquer M. and Trˆ an Thanh Vˆ an J. ´ (Editions Fronti`eres, Gif-sur-Yvette, 1994); edited by Reznikov M., et al., Superlattices and Microstructures 23 (1998) 901. [9] Khlus V.K., Sov. Phys. JETP 66 (1987) 1243. [10] Lesovik G.B., Pis’ma Zh. Eksp. Teor. Fiz. 49 (1989) 513; [JETP Lett. 49 (1989) 592]. [11] Landauer R., Physica D 38 (1989) 226; Landauer R., Phys. Rev. B 47 (1993) 16427. [12] Phys. Rev. Lett. 65 (1990) 2901. [13] Yurke B. and Kochanski G.P., Phys. Rev. B 41 (1990) 8184. [14] Martin Th. and Landauer R., Phys. Rev. B 45 (1992) 1742; Physica B 175 (1991) 167. [15] For quantum optics so-called non-classical light sources are now produced currently see the lectures on this topics in [7]. [16] Such approach can be found in Imry Y., Chapter 8, Introduction to Mesoscopic Physics (Oxford University Press, 1997), and is also used in [44]. [17] B¨ uttiker M., Physica B 175 (1991) 199; B¨ uttiker M., Phys. Rev B 46 (1992) 12485. [18] Landau L.D. and Lifschitz E.M., Statistical Mechanics (Pergamon Press, 1959). [19] Schottky W., Ann. Phys. (Leipzig) 57 (1918) 541. [20] Landauer R., IBM J. Res. Dev. 1 (1957) 223; 32 (1988) 306. [21] van Wees B.J., et al., Phys. Rev. Lett. 60 (1988) 848. [22] Wharam D.A., et al., J. Phys. C 21 (1988) L209. [23] Reznikov M., et al., Phys. Rev. Lett. 18 (1995) 3340. [24] Kumar A., et al., Phys. Rev. Lett. 76 (1996) 2778. [25] Li Y.P., et al., Appl. Phys. Lett. 57 (1990) 774. [26] Washburn S., et al., Phys. Rev. B 44 (1991) 3875. [27] Beenakker C.W.J. and B¨ uttiker M., Phys. Rev. B 43 (1992) 1889. [28] Nagaev K.E., Phys. Lett. A 169 (1992) 103. [29] Shimizu A. and Ueda M., Phys. Rev. Lett. 69 (1992) 1403. [30] Steinbach A., Martinis J.M. and Devoret M.H., Phys. Rev. Lett. 76 (1996) 3806. [31] Schoelkopf R.J., Burke P.J., Kozhevnikov A.A. and Prober D.E., Phys. Rev. Lett. 78 (1997) 3370. [32] Liu R.C., Odom B. and Yamamoto Y., Nature 391 (1998) 263. [33] Sukhorukov E.V. and Loss D., Phys. Rev. Lett. 80 (1998) 4959; Blanter Y.M. and B¨ uttiker M., Phys. Rev. B 56 (1997) 2127. [34] Lesovik G. and Levitov L.S., Phys. Rev. Lett. 72 (1994) 538. [35] H.E. van den Brom and van Ruitenbeek J.M., cond-mat/9810276 (preprint). [36] Beenakker C.W.J., in Mesoscopic Quantum Physics, edited by Akkermans E., Montambaux G., Pichard J.L. and Zinn-Justin J. (Elsevier Science, Amsterdam, 1994).

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[37] de Jong M.J.M. and Beenakker C.W.J., in Coulomb and Interference Eﬀects in Small Electronic Structures, edited by Glattli D.C., Sanquer M. and Trˆ an Thanh ´ Vˆ an J. (Editions Fronti`eres, Gif-sur-Yvette, 1994); Phys. Rev. B 49 (1994) 16070; de Jong M.J.M., Thesis, University of Leiden (1995). [38] Martin Th., Phys. Lett. A 220 (1996) 137. [39] Kane C.L. and Fisher M.P.A., Phys. Rev. Lett. 72 (1994) 724. [40] Chamon C. de C., Freed D.E. and Wen X.G., Phys. Rev. B 51 (1995) 2363. [41] Fendley P., Ludwig A.W.W. and Saleur H., Phys. Rev. Lett. 75 (1995) 2196. [42] Fendley P. and Saleur H., Phys. Rev B 54 (1996) 10845. [43] Sandler N.P., Chamon C. de C. and Fradkin E., cond-mat/9806335 (preprint). [44] Isakov S.B., Martin T. and Ouvry S., cond-mat/9811391 (preprint). [45] Schoutens K., Phys. Rev. Lett. 79 (1997) 2608; van Elburg R.A.J. and Schoutens K., to appear in Phys. Rev. B, cond-mat/9801272. [46] Saminadayar L., Glattli D.C., Jin Y. and Etienne B., Phys. Rev. Lett. 79 (1997) 2526; cond-mat/9706307. [47] de-Picciotto R., et al., Nature 389 (1997) 162; cond-mat/9707289. [48] von Klitzing K., Dorda G. and Pepper M., Phys. Rev. Lett. 45 (1980) 494. [49] B¨ uttiker M., Phys. Rev. Lett. 57 (1986) 1761. [50] Haug R.J., et al., Phys. Rev. Lett. 61 (1988) 2801. [51] See The Quantum Hall Eﬀect, edited by Prange R.E. and Girvin S.M. (SpringerVerlag, New York, 1987); Mac Donald A.H. in Mesoscopic Quantum Physics, edited by Akkermans E., Montambaux G., Pichard J.L. and Zinn-Justin J. (Elsevier Science, Amsterdam, 1994) and Girvin S.M., this book. [52] Leynaas J.M. and Myrheim L., Nuovo Cimento B 37 (1977) 1; Renn S.R. and Arovas D.P., Phys. Rev. B 51 (1995) 16832; Ouvry S., Phys. Rev. D 50 (1994) 5296; see also Myrheim L., this book for a comprehensive review. [53] Haldane F.D.M., Phys. Rev. Lett. 67 (1991) 937. [54] Wen X.G., Phys. Rev. Lett. 64 (1990) 2206. [55] See a review by Schulz H.J. in Mesoscopic Quantum Physics, edited by Akkermans E., Montambaux G., Pichard J.L. and Zinn-Justin J. (Elsevier Science, Amsterdam, 1994). [56] Wen X.G., Int. J. Mod. Phys. B 6 (1992) 1711. [57] Beenakker C.W.J., Phys. Rev. Lett. 64 (1990) 216. [58] Chklovskii D.B., Matveev K.A. and Schklovskii B.I., Phys. Rev. B 47 (1993) 12605. [59] Chang A.M., Pfeiﬀer L.N. and West K.W., Phys. Rev. Lett. 77 (1996) 2538. [60] Milliken F.P., Umbach C.P. and Webb R.A., Solid State Commun. 97 (1996) 309. [61] Reznokov M., et al., cond-mat/9901150 (preprint). [62] de-Picciotto R., cond-mat/980221. [63] Wu Y.S., Phys. Rev. Lett. 73 (1994) 922.

COURSE 7

MOTT INSULATORS, SPIN LIQUIDS AND QUANTUM DISORDERED SUPERCONDUCTIVITY

MATTHEW P.A. FISHER Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, U.S.A.

Contents 1 Introduction

577

2 Models and metals 579 2.1 Noninteracting electrons . . . . . . . . . . . . . . . . . . . . . . . . 579 2.2 Interaction effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 3 Mott insulators and quantum magnetism 583 3.1 Spin models and quantum magnetism . . . . . . . . . . . . . . . . 584 3.2 Spin liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 4 Bosonization primer 5 2 Leg Hubbard ladder 5.1 Bonding and antibonding bands . 5.2 Interactions . . . . . . . . . . . . 5.3 Bosonization . . . . . . . . . . . 5.4 d-Mott phase . . . . . . . . . . . 5.5 Symmetry and doping . . . . . .

588

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592 592 596 598 601 603

6 d-Wave superconductivity 604 6.1 BCS theory re-visited . . . . . . . . . . . . . . . . . . . . . . . . . 604 6.2 d-wave symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 6.3 Continuum description of gapless quasiparticles . . . . . . . . . . . 610 7 Effective field theory 612 7.1 Quasiparticles and phase flucutations . . . . . . . . . . . . . . . . . 612 7.2 Nodons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 8 Vortices 623 8.1 hc/2e versus hc/e vortices . . . . . . . . . . . . . . . . . . . . . . . 623 8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 9 Nodal liquid phase 9.1 Half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Doping the nodal liquid . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

628 628 632 634

Appendix

635

A Lattice duality 635 A.1 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 A.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

MOTT INSULATORS, SPIN LIQUIDS AND QUANTUM DISORDERED SUPERCONDUCTIVITY

Matthew P.A. Fisher

Abstract These introductory lecture notes describe recent results on novel Mott insulating phases which are “descendents” of superconductors – obtained by “quantum disordering”. After a brief overview of quantum magnetism, attention is focussed on the spin – liquid phase of the two-leg Hubbard ladder and the nodal liquid – a descendent of the dx2 −y 2 superconductor. These notes are self-contained and an effort has been made to keep them accessible.

1

Introduction

At the foundation of the quantum theory of metals is the theory of the noninteracting electron gas, in which the electrons move through the material interacting only with the periodic potential of the ions, and not with one another. Surprisingly, the properties of most metals are quite well described by simply ignoring the strong Coulomb repulsion between electrons, essentially because Pauli exlusion severely limits the phase space for electron collisions [1]. But in some cases electron interactions can have dramatic effects leading to a complete breakdown of the metallic state, even when the conduction band is only partially occupied. In the simplest such Mott insulator [2] there is only one electron per crystalline unit cell, and so a half-filled metallic conduction band would be expected. With the discovery of the cuprate superconductors in 1986 [3], there has been a resurgence of interest in Mott insulators. There are two broad classes of Mott insulators, distinguished by the presence or absence of magnetic order [4, 5]. More commonly spin rotational invariance is spontaneously broken, and long-range magnetic order, typically antiferromagnetic, is realized [43]. There are then low energy spin excitations, the spin one magnons. Alternatively, in a spin-liquid [4] Mott insulator there are no broken symmetries. Typically, the magnetic order is short-ranged and there is a gap to all spin excitations: a spin-gap. c EDP Sciences, Springer-Verlag 1999

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Topological Aspects of Low Dimensional Systems

In the cuprates the Mott insulator is antiferromagnetically ordered [7,8], but upon doping with holes the antiferromagnetism is rapidly destroyed, and above a certain level superconductivity occurs with dx2 −y2 pairing symmetry. But at intermediate doping levels between the magnetic and d-wave superconducting phases, there are experimental signs of a spin gap opening below a crossover temperature T ∗ (x) (see Fig. 1). The ultimate nature of the underlying quantum ground state in this portion of the phase diagram – commonly called the pseudo-gap regime – is an intriguing puzzle. More generally, the apparent connection between a spin-gap and superconductivity has been a source of motivation to search for Mott insulators of the spin-liquid variety. Generally, spin liquids are more common in low dimensions where quantum fluctuations can suppress magnetism. Quasi-one-dimensional ladder materials [9, 10] are promising in this regard and have received extensive attention, particularly the two-leg ladder [11]. The Mott insulating spinliquid phase of the two-leg ladder can be understood by mapping to an appropriate spin-model – the Heisenberg antiferromagnet. Spin-liquid behavior results from the formation of singlet bond formation across the rungs of the ladder [12, 13]. Almost without exception, theoretical studies of spin-liquids start by mapping to an appropriate spin-model, and the charge degrees of freedom are thereafter ignored. This represents an enormous simplification, since spin models are so much easier to analyze that the underlying interacting electron model. This approach to quantum magnetism has yielded tremendous progress in the past decade [5]. But is the simplification to a spinmodel always legitimate? A central goal of these lectures is to analyze a novel two-dimensional spin-liquid phase – called a nodal liquid [14, 15] – which probably cannot be described in terms of a spin model. Although the nodal liquid is a Mott insulator with a charge gap and has no broken symmetries, it possesses gapless Fermionic degrees of freedom which carry spin. The standard route to the spin-liquid invokes quantum fluctuations to suppress the magnetic order of a quantum spin-model [4]. The proximity of antiferromagnetism to d-wave superconductivity in the cuprates suggests an alternate route. Indeed, as we shall see, the nodal liquid phase results when a d-wave superconductor is “quantum disordered”. The gapless Fermionic excitations in the nodal liquid are descendents of the low energy quasiparticles of the d-wave superconductor. The spin-liquid phase of the two-leg ladder gives us a simpler example of a quantum disordered superconductor. To see this, we will revisit the two-leg ladder, employing a model of interacting electrons [16], rather than truncating to a spin-model. Retaining the charge degrees of freedom will enable us to show that the Mott-insulating phase of the two-leg

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

579

T T*(x)

Pseudo-Gap

AF

DSC

X

Fig. 1. Schematic phase diagram of a high-temperature superconductor as a function of doping x and temperature T .

ladder actually exhibits pairing, with an approximate d-wave symmetry. Moreover, upon doping, the two-leg ladder exhibits quasi-long-range superconducting (d-wave) pairing correlations. This behavior is reminiscent of that seen in the underdoped cuprate superconductors. These notes are organized as follows. In Section 2 a simple tight binding model of interacting electrons is introduced and it’s symmetry properties are discussed. Section 3 specializes to the Mott insulating state at half-filling, focussing on the magnetic properties employing the Heisenberg antiferromagnet spin-model. In Section 4 the method of Bosonization is briefly reviewed for the case of a one-dimensionless spinless electron gas. Section 5 is devoted to an analysis of the Mott insulating state of the twoleg Hubbard ladder, employing a weak coupling perturbative renormalization group approach. The remaining sections focus on the two-dimensional d-wave superconductor, and the nodal liquid phase which descends from it upon quantum disordering. Specifically, Section 6 briefly reviews BCS theory for a d-wave superconductor focussing on the gapless quasiparticles. An effective field theory for this state, including quantum phase fluctuations, is obtained in Section 7. Section 8 implements a duality transformation of this effective field theory, which enables a convenient description of the nodal liquid phase in Section 9. 2

Models and metals

2.1 Noninteracting electrons In metals the highest lying band of Bloch states is only partially occupied, and there are low energy electronic excitations which consist of

580

Topological Aspects of Low Dimensional Systems

exciting electrons from just below the Fermi energy into unoccupied states. These excitations can be thermally excited and contribute to thermodynamic properties such as the specific heat, as well as to electrical conduction [1]. Tight binding models give a particularly simple description of the conduction band. In the simplest case the states in the conduction band are built up from a single atomic orbital on each of the ions in the solid. The conduction electrons are presumed to be moving through the solid, tunnelling between ions. We denote the creation and annihilation operators for an electron with spin α =↑, ↓ on the ion at position x by c†α (x) and cα (x). These operators satisfy the canonical Fermionic anti-commutation relations, [cα (x), c†β (x0 )]− = δαβ δx,x0 .

(2.1)

If the orbitals in question form a simple Bravais lattice with, say, cubic symmetry, then the appropriate tight binding Hamiltonian is, X X n(x), (2.2) c†α (x)cα (x0 ) + h.c. − µ H0 = −t x hxx0 i where the first summation is over near neighbor sites. Here t is the tunnelling rate between neighboring ions and for simplicity we have ignored further neighbor tunnelling. The electron density n(x) = c†α (x)cα (x) can be adjusted by tuning the chemical potential, µ. In the Cuprate superconductors Copper and Oxygen atoms form two dimensional sheets [7], with the Copper atoms sitting at the sites of a square lattice and the Oxygen atoms sitting on the bonds, as depicted schematically in Figure 2. In the simplest one-band models the sites of the tight binding model are taken as the Copper atoms, and c† (x) removes an electron (adds a hole) from a Copper 3d orbital. In most of the materials the 3d shell is almost filled with roughly one hole per Copper atom, so that the tight binding model is close to half-filling with hn(x)i ≈ 1. The tight binding Hamiltonian is invariant under translations by an arbitrary real space lattice vector, R, cα (x) → cα (x + R).

(2.3)

This discrete symmetry implies the conservation of crystal momentum, up to a reciprocal lattice vector, G, with exp(iG · R) = 1. Being quadratic, the Hamiltonian can be diagonalized by transforming to (crystal) momentum space by defining, 1 X ckα eik·x . cα (x) = √ V k

(2.4)

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

581

ky

kx

Fig. 2. Schematic illustration of a single Copper-Oxygen plane, consisting of a square lattice of Copper atoms (solid points) and Oxygen atoms (open circles). Two-dimensional Brillouin zone for the 2d square lattice tight binding model with near neighbor hopping is shown at right. At half-filling all states in the Fermi sea (shaded) are occupied.

Here V denotes the “volume” of the system, equal to the total number of sites N with the lattice spacing set to unity, and the sum is over crystal momentum within the first Brillouin zone compatible with periodic boundary conditions. The momentum space creation and anihillation operators also satisfy canonical Fermion anticommutation relations: [ckα , c†k0 β ]− = δαβ δkk0 .

(2.5)

In momentum space the Hamiltonian takes the standard diagonal form, X k c†kα ckα , (2.6) H0 = kα

invariant under the discrete translation symmetry: ckα → eik·R ckα . For a 2d square lattice with near-neighbor hopping, the energy is simply k = −2t[cos kx + cos ky ] − µ.

(2.7)

The ground state consists of filling those states in momentum space with k negative, leaving the positive energy states unoccupied. The Fermi surface, separating the occupied from empty states, is determined by the condition k = 0. For the 2d square lattice at half-filling with energy dispersion equation (2.7) (at µ = 0), the Fermi surface is a diamond, as shown in Figure 2. Particle/hole excitations above the ground state consist of removing an electron from within the full Fermi sea, and placing it in an unoccupied positive energy state. In most metals the width of the conduction band (proportional to t) is of order an electron volt (roughly 104 K) so that even at room temperature only “low energy” particle/hole states confined within

582

Topological Aspects of Low Dimensional Systems

close proximity to the Fermi surface are thermally excited. In addition to being thermally active, these low energy particle/hole excitations can be excited by an electric field, and lead to metallic electrical conduction. In the band theory of solids, insulators occur whenever the highest lying energy band is fully occupied. Excited states then involve promoting electrons into the next available band which typically requires a very large energy (electron volts). Not surprisingly, such band insulators are very poor conductors of electricity. By constrast, in Mott insulators the highest band is only partially occupied, yet conduction is blocked by strong electron interactions. Before addressing the complications of electron interactions, it is instructive to briefly consider the symmetries of the above Hamiltonian, and the associated conserved quantities. There are only two continous symmetries, associated with conservation of charge and spin. The Hamiltonian is invariant under the global U (1) charge symmetry, cα (x) → eiθ0 cα (x),

(2.8)

for arbitrary (constant) angle θ0 . Conservation of spin is due to the global SU (2) symmetry, cα (x) → Uαβ cβ (x), with U = exp(iθ · σ) and Pauli matrices σ αβ . The Hamiltonian is invariant under this transformation, H0 → H0 , for arbitrary spin rotations θ. Here and below we ignore spinorbit effects which (usually weakly) break the continuous spin rotational symmetry. There are also a number of discrete symmetries. The Hamiltonian is real, H0∗ = H0 , a signature of time reversal invariance (for models with spin-independent interactions). For a square lattice the Hamiltonian is also invariant under reflection (or parity) symmetry, cα (x) → cα (−x). This implies that k = −k . On the square lattice, a discrete particle/hole transformation is implemented by p/h

cα (x) −→ eiπ ·x c†α (x),

(2.9)

with π = (π, π). At half-filling when µ = 0, H0 is invariant under this symmetry, but with further neighbor hopping terms the kinetic energy will generally not be particle/hole symmetric. In momentum space the particle/hole transformation is implemented via ckα → c†π −kα and invariance of the kinetic energy implies that k = −k+π . 2.2 Interaction effects Spin-independent density interactions can be included by adding an additional term to the Hamiltonian: 1 X v(x − x0 )n(x)n(x0 ). (2.10) H1 = 2 x,x0

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

583

For Coulomb interactions v(x) ∼ e2 /|x| is long-ranged. For simplicity the long-ranged interactions are often ignored. In the Hubbard model [2,5] only the on-site repulsive interaction is retained, X n↑ (x)n↓ (x), (2.11) Hu = u x with nα = c†α cα . This can be re-cast into a manifestly spin-rotationally invariant form: uX n(x)[n(x) − 1]. (2.12) Hu = 2 x Despite the deceptive simplicity of these effective models, they are exceedingly difficult to analyze. Even the Hubbard Hamiltonian, H = H0 + Hu , which is parameterized by just two energy scales, t and u, is largely intractable [4], except in one-dimension. Since the typical interaction scale u is comparable to the kinetic energy t there is no small parameter. Moreover, one is typically interested in phenomena occuring on temperature scales which are much smaller than both u and t. In most metals, the low energy properties are quite well described by simply ignoring the (strong!) interactions. This surprising fact can be understood (to some degree) from Landau’s Fermi-liquid theory [1], and more recent renormalization group arguments [17]. The key point is that the phase space available for collisions between excited particles and holes vanishes with their energy. In metals the phase space is evidently so restrictive that the surviving interactions do not change the qualitative behavior of the low energy particle/hole excitations. Indeed, the quasiparticle excitations within Landau’s Fermi liquid theory have the same quantum numbers as the electron (charge e spin 1/2 and momentum), but move with a “renormalized” velocity. But some materials such as the Cuprates are not metallic, even when band structure considerations would suggest a partially occupied conduction band. In these Mott insulators one must invoke electron interactions. 3

Mott insulators and quantum magnetism

The Hubbard model at half-filling is perhaps the simplest example of a Mott insulator. To see this, consider the behavior as the ratio u/t is varied. As discussed above, for u/t = 0 the model is diagonalized in momentum space, and exhibits a Fermi surface. But at half-filling the model is also soluble when u/t = ∞. Since the onsite Hubbard energy takes the form, u(n − 1)2 /2, in this limit the ground state consists simply of one electron on each site. The electrons are frozen and immobile, since doubly occupied

584

Topological Aspects of Low Dimensional Systems

and unoccupied sites cost an energy proportional to u. The state is clearly insulating – a Mott insulator. In this large u limit it is very costly in energy to add an electron, and the state exhibits a charge gap of order u. But there are many low energy spin excitations, which consist of flipping the spin of an electron on a given site. For infinite u this spin-one excitation costs no energy at all, and indeed the ground state is highly degenerate since the spins of each of the N localized electrons can be either up or down. For large but finite u/t one still expects a charge gap, but the huge spin degeneracy will be lifted. The fate of the spin degrees of freedom in the Mott insulator is enormously interesting. Broadly speaking, Mott insulators come in two classes, distinguished by the presence or absence of spontaneously broken symmetries. Often the spin rotational invariance is spontaneously broken and the ground state is magnetic, but SU (2) invariant spin structures which break translational symmetries are also possible. In the second class, usually referred to as spin liquid states there are no broken symmetries. 3.1 Spin models and quantum magnetism Traditionally, spin physics in the Mott insulating states have been analyzed by studying simple spin models. These focus on the electron spin operators: S(x) =

1 † c (x)σ αβ cβ (x), 2 α

(3.1)

where σ is a vector of Pauli matrices. These spin operators satisfy standard angular momentum commutation relations: [Sµ (x), Sν (x0 ] = iδxx0 µνλ Sλ .

(3.2)

They also satisfy, S 2 (x) =

3 n(x)[2 − n(x)]. 4

(3.3)

Within the restricted sector of the full Hilbert space with exactly one electron per site, these operators are bone fide spin 1/2 operators satisfying S 2 = s(s + 1) with s = 1/2. Their matrix elements in the restricted Hilbert space are identical to the Pauli matrices: σ/2. The simplest spin model consists of a (square) lattice of spin 1/2 operators coupled via a near neighbor exchange interaction, J: X S(x) · S(x0 ). (3.4) H =J 0 hxx i This spin model can be obtained from the half-filled Hubbard model [5], by working perturbatively in small t/u. For t/u = 0 the spins are decoupled,

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

585

but an antiferromagnetic exchange interaction J = 4t2 /u is generated at second order in t. Specifically, the matrix elements of the spin Hamiltonian in the restricted Hilbert space are obtained by using second order perturbation theory in t. The intermediate virtual states are doubly occupied, giving an energy denominator u. Mapping the Hubbard model to a spin model represents an enormous simplification. The complications due to the Fermi statistics of the underlying electrons have been subsumed into an exchange interaction. The spin operators are essentially bosonic, commuting at different sites. It should be emphasized that at higher order in t/u multi-spin exchange interactions will be generated, also between further separated spins. If t/u is of order one, then it is by no means obvious that it is legitimate to truncate to a spin model at all. A central focus of quantum magnetism during the past decade has been exploring the possible ground states and low energy excitations of such spin models [4, 5]. The above s = 1/2 square lattice Heisenberg antiferromagnet is, of course, only one member of a huge class of such models. These models can be generalized to larger spin s, to different lattices and/or dimensionalities, to include competing or frustrating interactions, to include multi-spin interactions, to “spins” in different groups such as SU (N ), etc. Not surprisingly, there is an almost equally rich set of possible ground states. The main focus of these notes is the 2d “nodal liquid”, a spin-liquid phase obtained by quantum disordering a d-wave superconductor. As we shall see in Section 9, in the nodal liquid the spin excitations are carried by Fermionic degrees of freedom and cannot be described by (Bosonic) spin operators. In truncating to the restricted Hilbert space with one electron per site, one has effectively “thrown out the baby with the bath water”. The nodal liquid phase probably requires retaining the charge degrees of freedom. But spin models are much simpler than interacting electron models, relevant to many if not most Mott insulators (as well as other localized spin systems) and extremely rich and interesting in their own right. So I would like to briefly summarize some of the possible ground states, focussing on spin 1/2 models on bi-partite lattices [5, 43]. Consider first those ground states with spontaneously broken symmetries. Most common is the breaking of spin-rotational invariance. If the spin operators are treated as classical fixed length vectors, which is valid in the large spin limit (s → ∞), the ground state of the near neighbor square lattice antiferromagnet is the Neel state (up on one sublattice, down on the other) which breaks the SU (2) symmetry. For finite s the Neel state is not the exact ground state, but the ground state is still antiferromagnetically ordered, even for s = 1/2. Quantum fluctuations play a role in reducing the sub-lattice magnetization, but

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Topological Aspects of Low Dimensional Systems

(for the 2d square lattice) do not drive it to zero. The low energy excitations are gapless spin-waves (i.e. magnons), as expected when a continuous symmetry is spontaneously broken. For some spin models the ground state is spin rotationally invariant but spontaneously breaks (discrete) translational symmetry. The classic example is the Majumdar – Ghosh Hamiltonian [5], X 1 (3.5) S(x) · S(x + 1) + S(x) · S(x + 2) , HMG = J 2 x which describes a one dimensional s = 1/2 Heisenberg antiferromagnetic spin chain with a second neighbor exchange interaction. The exact ground state of this model is a two-fold degenerate “spin-Peierls” state: Y [| ↑2x i| ↓2x±1 i − | ↓2x i| ↑2x±1 i]. (3.6) |Gi± = x

This state consists of a product of “singlet bonds” formed from neighboring pairs of spins, and breaks invariance under translations by one lattice spacing. Since the singlet bonds are rotationally invariant, the SU (2) symmetry remains unbroken. The second neighbor interaction has effectively suppressed the tendency towards antiferromagnetic order. 3.2 Spin liquids Spin liquid ground states in which no symmetries are broken generally occur more readily in low dimensions where quantum fluctuations are more effective at destroying magnetic order. The one-dimensional s = 1/2 chain with near neighbor antiferromagnetic exchange exhibits power law magnetic correlations at the antiferromagnetic wave vector π [43]. Although “almost” magnetically ordered the SU (2) symmetry is not broken in the ground state, which thus technically qualifies as a spin liquid. More dramatic is the behavior of the s = 1/2 antiferromagnetic two-leg ladder, shown in Figure 3. This model exhibits a featureless spin-rotationally invariant ground state with exponentially decaying spin correlation fuctions and a non-zero energy gap for all spin excitations [12, 13]. The physics can be best understood in the limit in which the exchange interaction across the rungs of the ladder greatly exceeds the intra-leg exchange: J⊥ J. When J = 0 the ground state consists of singlet bonds formed across the rungs of the ladder, with triplet excitations separated by an energy gap of order J⊥ . Perturbing in small J will cause these singlet bonds to “resonate”, but one expects the spin gap to survive at least for J J⊥ . It turns out that the ground state evolves adiabatically and smoothly with increasing J, and in fact the spin-liquid survives for arbitrarily large J⊥ /J.

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J J Fig. 3. Heisenberg spin model on a two-leg ladder. Spin 1/2 operators sit on the sites of the ladder, interacting via an antiferromagnetic exchange J along the ladder and J⊥ across the rungs.

There has been an enormous amount of theoretical effort expended searching for two-dimensional spin 1/2 models which exhibit spin-liquid ground states analogous to the two-leg ladder – but with little success. The original motivation soon after the discovery of superconductivity in the Cuprates was based on Anderson’s ideas [41] that a Mott insulating spin-liquid exhibits “pre-formed” Cooper pairing. Doping the Mott insulator would give the Cooper pairs room to move and to condense into a superconducting state, presumed to have s-wave pairing symmetry. But it soon became clear that the undoped Mott insulator in the Cuprates is not a spin-liquid, but actually antiferromagnetically ordered. Moreover, recent experiments have established that the pairing symmetry in the superconducting phase is d-wave rather than s-wave [19, 20]. However, recent theoretical work [16] (see Sect. 5 below) has established that the pairing in the spin-liquid phase of the two-leg ladder actually has (approximate) d-wave symmetry. Moreover, doping this Mott insulator does indeed give the pairs room to move [21, 22], and they form a onedimensional d-wave “superconductor” (with quasi-long-ranged pairing correlations). The nodal liquid phase [14, 15] discussed extensively below is a two-dimensional analog of this spin-liquid phase. Indeed, we shall explicitly construct the nodal liquid by quantum disordering a two-dimensional d-wave superconductor. As we shall see, the resulting 2d nodal liquid posesses gapless Fermionic excitations, which are descendents of the d-wave quasiparticles. These Fermions carry spin but no charge. The nodal liquid presumably cannot be the ground state of any (Bosonic) spin-model. To describe the nodal liquid one must employ the underlying interacting electron model which retains the charge degrees of freedom. Recent experiment has focussed attention on the underdoped regime of the Cuprate materials [8], occuring between the antiferromagnetic and superconducting phases (see Fig. 1). In this pseudo-gap regime insulating behavior is seen at low temperatures, and there are indications for a spin gap – behavior reminiscent of a Mott insulating spin-liquid. We have

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suggested [14] that this strange phase can perhaps be understood in terms of a doped nodal liquid. Before discussing further the 2d nodal liquid, it is instructive to revisit the spin liquid phase of the two-leg ladder and analyze it directly with a model of interacting electrons. Specifically, we consider weak interactions (small u/t), a limit in which truncation to a spin model is not possible. This analysis is greatly aided by “Bosonization” – a powerful method which enables an interacting electron model in one dimension to be re-formulated in terms of collective Bosonic degrees of freedom. See references [23–27] as well as Fradkin’s book [4] for useful reviews of Bosonization. First, in Section 4 we briefly review Bosonization for the simplest case of a spinless one-dimensional electron gas, before turning to the two-leg ladder in Section 5. 4

Bosonization primer

Consider the Hamiltonian for non-interacting spinless electrons hopping on a 1d lattice, X c† (x)c(x + 1) + h.c. (4.1) H = −t x

with hopping strength t. One can diagonalize this Hamiltonian by Fourier transforming to momentum space as in equation (2.4), giving X † k ck ck , (4.2) H= k

with energy dispersion k = −t cos(k) for momentum |k| < π, as shown in Figure 4. In the ground state all of the negative energy states with momentum |k| ≤ kF are occupied. At half-filling the Fermi wavevector kF = π/2. An effective low energy theory for these excitations can be obtained by focussing on momenta close to ±kF and defining continuum Fermi fields: ψR (q) = ckF +q ;

ψL (q) = c−kF +q .

(4.3)

Here the subscripts R/L refer to the right/left Fermi points, and q is assumed to be smaller than a momentum cutoff, |q| < Λ with Λ kF . One can then linearize the dispersion about the Fermi points, writing ±kF +q = ±vF q with vF the Fermi velocity. It is convenient to transform back to real space, defining fields 1 X iqx e ψP (q), ψP (x) = √ V |q|<Λ

(4.4)

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E

kF

k

Fig. 4. Schematic energy dispersion for the one-dimensional electron gas. The negative energy states are occupied, with momentum |k| < kF . The dispersion can be linearized around ±kF , leading to a continuum Dirac Fermion theory.

(with P = R, L) which vary slowly on the scale of the lattice spacing. This is equivalent to expanding the lattice electron operators in terms of continuum fields, c(x) ∼ ψR (x)eikF x + ψL (x)e−ikF x .

(4.5)

AfterR linearization, the effective low energy Hamiltonian takes the form, H = dxH, with Hamiltonian density, H = −vF [ψR† i∂x ψR − ψL† i∂x ψL ]. describing a one-dimensional relativistic Dirac particle. Lagrangian density is simply L = ψR† i∂t ψR + ψL† i∂t ψL − H.

(4.6) The associated

(4.7)

Consider a particle/hole excitation about the right Fermi point, where an electron is removed from a state with k < kF and placed into an unoccupied state with k + q > kF . For small momentum change q, the energy of this excitation is ωq = vF q. Together with the negative momentum excitations about the left Fermi point, this linear dispersion relation is identical to that for phonons in one-dimension. The method of Bosonization exploits this similarity by introducing a phonon displacement field, θ, to decribe this linearly dispersing density wave [23, 25]. We follow the heuristic development of Haldane [27], which reveals the important physics, dispensing with mathematical rigor. To this end, consider a Jordan-Wigner transformation [4] which replaces the electron operator, c(x), by a (hard-core) boson operator, # " X 0 n(x ) b(x). (4.8) c(x) = O(x)b(x) ≡ exp iπ x0 <x

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Topological Aspects of Low Dimensional Systems

where n(x) = c† (x)c(x) is the number operator. One can easily verify that the Bose operators commute at different sites. Moreover, the lattice Hamiltonian equation (4.1) can be re-expressed in terms of these Bosons, and takes the identical form with c0 s replaced by b0 s. This transformation, exchanging Fermions for Bosons, is a special feature of one-dimension. The Boson operators can be (approximately) decomposed in terms of an amplitude and a phase, √ (4.9) b(x) → ρeiϕ . We now imagine passing to the continuum limit, focussing on scales long compared to the lattice spacing. In this limit we decompose the total density as, ρ(x) = ρ0 + ρ˜, where the mean density, ρ0 = kF /π, and ρ˜ is an operator measuring fluctuations in the density. As usual, the density and phase are canonically conjugate quantum variables, taken to satisfy [ϕ(x), ρ(x ˜ 0 )] = iδ(x − x0 ).

(4.10)

Now we introduce a phonon-like displacement field, θ(x), via ρ˜(x) = ∂x θ(x)/π. The full density takes the form: πρ(x) = kF + ∂x θ. The above commutation relations are satisfied if one takes, [ϕ(x), θ(x0 )] = −iπΘ(x0 − x).

(4.11)

Here Θ(x) denotes the heavyside step function, not to be confused with the displacement field θ. Notice that ∂x ϕ/π is the momentum conjugate to θ. The effective (Bosonized) Hamiltonian density which describes the 1d density wave takes the form: H=

v [g(∂x ϕ)2 + g −1 (∂x θ)2 ]. 2π

(4.12)

This Hamiltonian describes a wave propagating at velocity v, as can be readily verified upon using the commutation relations to obtain the equations of motion, ∂t2 θ = v 2 ∂x2 θ, and similarly for ϕ. Clearly one should equate v with the Fermi velocity, vF . The additional dimensionless parameter, g, can be determined as follows. A small variation in density, ρ˜, will lead to a change in energy, E = ρ˜2 /2κ, where κ = ∂ρ/∂µ is the compressibility. Since ∂x θ = π ρ˜, one deduces from H that κ = g/πv. But for a non-interacting electron gas, πvκ = 1, so that g = 1. In the presence of (short-ranged) interactions between the (spinless) electrons, one can argue that the above Hamiltonian density remains valid, but with renormalized values of both g and v. This Hamiltonian would then describe a (spinless) Luttinger liquid [27, 28], rather than the free electron gas. The power of Bosonization relies on the ability to re-express the electron operator c(x) in terms of the Boson fields. Clearly c(x) must remove a unit

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charge (e) at x, and satisfy Fermion anticommutation relations. Consider first the Bose operator, b ∼ exp(iϕ), which removes unit charge. To see this, note that one can write, eiϕ(x) = eiπ

Rx

−∞

dx0 P (x0 )

,

(4.13)

where P = ∂x ϕ/π is the momentum conjugate to θ. Since the momentum operator is the generator of translations (in θ), this creates a kink in θ of height π centered at position x – which corresponds to a localized unit of charge since the density ρ˜ = ∂x θ/π. To construct the (Fermionic) electron operator requires multiplying this Bose operator by a Jordan-Wigner “string”: O(x) = eiπ

P

x0 <x

n(x0 )

→ eiπ

Rx

ρ(x0 )

= ei(kF x+θ) .

(4.14)

Since this string operator carries momentum kF , the resulting Fermionic operator Oeiϕ should be identified with the right moving continuum Fermi field, ψR . We have thereby identified the correct Bosonized form for the (continuum) electron operators: ψP (x) = eiφP (x) ;

φP = ϕ + P θ,

(4.15)

with P = R/L = ±. From equation (4.10) the chiral Boson fields φP can be shown to satisfy the so-called Kac-Moody commutation realtions: [φP (x), φP (x0 )] = [φR (x), φL (x0 )] =

iP π iπ.

sgn(x − x0 ),

(4.16) (4.17)

These commutation relations can be used to show that ψR and ψL anticommute. It is instructive to re-express the Bosonized Hamiltonian density in terms of the chiral boson fields, H = πvF [n2R + n2L ],

(4.18)

where we have defined right and left moving densities nP = P

1 ∂x φP , 2π

(4.19)

which sum to give the total density, nR + nL = ρ˜. These chiral densities can be expressed in terms of the chiral electron operators as, nP =: ψP† ψP :≡ ψP† ψP − hψP† ψP i.

(4.20)

Notice that the Bosonized Hamiltonian decouples into right and left moving sectors.

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An advantage of Bosonization is the ease with which electron interactions can be incorporated. Consider a (short-range) density-density interaction added to the original lattice Hamiltonian. Using equation (4.5) this can be decomposed into the continuum Dirac fields, and will be quartic and spatially local. Due to momentum conservation, only three terms are possible: Two chiral terms of the form (ψP† ψP )2 with P = R/L, and a right/left mixing term of the form, ψR† ψR ψL† ψL . Under Bosonization the chiral terms are proportional to (∂x φP )2 , and can be seen to simply shift the Fermi velocity in equation (4.18). The right/left mixing term also Bosonizes into a quadratic form proportional to (∂x θ)2 − (∂x ϕ)2 . When added to the Hamiltonian in equation (4.12), this term can be absorbed by shifting both the Fermi velcocity and the dimensionless Luttinger parameter, g, which is then no longer equal to one. For repulsive interactions g < 1, whereas g > 1 with attractive interactions. This innocuous looking shift in g has profound effects on the nature of the electron correlation functions. In fact, it leads to new chiral operators which have fractional charge, ge. The resulting onedimensional phase is usually called a “Luttinger liquid” [27]. For electrons with spin or for 1d models with multiple bands, the quartic Fermion operators can have even more dramatic consequences, for example opening up energy gaps as we shall see in Section 5. The Lagrangian density in the Bosonized representation takes the form of a free scalar field, g (4.21) L = κµ (∂µ ϕ)2 , 2 with g = 1 for the free Fermion gas, and g 6= 1 in the interacting Luttinger liquid. The Greek index µ runs over time and the spatial coordinate, µ = 0, 1 = t, x. Here κ0 = 1/πv and κ1 = −v 2 κ0 . When re-expressed in terms of θ the Lagrangian takes the identical form, except with g → 1/g for the Luttinger liquid. Changing from the ϕ to the θ representation can be viewed as a duality transformation. In Section 8 we will consider an analogous duality transformation in two spatial dimensions. 5

2 Leg Hubbard ladder

5.1 Bonding and antibonding bands We now consider electrons hopping on a two-leg ladder as shown in Figure 5. The kinetic energy takes the form, X X n(x), (5.1) c†α (x)cα (x0 ) + h.c. − µ H0 = −t x hxx0 i where n(x) = c†α (x)cα (x), and the summation is taken over near neighbors on the two-leg ladder, with y = 1, 2. Due to a parity symmetry under

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ε t

1L 1R 2L

t

2R

kx

Fig. 5. A two-leg ladder and its band structure. In the low-energy limit, the energy dispersion is linearized near the Fermi points. The two resulting relativistic Dirac Fermions are distinguished by pseudospin indices i = 1, 2 for the antibonding and bonding bands, respectively.

interchange of the two legs of the ladder, it is convenient to consider even and odd parity bonding and anti-bonding operators: 1 bα (x) = √ [cα (x, y = 1) + cα (x, y = 2)], 2

(5.2)

1 aα (x) = √ [cα (x, y = 1) − cα (x, y = 2)], 2

(5.3)

which depend only on the coordinate x along the ladder. The Hamiltonian splits into even and odd contributions, H0 = Ha (a) + Hb (b). Each is a one-dimensional tight binding model which can be readily diagonalized by transforming to momentum space, 1 X bk eikx , b(x) = √ N k

(5.4)

and similarly for the anti-bonding operator. Here N denotes the number of sites along the ladder. The diagonal form is X [ak a†kα akα + bk b†kα bkα ], (5.5) H0 = k a/b

= which describes two one-dimensional bands with dispersion k −2t cos k ± t − µ. These are sketched in Figure 5. Focussing on the case at half-filling with one electron per site (µ = 0), both bands intersect the Fermi energy, F = 0. There are four Fermi points at ±kF1 and ±kF2 , for the antibonding and bonding bands, respectively. Gapless particle/hole excitations exist at each of the four Fermi points.

594

Topological Aspects of Low Dimensional Systems ky 1R

2L

2R

kx

1L

Fig. 6. Fermi points for the two-leg ladder plotted in the two-dimensional Brillouin zone, with the antibonding band (denoted 1) at ky = π and the bonding band (2) at ky = 0. The shaded region represents the Fermi sea for a two-dimensional square lattice model at half-filling.

Due to particle/hole symmetry present with near neighbor hopping, ak + bk+π = 0, which implies that kF1 + kF2 = π. Moreover, the Fermi velocity in each band is the same, hereafter denoted as v. It is instructive to plot these Fermi points in two-dimensional momentum space, taking transverse momentum ky = 0, π for the two bands, as shown in Figure 6. The four Fermi points can be viewed as constant ky slices through a two-dimensional Fermi surface. As we shall see, with even weak electron interactions present the gapless Fermi points are unstable, and a gap opens in the spectrum. Of interest are the properties of the resulting Mott insulator. As discussed in Section 3, for strong interactions mapping to a spin model is possible, and the electron spins across the rungs of the ladder are effectively locked into singlets: 1 |RSi = √ [c†↑ (1)c†↓ (2) − c†↓ (1)c†↑ (2)]|0i, 2

(5.6)

where y = 1, 2 refers to the two legs of the ladder, and we have suppressed the rung position x. The state |0i denotes a rung with no electrons. It is extremely instructive to re-express this rung-singlet state in terms of the bonding and anti-bonding operators. One finds, 1 |RSi = √ [b†↑ b†↓ − a†↑ a†↓ ]|0i, 2

(5.7)

a linear combination of adding a singlet (Cooper) pair into the bonding and antibonding orbitals. This paired form is suggestive of superconductivity. Indeed, when viewed in momentum space, the ground state of a superconductor is a product of singlet pairs with zero center of mass momentum at different points around the Fermi surface. In an s-wave superconductor, the

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pairs are all added with the same sign, but if the pairs are formed with a relative angular momentum (e.g. d-wave) sign changes are expected. But notice the most important relative minus sign in the rung singlet state! The spin-liquid phase of the two-leg ladder is evidently related to a paired superconductor with non-zero angular momentum. Since pairing in the bonding band at ky = 0 has a positive sign and pairing in the anti-bonding band at ky = π is negative, in the two-dimensional Brillouin zone (see Fig. 6) the sign is proportional to kx2 − ky2 , consistent with a so-called dx2 −y2 pairing symmetry. If the interactions are weak, it is legitimate to focus on electronic states near the Fermi points. As in Section 4, the electron operators can be conveniently decomposed into continuum fields near the Fermi points which vary slowly on the scale of the lattice. Denoting c1 = a and c2 = b, the bonding and antibonding operators are expanded as, ciα ∼ ψRiα eikF i x + ψLiα e−ikF i x ,

(5.8)

with i = 1, 2. Upon linearizing the spectrum around the four Fermi points R the kinetic energy takes the form, H0 = dxH0 , with Hamiltonian density, X † [ψRiα i∂x ψRiα − ψL† iα i∂x ψLiα ]. (5.9) H0 = −v i,α

This Hamiltonian describes massless Dirac Fermions, with four flavors labelled by band and spin indices. Implicit in this theory is a momentum cutoff, Λ, whose inverse exceeds the lattice spacing. Only modes with momentum |k| < Λ are included in these continuum fields. Since the spectrum is massless, this simple theory is “critical” and scale invariant behavior is expected. This can be seen by considering the (Euclidian) action, written as a space-time integral of the Lagrangian density, Z (5.10) S = dτ dxL0 , L0 =

X

ψP† iα ∂τ ψP iα + H0 ,

(5.11)

Pα

with P = R/L, and τ denoting imaginary time. The partition function, Z = T r exp(−βH0 ), can be expressed as a (coherent state Grassman) path integral [29], Z ¯ ¯ −S(ψ,ψ) . (5.12) Z = [Dψ][Dψ]e A simple renormalization group can be implemented [17,30] by first integrating out fields ψ(k, ω) with momentum k lying in the interval Λ/b < |k| < Λ,

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with rescaling parameter b > 1. Since modes with different momentum and frequency are not coupled, the action takes the same form after this integration, except with a smaller momentum cutoff, Λ/b. The renormalization group transformation is completed by a rescaling procedure which returns the cutoff to it’s original value: x → bx; τ → bτ ; ψ → b−1/2 ψ.

(5.13)

The field rescaling has been chosen to leave the action invariant. This simple theory is at a renormalization group fixed point. 5.2 Interactions Electron-electron interactions scatter right-moving electrons into leftmoving electrons and vice-versa. We consider general finite-ranged spinindependent interactions, but assume that the typical interaction strength, u, is weak – much smaller than the bandwidth. We focus on the effects of the interactions to leading non-vanishing order in u. In this limit it is legitimate to keep only those pieces of the interactions which scatter the low energy Dirac Fermions. A general four Fermion interaction on the twoleg ladder (such as the Hubbard u) can be readily decomposed in terms of the continuum Dirac fields. It is instructive to see how these quartic terms in ψ(x) transform under the rescaling transformation equation (5.13). A simple quartic term with no spatial gradients is seen to be invariant, so that these operators are “marginal” under the renormalization group. The corresponding interaction strengths will “flow” under the renormalization group transformation due to non-linear interaction effects. On the other hand, a quartic term involving gradients such as u2 (ψ † ∂x ψ)2 , would rapidly scale to zero under rescaling: u2 → u2 /b2 , and can thus be ignored. Moreover, four-Fermion interactions which are chiral, say only scattering right movers, do not renormalize to lowest order in u and can thus also be neglected [21,22]. A As discussed in Section 4, these terms simply lead to small shifts in the Fermi velocity. All of the remaining four-Fermion interactions can be conveniently expressed in terms of currents, defined as 1 † ψ σ αβ ψjβ ; 2 iα 1 = ψiα (σ)αβ ψjβ , 2

† ψjα , Jij = ψiα

J ij =

(5.14)

Iij = ψiα αβ ψjβ ,

I ij

(5.15)

where the R, L subscript has been suppressed. Both J and I are invariant under global SU (2) spin rotations, whereas J and I rotate as SU (2) vectors. Due to Fermi statistics, some of the currents are (anti-)symmetric Iij = Iji

I ij = −I ji ,

(5.16)

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so that I ii = 0 (no sum on i). The full set of marginal momentum-conserving four-Fermion interactions can be written (1)

HI

= bρij JRij JLij − bσij J Rij · J Lij , + fijρ JRii JLjj − fijσ J Rii · J Ljj .

(5.17)

Here fij and bij denote the forward and backward (Cooper) scattering amplitudes, respectively, between bands i and j. Summation on i, j = 1, 2 is implied. To avoid double counting, we set fii = 0 (no sum on i). Hermiticity implies b12 = b21 and parity symmetry (R ↔ L) gives f12 = f21 , so that ρ,σ ρ,σ ρ,σ there are generally eight independent couplings bρ,σ 11 , b22 , b12 , and f12 . At half-filling with particle/hole symmetry b11 = b22 . Additional momentum non-conserving Umklapp interactions of the form (2)

HI

= uρij IR† ij ILˆiˆj − uσij I †Rij · I Lˆiˆj + h.c.

(5.18)

are also allowed, (here ˆ 1 = 2, ˆ 2 = 1). Because the currents (I ij ), Iij are (anti-)symmetric, one can always choose u12 = u21 for convenience. We also take uσii = 0 since I ii = 0. With particle/hole symmetry there are thus just three independent Umklapp vertices, uρ11 , uρ12 , and uσ12 . Together with the six forward and backward vertices, nine independent couplings are required to describe the most general set of marginal non-chiral fourFermion interactions for a two-leg ladder with particle/hole symmetry at half-filling. The renormalization group transformation described above can be implemented by working perturbatively for small interaction parameters [16, 21]. Upon systematically integrating out high-energy modes away from the Fermi points and then rescaling the spatial coordinate and Fermi fields, a set of renormalization group (RG) transformations can be derived for the interaction strengths. Denoting the nine interaction strengths as gi , and setting the rescaling parameter b = 1 + d` with d` infinitesimal, the leading order differential RG flow equations take the general form, ∂` gi = Aijk gj gk ,

(5.19)

valid up to order g 3 . The matrix of coefficients Aijk is given explicitly in reference [16]. These nine coupled non-linear differential equations are quite complicated, but can be integrated numerically starting with initial values appropriate to a lattice interaction (such as the Hubbard interaction). This integration reveals that some of the couplings remain small, while others tend to increase, sometimes after a sign change, and then eventually diverge. Quite surprisingly, though, the ratios of the growing couplings tend

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Topological Aspects of Low Dimensional Systems

to approach fixed constants, which are independent of the initial coupling strengths, at least over a wide range in the nine dimensional parameter space. These constants can be determined by inserting the Ansatz, gi (`) =

gi0 , (`d − `)

(5.20)

into the RG flow equations, to obtain nine algebraic equations quadratic in the constants gi0 . There are various distinct solutions of these algebraic equations, or rays in the nine-dimensional space, which correspond to different possible phases. But for generic repulsive interactions between the electrons on the two-leg ladder, a numerical integration reveals that the flows are essentially always attracted to one particular ray [16]. This is the spin-liquid phase of interest, which we refer to as a d-Mott σ , phase. In the d-Mott phase, two of the nine coupling constants, bρ11 and f12 remain small, while the other seven grow large with fixed ratios: 1 σ 1 ρ b = f12 = − bσ11 = 4 12 4 1 = 2uρ12 = uσ12 = g > 0. 2

bρ12 = 2uρ11

(5.21) (5.22)

Once the ratio’s are fixed, there is a single remaining coupling contant, denoted g, which measures the distance from the origin along a very special direction (or “ray”) in the nine dimensional space of couplings. The RG equations reveal that as the flows scale towards strong coupling, they are attracted to this special direction. If the initial bare interaction parameters are sufficiently weak, the RG flows have sufficient “time” to renormalize onto this special “ray”, before scaling out of the regime of perturbative validity. In this case, the low energy physics, on the scale of energy gaps which open in the spectrum, is universal, depending only on the properties of the physics along this special ray, and independent of the precise values of the bare interaction strengths. 5.3 Bosonization To determine the properties of the resulting d-Mott phase, it is extremely helpful to Bosonize the theory. As discussed in Section 4 the (continuum) electron fields can expressed in terms of Boson fields: ψP iα = κiα eiφP iα ;

φP iα = ϕiα + P θiα ,

(5.23)

with P = R/L = ±. The displacement field θiα and phase field ϕiα satisfy the commutation relations [ϕiα (x), θjβ (x0 )] = −iπδij δαβ Θ(x0 − x).

(5.24)

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Klein factors, satisfying {κiα , κjβ } = 2δij δαβ ,

(5.25)

have been introduced so that the Fermionic operators in different bands or with different spins anticommute with one another. When the Hamiltonian is Bosonized, the Klein factors only enter in the combination, Γ = κ1↑ κ1↓ κ2↑ κ2↓ . Since Γ2 = 1, one can take Γ = ±1. Hereafter, we will put Γ = 1. The Bosonized form for the kinetic energy equation (5.9) is v X [(∂x θiα )2 + (∂x ϕiα )2 ], (5.26) H0 = 2π i,α which describes density waves propagating in band i and with spin α. This expression can be conveniently separated into charge and spin modes, by defining √ (5.27) θiρ = (θi↑ + θi↓ )/ 2 √ (5.28) θiσ = (θi↑ − θi↓ )/ 2, √ and similarly for ϕ. The 2 ensures that these new fields satisfy the same commutators, equation (5.24). It is also convenient to combine the fields in the two bands into a ± combination, by defining √ (5.29) θµ± = (θ1µ ± θ2µ )/ 2, where µ = ρ, σ, and similarly for ϕ. The Hamiltonian density H0 can now be re-expressed in a charge/spin and flavor decoupled form, v X [(∂x θµ± )2 + (∂x ϕµ± )2 ]. (5.30) H0 = 2π µ,± The fields θρ+ and ϕρ+ describe the total charge and current fluctuations, since under Bosonization, ψP† iα ψP iα = 2∂x θρ+ /π and vP ψP† iα ψP iα = 2∂x ϕρ+ /π. While it is possible to Bosonize the interaction Hamiltonians in full generality [16], we do not reproduce it here. In addition to terms quadratic in gradients of the Boson fields (as in H0 ), the Bosonized interaction consists of terms bi-linear in cos 2θ and cos 2ϕ. More specifically, of the eight nonchiral Boson fields (θµ± and ϕµ± ) only five enter as arguments of cosine terms. In the momentum conserving terms these are θσ± , ϕρ− and ϕσ− . The Umklapp terms also involve the overall charge displacement field, via cos 2θρ+ . This can be understood by considering how the Boson fields transform under a spatial translation, x → x + x0 . The chiral electron operators

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transform as ψP i → ψP i eipkF i x0 , which is equivalent to θiα → θiα + kF i x0 . Three of the charge/spin and flavor fields are thus invariant under spatial translations, whereas θρ+ → θρ+ + πx0 . The momentum conserving terms are invariant under spatial translations, so cannot depend on cos 2θρ+ . The full interacting theory is invariant under spatially constant shifts of the remaining three Boson fields – ϕρ+ , ϕσ+ and θρ− . For the first two of these, the conservation law responsible for this symmetry is readily apparent. Specifically, the operators exp(iaQ) and exp(iaSz ), with Q the total electric charge and Sz the total z-component of spin, generate “translations” proportional to a in the two fields ϕρ+ and ϕσ+ . To see this, we note R that Q = dxρ(x) with ρ(x) = 2∂x θρ+ /π the momentum conjugate to ϕρ+ , whereas Sz can be expressed as an integral of the momentum conjugate to ϕσ+ . Since the total charge is conserved, [Q, H] = 0, the full Hamiltonian must therefore be invariant under ϕρ+ → ϕρ+ + a for arbitrary constant a, precluding a cosine term for this field. Similarly, conservation of Sz implies invariance under ϕσ+ → ϕσ+ + a. The five Boson fields entering as arguments of various cosine terms will tend to be pinned at the minima of these potentials. Two of these 5 fields, θσ− and ϕσ− , are dual to one another so that the uncertainty principle precludes pinning both fields. Since there are various competing terms in the potential seen by these 5 fields, minimization for a given set of bare interaction strengths is generally complicated. However, along the special ray in the nine dimensional space of interaction parameters the nine independent coupling constants can be replaced by a single parameter g. The resulting Bosonized theory is found to reduce to a very simple and highly symmetrical form when expressed in terms of a new set of Boson fields, defined by (θ, ϕ)1 = (θ, ϕ)3 =

(θ, ϕ)ρ+ , (θ, ϕ)σ− ,

(θ, ϕ)2 = (θ, ϕ)4 =

(θ, ϕ)σ+ , (ϕ, θ)ρ− .

(5.31)

The first three are simply the charge/spin and flavor fields defined earlier. However, in the fourth pair of fields, θ and ϕ have been interchanged. In terms of these new fields, the full interacting Hamiltonian density along the special ray takes an exceedingly simple form: H = H0 + HI , with H0 =

HI

=

v X [(∂x θa )2 + (∂x ϕa )2 ], 2π a g X [(∂x θa )2 − (∂x ϕa )2 ] 2π 2 a X cos 2θa cos 2θb . −4g a6=b

(5.32)

(5.33)

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5.4 d-Mott phase We now briefly discuss some of the general physical properties of the d-Mott phase which follow from this Hamiltonian. Ground state properties can be inferred by employing semi-classical considerations. Since the fields ϕa enter quadratically, they can be integrated out when the partition function is expressed as a path integral over Boson fields. This leaves an effective action in terms of the four fields θa . Since the single coupling constant g is marginally relevant and flowing off to strong coupling, these fields will be pinned in the minima of the cosine potentials. Specifically, there are two sets of semiclassical ground states with all θa = na π or all θa = (na + 1/2)π, where na are integers. It can be shown [16] that these different solutions actually correspond to the same physical state, so that the ground state is unique. Excitations will be separated from the ground state by a finite energy gap, since the fields are harmonically confined, and instanton excitations connecting different minima are also costly in energy. Consider first those fields which are pinned by momentum conserving interaction terms. Since both θσ± fields are pinned, so are the spin-fields in each band, θiσ (i = 1, 2). Since ∂x θiσ is proportional to the z-component of spin in band i, a pinning of these fields implies that the spin in each band vanishes, and excitations with non-zero spin are expected to cost finite energy: the spin gap. This can equivalently be interpreted as singlet pairing of electron pairs in each band. It is instructive to consider the pair field operator in band i: √

∆i = ψRi↑ ψLi↓ = κi↑ κi↓ ei

2(ϕiρ +θiσ )

.

(5.34)

With θiσ ≈ 0, ϕiρ can be interpreted as the phase of the pair field in band i. The relative phase of the pair field in the two bands follows by considering the product ∆1 ∆†2 = −Γei2θσ− ei2ϕρ− ,

(5.35)

with Γ = κ1↑ κ1↓ κ2↑ κ2↓ = 1. Since θ4 = ϕρ− the relative phase is also pinned by the cosine potential, with a sign change in the relative pair field, ∆1 ∆†2 < 0, corresponding to an approximate d-wave symmetry. To discuss the physics of the remaining overall charge mode (θρ+ ), it is convenient to first imagine “turning off” the Umklapp interactions. After pinning the other three fields to the minima of the cosine potentials, the pair field operator in band i becomes ∆i ∼ (−1)i eiϕρ+ ,

(5.36)

so that ϕρ+ is the phase of the pair field. In the absence of Umklapp scattering, the Lagrangian for this phase field is simply, 1 (5.37) L = κµ (∂µ ϕρ+ )2 . 2

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Being in one-spatial dimension, these gapless phase fluctuations lead to power law decay of the pair field spatial correlation function, ∆∗ (x)∆(0) ∼ 1/xη . A true superconductor (for d > 1) exhibits (off-diagonal) long-ranged order, and this correlation function would not decay to zero even as x → ∞. But in one-dimension a “superconductor” can at best exhibit power law decay, since true off-diagonal long-ranged order is not possible [5]. Thus, in the absence of Umklapp scattering the 2-leg ladder would be a onedimensional d-wave “superconductor”. But what is the effect of the momentum non-conserving Umklapp interactions? Once the other three fields are pinned in the minima of the cosine potentials in the above Hamiltonian equation (5.33), the Umklapp scattering terms take the simple form, Hu = −12g cos 2θρ+ .

(5.38)

This term tends to pin the field θρ+ . The pair field phase, ϕρ+ , being the conjugate field will fluctuate wildly. These quantum flucutations will destroy the power-law 1d “superconducting” phase, leading to an exponentially decaying pair-field correlation function. What is the fate of this one-dimensional “quantum disordered d-wave superconductor”? To see this, one simply has to consider the “dual” representation in terms of the θρ+ field, rather than ϕρ+ . A lattice version of this duality transformation is carried out in detail in the Appendix. Alternatively, one can obtain the dual theory directly from the Bosonized Hamiltonian equation (5.32). The appropriate Lagrangian dual to equation (5.37) above, is simply 1 κµ (∂µ θρ+ )2 , (5.39) 2 which describes gapless density waves. These density flucutations will be pinned by the Umklapp terms in Hu , leading to a Mott insulator with a gap to charge excitations. Since there is also a spin-gap this phase is equivalent to the spin-liquid, discussed at strong coupling in terms of the Heisenberg model in Section 3. But we now see that this spin-liquid phase exhibits superconducitng d-wave pairing correlations, despite being an insulator. The spin-liquid phase can thus be described as a quantum disordered one-dimensional d-wave “superconductor”. The Euclidian action associated with the phase Lagrangian in equation (5.37) is equivalent to the effective Hamiltonian in the low temperature phase of the classical 2d xy model, (with imaginary time playing the role of a second spatial coordinate). The 2d xy model can be disordered by introducing vortices into the phase of the order parameter [31]. For this it is convenient to go to a dual representation [32]. As shown explicitly in the Appendix, the dual represention is equivalent to the θρ+ representation, with the strength of the Umklapp term playing the role of a vortex L=

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fugacity. In Section 8, we will quantum disorder a two-dimensional d-wave superconductor, and it will be extremely convenient to consider a duality transformation – a three dimensional version of the 2d θ ↔ ϕ duality discussed here. The resulting nodal liquid phase will be particularly simple to analyze in the dual representation.

5.5 Symmetry and doping Due to the highly symmetric form of the Hamiltonian in equations (5.32) and (5.33), it is possible to make considerable further progress in analyzing it’s properties. Indeed, as shown in reference [16], under a re-Fermionization procedure this Hamiltonian is equivalent to the SO(8) Gross-Neveu model [33], which has been studied extensively by particle field theorists. The SO(8) Gross-Neveu model posesses a remarkable symmetry known as triality [34], which can be used to equate the energies of various excited states. In particular, the energy of the lowest excited state with the quantum numbers of an electron (charge e and s = 1/2) is equal to the energy of the lowest lying spinless charge 2e exited state (a Cooper pair). This beautifully demonstrates pairing in the insulating d-Mott phase: the energy to add two electrons of opposite spin far apart is twice as large as the energy to add them into a Cooper pair bound state. It turns out, moreover, that the Gross-Neveu model is integrable [35] so it is possible to fully enumerate the energies and quantum numbers of all the low energy excited states [16] (grouped into SO(8) multiplets) and compute exactly various correlation functions [36]. We finally briefly mention the effects of doping the d-Mott phase away from half-filling. This can be achieved by adding a chemical potential term to the Hamiltonian in equations (5.32) and (5.33), with Hµ = H − µQ, where Q is the total electric charge: Q=

2 π

Z ∂x θρ+ .

(5.40)

Since the field θρ+ is pinned in the cosine potential by the Umklapp interaction terms, Hu , for small µ the density will stay fixed at half-filling. Eventually, µ will pass through the Mott charge gap and the density will change. This occurs via π instantons in θρ+ , connecting adjacent minima of the cosine potential. Each instanton carries charge 2e, but no spin, so can be intepreted as a Cooper pair. In this doped phase, the Umklapp scattering terms will no longer we able to freeze the charge fluctuations, and one expects gapless excitations in the density and pair field phase, ϕρ+ . This doped phase will exhibit power-law d-wave superconducting correlations [21].

604

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d-Wave superconductivity

We now turn to the case of a two-dimensional superconductor which exhibits a particular type of d-wave pairing (denoted dx2 −y2 ) appropriate to the Cuprates. Our ultimate goal is to quantum disorder this state to obtain a description of the “nodal liquid”. There are two main distinctions between the 2d d-wave superconductor and it’s one-dimensional counterpart considered above. Firstly, a 2d superconductor exhibits true (off-diagonal) long-ranged order at T = 0. But more importantly, due to sign changes in the pair wave function, the dx2 −y2 superconductor exhibits gapless quasiparticle excitations. We first briefly review BCS theory which gives one a powerful framework to describe d-wave pairing and the gapless quasiparticles. In Section 7 below we incorporate quantum flucutations of the order parameter phase to obtain a complete effective low-energy theory of the dx− y2 phase. In Section 8 a dual represention is derived, and used to quantum disorder the superconductor in Section 9. 6.1 BCS theory re-visited It is instructive to briefly review BCS theory [37], focussing on the symmetries of the pair wave function and the superconducting order parameter. In particular, it is important to emphasize the important distinction between the wave function for the center of mass of the Cooper pair (often ignored) and the wavefunction for the relative coordinate. Consider a Hamiltonian expressed as a sum of kinetic energy and interaction terms, H = H0 + Hint , with H0 given in equation (2.2). We consider a rather general form for the electron interactions: Hint =

1 X vq (k, k0 )c†k+qα c†−k+qβ c−k0 +qβ ck0 +qα , 2V 0

(6.1)

k,k q

which is invariant under global charge U (1) and spin SU (2) symmetries. For simplicity Umklapp interaction terms have been ignored, so that the crystal momentum is conserved. The interaction term describes a two electron scattering process with 2q the total conserved momentum of the pair. For a density-density interaction in real space, such as the Coulomb interaction, vq (k, k0 ) = v(|k − k0 |), so is independent of q. Superconductivity within BCS theory requires an attractive interaction (in the appropriate angular momentum channel) between electrons. But the bare Coulomb interaction is of course strongly repulsive. In traditional low temperature superconductors, phonons are believed to drive the pairing, inducing a retarded attractive interaction at low energies below the deBye energy. Superconductivity in the high temperature Cuprates is probably of electronic origin. In this case, retardation leading to an attractive

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interaction at low energies would be due to virtual interactions via high energy electron states well away from EF . These processes can be studied via a renormalization group procedure [17], which consists of “integrating out” high energy electron states, and seeing how the remaining interactions between those electrons near the Fermi energy are modified. This is precisely what we implemented in detail for the two-leg ladder in Section 5. One thereby arrives at an effective low energy theory involving electron states within a small energy range of width 2Λ around EF , scattering off one another with an effective (or renormalized) interaction potential. In the following, we view vq (k, k0 ) as an effective low energy interaction. For the two-leg ladder the renormalized potential is given by putting the nine coupling contants equal to their values along the special ray. Upon Bosonization, the effective potential is given explicitly in equation (5.33). More generally, the form of the renormalized potential will be constrained by the original symmetries of the Hamiltonian. Specifically, time reveral and parity symmetries imply that vq (k, k0 ) is real, and odd in it’s arguments: vq (k, k0 ) = v−q (−k, −k0 ). Hermiticity implies vq (k, k0 ) = vq (k0 , k). The summation over momentum is now understood to be constrained, involving only electron operators with energy in a shell of width 2Λ about EF . BCS theory can be implemented by considering the operator, Pkαβ (q) = c−k+qα ck+qβ ,

(6.2)

which destroys a pair of electrons, with total momentum 2q. For k near the Fermi surface, and |q| kF , [Pk (q), Pk† (q0 )] = 0 for q 6= q0 , so that the pair operator resembles a boson operator, b(q). By analogy with Bose condensation, in the superconducting phase one expects a non-zero expectation value for the pair operator: hP i 6= 0. The pair operators entering into Hint are expressed as P = hP i + δP , and the fluctuations δP = P − hP i are presumed to be small. Upon ignoring terms quadratic in δP , Hint can be written (dropping additive constants), H1 =

1 X † [ck+qα c†−k+qβ ∆βα k (q) + h.c.], 2V

(6.3)

k,q

where we have introduced the (complex) superconducting order parameter (or “gap”), ∆, defined as, X vq (k, k0 )hc−k0 +qα ck0 +qβ i. (6.4) ∆αβ k (q) = k0

BCS is a self-consistent mean field theory: the full mean field (or quasiparticle) Hamiltonian, Hqp = H0 + H1 , which depends on ∆, is employed to compute the expectation value hc−k0 +qα ck0 +qβ i. Upon insertion in equation (6.4) one obtains a self-consistent equation which determines ∆ – the

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Topological Aspects of Low Dimensional Systems

celebrated BCS gap-equation. Notice that Hqp is bi-linear in electron operators and hence tractable, although it does involve “anomalous” terms involving pairs of creation or annihilation operators. Before carrying through this procedure, it is instructive to consider the form for the pair wavefunction which follows from a non-zero expectation value of the pair operator hP i = 6 0. Consider removing a pair of electrons, at positions R ± r/2, with R the center of mass position and r the relative coordinate. The pair wave function can be defined as, Φαβ (R, r) = hcα (R − r/2)cβ (R + r/2)i,

(6.5)

which depends on the spin of the electrons as well as the (center of mass and relative) positions. Upon transforming the electron operators into momentum space, one finds that X eiQ·R Φαβ (Q, r), (6.6) Φαβ (R, r) = Q with Q the center of mass momentum and Φαβ (Q, r) =

1 X αβ hPk (Q/2)ieik·r . N

(6.7)

k

Notice that the wavefunction in the relative coordinate, involves a Fourier transform with respect to the relative pair momentum, k. It is also instructive to define a spatially varying superconducting order parameter by Fourier transforming the gap function, ∆k (q): X (x) = eiQ·x ∆k (Q/2). (6.8) ∆αβ k Q In the superconducting phase one can often ignore the spatial dependence of the complex order parameter ∆k (x), and indeed in BCS theory this x dependence is dropped. However, if one wishes to include the effects of quantum fluctuations (to quantum disorder the superconductor) it is necessary to consider a spatially varying order parameter as discussed in Section 7 below. By analogy with Bose condensation, one expects the Cooper pairs to be condensed into a state of zero momentum, Q = 0. This requires hPkαβ (q)i = δq,0 hc−kα ckβ i,

(6.9)

which gives a relative pair wavefunction, Φ(r) ≡ Φ(Q = 0, r) of the form, Φαβ (r) =

1 X ik·r αβ Φk ; e N k

Φαβ k = hc−kα ckβ i.

(6.10)

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βα Due to the electron anticommutation relations one has Φαβ k = −Φ−k , which implies that the pair wavefunction is antisymmetric under exchange of the two electrons: Φαβ (r) = −Φβα (−r). When the Cooper pairs are condensed into a state with zero momentum, the superconducting order parameter becomes spatially uniform: ∆αβ k (x) ≡ , as seen from equation (6.4). The mean field Hamiltonian then takes ∆αβ k a rather simpler form:

H1 =

1X † † [ckα c−kβ ∆βα k + h.c.], 2

(6.11)

k

whereas the self-consistentcy condition becomes, ∆αβ k =

1 X v0 (k, k0 )hc−k0 α ck0 β i · V 0

(6.12)

k

Since the full model has a conserved SU (2) spin symmetry, the relative pair wavefunction can be expressed as the product of an orbital and a spin wavefunction: Φαβ k = φαβ Φk . The spin piece can be chosen as an eigenfunction of the total spin of the pair, that is a singlet with S = 0 or a triplet with S = 1. In conventional low temperature superconductors and in the Cuprates the Cooper pairs are singlets with, φαβ = δα↓ δβ↑ − δα↑ δβ↓ ,

(6.13)

in which case the orbital wavefunction is symmetric: Φk = Φ−k = hc−k↓ ck↑ i. (In the superfluid phases of 3 − He on the other hand, the Cooper pairs have S = 1.) The superconducting order parameter is then also a singlet; ∆αβ k ≡ φαβ ∆k , with ∆k = ∆−k satisfying ∆k =

1 X v0 (k, k0 )hc−k0 ↓ ck0 ↑ i · V 0

(6.14)

k

For singlet pairing, the final mean field (quasiparticle) Hamiltonian becomes, Hqp = H0 + H1 with, X [∆k c†k↑ c†−k↓ + ∆∗k c−k↓ ck↑ ]. (6.15) H1 = k

To complete the self-consistency requires diagonalizing the quasiparticle Hamiltonian. This is usually done in a way which masks the spin rotational invariance [37]. We prefer to keep the spin rotational invariance explicit, by defining a new set of Fermion operators, for ky > 0: χ1α (k) = ckα ;

y χ2α (k) = iσαβ c†−kβ ,

(6.16)

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which satisfy canonical Fermion anti-commutation relations: [χaα (k), χ†bβ (k0 )]− = δab δαβ δkk0 .

(6.17)

The first index a, b = 1, 2 acts in the particle/hole subspace. The σ y in the definition of χ2α has been introduced so that these new operators transform like SU (2) spinors under spin rotations: χaα → Uαβ χaβ , with U = exp(iθ · σ) a global spin rotation. In these variables, the quasiparticle Hamiltonian becomes X0 χ† (k)[τ z k + τ + ∆k + τ − ∆∗k ]χ(k), (6.18) Hqp = k

where the prime on the summation denotes over ky positive, only, and we have introduced a vector of Pauli matrices, ~τab acting in the particle/hole subspace. Also, we are employing the notation τ ± = (τ x ± iτ y )/2. To evaluate the self-consistency condition equation (6.14) we need the anomalous average of two electron fields (the orbital piece of the relative pair wavefunction), which is re-expressed as, Φk ≡

1X 1 hc∓k↓ c±k↑ i = hχ† (k)τ + χ(k)i · 2 ± 2

(6.19)

Diagonalization is now achieved by performing an SU (2) rotation in the particle/hole subspace, by defining rotated Fermion fields: χ(k) ≡ U (k)χ(k), ˜ with U (k) = e−iθ k ·τ . Assuming for simplicity that ∆k is real, the apy propriate rotation is around the y-axis by an angle θk , U (k) = e−iθk τ /2 , with q ∆k ; Ek = 2k + ∆2k . (6.20) sin(θk ) = Ek In terms of the rotated Fermion fields, χ, ˜ the quasiparticle Hamiltonian is diagonal, X0 Ek χ ˜† (k)τ z χ(k), ˜ (6.21) Hqp = k

with Ek the quasiparticle energy. Finally, we define a set of rotated electron operators via y χ ˜2α (k) = iσαβ a†−kβ ,

χ ˜1α (k) = akα ;

(6.22)

and the quasiparticle Hamiltonian can be re-expressed in standard form, X Ek a†kα akα , (6.23) Hqp = k

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where we have dropped an additive constant. Notice that the quasiparticle energy Ek ≥ 0 for all momentum. On the Fermi surface, k = 0 and the quasiparticle energy is given by |∆k | – the energy gap. To complete the self-consistentcy, the anomalous electron average (or relative orbital pair wavefunction from Eq. (6.19)) is expressed in terms of the quasiparticle operators. Upon using the fact that [U † τ + U ]diag = sin(θ)τ z /2 one obtains, Φk =

∆k [ha† akα i − 1], 2Ek kα

(6.24)

which reduces to Φk = −∆k /2Ek at zero temperature. At finite temperature the number of quasiparticles is simply a Fermi function: ha†kα akα i = 2f (Ek ), with f (E) = [exp(βE) + 1]−1 . One thereby obtains the celebrated BCS gap equation: ∆k = −

1 X ∆k0 v0 (k, k0 ) [1 − 2f (Ek0 )]. V 0 2Ek0

(6.25)

k

6.2 d-wave symmetry In a system with rotational invariance the orbital piece of the pair wavefunction, proportional to ∆k from equation (6.24), can be chosen as an eigenstate of angular momentum, a spherical harmonic Ylm in three dimensions. The simplest case is s-wave, with ∆k a constant over the (spherical) Fermi surface. Real materials of course do not share the full continuous rotational symmetry of free space. Nevertheless, a superconductor in which ∆k is everywhere positive over the Fermi surface is (loosely) referred to as having s-wave pairing – a property of all conventional low temperature superconductors. Since |∆k | is the quasiparticle energy on the Fermi surface, there are no low energy electronic excitations in an s-wave superconductor – the Fermi surface is fully gapped. Within BCS theory the magnitude of the (zero temperature) energy gap is related to the superconducting transition temperature: 2|∆| ≈ 3.5kB Tc . The presence of an energy gap leads to thermally activated behavior for various low temperature properties, such as the electronic specific heat and the magnetic penetration length. It is clear from the self-consistent gap equation (Eq. (6.25)) that a purely repulsive effective interaction, v0 (k, k0 ) > 0, precludes s-wave pairing within BCS theory (since 1 − 2f (Ek ) ≥ 0). In conventional superconductors, phonons are believed to drive s-wave pairing [37], generating an effective attractive interaction at low energies. Recent experiment [19, 20] has established that in the high temperature superconductors the orbital pairing symmetry is a particular form of d-wave, usually denoted as dx2 −y2 . Here x and y refer to the directions along the crystalline axis of a single Cu-O sheet, within which the Cu atoms form a

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Topological Aspects of Low Dimensional Systems

qy qx K 2

K1

b a

Fig. 7. In the dx2 −y 2 superconductor the quasiparticle energy vanishes at four points (±K1 and ±K2 ) in the Brillouin zone. The dotted line represents the Fermi surface. The wavevector q is rotated with respect to the a and b axis of the square lattice.

square lattice. In terms of the corresponding two dimensional momentum, k = (kx , ky ), the angular dependence of the gap function in this state is ∆k ∼ kx2 − ky2 , and from equation (6.24) the orbital piece of the relative pair wave function has the same d-wave symmetry. A novel feature of the dx2 −y2 state is that the gap function ∆k vanishes along lines in k − space with kx = ±ky , corresponding to nodes in the relative pair wave function. These lines intersect the (two-dimensional) Fermi surface at four points in momentum space. Near these four points (or “nodes”) in momentum space there are electronic excitations with arbitrary low energy, in striking constast to the fully gapped s-wave case. These low energy quasiparticle excitations dominate the physics of the dx2 −y2 superconductor at temperatures well below Tc , leading to power law temperature corrections in such quantities as the electronic specific heat and the magnetic penetration length. 6.3 Continuum description of gapless quasiparticles It is convenient to obtain a continuum description of the gapless d-wave quasiparticles, analogous to the Dirac theory description of the low energy properties of the 1d free Fermions employed in Section 4. A continuum form can be obtained directly from the general quasiparticle Hamiltonian equation (6.18) by specializing to dx2 −y2 symmetry and then focussing on those momenta close to the four nodes where the quasiparticle energy Ek = 0 (see Fig. 7). For a model with particle/hole symmetry k = −k+π , which together with parity symmetry implies that the four nodes occur at the

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special wavevectors ±Kj , with K1 = (π/2, π/2) and K2 = (−π/2, π/2). It is convenient to introduce two continuum fields Ψj , one for each pair of nodes, expanded around ±K1 , ±K2 : Ψjaα (q) = χaα (Kj + q).

(6.26)

Here, the wavevectors q are assumed to be small, within a circle of radius Λ around the origin. With this definition, the particle/hole transformation is extremely simple, Ψ → Ψ† .

(6.27)

For this reason it is convenient to always define the continuum fields Ψ around ±Kj , and account for deviations of the node momenta from these values by a particle/hole symmetry-breaking parameter λ. Once we have restricted attention to the momenta near the nodes, it is legitimate to linearize in the quasiparticle Hamiltonian. The resulting theory is more conveniently written in coordinates perpendicular and √ parallel to the Fermi√surface, so we perform the rotation via x → (x − y)/ 2 and y → (x + y)/ 2, correspondingly transforming the momenta qx and qy (see Fig. 7). Linearizing near the nodes, we put K1 +q = vF qx where vF is the Fermi velocity and ˜ y + O(q 2 ), ∆K1 +q = ∆q

(6.28)

˜ has dimensions of a velocity. An identical linearization is possible where ∆ around the second pair of nodes, except with qx ↔ qy . It is finally convenient to Fourier transform back into real space by defining, 1 X iq·x e Ψj (q), Ψj (x) = √ V q

(6.29)

where the momentum summation is for q < Λ. The continuum fields Ψ(x) describe long lengthscale variations of the quasiparticles, on scales greater than Λ−1 . We thereby arrive at a compact form for the continuum quasiparticle Hamiltonian in a dx2 −y2 superconductor: Hqp = HΨ + Hλ with HΨ

=

˜ + +∆ ˜ ∗ τ − )i∂y ]Ψ1 Ψ†1 [vF τ z i∂x + (∆τ +(1 ↔ 2; x ↔ y),

(6.30)

and the particle/hole symmetry breaking term, Hλ = λΨ†j τ z Ψj .

(6.31)

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Topological Aspects of Low Dimensional Systems

The quasiparticle Hamiltonian takes the form of (four) Dirac equations in 2 + 1 space-time dimensions, and can be readily diagonalized. For the first pair of nodes one obtains the relativistic dispersion, E1 (q) =

q ˜ 2 qy2 , (vF qx + λ)2 + |∆|

(6.32)

and a similar expression is obtained for E2 except with qx and qy interchanged. As usual in Dirac theory, the negative energy single particle states with energy −Ej (q) are filled but positive energy holes states can be created. As expected, the quasiparticle energy vanishes at the nodes (q = 0 with particle/hole symmetry λ = 0), so the “relativistic” particle is massless. Notice that non-zero λ indeed shifts the positions of the nodes. ˜ serves as a complex superconducting In this continuum description ∆ ˜ = 0 one recovers the order parameter for the dx2 −y2 state. Indeed, when ∆ metallic Fermi surface and the quasiparticle Hamiltonian describes gapless excitations for all qy . Below we will include quantum fluctuations by al˜ to depend on space and time. Before doing so, it is convenient lowing ∆ ˜ transforms under a particle/hole transformation. From the to see how ∆ transformation properties of the electron fields one deduces that the gap transforms as, ∆k → −∆∗−k+π , which is equivalent to complex conjugation for the (linearized) order parameter, ˜ →∆ ˜ ∗. ∆

(6.33)

Together with equation (6.27) this implies that the quasiparticle Hamiltonian in equation (6.30) is indeed particle/hole symmetric: HΨ → HΨ . 7

Effective field theory

7.1 Quasiparticles and phase flucutations Our goal in this section is to obtain a complete low-energy effective theory for the dx2 −y2 superconductor. This task is complicated by the existence of additional gapless excitations, besides the quasiparticles. Specifically, since the global U (1) charge conservation symmetry (cα → eiθ0 cα ) is spon˜ 6= 0, taneously broken by the existence of a non-zero order parameter, ∆ gapless Goldstone modes are expected. (In a three-dimensional superconductor these modes are actually gapped, due to the presence of long-ranged Coulomb interactions, but would be gapless for a thin 2d film.) These modes propogate in the phase of the complex order parameter. Thus a correct low energy theory for the dx2 −y2 state requires consideration of a spatially vary˜ ing order parameter, ∆(x). Generally, both the magnitude and the phase

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of the complex order parameter can vary, but we will focus exclusively on the phase fluctuations, writing ˜ ∆(x) = |∆|eiϕ(x) ,

(7.1)

with |∆| a (real) constant. Since amplitude fluctuations are costly in energy this should suffice in the superconducting phase, and will also allow us to describe the nodal liquid in which superconductivity is destroyed by phase fluctuations. The desired low energy effective theory can be obtained from symmetry considerations, and the form of the continuum quasiparticle Hamiltonian. A more microscopic approach, discussed briefly below, would entail integrating out high energy degrees of freedom in a functional integral representation. 7.1.1 Symmetry considerations Since the BCS gap equation has a degenerate manifold of solutions for arbitrary phase ϕ, the energy should only depend on gradients of ϕ(x). The appropriate Lagrangian which describes the fluctuations of the phase of the superconducing order parameter can thus be developed as a gradient expansion, with lowest order terms of the form, Lϕ =

1 κµ (∂µ ϕ)2 , 2

(7.2)

where the Greek index µ runs over time and two spatial coordinates: µ = 0, 1, 2 = t, x, y. Here κ0 is equal to the compressibility of the condensate (ignoring for the moment long-ranged Coulomb forces) and κj = −vc2 κ0 (for j = 1, 2 = x, y) with vc the superfluid sound velocity. This form is largely dictated by symmetry. Parity and four-fold rotational symmetry determine the form of the spatial gradient terms. The stiffness coefficients, κµ , can be estimated as follows. The pair compressibility κ0 should be roughly one half the electron compressibility – at least for weak interactions. If the pairing is electronic in origin, the Fermi velocity sets the scale for vc . In general a Berry’s phase term [4] linear in ∂t ϕ is allowed, LBerry = n0 ∂t ϕ,

(7.3)

where n0 is a two-dimensional number density. For a model with particle/hole symmetry which must be invariant under ϕ → −ϕ,

(7.4)

(which follows from the particle/hole transformation properties of the order ˜ ∗ ) it naively appears that the number density n0 ˜ ∼ eiϕ → ∆ parameter ∆ must vanish. However, this is not the case [15]. To see this it is necessary to

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Topological Aspects of Low Dimensional Systems

return to the lattice where the term in the (Euclidian) action which follows from LBerry is, Z SBerry = in0

β¯ h

dτ 0

X

∂τ ϕi ,

(7.5)

i

where i labels the sites of a square lattice with lattice spacing set to one and β = 1/kB T . The partition function is expressed as a functional integral of exp(−S) over configurations ϕi (τ ), with β periodic boundary conditions on the fields eiϕ . This implies the boundary conditions, ϕi (β) = ϕi (0) + 2πNi ,

(7.6)

with integer winding numbers Ni . We thus see that the Berry’s phase term contributes a multiplicative piece to the partition function (in each winding sector) of the form; exp(−SBerry ) = ei2πn0 NW ,

(7.7)

P with integer NW = i Ni . Under the particle/hole transformation equation (7.4), the winding numbers change sign, NW → −NW . The Berry’s phase term is thus invariant under the particle/hole transformation provided n0 is integer or half-integer. The appropriate value for n0 can be readily determined by obtaining the lattice Hamiltonian associated with the Lagrangian density Lϕ +LBerry . The first contribution can be conveniently regularized on the lattice as, Lϕ = −t

X

cos(ϕi − ϕj ) −

1X (∂t ϕi )2 . u i

(7.8)

Upon inclusion of the Berry’s phase term this gives the lattice Hamiltonian, X X cos(ϕi − ϕj ) + u (ni − n0 )2 . (7.9) Hϕ = −t

i

Here ni denotes a Cooper-pair number operator, canonically conjugate to the phase fields: [ϕi , nj ] = iδij .

(7.10)

The first term in Hϕ describes the hopping of charge 2e (spinless) Cooper pairs between neighboring sites of the lattice, and the second term is an onsite repulsive interaction. The parameter n0 plays the role of an “off-set” charge and determines the average number of Cooper pairs per site. For the Hubbard model at half-filling with one electron per site, the number of Cooper pairs clearly equals one-half the number of lattice sites. This is

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

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especially apparent in the limit of very large attractive Hubbard interaction when the electrons pair into on-site singlets, but is expected to be more generally valid. Thus, it is clear that one should take n0 = 1/2. Tuning away from particle/hole symmetry with a chemical potantial µ, corresponds to changing n0 away from one-half. In the superconducting phase one expects that the winding numbers will all vanish, since the phase field ϕ is essentially constant in space and time, and the Berry’s term plays no role. But when the superconductor is “quantum disordered”, the phase field fluctuates wildly with signifigant winding, and inclusion of the Berry’s phase term is expected to be important (but see Sect. 8 below). It remains to couple these phase fluctuations to the gapless quasiparti˜ directly enters the quasiparticle cles. Since the order parameter ∆ Hamiltonian equation (6.30), one can readily guess the appropriate coupling. ˜ → v∆ eiϕ with v∆ real. Since ϕ varies spatially, We should simply replace ∆ some care is required. In the quasiparticle Hamiltonian we let, ˜ + i∂y → v∆ τ + eiϕ/2 (i∂y )eiϕ/2 , ∆τ

(7.11)

and similarly for the τ − term. This “symmetric” form leads to an hermitian Hamiltonian, physical currents, and respects the symmetries of the problem. A careful derivation of equation (7.11) is given below. With this prescription, the quasiparticle Hamiltonian becomes X † Ψ1 [vF τ z i∂x + v∆ τ s eisϕ/2 (i∂y )eisϕ/2 ]Ψ1 Hqp = s=±

+(1 ↔ 2; x ↔ y).

(7.12)

Since ϕ can also fluctuate with time, it will convenient to consider the time dependence via a Lagrangian formulation. The Lagrangian density is Lqp = Ψ†j i∂t Ψj − Hqp .

(7.13)

The full low-energy effective Lagrangian in the d-wave superconductor is obtained by adding the two contributions: Lϕ + Lqp . 7.1.2 Microscopic approach To illustrate how one might try to “derive” this effective theory from a more microscopic starting point, we briefly consider a simple model Hamiltonian, X c†α (~x)c†β (~x0 )cβ (~x0 )cα (~x), (7.14) H = H0 − V h~ x~ x0 i

where H0 is the usual kinetic energy describing hopping on a 2d square lattice and we have added an attractive near-neighbor interaction with strength

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Topological Aspects of Low Dimensional Systems

V . To derive the effective field theory, it is convenient to express the partition function Z = T r e−βH , as an imaginary time coherent state path integral [29], Z (7.15) Z = [Dc][Dc]e−S , where c and c are Grassman fields and the Euclidean action is simply ) ( Z X cα (~x)∂τ cα (~x) + H[c, c] · (7.16) S = dτ ~ x

We consider here only T = 0, for which the τ integration domain is infinite. The possibility of a d-wave superconducting phase can be entertained by decoupling the above action using a Hubbard-Stratonovich transformation: Z (7.17) Z = [Dc][Dc][D∆][D∆∗ ]e−S1 , R P with S1 = dτ [ ~x cα (~x)∂τ cα (~x) + Heff ]. The effective Hamiltonian can be decomposed into Heff = H0 + Hint + H∆ , with i X h αβ (7.18) ∆~x~x0 cα (~x)cβ (~x0 ) + h.c. , Hint = h~ x~ x0 i

H∆ =

1 X αβ 2 |∆~x~x0 | . V 0

(7.19)

h~ x~ xi

Equations (7.18- 7.19) form a basis for studying the original electron model. At this stage BCS mean field theory could be implemented by integrating out the electron degrees of freedom to obtain an effective action only depending on ∆, Seff (∆). Minimizing this action with respect to ∆ would give the gap equation. One could imagine including fluctuations by expanding about the saddle point solution. But for a d-wave superconductor this procedure is problematic, since integrating out gapless quasiparticles will generate singular long-ranged interactions in Seff (∆). It is preferable to retain the gapless quasiparticles in the effective theory, and only integrate out the high frequency electron modes which are well away from the nodes. In this way, the dynamics and interactions generated for the order parameter ∆ will be local. Rather than trying to implement this procedure, we content ourselves with arguing that the “symmetric” prescription adopted above indeed gives the correct form for the phase-quasiparticle coupling term. To this end we focus on singlet pairing, defining x, ~x0 )(δα↑ δβ↓ − δα↓ δβ↑ ). ∆~αβ x~ x0 = ∆(~

(7.20)

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The triplet pieces of ∆ are presumed to be massive, so that they can be safely integrated out. Since ∆ lives on the bonds, it is convenient to associate two such fields with each site on the square lattice, i.e. ∆1 (~x) ≡

∆(~x, ~x + eˆ1 ),

(7.21)

∆2 (~x) ≡

∆(~x, ~x + eˆ2 ),

(7.22)

where eˆ1 , eˆ2 are unit vectors along the a and b axes of the square lattice, respectively. The interaction Hamiltonian becomes, ) ( h i X † † (7.23) ∆j (~x) c↑ (~x)c↓ (~x + eˆj )− ↑↔↓ + h.c. , Hint = j,~ x

where the sum includes all lattice sites and j = 1, 2. The magnitudes of ∆1 and ∆2 , as well as their relative sign, are determined by the effective action generated upon integrating out the high-energy modes. For a d-wave superconductor the effective action will be minimized for ∆1 = −∆2 = For ∆0 eiϕ , up to massive modes. We can now√take the continuum limit. √ ˜ 2. agreement with Section 6, we define v∆ = 2 2∆0 , or ∆1 = −∆2 = ∆/2 In addition, we take the continuum limit of the electron fields, using the decompositions c†↑

∼

Ψ†111 ix+y −Ψ122 (−i)x+y +Ψ†211 (−i)x−y −Ψ222 ix−y ,

c†↓

∼

Ψ†112 ix+y +Ψ121 (−i)x+y +Ψ†212 (−i)x−y +Ψ221 ix−y ,

and the hermitian conjugates of these equations. Inserting these into equation (7.23), gradient-expanding the Ψ fields, and rotating 45 degrees to coordinates along the (π, π) and (−π, π) directions, one obtains Hint = Rx−y d2 xHint , with " # ˜ † ∆ Ψ1 τ + i∂y Ψ1 − (i∂y Ψ†1 )τ + Ψ1 + h.c. Hint = 2 +(1 ↔ 2, x ↔ y).

(7.24)

˜ term in equation (6.30) when the order This form is identical to the ∆ ˜ parameter ∆ is constant, but the symmetric placement of derivatives is ˜ = important in the presence of phase gradients. In particular, now let ∆ iϕ v∆ e and integrate by parts to transfer the derivative in the second term ˜ combination. Upon using the operator identity from the Ψ† to the ∆Ψ 1 iϕ e i∂y + i∂y eiϕ = eiϕ/2 i∂y eiϕ/2 , 2

(7.25)

this becomes identical to the symmetrized form of the phase-quasiparticle interaction hypothesized in equation (7.12).

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Topological Aspects of Low Dimensional Systems

7.2 Nodons Treatment of quantum phase fluctuations is complicated by the coupling between the quasiparticle Fermion operators, Ψ, and exponentials of the phase ϕ, as seen explicitly in Hqp in equation (7.12). The form of the coupling is determined by the electric charge carried by Ψ, which is uncertain – being built from electron and hole operators. To isolate the uncertain charge of Ψ it is extremely convenient to perform a change of variables [14], defining a new set of fermion fields ψj via ψj = exp(−iϕτ z /2)Ψj .

(7.26)

In the superconducting phase, and in the absence of quantum flucutations of the order-paramater phase, one can set ϕ = 0, and these new fermions are simply the d-wave quasiparticles. However, when the field ϕ is dynamical and fluctuates strongly this change of variables is non-trivial. In particular, the new fermion fields ψ are electrically neutral, invariant under a global U (1) charge transformation (since ϕ → ϕ + 2θ0 under the U (1) charge transformation in Eq. (2.8)). As we shall see, when the d-wave superconductivity is quantum disordered, these new fields will play a fundamental role, describing low energy gapless excitations, centered at the former nodes. For this reason, we refer to these fermions as nodons. For completeness, we quote the symmetry properties of the nodon field under a particle/hole transformation. Since ϕ → −ϕ, one has simply ψ → ψ† .

(7.27)

The full Lagrangian in the d-wave superconductor, L = Lϕ + Lqp , can be conveniently re-expressed in terms of these nodon fields since Lqp = Lψ + Lint + Lλ with a free nodon piece, Lψ

=

ψ1† [i∂t − vF τ z i∂x − v∆ τ x i∂y ]ψ1 +(1 ↔ 2, x ↔ y),

(7.28)

interacting with the phase of the order-parameter: Lint = ∂µ ϕJµ .

(7.29)

Here the electrical 3-current Jµ is given by J0 =

Jj =

1 † z ψ τ ψj , 2 j

(7.30)

vF † ψ ψ . 2 j j

(7.31)

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

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Because the transformation in equation (7.26) is local, identical expressions hold for these currents in terms of the quasiparticle fields, Ψ. The form of the particle/hole asymmetry term remains the same in terms of the nodon fields: Lλ = λψj† τ z ψj .

(7.32)

It is instructive to re-express the components of the currents Jµ back in terms of the original electron operators. One finds 1 † cKj cKj + c†−Kj c−Kj , (7.33) J0 = 2 (with an implicit spin summation) which corresponds physically to the total electron density living at the nodes, in units of the Cooper pair charge. Similarly, vF † cK cK − c†−K c−K (7.34) Jj = j j j j 2 corresponds to the current carried by the electrons at the nodes. Thus, Jµ can be correctly interpreted as the quasiparticles three-current. To complete the description of a quantum mechanically fluctuating order parameter phase interacting with the gapless fermionic excitations at the nodes, we minimally couple to an external electromagnetic field, Aµ . Since the nodon fermions are neutral, the only coupling is to the order-parameter phase, via the substitution ∂µ ϕ → ∂µ ϕ− 2Aµ . Here we have set the electron charge e = 1, with a factor of 2 appropriate for Cooper pairs. The final Lagrangian then takes the form L = Lϕ + Lψ + Lint + Lλ , with Lϕ =

1 κµ (∂µ ϕ − 2Aµ )2 , 2

Lint = (∂µ ϕ − 2Aµ )Jµ ,

(7.35) (7.36)

and Lψ still given by equation (7.28). Here we have dropped the Berry’s phase term, which is not expected to play an important role in the superconducting phase. Long-ranged Coulomb interactions could be readily incorporated at this stage by treating A0 as a dynamical field and adding a term to the Lagrangian of the form, Lcoul = (1/2)(∂j A0 )2 . The spatial components of the electromagnetic field, Aj , have been included to keep track of the current operator. 7.2.1 Symmetries and conservation laws If the full effective Lagrangian L is to correctly describe the low energy physics it must exhibit the same symmetries as the original electron Hamiltonian – the most important being charge and spin conservation. Since the

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Topological Aspects of Low Dimensional Systems

ψ operators are electrically neutral the full U (1) charge transformation is implemented by ϕ → ϕ + 2θ0 for constant θ0 , and L is indeed invariant. Moreover, the Lagrangian is invariant under ψα → Uαβ ψβ for arbitrary (global) SU (2) spin rotations U = exp(iθ · σ). Since the Cooper pairs are in spin singlets, all of the spin is carried by the nodons. As usual, associated with each continuous symmetry is a conserved “charge” which satisfies a continuity equation (Noether’s theorem). Since the Lagrangian only depends on gradients of ϕ, the Euler-Lagrange equation of motion reduces to the continuity equation, ∂µ Jµtot = 0,

(7.37)

where the total electric 3-current is given by Jµtot = ∂L/∂(∂µ ϕ) = −∂L/∂Aµ . This gives, Jµtot = κµ (∂µ ϕ − Aµ ) + Jµ ,

(7.38)

where the first term is the Cooper pair 3-current and the second the quasiparticles current. The analogous conserved spin currents can be obtained by considering infinitesimal spin rotations, U = 1 + iθ(x, t) · σ,

(7.39)

for slowly varying θ(x, t). Under this spin rotation the Lagrangian transforms as, L → L + ∂µ θ · j µ ,

(7.40)

with j µ given below. After an integration by parts, invariance of the action S under global spin rotations implies continuity equations ∂µ j µ = 0 for each of the three spin polarizations, j. The space-time components of the conserved spin currents are given explicitly by, 1 † ψ σψ1 + (1 → 2), 2 1

(7.41)

1 1 vF ψ1† στ z ψ1 + v∆ ψ2† στ x ψ2 , 2 2

(7.42)

j0 =

jx =

and j y the same as j x except with ψ1 ↔ ψ2 . Notice that in contrast to the electrical current, the spin current operator has a contribution which is proportional to the velocity tangential to the Fermi surface, v∆ , which is anomalous when re-expressed in terms of the original electron operators. Surprisingly, the effective Lagrangian exhibits additional continuous symmetries, not present in the original Hamiltonian. Firstly, L is invariant under separate SU (2) spin rotations on the two pairs of nodes, ψj for

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j = 1, 2. Moreover, the Lagrangian is also invariant under two additional U (1) transformations ψj → eiθj ψj for arbitrary constant phases, θj . These latter symmetries imply two new conserved “charges”, ψj† ψj (no sum on j). We refer to these conserved quantities as “nodon charges”. The associated conserved nodon 3-currents take the same form as the spin currents above, except replacing σ/2 by the identity. As seen from equation (7.31), the conserved nodon charges are proportional to the quasiparticle electrical current, since Jj = (vF /2)ψj† ψj . It is possible to add to L additional interaction terms which are consistent with the original U (1) and SU (2) symmetries, but do not conserve the “nodon charge”. Specifically, anomalous quartic interaction terms of the form ψ 4 arise from Umklapp scattering processes in the original electron Hamiltonian and clearly change the nodon charge. However, such interactions are unimportant at low energies due to severe phase space reR strictions. To see this, consider how the action, S = d2 xdtL transforms under a renormalization group (RG) rescaling transformation, xµ → bxµ ;

ψ → b−1 ψ;

ϕ → b−1/2 ϕ,

(7.43)

with rescaling parameter b > 1. By construction, this leaves the quadratic pieces Sψ and Sϕ invariant, but interaction terms such as uψ 4 scale to zero under the RG (b → ∞) since u → u/b. It is the T = 0 “fixed point” theory described by the quadratic terms which exhibits the additional symmetries. Incidentally, the coupling term Lint above also scales to zero (as b−1/2 ) under the renormalization group. In the resulting quadratic theory the quasiparticles and phase fluctuations actually decouple. 7.2.2 Superfluid stiffness The above effective theory is particularly convenient for examining very low temperatures properties of the dx2 −y2 state. Of interest are charge response functions such as the electrical conductivity and the superfluid stiffness (measureable via the penetration length). The spin excitations (carried by the quasiparticles) can also be probed via resonance techniques, such as NMR and ESR. Impurity scattering can be readily incorporated by coupling a random potential to the electron density (which can be re-expressed as a nodon bi-linear). For illustrative purposes we briefly consider the quasiparticle contribution to the low temperature superfluid stiffness and extract the famous T -linear dependence. For a Galilean invariant system of mass m bosons the superfluid stiffness Ks equals the superfluid density divided by m. But more generally Ks can be extracted rather directly by considering the response of the system to a transverse vector potential [38]. We set A0 = 0 and decompose the static vector potential Aj into longitudinal and transverse

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Topological Aspects of Low Dimensional Systems

pieces: Aj = A`,j + At,j ,

(7.44)

with ∂j At,j = 0 and ij A`,j = 0. The superfluid stiffness is then given by, Ks =

1 ∂ 2F , V ∂A2t,x

(7.45)

where F = −kB T lnZ is the Free energy and V → ∞ is the area of the 2d system. Here At,x can be taken spatially constant. To extract F the partition function can be written as an imaginary time coherent state path integral [29], Z (7.46) Z = [Dϕ][Dψ][Dψ]exp(−SE ), R with Euclidian action SE = d2 xdτ LE . The longitudinal vector potential, which can be expressed as a gradient of a scalar field A`,j = ∂j Λ, can be eliminated entirely by shifting ϕ → ϕ+ Λ. Moreover, the crossterm between ∂j ϕ and At,j vanishes since At is divergenceless. The Gaussian integral over ϕ can then be readily perfomed and simply generates an irrelevant interaction term (J ∼ (ψ † ψ)2 ) which can be ignored. One thereby arrives at an effective action depending only on ψ and Aj with associated Hamiltonian density of the form: Heff = Hψ + HA , with Hψ the free nodon Hamiltonian and HA =

1 0 2 K A + At,j Jj . 2 s t,j

(7.47)

Here Ks0 = κ0 vc2 is the superfluid stiffness from the Cooper pairs, and Jj = (vF /2)ψj† ψj . Notice that the (transverse) vector potential acts as an effective chemical potential for the “nodon charge” density, ρn = ψj† ψj . Thus, the superfluid stiffness can be expressed in terms of the nodon “compressibility” as Ks = Ks0 − (vF /2)2 κn ,

(7.48)

where κn = ∂ρn /∂µn and µn = (vF /2)At,x is the nodon “chemical potential”. The nodon compressibility can be extracted by diagonalizing the Hamiltonian, Hψ . From the first pair of nodes one obtains the free Fermion form, X E1 (q)[a†q aq + b†q bq ], (7.49) Hψ = q

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where E1 (q) is given in equation (6.32) and we have suppressed the spin index. Here a and b are particle and hole operators, respectively. The nodon charge is simply, ρn =

1 X † [haq aq i − hb†q bq i], V q

(7.50)

where the averages are taken with Hψ − µn ρn . At finite temperatures one obtains Z dq [f (E1 (q) − µn ) − f (E1 (q) + µn )], (7.51) ρn = 2 (2π)2 where f (E) are Fermi functions, and the factor of 2 is from the spin sum. Finally, upon differentiating with respect to µn and performing the momentum integral one extracts the desired result for the low temperature superfluid stiffness: Ks (T ) = Ks0 − c

vF kB T, v∆

(7.52)

with the dimensionless constant c = (ln 2/2π). 8

Vortices

8.1 hc/2e versus hc/e vortices Having successively incorporated phase fluctuations into the effective low energy description of the dx2 −y2 state, we now turn to a more interesting task – quantum disordering the superconductivity to obtain the nodal liquid phase, a novel Mott insulator. The superconductivity is presumed to be destroyed by strong quantum fluctuations of the order parameter phase ϕ driven by vortex excitations. In two-dimensions vortices are simply whorls of current swirling around a core region. But in a superconductor the circulation of such vortices is quantized, since upon encircling the core the phase ϕ can only change by integer multiples of 2π. Inside the core of a vortex the ˜ vanishes, but is essentially magnitude of the complex order parameter |∆| constant outside. In the superconducting phase, the size of the core is set by the coherence length - roughly 10 ˚ A in the Cuprate materials. Such vortices are thus tiny “point-like” objects, with a truly microscopic size in the Cuprate materials. The “elementary” vortex has a phase winding of ±2π. When a superconductor is placed in an external magnetic field, the currents circulating around the core of a vortex tend to screen out the magnetic field, except within a region of the penetration length, λ, from the vortex core. (In the cuprate materials λ is in the range of a thousand angstr¨ oms.) In addition

624

Topological Aspects of Low Dimensional Systems

to the circulation, the total magnetic flux near a vortex is quantized – in units of the flux quantum hc/2e. An “elementary” vortex quantizes precisely hc/2e of magnetic flux, and will thus henceforth be referred to as an hc/2e vortex. As we shall argue [15], to obtain the nodal liquid phase it will be necessary to “liberate” double-strength hc/e vortices, keeping the hc/2e vortices “confined”. Generally, the position of these “point-like” vortices can change with time, and their dynamics requires a quantum mechanical description. Thus a collection of many vortices can be viewed as a many body system of “pointlike” particles. Since positive (+1) and negative (−1) circulation vortices can annihilate – and disappear (just as for real elementary particles like electrons and positrons), they behave as “relativistic” particles. There is a conserved vortex “charge” in this process, namely the total circulation, and an associated current. Since the Cooper pairs are Bosons, one anticipates that the “dual particles” – the vortices – are also Bosonic forming a relativistic Boson system, and this is indeed the case [39]. However, in the superconducting phase at zero temperature there are no vortices present – this phase constitutes a “vacuum” of vortices. More precisely, due to quantum fluctuations vortices are present as short-lived “virtual” fluctuations, popping out of the “vacuum” in the form of small tightly bound (neutral) pairs. For the low energy properties of the superconductor these fluctuations can be largely ignored. But what happens if these virtual pairs unbind into a proliferation of free mobile vortices? Vortex motion is very effective at scrambling the phase ϕ of the superconducting order, so that mobile vortices will in fact destroy the superconductivity. Since the vortices are Bosonic, once they are free and mobile they will “Bose condense”, at least at zero temperature. One thereby obtains a nonsuperconducting insulating state, with the “vortex-condensate” serving as an appropriate order parameter. As we shall see, it will be extremely convenient to pass to a “dual” representation [39, 40] in which the vortices are the basic “particles” – rather than the Cooper pairs. Consider first unbinding and condensing the “elementary” hc/2e vortices [15]. When a Cooper pair is taken around such a vortex it’s wave function acquires a ±2π phase change. Likewise, when an hc/2e vortex is taken around a Cooper pair, the vortex wavefunction acquires the 2π phase change. Thus, hc/2e vortices “see” Cooper pairs as a source of “dual flux”, each carrying one unit. (This notion can be made precise by performing a duality transformation – see below and the Appendix.) For a Hubbard model of electrons at half-filling, on average there is one-half of a Cooper pair per site, as seen explicitly in the effective lattice Cooper pair Hamiltonian, equation (7.9), which has offset charge n0 = 1/2. Thus, these elementary vortices “see” a dual “magnetic field”, with one-half of a dual flux-quantum per plaquette. When the hc/2e vortices unbind and condense, they will

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quantize this dual flux, in precisely the same way that the condensation of Cooper pairs in a real superconductor will quantize an applied magnetic field – forming an Abrikosov flux-lattice (if Type II). The analog of the Abrikosov flux-lattice for the hc/2e vortex condensate is an ordered lattice of Cooper pairs. In this “crystal” state at half-filling, the Cooper pairs will preferentially sit on one of the two equivalent sub-lattices of the square lattice. This state can be described as a commensurate charge-density-wave with ordering wavevector (Q = π, π), which spontaneously breaks the discrete symmetry under translation by one lattice spacing. Such ordering implies a considerable degree of double occupancy for the electrons, and thus seems most reasonable for a Hubbard type model with an attractive on-site interaction (negative u). In the Cuprate materials there is a strong on-site repulsion, and moreover there is no evidence for “charge-ordering” near Q. Thus, for a description of the pseudo-gap regime in the Cuprate materials, we can rule out the hc/2e vortex-condensate on phenomenological grounds. Instead, we consider the possibility of unbinding and condensing doublestrength hc/e vortices, keeping the elementary hc/2e vortices confined [15]. When an hc/e vortex is taken around a Cooper pair it acquires a 4π phase change. A 2π phase change corresponds to taking such an hc/e vortex around “half” of a Cooper pair – which has charge e. Thus, a condensation of hc/e vortices should correspond to a “crystal” of such charge e objects. But at half-filling with charge e per lattice site, this should correspond to a state without charge ordering or translational symmetry breaking. As we shall see, for a dx2 −y2 superconductor the resulting hc/e “vortex-condensate” gives a description of the nodal liquid phase. This procedure – keeping the elementary hc/2e vortices confined and only liberating the hc/e vortices – is responsible for the remarkable properties of the nodal liquid [15]. To see why, consider first the Berry’s phase term in equation (7.3). With only hc/e vortices present, the Cooper pair phase, ϕ, only winds by integer multiples of 4π – not 2π. At half-filling (with n0 = 1/2) the Berry’s phase term will not contribute to the partition function (see Eq. (7.7)) and can thus be dropped entirely in the description of the nodal liquid. This can be implemented by defining a new phase field: φ = ϕ/2,

(8.1)

and only allowing vortices in φ(x) with circulation 2π times an integer. This restriction precludes hc/2e vortices, and guarantees that the field b = eiφ ,

(8.2)

is single-valued. As an operator, b creates a spinless excitation with charge e. When re-written in terms of φ, the effective Lagrangian for a d-wave superconductor with quantum phase fluctuations (from Eqs. (7.35, 7.36))

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Topological Aspects of Low Dimensional Systems

becomes L = Lφ + Lint + Lψ with Lφ + Lint =

1 2 κµ (∂µ φ − Aµ + κ−1 µ Jµ ) , 2

(8.3)

and Lψ given in equation (7.28). The Berry’s phase term has been dropped, since it plays no role when exp(iφ) is a single valued field. Here, we have absorbed a factor of two into κµ and also completed the square with the nodon current, Jµ , dropping order Jµ2 terms which are irrelevant as discussed after equation (7.43). Notice that the coefficient of Aµ is one – as expected for a charge e operator exp(iφ). By precluding hc/2e vortices, we see the emergence of a new bosonic field, exp(iφ), with exotic quantum numbers – charge e but spin zero – which will be referred to as a “holon”. This is the first hint of spin-charge separation [41–43] in the nodal liquid. As we shall see, another remarkable consequence of precluding hc/2e vortices, is that the charge neutral spin one-half nodons survive under hc/e vortex condensation into the nodal liquid. To see why this is not the case if elementary hc/2e vortices are condensed [15] (as in the charge-densitywave), it is very instructive to consider the transformation which relates the nodons to the d-wave quasiparticles, equation (7.26), which can be written in terms of the new field φ (= ϕ/2) as: ψ = exp(−iτz φ)Ψ.

(8.4)

In the presence of vortices, the nodon field ψ only remains single-valued if hc/2e vortices are excluded (so that exp(±iφ) is single valued). Indeed, when a nodon is taken around an hc/2e vortex, it’s wavefunction changes sign, since φ winds by π. This implies a very strong and long-ranged “statistical” interaction between nodons and hc/2e vortices. If hc/2e vortices proliferate and condense, it will clearly be very difficult for the nodons to propogate coherently. In fact, we have argued recently [15] that in this case the nodons are bound (actually “confined”) to the holons, leaving only the electron in the spectrum of the charge-density-wave. 8.2 Duality We now consider implementing the procedure of unbinding and condensing hc/e vortices in the dx2 −y2 superconductor. To this end, it is extremely convenient to pass to the “dual” representation [39, 40] in which the vortices are the basic “particles”, rather than the Cooper pairs. The most straightforward way to incorporate hc/e vortices is by placing the (singlevalued) field exp(iφ) on the sites of a lattice [39], so that vortices can exist in the plaquettes. A lattice duality transformation can be implemented in which the phase φ is replaced by a dual field, θ, which is the phase of a vortex complex field, Φ ∼ eiθ . In a Hamiltonian description, Φ and Φ† can

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be viewed as vortex quantum field operators – which destroy and create hc/e vortices. On a 2 + 1-dimensional Euclidian space-time lattice, the appropirate model corresponding to the phase Lagrangian equation (8.3) is essentially a classical 3d-xy model with an effective gauge field: −1 Aeff µ = Aµ − κµ Jµ .

(8.5)

The lattice duality transformation for the 3d-xy model with gauge field is implemented in some detail in the Appendix. An alternative method which we sketch below, involves implementing the duality transformation directly in the continuum [40]. To this end we introduce a vortex 3-current, jµv , which satisfies, jµv = µνλ ∂ν ∂λ φ.

(8.6)

In the presence of hc/e vortices, φ is multi-valued, ∂µ φ is not curl-free, and jµv is non-vanishing. Even in the dual vortex representation the total electrical charge must be conserved. This can be achieved by expressing the total electrical 3-current (in units of the electron charge e) as a curl, Jµtot = µνλ ∂ν aλ ,

(8.7)

where we have introduced a “fictitious” dynamical gauge field, aµ . (In the Appendix the electrical 3-current is expressed as a lattice curl of aµ .) Upon combining equation (7.38) with (8.6) and (8.7), one can eliminate the phase field, φ, and relate aµ to the vortices: −1 jµv = µνλ ∂ν [κ−1 λ λαβ ∂α aβ + Aλ − κλ Jλ ],

(8.8)

where Jµ is the quasiparticle 3-current defined earlier in equations (7.307.31). In this continuum approach to duality, a dual description is obtained by constructing a Lagrangian, LD , depending on aµ , Jµ and jµv , whose equation of motion, obtained by differentiating the action with respect to aµ , leads to the above equation. It is convenient to first express the vortex 3-current in terms of a complex field, Φ, which can be viewed as an hc/e vortex destruction operator. The dual Lagrangian is constructed to have an an associated U (1) invariance under Φ → eiα Φ, which guarantees that jµv is indeed conserved. When an hc/e vortex is taken around a Cooper pair it aquires a 4π phase change (2π around a charge e “holon”). In the dual representation the vortex wavefunction Φ should acquire a 4π phase change (or 2π for a “holon”). This can be achieved by minimally coupling derivatives af Φ to the “fictitious” vector potential aµ . The appropriate dual Lagrangian can be conveniently decomposed as LD = Lψ + Lv + La , where Lψ is given in equation (7.28). The vortex piece

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Topological Aspects of Low Dimensional Systems

has the Ginzburg-Landau form [44], Lv =

κµ |(∂µ − iaµ )Φ|2 − VΦ (|Φ|), 2

(8.9)

as constructed explicitly with lattice duality in the Appendix. The vortex 3-current, following from jµv = −∂Lv /∂aµ , is jµv = κµ Im[Φ∗ (∂µ − iaµ )Φ].

(8.10)

For small |Φ| (appropriate close to a second order transition) one can expand the potential as, VΦ (X) = rΦ X 2 + uΦ X 4 . The remaining piece of the dual Lagrangian is La =

1 2 (e − b2 ) + aµ µνλ ∂ν (Aλ − κ−1 λ Jλ ), 2κ0 j

(8.11)

with dual “magnetic” and “electric” fields: b = ij ∂i aj and ej = vc−1 (∂j a0 − ∂0 aj ). It can be verified that the dual Lagrangian has the desired property that equation (8.8) follows from the equation of motion δSD /δaµ = 0. 9

Nodal liquid phase

In this section we employ the dual representation of the dx2 −y2 superconductor to analyze the quantum disordered phase - the nodal liquid. The dual representation comprises a complex vortex field, which is minimally coupled to a gauge field, as well as a set of neutral nodon fermions. Without the nodons and in imaginary time, the dual Lagrangian is formally equivalent to a classical three-dimensional superconductor at finite temperature, coupled to a fluctuating electromagnetic field. To disorder the d-wave superconductor, we must order the dual “superconductor” – that is, condense the hc/e vortices. The nature of the resulting phase will depend sensitively on doping, since upon doping, the dual “superconductor” starts seeing an applied “magnetic field”. Below, we first consider the simpler case of half-filling. We then turn to the doped case, where two scenarios are possible depending on whether the dual “superconductor” is Type I or Type II [44]. 9.1 Half-filling Specialize first to the case of electrons at half-filling, with particle-hole symmetry. In the dual representation, the “magnetic field”, b, is equal to the deviation of the total electron density from half-filling. Thus at half-filling hbi = 0 and the dual Ginzburg-Landau theory is in zero applied field. The quantum disordered phase corresponds to condensing the hc/e vortices, setting hΦi = Φ0 6= 0. In this dual Meissner phase the vortex Lagrangian

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becomes Lv =

1 κµ Φ2o (atµ )2 , 2

(9.1)

where at represents the transverse piece of aµ . It is then possible to integrate out the field aµ which now enters quadratically in the Lagrangian. Equivalently, aµ can be eliminated using the equation of motion which follows from δSD /δaµ = 0. The full Lagrangian in the nodal liquid phase is then Lnl = Lψ + Aµ Iµ +

0 2 B2 Ej − + O (∂J)2 , 2 2µ0

(9.2)

where we have introduced the physical magnetic and electric fields: B = ij ∂i Aj and Ej = ∂j A0 −∂t Aj . The last two terms describe a dielectric, with magnetic permeability µ0 = κ0 Φ20 and dielectric constant 0 = (µ0 vc2 )−1 , with the sound velocity entering, rather than the speed of light. The external electromagnetic field is coupled to the 3-current Iµ , which can be expressed as a bi-linear of the nodon fermions as, Iµ =

0 [κν ∂ν2 Jµ − κµ ∂µ (∂ν Jν )]. κ20 vc2

(9.3)

Notice that this 3-current is automatically conserved: ∂µ Iµ = 0. The order (∂J)2 terms which we have not written out explicitly are quartic in the fermion fields, and also involve two derivatives. Since Lψ describes Dirac fermions in 2 + 1 space-time dimensions, these quartic fermion terms are highly irrelevant, and rapidly vanish under the rescaling transformation in equation (7.43). Thus, the low energy description of the nodal liquid phase is exceedingly simple. It consists of four neutral Dirac fermion fields – two spin polarizations (α = 1, 2) for each of the two pairs of nodes. Despite the free fermion description, the nodal liquid phase is highly non-trivial when re-expressed in terms of the underlying electron operators. Indeed, the ψ fermion operators are built from the quasiparticle operators Ψ in the d-wave superconductor, but are electrically neutral, due to the “gauge transformation” in equation (7.26). In addition to the gapless nodons, one expects exotic charged excitations at finite energy in the nodal liquid. To see this, imagine applying an external dual “magnetic field” to the Ginzburg-Landau “superconductor”, which corresponds to a non-zero chemical potential for the electrons. Being in the Meissner state, this “field” will be screened out, so that the internal field, b, which corresponds to deviations in the electron charge density from half-filling, will vanish. Clearly, this corresponds to a Mott insulator [45] with the Mott gap being proportional to the dual critical field. In a Type II superconductor, an internal magnetic field will be “quantized” into

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Topological Aspects of Low Dimensional Systems

flux-tubes carrying a quantum of flux [44]. For the dual Ginzburg-Landau theory, this corresponds to a quantization of electric charge, with a flux tube corresponding to charge e. Thus, in the nodal liquid one expects the presence of gapped finite energy excitations with charge e. These “holon” excitations are exotic since they carry no spin. The holon is the basic topological excitation that can be created in the hc/e vortex-condensate. The existence of a spin one-half neutral nodon excitation and a spinless charge e holon excitation in the nodal liquid, is a dramatic demonstration of spincharge separation [41–43]. The excitations in the nodal liquid have the same quantum numbers as in the spin-charge separated gauge theories [46], but are weakly interacting, rather than strongly coupled by a gauge field. 9.1.1 Spin response Although the nodons are electrically neutral they do carry spin, so the lowenergy spin response in the nodal liquid can be computed from the Dirac Lagrangian Lψ . Moreover, since Lψ was not altered under the duality transformation, the spin properties of the nodal liquid are essentially identical to those in the dx2 −y2 superconducting phase. As a simple example, consider the uniform magnetic spin susceptibility, χ. The uniform part of the electron spin operator is given as the conserved spin density in equation (7.41): S(x) =

1 † ψ (x)σψja (x). 2 ja

(9.4)

Being bi-linear in nodon operators spin correlation and response functions can be readily computed from the free nodon theory. For example, the uniform spin susceptibility is given by Z ∞ dE(−∂f /∂E)ρn(E), (9.5) χ= 0

where the nodon density of states is ρn (E) = (const)E/vF v∆ , and f (E) is a Fermi function. One finds χ ∼ T /vF v∆ . There are also low energy spin excitations at wavevectors which span between two different nodes. The associated spin operators can be obtained by re-expressing the electron spin operator, Sq =

1X † ck+q σck , 2

(9.6)

k

in terms of the nodons. For example, the staggered magnetization operator, Sπ , is found to be Sπ =

1 † y y † ψ (τ σσ )ψ + h.c. . 2

(9.7)

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Notice that this operator is actually “anomalous” in terms of the conserved nodon charge. In addition to carrying spin, the nodons carry energy, and so will contribute to the thermal transport. In the absence of scattering processes (such as Umklapp) the finite temperature nodon thermal conductivity is infinite. In practive, impurities will scatter the nodons and lead to a finite thermal conductivity. In fact, impurity scattering should also play an important role in modifying the spin response of the nodal liquid. 9.1.2 Charge response The electrical charge properties of the nodal liquid are of course very different than in the superconductor. To see this, imagine changing the chemical potential away from µ = 0 which corresponds to applying an external “magnetic” field to the dual Ginzburg-Landa theory: Lµ = −µb. Being in the “Meissner” phase, the electron density will stay “pinned” at half-filling for µ ≤ µc , with µc the Ginzburg-Landau critical field. Despite the presence of this charge gap, there are low energy current fluctuations in the nodal liquid. Indeed, in this phase the electrical current operator is Iµ , which is bi-linear in the nodon fermions, ψ. To compute the electrical conductivity in the nodal liquid requires computing a two-point correlator of Iµ at zero wavevector (say in the x−direction) Ix (q = 0) = (0 /κo vs2 )∂t2 Jx (q = 0). But notice that Jx (q = 0) is proportional to a globally conserved nodon charge, since Jx (x) = (vF /2)ψ1† ψ1 . Thus, when the nodon number is conserved one has Ix (q = 0) = 0, and the nodons do not contribute to the electrical conductivity. When impurity (or Umklapp) scattering is present, however, the nodon number is no longer conserved, and the nodons will contribute to the real part of the electrical conductivity, but only at finite frequencies. It is instructive to briefly consider the behavior of the electron Green’s function, which can be accessed in photo-emission and tunneling experiments. The electron operator cα (x) can be decomposed as a product of nodon and holon operators. For example, near the node at K j one can write, cα (x) = eiKj ·x eiφ(x) ψj1α (x) + ...

(9.8)

where ψ is a nodon operator and exp(iφ) can be interpreted as a holon destruction operator. In the nodal liquid phase, the electron Green’s function, G(x, t) = hc† (x, t)c(0, 0)i factorizes as, G(x, t) = eiKj ·x he−iφ(x,t) eiφ(0,0) iGn (x, t),

(9.9)

where the nodon Green’s function is, † (x, t)ψj1α (0, 0)i · Gn (x, t) = hψj1α

(9.10)

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Topological Aspects of Low Dimensional Systems

Although Gn (x, t) decays as a power law |x|−2 and t−2 , since creating a holon costs a finite energy the holon Green’s function is expected to be short-ranged, decaying exponentially in space and time. This indicates a gap in the electron spectral function at the Fermi energy. 9.2 Doping the nodal liquid We briefly discuss the effects of doping charge into the nodal liquid phase. In a grand canonical ensemble this is achieved by changing the chemical potential, µ = A0 . In the dual Ginzburg-Landau description of the vortices, a chemical potential acts as an applied dual field, as seen from equation (8.11), since Lµ = −µb.

(9.11)

The dual magnetic field, b = ij ∂i aj , is the total electric charge in units of e. Provided the applied dual field, µ, is smaller than the critical field (µc ) of the Ginzburg-Landau theory, the dual superconductor stays in the Meissner phase – which is the nodal liquid phase at half-filling. But for µ ≥ µc dual flux will penetrate the Ginzburg-Landau superconductor, which corresponds to doping the nodal liquid. The form of the dual flux penetration will depend critically on whether the dual Ginzburg-Landau theory is Type I or Type II. Within a mean-field treatment this is determined by the ratio of the dual penetration length, λv , to the dual coherence length, ξv (where the subscript v denotes vortices). In particular, Type II behavior is expected √ if λv /ξv ≥ 1/ 2, and Type I behavior otherwise. In the Ginzburg-Landau description λv determines the size of a dual flux tube, which is essentially the size of a Cooper pair. We thus expect that λv will be roughly equal to the superconducting coherence length, ξ, which is perhaps 10 − 15 ˚ A in the cuprates. On the other hand, ξv is the size of the “vortex-core” in the dual vortex field, and presumably can be no smaller than the microscopic crystal A. This reasoning suggest that λv /ξv is probably lattice spacing, ξv ≥ 3 − 5 ˚ close to unity in the cuprates, so that either Type I or Type II behavior might be possible – and could be material dependent. We first consider such Type II doping, returning below to the case of a Type I Ginzburg-Landau theory. 9.2.1 Type II behavior The phase diagram of a clean three-dimensional Type II superconductor is well understood [44]. Above the lower critical field, Hc1 , flux tubes penetrate, and form an Abrikosov flux lattice – usually triangular. As the applied field increases the flux tubes start overlapping, when their separation is closer than the penetration length. Upon approaching the upper

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critical field Hc2 their cores start overlapping, the Abrikosov flux lattice disappears, and the superconductivity is destroyed. These results hold equally well for our dual Ginzburg-Landau superconductor, except that now the direction parallel to the applied field is actually imaginary time. Moreover, the Ginzburg-Landau order parameter describes quantum (hc/e) vortices, and the penetrating flux tubes are spinless charge e holons. Upon doping the nodal liquid with µ > µc1 , charge is added to the 2d system, which corresponds to the penetration of dual magnetic flux. In this dual transcription, the resulting Abrikosov flux-lattice phase is a Wigner crystal of holons, with one holon per real space unit cell of the lattice. Upon further doping, at µ = µc2 , the crystal of holons melts, and they condense – this is the d-wave superconductor. In the holon Wigner crystal phase, translational symmetry is spontaneously broken. However, in a real material the Wigner crystal will have a preferred location, determined by impurities and perhaps crystal fields, which will tend to pin and immobilize the crystal. The resulting phase should be an electrical insulator. A striking and unusual feature of the holon Wigner crystal is that it coexists with the nodal liquid. We thereby arrive at a description of a rather remarkable new phase of matter. A Wigner crystal of doped holons co-exists with neutral gapless fermionic excitations – the nodons. In this co-existing phase, low energy spin and thermal properties will be dominated by the nodons. The behavior will be qualitatively similar to that in the undoped nodal liquid phase. It is possible that this phase underlies the physics of the pseudo-gap region of the high Tc cuprates. 9.2.2 Type I behavior In a Type I superconductor, the applied field is expelled until the critical Hc is exceeded [44]. At this point there is a first order phase transition from the Meissner phase with all the flux expelled, to a normal metal phase in which (essentially) all the field penetrates. If our dual Ginzburg-Landau theory is I Type I, then analogous properties are expected. Specifically, as the chemical potential increases, the dual field – which is the holon density – remains at zero until a critical chemical potential µc is reached. At this point there is a first order phase transition, between the nodal liquid phase at halffilling, and a d-wave superconductor at finite doping, xc . At fixed doping x < xc , phase separation is impeded by long-ranged Coulomb interactions between the holons. The system will break apart into co-existing “microphases” of nodal liquid and d-wave superconductor. The configuration of the “micro-phases” will be determined by a complicated competition between the Coulomb energy and the (positive) energy of the domain walls. In practice, impurities will also probably play a very important role.

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Topological Aspects of Low Dimensional Systems

9.3 Closing remarks The theoretical framework described above gives a skeletal description of the nodal liquid and, upon doping, the holon Wigner crystal. There are many important issues which will need to be addressed in detail to see if this novel Mott insulating phase gives a correct description of the low temperature pseudo-gap regime in the cuprates. At very low doping the cuprates are antiferromagnetic so it will clearly be necessary to incorporate magnetism into the theoretical framework. Perhaps even more important is assessing the role of impurities, which are expected to have rather dramatic effects both on the holon Wigner crystal and the gapless nodons. Impurities will tend to disorder the Wigner crystal and will scatter the nodons probably leading to a finite density of states and diffusive rather than ballistic motion. Since the nodons carry spin but no charge, a rather exotic “spin metal” phase is possible with a finite “spin conductivity” (but zero electrical conductivity) even at zero temperature. It is also possible that the impurities will localize the nodons, perhaps leading to a random singlet phase or a spin glass. An additional complication is that some materials might exhibit phase separation upon doping (Type I rather than Type II behavior) exhibiting micro-phase co-existence between the antiferromagnet and the d-wave superconductor, preempting the nodal liquid phase. It clearly remains as a future challenge to fully sort out the mysteries of the pseudo-gap regime. A more general theme of these notes is that novel spin liquid phases can sometimes be more conveniently viewed as descendents of superconductors – rather than the more traditional route via magnetism. One can imagine quantum disordering other exotic superconducting phases besides the dx2 −y2 state, to obtain new spin liquid phases. Perhaps some of these phases will appear in other systems which exhibit finite angular momentum pairing, such as 3 − He and the heavy Fermion materials. It gives me genuine pleasure to acknowledge my wonderful collaborators on the research described above. The renormalization group analysis of the two-leg ladder was carried out in collaboration with Hsiu-hau Lin and Leon Balents. The nodal liquid phase was introduced and analyzed in a collaboration with Chetan Nayak and Leon Balents. This research has been a true collective phenomena, to which I am deeply appreciative. I am also extremely grateful to Doug Scalapino for stimulating my interest in strongly correlated d-wave superconductors and for numerous discussions about Hubbard ladders. I would like to thank T. Senthil for sharing his insights about the effects of impurities in d-wave superconductors. This work has been supported by the National Science Foundation under grants Nos. PHY94-07194, DMR94-00142 and DMR95-28578.

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Appendix A

Lattice duality

Duality plays a key role in understanding how to quantum disorder a superconductor, both in 1 + 1 space-time dimensions (Sect. 5) and in 2 + 1 (Sect. 8). The key idea involves exchanging the order parameter phase φ for vortex degrees of freedom. In 1 + 1 dimensions these are point-like spacetime vortices [31], whereas in 2 + 1 there are point like vortices in space which propogate in time [39]. In Section 8 we chose to work directly in the continuum in implementing the 2 + 1 duality transformation. However, the physics of duality is perhaps more accessible when carried out on the lattice. In this Appendix we show in some detail how lattice duality is implemented in both 1 + 1 and 2 + 1 dimensions [31, 39]. For simplicity we first Wick rotate to Euclidian space, and rescale imaginary time to set the charge velocity to one. The appropriate lattice model is then simply a 2d square lattice or 3d cubic lattice xy model. In the latter case, we also want to include a gauge-field, A, which is a sum of the physical electromagnetic field and the nodon current, as discussed in Section 8 – see equation (8.5). The degrees of freedom which live on the sites of the square or cubic lattice (denoted by a vector of integers ~x) are the phases φx ∈ [0, 2π]. As usual, the gauge field lives on the links. Discrete lattice derivatives are denoted by 4µ φx = φx+µ − φx ,

(A.1)

where µ = x, y for the square lattice and µ = x, y, z for the cubic lattice and x + µ denotes the nearest neighbor site to ~x in the µ ˆ direction. The gauge field is minimally coupled via, 4µ φx → 4µ φx + Aµx .

(A.2)

Consider the partition function, " # Z 2π Y X dφx exp Vκ (4µ φx ) . Z= 0

x

(A.3)

x,µ

Here the periodic “Villain” potential Vκ is given by, exp[Vκ (4φ)] =

∞ X

e−κJ

2

/2 iJ4φ

e

,

(A.4)

J=−∞

with integer J. When κ 1 only the terms with J = 0, ±1 contribute appreciably in the sum and this reduces to the more familiar form: Vκ (4φ) = K cos(4φ),

(A.5)

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Topological Aspects of Low Dimensional Systems

with K = 2exp(−κ/2). The partition function can thus be expressed as a sum over both φ and a vector of integers, J~x , with components Jxµ living on the links of the lattice: Z Y X dφ e−S ≡ T rφ,J~ e−S , (A.6) Z= x

~ [J]

with action S = S0 +

X

~ · J~x )φx , i(4

(A.7)

x

S0 =

κX ~ 2 |Jx | . 2 x

(A.8)

In this form the integration over φ can be explicitly performed giving Z = T rJ0~ e−S0 ,

(A.9)

where the prime on the trace indicates a divergenceless constraint at each site of the lattice: ~ · J~x = 0. 4

(A.10)

In the presence of a gauge field there is an additional term in the action of the form, X ~x. J~x · A (A.11) SA = i x

It is thus clear that the integer of vectors J~ can be interpreted as a conserved electrical current flowing on the links of the lattice. The divergenceless constraint on this electrical 3-current can be imposed automatically by reexpressing J~ as a curl of an appropriate dual field. Consider first the 2d case. A.1

Two dimensions

To guarantee divergenceless we set the current equal to the (2d) curl of a scalar field, θx : 2πJxµ = µν 4ν θx , so that the action becomes S0 (θ) =

κ X (4µ θx )2 . 8π 2 x,µ

(A.12)

(A.13)

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637

To insure that J~ is an integer field, θ must be constrained to be 2π times an integer. This additional constraint can be imposed by introduction of yet another integer field, nx , which will be interpreted as the (space-time) vortex density. The partition is thereby re-expressed as (dropping an unimportant multiplicative constant), Z ∞Y X dθx e−S , (A.14) Z˜ = −∞ x

with S = S0 (θ) +

[nx ]

X κ ˜ x

2

n2x

+ inx θx .

(A.15)

For κ ˜ = 0 the summation over nx gives a sum of delta functions restricting θx /2π to be integer. But we have softened this constraint, introducing a vortex “core” energy κ ˜ 6= 0. At this stage one could perform the Gaussian integral over θ, to obtain a logarithmically interacting plasma of (space-time) vortices. Alternatively, for κ ˜ 1 the summation over nx can be performed giving, X cos(θx ), (A.16) S = S0 (θ) − u x

with u = 2exp(−˜ κ/2). Upon taking the continuum R limit, θx → θ(x), one recovers the (Euclidian) sine-Gordon theory, S = d2 xL with L=

κ ~ 2 (∇θ) − u cos(θ). 8π 2

(A.17)

After Wick rotating back to real time and restoring the velocity this takes the identical form to the dual Lagrangian considered for the 2-leg ladder in Section 5. A.2

Three dimensions

In three dimensions the divergenceless integer 3-current J~ can be written as the curl of a vector field, ~a: ~ × ~ax . 2π J~x = 4

(A.18)

As in 2d one imposes the integer constraint (softly) by introducing an integer vortex field, in this case a 3-vector ~j, to express the partition function as, Z ∞ Y X ˜ d~ax e−S , (A.19) Z= −∞ x

[~jx ]

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Topological Aspects of Low Dimensional Systems

with S = S0 (~a) +

X κ ˜ x

S0 (~a) =

2

|~jx |2 − i~jx · ~ax ,

κ X ~ |4 × ~ax |2 . 8π 2 x

(A.20)

(A.21)

The integer vector field ~j is the vortex 3-current, “minimally” coupled to ~a. To see that the vortex 3-current is conserved, it is convenient to decompose ~ with the vector field ~a into transverse and longitudinal pieces: ~a = ~at − 4θ, θx a scalar field. The action becomes, Xκ ˜~ 2 ~ ~ |jx | + ijx · (4θx − ~ax ) , (A.22) S = S0 (~a) + 2 x where we have dropped the subscript “t” on ~a. The partition function follows from integrating over both ~a and θ and summing over integer ~j. ~ · ~j = 0. Alternatively, Integrating over θ leads to the expected condition: 4 for κ ˜ 1 one can perform the summation over ~j to arrive at an action depending on θ and ~a: X cos(4µ θx − aµx ), (A.23) S = S0 (~a) − K x,µ

with K = 2exp(−˜ κ/2). In the presence of a gauge field Aµ there is an additional term in the action of the form, SA =

i X ~ ~x, (4 × ~ax ) · A 2π x

(A.24)

which follows directly from equations (A.11) and (A.18). At this stage one can take the continuum limit, letting ~ax → ~a(x) and θx →R θ(x). Upon expanding the cosine for small argument one obtains S = d3 xL with (Euclidian) Lagrangian L=

K ~ κ ~ − ~a)2 . (∇ × ~a)2 + (∇θ 2 8π 2

(A.25)

In this dual representation, the vortex 3-current (which follows from ∂L/∂~a) ~ −~a). Notice that the vortices are minimally coupled is given by ~j v = K(∇θ to the “vector potential” ~a, whose curl equals the electrical 3-current. The field θ can be interpreted as the phase of a vortex operator. In fact it

M.P.A. Fisher: Mott Insulators, Spin Liquids and Quantum

639

is convenient to introduce such a complex vortex field before taking the continuum limit: eiθx → Φ(~x).

(A.26)

The continuum limit can then be taking retaining the full periodicity of the cosine potential. The appropriate vortex Lagrangian replacing the second term in equation (A.25) is, Lv =

K ~ |(∇ − i~a)Φ|2 + VΦ (|Φ|). 2

(A.27)

The vortex current operator becomes, ~ − i~a)Φ]. ~j v = KIm[Φ∗ (∇

(A.28)

If the potential is expanded for small Φ as VΦ (X) = rΦ X 2 + uΦ X 4 , the full dual theory is equivalent to a Ginzburg-Landau theory for a classical three-dimensional superconductor. Inclusion of the original gauge field Aµ leads to an additional term in the dual Lagrangian: LA =

i ~ ~ (∇ × ~a) · A. 2π

(A.29)

After Wick rotating back to real time and restoring the velocity, L + LA becomes identical to the dual vortex Lagrangian in Equations (8.9) and (8.11). References [1] See for example Solid State Physics, edited by N. Ashcroft and N. Mermin (Harcourt Brace, 1976). [2] Mott N., Metal-Insulator Transitions (Taylor and Francis, London, 1997). [3] Bednorz J.G. and M¨ uller K.A., Z. Phys. B 64 (1986) 189. [4] Field Theories of Condensed Matter Systems, by edited E. Fradkin (Addison-Wesley, 1991), and references therein, for a discussion of spin-liquids. [5] See Interacting Electrons and Quantum Magnetism, edited by A. Auerbach (Springer-Verlag, New York, 1994), and references therein for recent progress on quantum magnetism. [6] Affleck I., in Strings, Fields and Critical Phenomena, Les Houches Summer school, Session XLIX, edited by E. Brezin and J. Zin-Justin (North Holland, 1990). [7] See for example Physical Properties of High Temperature Superconductivity I-V, edited by D.M. Ginsberg (World Scientific, Singapore, 1989-1996). [8] For a more recent review on high temperature superconductors, see Maple M.B., cond-mat/9802202 (unpublished). [9] Kojima K., Keren A., Luke G.M., Nachumi B., Wu W.D., Uemure Y.J., Azuma M. and Takano M., Phys. Rev. Lett. 74 (1995) 2812. [10] Uehara M., Nagata T., Akimitsu J., Takahashi H., Mori N. and Kinoshita K., J. Phys. Soc. Jpn. 65 (1996) 2764.

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Topological Aspects of Low Dimensional Systems

[11] For a review of two-leg ladders, see Dagotto E. and Rice T., Science 271 (1996) 618; and references therein. [12] Schulz H.J., Phys. Rev. B 34 (1986) 6372. [13] Dagotto E., Riera J. and Scalapino D., Phys. Rev. B 45 (1992) 5744. [14] The nodal liquid was introduced and analyzed in a recent paper, edited by L. Balents, M.P.A. Fisher and C. Nayak, Int. J. Mod. Phys. B 12 (1998) 1033. Sections VI-IX of these notes closely parallels this paper. [15] Balents L., Fisher M.P.A. and Nayak C., cond-mat/9811236 for a careful discussion of the dual order parameter for the nodal liquid. [16] Lin H., Balents L. and Fisher M.P.A., Phys. Rev. B 58 (1998) 1794, and reference therein for a discussion of the weakly interacting two-leg ladder. Section V of these notes is based on early sections of this paper. [17] Shankar R., Rev. Mod. Phys. 66 (1994) 129. [18] Anderson P., Science 235 (1987) 1196. [19] Wollman D.A., Van Harlingen D.J., Giapintzakis J. and Ginsberg D.M., Phys. Rev. Lett. 74 (1995) 797. [20] Kirtley J.R., Tsuei C.C., Sun J.Z., Chi C.C., Yu-Jahnes L.S., Gupta A., Rupp M. and Ketchen M.B., Nature 373 (1995) 225. [21] Balents L. and Fisher M.P.A., Phys. Rev. B 53 (1996) 12133. [22] Lin H.H., Balents L. and Fisher M.P.A., Phys. Rev. B 56 (1997) 6569. [23] Emery V., in Highly conducting one-dimensional solids, edited by J. Devreese, R. Evrard and V. Van Doren (Plenum Press, New York, 1979) p. 247. [24] Ludwig A.W.W., Int. J. Mod. Phys. B 8 (1994) 347, for a thorough discussion of Abelian Bosonization. [25] Shankar R., Acta Phys. Polonica B 26 (1995) 1835. [26] For a very recent and detailed discussion of Bosonization see, vonDelft J. and Schoeller H., cond-mat/9805275. [27] Haldane F.D.M., J. Phys. Colloq 14 (1981) 2585; Phys. Rev. Lett. 47 (1981) 1840. [28] Tomonaga S., Prog. Theor. Phys. (Kyoto) 5 (1950) 544; Luttinger J.M., J. Math. Phys. N.Y. 4 (1963) 1154. [29] Quantum Many Particle Systems, edited by J. Negele and H. Orland (AddisonWesley, 1987). [30] Phase Transitions and the Renormalization Group, edited by N. Goldenfeld (Addison-Wesley, 1992). [31] Jose J.V., Kadanoff L.P., Kirkpatrick S. and Nelson D.R., Phys. Rev. B 16 (1978) 1217, and references therein. [32] Amit D.J., Goldschmidt Y.Y. and Grinstein G., J. Phys. A 13 (1980) 585. [33] Gross D. and Neveu A., Phys. Rev. D 10 (1974) 3235. [34] Shankar R., Phys. Lett. B 92 (1980) 333; Phys. Rev. Lett. 46 (1981) 379. [35] Zamolodchikov A. and Zamolodchikov A., Ann. Phys. 120 (1979) 253. [36] Konik R., Ludwig A.W.W., Lesage F. and Saleur H., (1998) unpublished. [37] Theory of Superconductivity, edited by J. Schrieffer (Benjamin-Cummings, 1983). [38] The Theory of Quantum Liquids Vol. II, edited by P. Nozieres and D. Pines (Addison-Wesley, 1990). [39] Dasgupta C. and Halperin B.I., Phys. Rev. Lett. 47 (1981) 1556; Fisher M.P.A. and Lee D.H., Phys. Rev. B 39 (1989) 2756. [40] Peskin M., Ann. Phys. 113 (1978) 122; Thomas P.O. and Stone M. , Nucl. Phys. B 144 (1978) 513; Wen X.G. and Zee A. , Int. J. Mod. Phys. B 4 (1990) 437. [41] Anderson P.W., Science 235 (1987) 1196.

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[42] Kivelson S.A., Rokhsar D.S. and Sethna J.P., Phys. Rev. B 35 (1987) 8865. [43] Affleck I. and Marston J.B., Phys. Rev. B 37 (1988) 3774; Phys. Rev. B 39 (1989) 11538. [44] Introduction to Superconductivity, edited by M. Tinkham (Mc-Graw Hill, New York, 1996). [45] Fisher M.P.A., Weichman P.B., Grinstein G. and Fisher D.S., Phys. Rev. B 40 (1989) 546. [46] Wen X.G. and Lee P.A., Phys. Rev. Lett. 76 (1996) 503; Kim D.H. and Lee P.A., cond-mat/9810130, and references therein.

COURSE 8

STATISTICS OF KNOTS AND ENTANGLED RANDOM WALKS

S. NECHAEV UMR 8626, CNRS-Universit´ e Paris XI, LPTMS, bˆ atiment 100, Universit´e Paris Sud, 91405 Orsay Cedex, France and L D Landau Institute for Theoretical Physics, 117940 Moscow, Russia

Contents 1 Introduction

645

2 Knot diagrams as disordered spin systems 2.1 Brief review of statistical problems in topology . . . . . . . . . . . 2.2 Abelian problems in statistics of entangled random walks and incompleteness of Gauss invariant . . . . . . . . . . . . . . . . . . . 2.3 Nonabelian algebraic knot invariants . . . . . . . . . . . . . . . . . 2.4 Lattice knot diagrams as disordered Potts model . . . . . . . . . . 2.5 Notion about annealed and quenched realizations of topological disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

647 647

3 Random walks on locally non-commutative groups 3.1 Brownian bridges on simplest non-commutative groups and knot statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Random walks on locally free groups . . . . . . . . . . . . . . . . . 3.3 Analytic results for random walks on locally free groups . . . . . . 3.4 Brownian bridges on Lobachevskii plane and products of non-commutative random matrices . . . . . . . . . . . . . . . . .

675

4 Conformal methods in statistics of random walks with topological constraints 4.1 Construction of nonabelian connections for Γ2 and P SL(2, ZZ) from conformal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Random walk on double punctured plane and conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Statistics of random walks with topological constraints in the two–dimensional lattices of obstacles . . . . . . . . . . . . . . . . .

651 656 663 669

676 689 692 697

701 702 707 709

5 Physical applications. Polymer language in statistics of entangled chain–like objects 715 5.1 Polymer chain in 3D-array of obstacles . . . . . . . . . . . . . . . . 716 5.2 Collapsed phase of unknotted polymer . . . . . . . . . . . . . . . . 719 6 Some “tight” problems of the probability physics 6.1 Remarks and comments to Section 2 . . . . 6.2 Remarks and comments to Sections 3 and 4 6.3 Remarks and comments to Section 5 . . . .

theory and statistical 727 . . . . . . . . . . . . . 728 . . . . . . . . . . . . . 728 . . . . . . . . . . . . . 729

STATISTICS OF KNOTS AND ENTANGLED RANDOM WALKS

S. Nechaev

Abstract The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) we estimate the probability of a trivial knot formation on the lattice using the Kauffman algebraic invariants and show the connection of this problem with the thermodynamic properties of 2D disordered Potts model; (ii) we investigate the limit behavior of random walks in multi-connected spaces and on non-commutative groups related to the knot theory. We discuss the application of the above mentioned problems in statistical physics of polymer chains. On the basis of non-commutative probability theory we derive some new results in statistical physics of entangled polymer chains which unite rigorous mathematical facts with more intuitive physical arguments.

1

Introduction

It wouldn’t be an exaggeration to say that contemporary physical science is becoming more and more mathematical. This fact is too strongly manifested to be completely ignored. Hence I would permit myself to bring forward two possible conjectures: (a) On the one hand there are hardly discovered any newly physical problem which would be beyond the well established methods of the modern theoretical physics. This leads to the fact that nowadays real physical problems seem to be less numerous than mathematical methods of their investigation. (b) On the other hand the mathematical physics is a fascinating field which absorbs new ideas from different branches of modern mathematics, translates them into the physical language and hence fills the abstract mathematical constructions by the new fresh content. This ultimately leads to creating new concepts and stimulates seeking for newel deep conformities to natural laws in known physical phenomena. c EDP Sciences, Springer-Verlag 1999

646

Topological Aspects of Low Dimensional Systems

The penetration of new mathematical ideas in physics has sometimes rather paradoxical character. It is not a secret that difference in means (in languages) and goals of physicists and mathematicians leads to mutual misunderstanding, making the very subject of investigation obscure. What is true for general is certainly true for particular. To clarify the point, let us turn to statistics of entangled uncrossible random walks–the wellknown subject of statistical physics of polymers. Actually, since 1970s, after Conway’s works, when the first algebraic topological invariants–Alexander polynomials–became very popular in mathematical literature, physicists working in statistical topology have acquired a much more powerful topological invariant than the simple Gauss linking number. The constructive utilization of algebraic invariants in statistical physics of macromolecules has been developed in the classical works of Vologodskii, Frank-Kamenetskii and their co-workers [1]. However until recently in overwhelming majority of works the authors continue using the commutative Gauss invariant invariant just making references to its imperfectness. One of the reasons of such inertia consists in the fact that new mathematical ideas are often formulated as “theorems of existence” and it takes much time to retranslate them into physically acceptable form which may serve as a real computational tool. We intend to use some recent advances in algebraic topology and theory of random walks on non-commutative groups for reconsidering the old problem–evaluating of the entropy of randomly generated knots and entangled random walks in a given homotopic state. Let us emphasize that this is a real physical course of lectures and when it is possible the rigorous statements are replaced by some physically justified conjectures. Generally speaking, the course is devoted to an analysis of probabilistic problems in topology and their applications in statistical physics of polymer systems with topological constraints. Let us formulate briefly the main results. 1. The probability for a long random walk to form randomly a knot with specific topological invariant is computed. This problem is considered using the Kauffman algebraic invariants and the connection with the thermodynamic properties of 2D Potts model with “quenched” and “annealed” disorder in interaction constants is discussed. 2. The limit behavior of random walks on the non-commutative groups related to the knot theory is investigated. Namely, the connection between the limit distribution for the Lyapunov exponent of products of non-commutative random matrices–generators of “braid group”–and the asymptotic of powers (“knot complexity”) of algebraic knot invariants is established. This relation is applied for calculating the knot entropy. In particular, it is shown that the “knot complexity” corresponds to the well known topological invariant, “primitive path”, repeatedly used is statistics

S. Nechaev: Statistics of Knots and Entangled Random Walks

647

Spectral and quantum problems of dynamic systems on hyperbolic manifols

Lattice random walk in regular arrays of obstacles

Diffusion on double (multi) punctured plane

Limit behavior of random walks on Riemann surfaces of constant negative curvature

Random walks on free and "local" groups

Topological invariants from conformal methods

Statistics of lattice knots and thermodynamic behavior of Potts spin glasses

Random walk on braid groups

Monodromy transformatios and correlation functions in CFT

Limit distribution for powers of algebraic invariants of randomly generated knots

Knot invariants and nonabelian ChernSimons field theory

Fig. 1. Links between topologically-probabilistic problems. Solid boxes – problems, discussed in the course; dashed boxes – problems not included in the consideration.

of entangled polymer chains. 3. The random walks on multi-connected manifolds is investigated using conformal methods and the nonabelian topological invariants are constructed. It is shown that many nontrivial properties of limit behavior of random walks with topological constraints can be explained in context of random walks on hyperbolic groups. The knowledge of the limit behavior of entangled random paths established above is applied for investigation of the statistical properties of socalled “crumpled globule” (trivial ring without self-intersections in strongly contracted state). The connection between all these problems is shown in Table 1. 2

Knot diagrams as disordered spin systems

2.1 Brief review of statistical problems in topology The interdependence of such branches of modern theoretical and mathematical physics as theory of integrable systems, algebraic topology and

648

Topological Aspects of Low Dimensional Systems

conformal field theory has proved to be a powerful catalyst of development of the new direction in topology, namely, of analytical topological invariants construction by means of exactly solvable statistical models. Today it is widely believed that the following three cornerstone findings have brought the fresh stream in topology: – It has been found the deep relation between the Temperley-Lieb algebra and the Hecke algebra representation of the braid group. This fact resulted in the remarkable geometrical analogy between the Yang-Baxter equations, appearing as necessary condition of the transfer matrix commutativity in the theory of integrable systems on the one hand, and one of Reidemeister moves, used in the knot invariant construction on the other hand. – It has been discovered that the partition function of the Wilson loop with the Chern-Simons action in the topological field theory coincides with the representation of the known nonabelian algebraic knot invariants written in terms of the time-ordered path integral. – The need for new solutions of the Yang-Baxter equations has given a power impetus to the theory of quantum groups. Later on the related set of problems was separated in the independent branch of mathematical physics. Of course the above mentioned findings do not exhaust the list of all brilliant achievements in that field during the last decade, but apparently these new accomplishments have used profound “ideological” changes in the topological science: now we can hardly consider topology as an independent branch of pure mathematics where each small step forward takes so much effort that it seems incidental. Thus in the middle of the 80s the “quantum group” gin was released. It linked by common mathematical formalism classical problems in topology, statistical physics and field theory. A new look at the old problems and the beauty of the formulated ideas made an impression on physicists and mathematicians. As a result, in a few last years the number of works devoted to the search of the new applications of the quantum group apparatus is growing exponentially going beyond the framework of original domains. As an example of persistent penetrating of the quantum group ideas in physics we can name the works on anyon superconductivity [2], intensively discussing problems on “quantum random walks” [3], the investigation of spectral properties of “quantum deformations” of harmonic oscillators [4] and so on. The time will show whether such “quantum group expansion” is physically justified or it merely does tribute to today’s fashion. However it is clear that physics has acquired new convenient language allowing to construct new “nonabelian objects” and to work with them.

S. Nechaev: Statistics of Knots and Entangled Random Walks

649

Among the vast amount of works devoted to different aspects of the theory of integrable systems, their topological applications connected to the construction of knot and link invariants and their representation in terms of partition functions of some known 2D-models deserve our special attention. There exist several reviews [5] and books [6] on that subject and our aim by no means consists in re-interpretation or compilation of their contents. We make an attempt of consecutive account of recently solved probabilistic problems in topology as well as attract attention to some interesting, still unsolved, questions lying on the border of topology and the probability theory. Of course we employ the knowledges acquired in the algebraic topology utilizing the construction of new topological invariants done by Jones [5] and Kauffman [7]. Besides the traditional fundamental topological issues concerning the construction of new topological invariants, investigation of homotopic classes and fibre bundles we mark a set of adjoint but much less studied problems. First of all, we mean the problem of so-called “knot entropy” calculation. Most generally it can be formulated as follows. Take the lattice ZZ3 embedded in the space IR3 . Let ΩN be the ensemble of all possible closed nonselfintersecting N -step loops with one common fixed point on ZZ3 ; by ω we denote the particular trajectory configuration. The question is: what is the probability PN of the fact that the trajectory ω ∈ ΩN belongs to some specific homotopic class. Formally this quantity can be represented in the following way PN {Inv} =

1 X δ [Inv{ω} − Inv] ΩN {ω}

1 ≡ ΩN

X

δ [Inv{r1 , . . . , rN } − Inv] 1 − δ [ri − rj ] δ [rN ]

(2.1)

{r1 , ..., rN }

where Inv{ω} is the functional representation of the knot invariant corresponding to the trajectory with the bond coordinates {r1 . . . , rN }; Inv is the topological invariant characterizing the knot of specific homotopic type and δ(x) is the Kronecker function: δ(x = 0) = 1 and δ(x 6= 0) = 0. The first δ-function in equation (2.1) cuts the set of trajectories with the fixed topological invariant while the second and the third δ-functions ensure the N -step trajectory to be nonselfintersecting and to form a closed loop respectively. The distribution function PN {Inv} satisfies the normalization condition X PN {Inv} = 1. (2.2) all homotopic classes

The entropy SN {Inv} of the given homotopic state of the knot represented

650

Topological Aspects of Low Dimensional Systems

by N -step closed loop on ZZ3 reads SN {Inv} = ln [ΩN PN {Inv}] .

(2.3)

The problem concerning the knot entropy determination has been discussed time and again by the leading physicists. However the number of new analytic results in that field was insufficient till the beginning of the 80s: in about 90 percents of published materials their authors used the Gauss linking number or some of its abelian modifications for classification of a topological state of knots and links while the disadvantages of this approach were explained in the rest 10 percent of the works. We do not include in this list the celebrated investigations of Vologodskii et al. [1] devoted to the first fruitful usage of the nonabelian Alexander algebraic invariants in the numerical simulations in the statistical biophysics. We discuss physical applications of these topological problems at length in Section 5. Despite of the clarity of geometrical image, the topological ideas are very hard to formalize because of the non-local character of topological constraints. Besides, the main difficulty in attempts to calculate analytically the knot entropy is due to the absence of convenient analytic representation of the complete topological invariant. Thus, to succeed, at least partially, in the knot entropy computation we simplify the general problem replacing it by the problem of calculating the distribution function for the knots with defined topological invariants. That problem differs from the original one because none of the known topological invariants (Gauss linking number, Alexander, Jones, HOMFLY) are complete. The only exception is Vassiliev invariants [8], which are beyond the scope of the present book. Strictly speaking we are unable to estimate exactly the correctness of such replacement of the homotopic class by the mentioned topological invariants. Thus under the definition of the topological state of the knot or entanglement we simply understand the determination of the corresponding topological invariant. The problems where ω (see Eq. (2.1)) is the set of realizations of the random walk, i.e. the Markov chain are of special interest. In that case the probability to find a closed N -step random walk in IR3 in some prescribed topological state can be presented in the following way Z PN {Inv}

=

...

Z Y N j=1

drj

N −1 Y

g (rj+1 − rj )

j=1

×δ [Inv{r1 . . . , rN } − Inv] δ [rN ]

(2.4)

where g (rj+1 − rj ) is the probability to find j + 1th step of the trajectory in the point rj+1 if jth step is in rj . In the limit a → 0 and N → ∞ (N a = L = const) in three-dimensional space we have the following expression for

S. Nechaev: Statistics of Knots and Entangled Random Walks

651

g (rj+1 − rj )

g (rj+1 − rj ) = '

3(rj+1 − rj )2 exp − 2a2 ( 2 ) 3/2 3 dr(s) 3 exp 2πa2 2a ds 3 2πa2

3/2

(2.5)

where we have introduced the “time”, s, along the trajectory. Rewrite now the distribution function PN {Inv} (Eq. (2.4)) in the path integral form with the Wiener measure density ( 2 ) Z Z Z L dr(s) 1 3 ds . . . D{r} exp − PN {Inv} = Z 2a 0 ds δ[Inv{r(s)} − Inv] and the normalization condition Z=

X

(2.6)

PN {Inv}·

all different knot invariants

The form of equation (2.6) up to the Wick turn and the constants coincides with the scattering amplitude α of a free quantum particle in the multiconnected phase space. Actually, for the amplitude α we have Z X i r˙ 2 (s)ds · exp (2.7) α∼ h all paths from given topological class

If phase trajectories can be mutually transformed by means of continuous deformations, then the summation in Eq. (2.7) should be extended to all available paths in the system, but if the phase space consists of different topological domains, then the summation in Eq. (2.7) refers to the paths from the exclusively defined class and the “knot entropy” problem arises. 2.2 Abelian problems in statistics of entangled random walks and incompleteness of Gauss invariant As far back as 1967 Edwards had discovered the basis of the statistical theory of entanglements in physical systems. In [9] he proposed the way of exact calculating the partition function of self-intersecting random walk topologically interacting with the infinitely long uncrossible string (in 3D case) or obstacle (in 2D-case). That problem had been considered in mathematical literature even earlier–see for instance the paper [10] –but Edwards

652

Topological Aspects of Low Dimensional Systems

n=1 O q

rL r0

Fig. 2. Random walk on the plane near the single obstacle.

was apparently the first to recognize the deep analogy between abelian topological problems in statistical mechanics of the Markov chains and quantummechanical problems (like Bohm-Aharonov) of the particles in the magnetic fields. The review of classical results is given in [12], whereas some modern advantages are discussed in [11]. The 2D version of the Edwards’ model is formulated as follows. Take a plane with an excluded origin, producing the topological constraint for the random walk of length L with the initial and final points r0 and rL respectively. Let trajectory make n turns around the origin (Fig. 2). The question is in calculating the distribution function Pn (r0 , rL , L). In the said model the topological state of the path C is fully characterized by number of turns of the path around the origin. The corresponding abelian topological invariant is known as Gauss linking number and when represented in the contour integral form, reads Z Inv{r(s)} ≡ G{C} = C

ydx − xdy = x2 + y 2

Z A(r)dr ≡ 2πn + ϑ

(2.8)

C

where A(r) = ξ ×

r ; r2

ξ = (0, 0, 1)

(2.9)

and ϑ is the angle distance between ends of the random walk. Substituting equation (2.8) into equation (2.6) and using the Fourier transform of the δ-function, we arrive at 2 2 r0 + rL 1 exp Pn (r0 , rL , L) = πLa La Z ∞ 2r0 rL I|λ| (2.10) eiλ(2πn+ϑ) dλ × La −∞

S. Nechaev: Statistics of Knots and Entangled Random Walks

653

which reproduces the well known old result [9] (some very important generalizations one can find in [11]). Physically significant quantity obtained on the basis of equation (2.10) is the entropic force ∂ ln Pn (ρ, L) (2.11) fn (ρ) = − ∂ρ which acts on the closed chain (r0 = rL = ρ, ϑ = 0) when the distance between the obstacle and a certain point of the trajectory changes. Apparently the topological constraint leads to the strong attraction of the path to the obstacle for any n 6= 0 and to the weak repulsion for n = 0. Another exactly solvable 2D-problem closely related to the one under discussion deals with the calculation of the partition function of a random walk with given algebraic area. The problem concerns the determination of the distribution function PS (r0 , rL , L) for the random walk with the fixed ends and specific algebraic area S. As a possible solution of that problem, Khandekar and Wiegel [13] again represented the distribution function in terms of the path integral equation (2.6) with the replacement δ[Inv{r(s)} − Inv] → δ[S{r(s)} − S] where the area is written in the Landau gauge: Z Z 1 1 ˜ ˜ =ξ×r ydx − xdy = A{r} r˙ ds; A S{r(s)} = 2 C 2 C (compare to Eqs. (2.8-2.9)). The final expression for the distribution function reads ([12]) Z ∞ 1 dg eiqS Pq (r0 , rL , L) PS (r0 , rL , L) = 2π −∞ where Pq (r0 , rL , L) =

(2.12)

(2.13)

(2.14)

λ (x0 yL − y0 xL ) 2 Laλ λ 2 2 (2.15) − (xL − x0 ) + (yL − y0 ) cot 4 4 λ × exp 4π sin Laλ 4

and λ = −iq. For closed trajectories equations (2.14-2.15) can be simplified essentially, giving 2πS 2 cl · (2.16) PS (N ) = 2La cosh La Different aspects of this problem have been extensively studied in [11].

654

Topological Aspects of Low Dimensional Systems

There is no principal difference between the problems of random walk statistics in the presence of a single topological obstacle or with a fixed algebraic area–both of them have the “abelian” nature. Nevertheless we would like to concentrate on the last problem because of its deep connection with the famous Harper-Hofstadter model dealing with spectral properties of the 2D electron hopping on the discrete lattice in the constant magnetic field [14]. Actually, rewrite equation (2.4) with the substitution equation (2.12) in form of recursion relation in the number of steps, N : Z iq ξ(rN × rN +1 ) Pq (rN +1 , N + 1) = drN g (rN +1 − rN ) exp 2 (2.17) ×Pq (rN , N ). For the discrete random walk on ZZ2 we use the replacement Z X w (rN +1 − rN ) (. . .) drN g (rN +1 − rN ) (. . .) →

(2.18)

{rN }

where w (rN +1 − rN ) is the matrix of the local jumps on the square lattice; w is supposed to be symmetric: ( 1 for (x, y) → (x, y ± 1) and (x, y) → (x ± 1, y) 4 (2.19) w= 0 otherwise. Finally, we get in the Landau gauge: 4 W (x, y, q, ε) = ε

1

1

1

1

e 2 iqx W (x, y − 1, q) + e− 2 iqx W (x, y + 1, q) + e 2 iqy W (x − 1, y, q) + e− 2 iqy W (x + 1, y, q) (2.20)

where W (x, y, q, ε) is the generating function defined via relation W (x, y, q, ε) =

∞ X

εN Pq (rN , N )

N =0

and q plays a role of the magnetic flux through the contour bounded by the random walk on the lattice. There is one point which is still out of our complete understanding. On the one hand the continuous version of the described problem has very clear abelian background due to the use of commutative “invariants” like algebraic area equation (2.13). On the other hand it has been recently discovered ([15]) that so-called Harper equation, i.e. equation (2.20) written R in the gauge S{r} = C ydx, exhibits the hidden quantum group symmetry related to the so-called C ∗ –algebra ([16]) which is strongly nonabelian.

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655

Fig. 3. Pochhammer contour entangled with two obstacles together but not entangled with every one.

Usually in statistical physics we expect that the continuous limit (when lattice spacing tends to zero with corresponding rescaling of parameters of the model) of any discrete problem does not change the observed physical picture, at least qualitatively. But for the considered model the spectral properties of the problem are extremely sensitive to the actual physical scale of the system and depend strongly on the lattice geometry. The generalization of the above stated problems concerns, for instance, the computation of the partition function for the random walk entangled with k > 1 obstacles on the plane located in the points {r1 , . . . , rk }. At first sight, approach based on usage of Gauss linking number as topological invariant, might allow us to solve such problem easily. Let us replace the vector potential A(r) in equation (2.8) by the following one A(r1 , . . . , rk ) = ξ ×

k X r − rj · |r − rj |2 j=1

(2.21)

The topological invariant in this case will be the algebraic sum of turns around obstacles, which seems to be a natural generalization of the Gauss linking number to the case of many-obstacle entanglements. However, the following problem is bound to arise: for the system with two or more obstacles it is possible to imagine closed trajectories entangled with a few obstacles together but not entangled with every one. In Figure 3 the so-called “Pochhammer contour” is shown. Its topological state with respect to the obstacles cannot be described using any abelian version of the Gauss-like invariants. To clarify the point we can apply to the concept of the homotopy group [17]. Consider the topological space R = IR2 − {r1 , r2 } where {r1 , r2 } are the coordinates of the removed points (obstacles) and choose an arbitrary reference point r0 . Consider the ensemble of all directed trajectories starting and finishing in the point r0 . Take the basis loops γ1 (s) and γ2 (s) (0 < s < L) representing the right-clock turns around the points r1 and r2 respectively. The same trajectories passed in the counter-clock direction

656

Topological Aspects of Low Dimensional Systems

are denoted by γ1−1 (s) and γ2−1 (s). The multiplication of the paths is their composition: for instance, γ1 γ2 = γ1 ◦ γ2 . The unit (trivial) path is the composition of an arbitrary loop with its inverse: i = {1, 2}· (2.22) e = γi γi−1 = γi−1 γi The loops γi (s) and γ˜i (s) are called equivalent if one can be transformed into another by means of monotonic change of variables s = s(˜ s). The homotopic classes of directed trajectories form the group with respect to the paths multiplication; the unity is the homotopic class of the trivial paths. This group is known as the homotopy group π1 (R, r0 ). Any closed path on R can be represented by the “word” consisting of set of letters {γ1 , γ2 , γ1−1 , γ2−1 }. Taking into account equation (2.22), we can reduce each word to the minimal irreducible representation. For example, the word W = γ1 γ2−1 γ1 γ1 γ1−1 γ2−1 γ2 γ1−1 γ2−1 can be transformed to the irreducible form: W = γ1 γ2−1 γ2−1 . It is easy to understand that the word W ≡ e represents only the unentangled contours. The entanglement in Figure 3 corresponds to the irreducible word W = γ1−1 γ2 γ1 γ2−1 . The non-abelian character of the topological constraints is reflected in the fact that different entanglements do not commute: γ1 γ2 6= γ2 γ1 . At the same time, the total algebraic number of turns (Gauss linking number) for the path in Figure 3 is equal to zero, i.e. it belongs to the trivial class of cohomology. Speaking more formally, the mentioned example is the direct consequence of the well known fact in topology: the classes of cohomology of knots (of entanglements) do not coincide in general with the corresponding homotopic classes. The first ones for the group π1 can be distinguished by the Gauss invariant, while the problem of characterizing the homotopy class of a knot (entanglement) by an analytically defined invariant is one of the main problems in topology. The principal difficulty connected with application of the Gauss invariant is due to its incompleteness. Hence, exploiting the abelian invariants for adequate classification of topologically different states in the systems with multiple topological constraints is very problematic. 2.3 Nonabelian algebraic knot invariants The most obvious topological questions concerning the knotting probability during the random closure of the random walk cannot be answered using the Gauss invariant due to its weakness. The break through in that field was made in 1975-1976 when the algebraic polynomials were used for the topological state identification of closed random walks generated by the Monte-Carlo method [1]. It has been recognized that the Alexander polynomials being much stronger invariants than the Gauss linking number, could serve as a convenient tool for the

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calculation of the thermodynamic properties of entangled random walks. That approach actually appeared to be very fruitful and the main part of our modern knowledge on knots and links statistics is obtained with the help of these works and their subsequent modifications. In the present section we develop the analytic approach in statistical theory of knots considering the basic problem–the probability to find a randomly generated knot in a specific topological state. We would like to reiterate that our investigation would be impossible without utilizing of algebraic knot invariants discovered recently. Below we reproduce briefly the construction of Jones invariants following the Kauffman approach in the general outline. 2.3.1 Disordered Potts model and generalized dichromatic polynomials The graph expansion for the Potts model with the disorder in the interaction constants can be defined by means of slight modification of the well known construction of the ordinary Potts model [18,19]. Let us recall the necessary definitions. Take an arbitrary graph L with N vertices. To each vertex of the given graph we attribute the “spin” variable σi (i ∈ [1, N ]) which can take q states labelled as 1, 2, . . . , q on the simplex. Suppose that the interaction between spins belonging to the connected neighboring graph vertices only contributes to the energy. Define the energy of the spin’s interaction as follows Jkl σk = σl , (σk , σl ) – neighbors (2.23) Ekl = Jkl δ(σk , σl ) = 0 otherwise where Jkl is the interaction constant which varies for different graph edges and the equality σk = σl means that the neighboring spins take equal values on the simplex. The partition function of the Potts model now reads   X J  X kl δ(σk , σl ) exp (2.24) Zpotts =   T {σ}

{kl}

where T is the temperature. Expression (2.24) gives for q = 2 the well-known representation of the Ising model with the disordered interactions extensively studied in the theory of spin glasses [20]. (Later on we would like to fill in this old story by a new “topological” sense.) To proceed with the graph expansion of the Potts model [19], rewrite the partition function (2.24) in the following way XY [1 + vkl δ(σk , σl )] (2.25) Zpotts = {σ} {kl}

658

Topological Aspects of Low Dimensional Systems

where

vkl = exp

Jkl T

− 1.

(2.26)

If the graph L has N edges then the product (2.25) contains N multipliers. Each multiplier in that product consists of two terms {1 and vkl δ(σk , σl )}. Hence the partition function (2.25) is decomposed in the sum of 2N terms. Each term in the sum is in one-to-one correspondence with some part of the graph L. To make this correspondence clearer, it should be that an arbitrary term in the considered sum represents the product of N multipliers described above in ones from each graph edge. We accept the following convention: (a) if for some edge the multiplier is equal to 1, we remove the corresponding edge from the graph L; (b) if the multiplier is equal to vkl δ(σk , σl ) we keep the edge in its place. After repeating the same procedure with all graph edges, we find the unique representation for all terms in the sum (2.25) by collecting the components (either connected or not) of the graph L. Take the typical graph G consisting of m edges and C connected components where the separated graph vertex is considered as one component. The presence of δ-functions ensures the spin’s equivalence within one graph component. As a result after summation of all independent spins and of all possible graph decompositions we get the new expression for the partition function of the Potts system (2.24)

Zpotts =

X {G}

qC

m Y

vkl

(2.27)

{kl}

where the product runs over all edges in the fixed graph G. It should be noted that the graph expansion equation (2.27) where vkl ≡ v for all {k, l} coincides with the well known representation of the Potts system in terms of dichromatic polynomial (see, for instance [18, 19]). Another comment concerns the number of spin states, q. As it can be seen, in the derivation presented above we did not account for the fact that q has to take positive integer values only. From this point of view the representation (2.27) has an advantage with respect to the standard representation (2.24) and can be considered an analytic continuation of the Potts system to the non-integer and even complex values of q. We show in the subsequent sections how the defined model is connected to the algebraic knot invariants.

659

ambient isotopy

regular isotopy

S. Nechaev: Statistics of Knots and Entangled Random Walks

Fig. 4. Reidemeister moves of types I, II and III.

2.3.2 Reidemeister moves and state model for construction of algebraic invariants Let K be a knot (or link) embedded in the 3D-space. First of all we project the knot (link) onto the plane and obtain the 2D-knot diagram in the socalled general position (denoted by K as well). It means that only the pair crossings can be in the points of paths intersections. Then for each crossing we define the passages, i.e. parts of the trajectory on the projection going “below” and “above” in accordance with its natural positions in the 3Dspace. For the knot plane projection with defined passages the following theorem is valid: (Reidemeister [22]): Two knots embedded in IR3 can be deformed continuously one into the other if and only if the diagram of one knot can be transformed into the diagram corresponding to another knot via the sequence of simple local moves of types I, II and III shown in Figure 4. The work [22] provides us with the proof of this theorem. Two knots are called regular isotopic if they are isotopic with respect to two last Reidemeister moves (II and III); meanwhile, if they are isotopic with respect to all moves, they are called ambient isotopic. As it can be seen from Figure 4, the Reidemeister move of type I leads to the cusp creation on

660

Topological Aspects of Low Dimensional Systems

the projection. At the same time it should be noted that all real 3D-knots (links) are of ambient isotopy. Now, after the Reidemeister theorem has been formulated, it is possible to describe the construction of polynomial “bracket” invariant in the way proposed by Kauffman [7,23]. This invariant can be introduced as a certain partition function being the sum over the set of some formal (“ghost”) degrees of freedom. Let us consider the 2D-knot diagram with defined passages as a certain irregular lattice (graph). Crossings of path on the projection are the lattice vertices. Turn all these crossings to the standard positions where parts of the trajectories in each graph vertex are normal to each other and form the angles of ±π/4 with the x-axis. It can be proven that the result does not depend on such standardization. There are two types of vertices in our lattice–a) and b) which we label by the variable bi = ±1 as it is shown below:

The next step in the construction of algebraic invariant is introduction of two possible ways of vertex splittings. Namely, we attribute to each way of graph splitting the following statistical weights: A to the horizontal splitting and B to the vertical one for the vertex of type a); B to the horizontal splitting and A to the vertical one for the vertex of type b). The said can be schematically reproduced in the following picture:

the constants A and B to be defined later. For the knot diagram with N vertices there are 2N different microstates, each of them representing the set of splittings of all N vertices. The entire microstate, S, corresponds to the knot (link) disintegration to the system of disjoint and non-selfintersecting circles. The number of such circles for the given microstate S we denote as S. The following statement belongs to Kauffman [7]. Consider the partition function X dS−1 Ai B j , (2.28) hKi = {S}

S. Nechaev: Statistics of Knots and Entangled Random Walks

661

P where {S} means summation over all possible 2N graph splittings, i and j = N − i being the numbers of vertices with weights A and B for the given realization of all splittings in the microstate S respectively. The polynomial in A, B and d represented by the partition function (2.28) is the topological invariant of knots of regular isotopy if and only if the following relations among the weights A, B and d are fulfilled: AB = 1 ABd + A2 + B 2 = 0.

(2.29)

The sketch of the proof is as follows. Denote with h. . .i the statistical weight of the knot or of its part. The hKi-value equals the product of all weights of knot parts. Using the definition of vertex splittings, it is easy to test the following identities valid for unoriented knot diagrams

(2.30) completed by the “initial condition” D K

D E [ E O =d K ;

K is not empty

(2.31)

where O denotes the separated trivial loop. The skein relations (2.30) correspond to the above defined weights of horizontal and vertical splittings while the relation (2.31) defines the statistical weights of the composition of an arbitrary knot and a single trivial ring. These diagrammatic rules are well defined only for fixed “boundary condition” of the knot (i.e., for the fixed part of the knot outside the brackets). Suppose that by convention the polynomial of the trivial ring is equal to the unity; D E O = 1. (2.32) Now it can be shown that under the appropriate choice of the relations between A, B and d, the partition function (2.28) represents the algebraic invariant of the knot. The proof is based on direct testing of the invariance of hKi-value with respect to the Reidemeister moves of types II and III.

662

Topological Aspects of Low Dimensional Systems

For instance, for the Reidemeister move of type II we have:

(2.33) Therefore, the invariance with respect to the Reidemeister move of type II can be obtained immediately if we set the statistical weights in the last line of equation (2.33) as it is written in equation (2.29). Actually, the topological equivalence of two knot diagrams is restored with respect to the Reidemeister move of type II only if the right- and left-hand sides of equation (2.33) are identical. It can also be tested that the condition of obligatory invariance with respect to the Reidemeister move of type III does not violate the relations (2.29). The relations equation (2.29) can be converted into the form B = A−1 ,

d = −A2 − A−2

(2.34)

which means that the Kauffman invariant (2.28) is the Laurent polynomial in A-value only. Finally, Kauffman showed that for oriented knots (links) the invariant of ambient isotopy (i.e., the invariant with respect to all Reidemeister moves) is defined via relation: f [K] = (−A)3T w(K) hKi

(2.35)

here T w(K) is the twisting of the knot (link), i.e. the sum of signs of all crossings defined by the convention:

(not to be confused with the definition of the variable bi introduced above). Equation (2.35) follows from the following chain of equalities

The state model and bracket polynomials introduced by Kauffman seem to be very special. They explore only the peculiar geometrical rules such

S. Nechaev: Statistics of Knots and Entangled Random Walks

663

as summation over the formal “ghost” degrees of freedom–all possible knot (link) splittings with simple defined weights. But one of the main advantages of the described construction is connected with the fact that Kauffman polynomials in A-value coincide with Jones knot invariants in t-value (where t = A1/4 ). Jones polynomial knot invariants were discovered first by Jones during his investigation of topological properties of braids (see Sect. 3 for details). Jones’ proposition concerns the establishment of the deep connection between the braid group relations and the Yang-Baxter equations ensuring the necessary condition of transfer matrix commutativity [6]. The Yang-Baxted equations play an exceptionally important role in the statistical physics of integrable systems (such as ice, Potts, O(n), 8-vertex, quantum Heisenberg models [19]). The attempt to apply Kauffman invariants of regular isotopy for investigation of statistical properties of random walks with topological constraints in a thin slit has been made in [24]. Below we extend the ideas of the work [24] considering the topological state of the knot as a special kind of a quenched disorder. 2.4 Lattice knot diagrams as disordered Potts model Let us specify the model under consideration. Take a square lattice M turned to the angle π/4 with respect to the x-axis and project a knot embedded in IR3 onto M supposing that each crossing point of the knot diagram coincides with one lattice vertex without fall (there are no empty lattice vertices)–see Figure 5. Define the passages in all N vertices and choose such boundary conditions which √ ensure the lattice to form a single closed path; that is possible when N is an odd number. The frozen pattern of all passages {bi } on the lattice together with the boundary conditions fully determine the topology of some 3D knot. Of course, the model under consideration is rather rough because we neglect the “space” degrees of freedom due to trajectory fluctuations and keep the pure topological specificity of the system. Later on in Section 4 we discuss the applicability of such model for real physical systems and produce arguments in support of its validity. The basic question of our interest is as follows: what is the probability PN {f [K]} to find a knot diagram on our lattice M in a topological state characterized by some specific Kauffman invariant f [K] among all 2N microrealizations of the disorder {bi } in the lattice vertices. That probability distribution reads (compare to Eq. (2.1)) PN {f [K]} =

i 1 X h δ f [K{b , b , . . . , b }] − f [K] 1 2 N 2N {bi }

(2.36)

664

Topological Aspects of Low Dimensional Systems

Fig. 5. Lattice knot with topological disorder realized in a quenched random pattern of passages.

where f [K{b1 , . . . , bN }] is the representation of the Kauffman invariant as a function of all passages {bi } on the lattice M. These passages can be regarded as a sort of quenched “external field” (see below). Our main idea of dealing with equation (2.36) consists in two steps: (a) at first we convert the Kauffman topological invariant into the known and well-investigated Potts spin system with the disorder in interaction constants; (b) then we apply the methods of the physics of disordered systems to the calculation of thermodynamic properties of the Potts model. It enables us to extract finally the estimation for the requested distribution function. Strictly speaking, we could have disregarded point (a), because it does not lead directly to the answer to our main problem. Nevertheless we follow the mentioned sequence of steps in pursuit of two goals: 1) we would like to prove that the topologically-probabilistic problem can be solved within the framework of standard thermodynamic formalism; 2) we would like to employ the knowledges accumulated already in physics of disordered Potts systems to avoid some unnecessary complications. Let us emphasize that the mean–field approximation and formal replacement of the model with short–range interactions by the model with infinite long–range ones serves to be a common computational tool in the theory of disordered systems and spin glasses. 2.4.1 Algebraic invariants of regular isotopy The general outline of topological invariants construction deals with seeking for the functional, f [K{b1 , ... bN }], which is independent on the knot shape

S. Nechaev: Statistics of Knots and Entangled Random Walks

665

i.e. is invariant with respect to all Reidemeister moves. Recall that the Potts representation of the Kauffman polynomial invariant (2.28) of regular isotopy for some given pattern of “topological disorder”, {bi }, deals with simultaneous splittings in all lattice vertices representing the polygon decomposition of the lattice M. Such lattice disintegration looks like a densely packed system of disjoint and non-selfintersecting circles. The collection of all polygons (circles) can be interpreted as a system of the so-called Eulerian circuits1 completely filling the square lattice. Eulerian circuits are in one-to-one correspondence with the graph expansion of some disordered Potts system introduced in Section 2.3.1 (see details below and in [27]). Rewrite the Kauffman invariant of regular isotopy, hKi, in form of disordered Potts model defined in the previous section. Introduce the two-state “ghost” spin variables, si = ±1 in each lattice vertex independent on the crossing in the same vertex

si = +1

si = −1.

Irrespective of the orientation of the knot diagram shown in Figure 5 (i.e. restricting with the case of regular isotopic knots), we have hK{bi }i =

X {S}

2

−2 S−1

A +A

N X exp ln A b i si

! .

(2.37)

i=1

Written in such form the partition function hK{bi }i represents the weihgted sum of all possible Eulerian circuits on the lattice M. Let us show explicitly that the microstates of the Kauffman system are in one-toone correspondence with the microstates of some disordered Potts model on a lattice. Apparently for the first time the similar statement was expressed in the paper [7]. To be careful, we would like to use the following definitions: (i) Let us introduce the lattice L dual to the lattice M, or more precisely, one of two possible (odd and even) diagonal dual lattices, shown in Figure 6. It can be easily noticed that the edges of the lattice L are in one-to-one correspondence with the vertices of the lattice M. Thus, the disorder on the dual lattice L is determined on the edges. In turn, the edges of the lattice L can be divided into the subgroups of vertical and horisontal bonds. Each kl-bond of the lattice L carries the “disorder variable” bkl being a function of the variable bi located in the corresponding i-vertex of the lattice M. The simplest and most sutable choice of the function bkl (bi ) is as in equation 1 Eulerian circuit is a trajectory on the graph which visits once and only once all graph edges.

666

Topological Aspects of Low Dimensional Systems

polygon decomposition of lattice M backbone graph on dual lattice L

Fig. 6. Disintegration of the knot diagram on the M-lattice into ensemble of nonselfintersecting loops (Eulerian circuits) and graph representation of the Potts model on the dual L-lattice.

(2.48) below (or vice versa for another choice of dual lattice); i is the vertex of the lattice M belonging to the kl-bond of the dual lattice L. (ii) For the given configuration of splittings on M and chosen dual lattice L let us accept the following convention: we mark the edge of the L-lattice by the solid line if this edge is not intersected by some polygon on the M-latice and we leave the corresponding edge unmarked if it is intersected P by any polygon–as it is shown in Figure 6. Similarly, the sum si bi in equation (2.37) can be rewritten in terms of marked and unmarked bonds on the L-lattice X X X si b i = si b i + si b i i

mark

=

horiz X

nonmark

si b i +

mark

= −

horiz X

X

si b i +

mark

bkl −

mark

=

vertic X

where we used the relation

X

bkl +

P nonmark

horiz X

X

bkl +

P mark

si b i

bkl =

vertic X

bkl

nonmark

bkl − 2

all edges

bkl +

vertic X nonmark

nonmark

bkl =

mark

si b i +

nonmark

mark

bkl −

nonmark

vertic X

horiz X

X

bkl

mark

P all edges

bkl .

(2.38)

S. Nechaev: Statistics of Knots and Entangled Random Walks

667

(iii) Let ms be the number of marked edges and Cs be the number of connected components of marked graph. Then the Euler relation reads: S = 2Cs + ms − N + χ.

(2.39)

The equation (2.39) can be proved directly. The χ-value depends on the genus of the surface, which can be covered by the given lattice, (i.e. χ depends on the boundary conditions). In the thermodynamic limit N 1 the χ-dependence should disappear (at least for the flat surfaces), so the standard equality S = 2Cs + ms − N will be assumed below. By means of definitions (i)-(iii), we can easily convert equation (2.37) into the form: −2 −(N +1)

hK{bkl }i = (A + A 2

×

X

)

N Y

b A kl

all edges 2

−2 2Cs

(A + A

)

{G}

ms Y −2bkl (−A2 − A−2 ) A

(2.40)

mark

where we used equation (2.38) and the fact that N + 1 is even. Comparing equation (2.40) with equation (2.25) we immediately conclude that X

(A2 + A−2 )2Cs

{G}

ms Y

XY (1 + vkl δ(σk , σl )) A−2bkl (−A2 − A−2 ) ≡

mark

{σ} {kl}

(2.41) what coincides with the partition function of the Potts system written in the form of dichromatic polynomial. Therefore, we have def

vkl = A−2bkl (−A2 − A−2 ) = −1 − A−4bkl q = (A2 + A−2 )2 .

(2.42)

Since the “disorder” variables bkl take the discrete values ±1 only, we get the following expression for the interaction constant Jkl (see Eq. (2.26)) Jkl = ln 1 − (A2 + A−2 )A−2bkl = ln[−A−4bkl ]. T

(2.43)

Combining equations (2.40-2.43) we obtain the following statement. (a) Take N -vertex knot diagram on the lattice M with given boundary conditions and fixed set of passages {bi }. (b) Take the dual lattice L in one-to-one correspondence with M where one vertex of M belongs to one edge of L. The Kauffman topological invariant hK(A)i of regular isotopy for knot diagrams on M admits representation in form of 2D Potts system on the dual lattice L: (2.44) hK(A)i = H A, {bkl } Zpotts q(A), Jkl bkl , A

668

Topological Aspects of Low Dimensional Systems

where:   X −(N +1) exp ln A bkl  H A, {bkl } = A2 + A−2

(2.45)

{kl}

is the trivial multiplier (H does not depend on Potts spins);    X J b , A  X kl kl δ(σk , σl ) = exp Zpotts q(A), Jkl bkl , A   T {σ}

(2.46)

{kl}

is the Potts partition function with interaction constants, Jkl , and number of spin states, q, defined as follows Jkl = ln[−A−4bkl ]; T

q = A2 + A−2

2

(2.47)

and the variables bkl play a role of disorder on edges of the lattice L dual to the lattice M. The connection between bkl and bi is defined by convention ( −bi if (kl)-edge is vertical (2.48) bkl = if (kl)-edge is horizontal. bi Equation (2.41) has the sense of partition function of the 2D disordered Potts system with the random nearest-neighbor interactions whose distribution remains arbitrary. The set of passages {bkl } uniquely determines the actual topological state of the woven carpet for the definite boundary conditions. Therefore the topological problem of the knot invariant determination is reduced to usual statistical problem of calculation of the partition function of the Potts model with the disorder in the interaction constants. Of course, this correspondence is still rather formal because the polynomial variable A is absolutely arbitrary and can take even complex values, but for some regions of A that thermodynamic analogy makes sense and could be useful as we shall see below. The specific feature of the Potts partition function which gives the representation of the Kauffman algebraic invariant is connected with the existence of the relation between the temperature T and the number of spin states q (see Eq. (2.42)) according to which T and q cannot be considered anymore as independent variables. 2.4.2 Algebraic invariants of ambient isotopy The invariance of the algebraic topological invariant, f [K], with respect to all Reidemeister moves (see Eq. (2.35)) for our system shown in

S. Nechaev: Statistics of Knots and Entangled Random Walks

669

Figure 5 is related to the oriented Eulerian circuits called Hamiltonian walks2 . Let us suppose that the orientation of the knot diagram shown in Figure 5 is chosen according to the natural orientation of the path representing a knot K in IR3 . For the defined boundary conditions we get the so-called Manhattan lattice consisting of woven threads with alternating directions. It follows from the definition of twisting T w(K) (see Sect. 1.3.2) that T w(K) changes the sign if the direction of one arrow in the vertex is changed to the inverse. Reversing the direction of any arrows in the given vertex even times we return the sign of twisting to the initial value. We define groups of “even” and “odd” vertices on the lattice M as follows. The vertex i is called even (odd) if it belongs to the horizontal (vertical) bond (kl) of the dual lattice L. Now it is easy to prove that the twisting of the knot on the Manhattan lattice M can be written in terms of above defined variables bkl . Finally the expression for the algebraic invariant of ambient isotopy f [K] on the lattice L reads   X bkl  hK ({bkl }, A)i (2.49) f [K] = exp 3 ln[−A] {kl}

where hK ({bkl }, A)i is defined by equation (2.44). 2.5 Notion about annealed and quenched realizations of topological disorder Fixed topological structure of a trajectory of given length fluctuating in space is a typical example of a quenched disorder. Actually, the knot structure is formed during the random closure of the path and cannot be changed without the path rupture. Because of the topological constraints the entire phase space of ensemble of randomly generated closed loops is divided into the separated domains resembling the multi-valley structure of the spin glass phase space. Every domain corresponds to the sub-space of the path configurations with the fixed value of the topological invariant. The methods of theoretical description of the systems with quenched disorder in interaction constants are rather well developed, especially in regard to the investigation of spin glass models [20]. Central for these methods is the concept of self-averaging which can be explained as follows. Take some additive function F (the free energy, for instance) of some disordered spin system. The function F is the self-averaging quantity if the observed value, 2 A Hamiltonian walk is a closed path which visits once and only once all vertices of the given oriented graph.

670

Topological Aspects of Low Dimensional Systems

Fobs , of any macroscopic sample of the system coincides with the value Fav averaged over the ensemble of disorder realizations: Fobs = hF iav . The central technical problem is in calculation of the free energy F = −T ln Z averaged over the randomly distributed quenched pattern in the interaction constants. In this section we show that this famous thermodynamic problem of the spin glass physics is closely related to the knot entropy computation. Another problem arises when averaging the partition function Z (but not the free energy) over the disorder. Such problem is much simpler from computational point of view and corresponds to the case of annealed disorder. Physically such model describes the situation when the topology of the closed loop can be changed. It means that the topological invariant, i.e. the Potts partition function, has to be averaged over all possible realizations of the pattern disorder in the ensemble of open (i.e. unclosed) loops on the lattice. It has been shown in [26] that the calculation of the mean values of topological invariants allows to extract rather rough but nontrivial information about the knot statistics. 2.5.1 Entropy of knots. Replica methods Our main goal is the computation of the probability distribution PN {f [K]} (see Eq. (2.36)). Although we are unable to evaluate this function exactly, the representation of PN {f [K]} in terms of disordered Potts system enable us to give an upper estimation for the fraction of randomly generated paths belonging to some definite topological class (in particular, to the trivial one). We use the following chain of inequalities restricting ourselves with the case of regular isotopic knots for simplicity ([24]): Probability PN of knot Probability PN {K(A)} of knot formation in a given ≤ formation with specific topological topological state invariant hK(A)i for all A Probability PN {K(A∗ )} of knot formation for specific value of A∗ ≤ minimizing the free energy of associated Potts system

(2.50)

The first inequality is due to the fact that Kauffman invariant of regular isotopic knots is not a complete topological invariant, whereas the last probability in the chain can be written as follows h i X Θ{bkl }δ hK{bkl , A∗ }i − hK(A∗ )i (2.51) PN {K(A∗ )} = {bkl }

S. Nechaev: Statistics of Knots and Entangled Random Walks

671

P where means summation over all possible configurations of the “crossing field” {bkl }, δ-function cuts out all states of the field {bkl } with specific value of Kauffman invariant hK{bkl , A∗ }i ≡ hK(A∗ )i and Θ{bkl } is the probability of realization of given configuration of crossings. In principle the distribution Θ{bi } depends on statistics of the path in underlying 3D space and is determined physically by the process of the knot formation. Here we restrict ourselves to the following simplest suppositions: (i) We regard crossings {bi } in different vertices of M-lattice as completely uncorrelated variables (or, in other words, we assume that the variables {bkl } defined on the edges of the L-lattice are statistically independent): N Y P (bi ). (2.52) Θ{bi } = i

(ii) We suppose variable bi (or bkl ) to take values ±1 with equal probabilities, i.e.: 1 1 (2.53) P (bi ) = δ(bi − 1) + δ(bi + 1). 2 2 The probability of trivial knot formation can be estimated now as follows h i X (0) Θ{bkl }δ ln hK{bkl , A∗ }i PN (A∗ ) ≤ {bkl }

'

1 2π

Z

Z

∞

dy −∞

...

Z Y

P (bkl )dbkl K iy {bkl , A∗ }

(2.54)

kl

where hK(A∗ )i ≡ 1 for trivial knots. Thus our problem is reduced to the calculation of non-integer complex moments of the partition function, i.e., values of the type hK iy {bkl , A∗ }i. An analogous problem of evaluation of non-integer moments is well known in the spin-glass theory. Indeed, the averaging of the free energy of the system, F , over quenched random field is widely performed via so-called replica-trick [28]. The idea of the method is as follows. Consider the identity Z n ≡ en ln Z and expand the right-hand side up to the first order in n. We get Z n = 1 + n ln Z + O(n2 ). Now we can write Zn − 1 · n→0 n

F = − ln Z = − lim

We proceed with the calculation of the complex moments of the partition function hK{bkl }i. In other words we would like to find the averaged value hK n i for integer values of n. Then we put n = iy and compute the remaining integral in equation (2.54) over y-value. Of course, this procedure needs to be verified and it would be of most desire to compare our predictions with the data obtained in numerical simulations. However let us stress that

672

Topological Aspects of Low Dimensional Systems

our approach is no more curious than replica one, it would be extremely desirable to test the results obtained by means of computer simulations. The outline of our calculations is as follows. We begin by rewriting the averaged Kauffman invariant using the standard representation of the replicated Potts partition function and extract the corresponding free energy F (A) in the frameworks of the infinite–range mean–field theory in two dimensions. Minimizing F (A) with respect to A we find the equilibrium value A∗ . Then we compute the desired probability of trivial knot forma(0) tion PN (A∗ ) evaluating the remaining Gaussian integrals. Averaging the nth power of Kauffman invariant over independent values of the “crossing field” bkl = ±1 we get Z Z Y −2n(N +1) hK n (A)i = . . . P (bkl )dbkl K 2n {bkl } = [2 cosh(2β)] ×

XY {σ} kl

( exp iπ

kl

X

δ

σkα , σlα

" + ln cosh β

n X

4δ

σkα , σlα

−1

#)

α=1

kl

(2.55) where β = ln A. Let us break for a moment the connection between the number of spin states, q, and interaction constant and suppose |β| 1. Later on we shall verify the selfconsistency of this approximation. Now the exponent in the last expression can be expanded as a power series in β. Keeping the terms of order β 2 only, we rewrite equation (2.55) in the standard form of n-replica Potts partition function 1 2 2 −2n(N +1) n β n hK (A)i = [2 cosh(2β)] exp N 2 ( n N X J2 X X α β α β exp σka σkb σla σlb × (2.56) 2 kl α6=β {σ1 ... σn } ) 2 P P N n α β + J2 (q − 2) + J¯0 kl α=1 σka σlb where spin indexes a, b change in the interval [0, q − 1], β 2 1 and J2

= 16β 2

J¯0

= iπ − 4β 2 n

q

(2.57)

2

= 4 + 16β > 4.

According to the results of Cwilich and Kirkpatrick [29] and later works (see, for instance, [30]), the spin-glass ordering takes place and the usual ferromagnetic phase makes no essential contribution to the free energy under

S. Nechaev: Statistics of Knots and Entangled Random Walks

the condition

q−4 J¯0 < · J 2

673

(2.58)

Substituting equation (2.57) into equation (2.58) it can be seen that <(l.h.s.) < <(r.h.s) in equation (2.58) for all β. Thus, we expect that the spin-glass ordering (in the infinite-range model) corresponds to the solutions D E = qδ(σkα , a) − 1 = 0 mα a D ED E = qδ(σkα , a) − 1 qδ(σkβ , b) − 1 6= 0 Qαβ ab αβ where mα a and Qab are the ferromagnetic and spin-glass order parameters respectively. If it is so, we can keep the term in the exponent (Eq. (2.56)) corresponding to inter-replica interactions only. We follow now the standard scheme of analysis of Potts spin glasses partition function exhaustively described in [29–31]; main steps of this analysis are shortly represented below. Performing the Hubbard-Stratonovich transformation to the scalar fields Qαβ iab and implying the homogeneous isotropic αβ n solution of the form Qiab = Qαβ i δab , we can write down the value hK i (Eq. (2.56)) as follows ([29]): #) ( " 2 2 π J n J hK n i = exp N ln 2 n(n − 1)(q − 1)2 − ln 2 cosh + J 2 32 Z XZ Y 2 dQαβ exp − H{Qαβ × i i }d x

where H{Qαβ } =

{σ}

i

(2.59)

" X 1 2 (q − 1) −1 (Qαβ )2 4 J2 α6=β

1 X q − 2 X αβ 3 − Qαβ Qβγ Qγα − (Q ) 6 12 α6=β6=γ

α6=β

q−2 X 1 X − (Qαβ )2 Qβγ Qγα − Qαβ Qβγ Qγδ Qδα 4 8 α6=β6=γ α6=β6=γ6=δ # q 2 − 6q + 6 X αβ 4 − (Q ) . (2.60) 48 αβ

In [29, 31] it was shown, that the mean-field replica symmetric solution of the mean-field Potts spin glass is unstable for q ≥ 2 and the right ansatz of equations (2.59–2.60) corresponds to the first level of Parisi replica breaking

674

Topological Aspects of Low Dimensional Systems

scheme for spin glasses. Hence, we have ( Q if α and β belong to the same group of m replicas αβ Q = 0 otherwise.

(2.61)

Analysis shows that for q > 4 (our case) the transition to the glassy state corresponds to m = 1 which implies the accessory condition Fpm = Fsg , where Fpm and Fsg are the free energies of paramagnetic and spin-glass phases respectively. The transition occurs at the point 1−

(q − 4)2 2 · = J2 3(q 2 − 18q + 42)

(2.62)

Substituting equation (2.57) into equation (2.62) we find the self-consistent value of reverse temperature of a spin-glass transition, βtr : βtr ≈ 0.35.

(2.63)

2 1 implied above This numerical value is consistent with the condition βtr in the course of expansion of equation (2.56). According to the results of the work [29] the n-replica free energy near the transition point has the following form 2 1 1 1 N n(q − 1)2 Qtr − (2.64) F ' 2 64 β2 βtr

with the following expression of the spin-glass order parameter Qtr =

q2

2(4 − q) > 0. − 18q + 42

(2.65)

From equation (2.64) we conclude that the free energy F reaches its minimum as a function of A = exp (β) just at the point A∗ = exp(βtr ). Using equations (2.64) and (2.65) we rewrite the expression for the averaged nreplica Kauffman invariant hK n i in the vicinity of βtr as follows (compare to [29]): # ( " 2 π β 2 hK 2n i ' exp N n2 3 + 16β 2 ln + 16β 2 2 " 2 π β2 (2.66) + ln 2 + −N n 3 + 16β 2 ln 16β 2 2 #) 2 −2 −2 2 2 3 + 16β 2 βtr β − βtr · − 2 )2 − 18 (4 + 16β 2 ) + 42 (4 + 16βtr tr Substituting equation (2.66) into equation (2.54) and bearing in mind, that n = iy, we can easily evaluate the remaining Gaussian integral over y-value

S. Nechaev: Statistics of Knots and Entangled Random Walks

675

(0)

and obtain the result for PN (A). As it has been mentioned above, to get the simplest estimation for probability of trivial knot formation, we use the last inequality in the chain of equations (2.50) corresponding to the choice A = A∗ ≡ exp(βtr ): PN (A∗ ) ' exp(c N ); (0)

c ≈ 1.

(2.67)

This dependence it is not surprising from the point of view of statistical (0) mechanics because the value η = PN (A∗ ) is proportional to the free energy of the Potts system. But from the topological point of view the value η has the sense of typical “complexity” of the knot (see also Sect. 3). The fact that η grows linearly with N means that the maximum of the distribution function P (η, N ) is in the region of very “complex” knots, i.e. knots far from trivial. This circumstance directly follows from the non-commutative nature of topological interactions. 3

Random walks on locally non-commutative groups

Recent years have been marked by the emergence of more and more problems related to the consideration of physical processes on non-commutative groups. In trying to classify such problems, we distinguish between the following categories in which the non-commutative origin of phenomena appear with perfect clarity: 1. Problems connected with the spectral properties of the Harper– Hofstadter equation [14] dealing with the electron dynamics on the lattice in a constant magnetic field. We mean primarily the consideration of groups of magnetic translations and properties of quantum planes [15, 32]. 2. Problems of classical and quantum chaos on hyperbolic manifolds: spectral properties of dynamical systems and derivation of trace formulae [33–35] as well as construction of probability measures for random walks on modular groups [36]. 3. Problems giving rise to application of quantum group theory in physics: deformations of classical abelian objects such as harmonic oscillators [4] and standard random walks [3]. 4. Problems of knot theory and statistical topology: construction of nonabelian topological invariants [5,23], consideration of probabilistic behavior of the words on the simplest non-commutative groups related to topology (such as braid groups) [37], statistical properties of “anyonic” systems [38]. 5. Classical problems of random matrix and random operator theory and localization phenomena: determination of Lyapunov exponents for products of random non-commutative matrices [39–41], study of the spectral properties and calculation of the density of states of large random matrices [21,42]. Certainly, such a division of problems into these categories is very speculative and reflects to a marked degree the authors’ personal point of view.

676

Topological Aspects of Low Dimensional Systems

However, we believe that the enumerated items reflect, at least partially, the currently growing interest in theoretical physics of the ideas of noncommutative analysis. Let us stress that we do not touch upon the pure mathematical aspects of non-commutative analysis in this paper and the problems discussed in the present work mainly concern the points 4 and 5 of the list above. In the present section we continue analyzing the statistical problems in knot theory, but our attention is paid to some more delicate matters related to investigation of correlations in knotted random paths caused by the topological constraints. The methods elaborated in Section 2 allow us to discuss these questions but we find it more reasonable to take a look at the problems of knot entropy estimation in terms of conventional random matrix theory. We believe that many non-trivial properties of the knot entropy problem can be clearly explained in context of the limit behavior of random walks over the elements of some non-commutative (hyperbolic) groups [46]. Another reason which forces us to consider the limit distributions (and conditional limit distributions) of Markov chains on locally non-commutative discrete groups is due to the fact that this class of problems could be regarded as the first step in a consistent harmonic analysis on the multiconnected manifolds (like Teichm¨ uller space); see also Section 4. 3.1 Brownian bridges on simplest non-commutative groups and knot statistics As it follows from the said above the problems dealing with the investigation of the limit distributions of random walks on non-commutative groups is not a new subject in the probability theory and statistical physics. However in the context of “topologically-probabilistic” consideration the problems dealing with distributions of non-commutative random walks are practically out of discussion, except for very few special cases [41, 43, 49]. Particularly, in these works it has been shown that statistics of random walks with the fixed topological state with respect to the regular array of obstacles on the plane can be obtained from the limit distribution of the so-called “Brownian bridges” (see the definition below) on the universal covering– the graph with the topology of Cayley tree. The analytic construction of nonabelian topological invariant for the trajectories on the double punctured plane and statistics of simplest nontrivial random braid B3 was shortly discussed in [44]. Below we calculate the conditional limit distributions of the Brownian bridges on the braid group B3 and derive the limit distribution of powers of Alexander polynomial of knots generated by random B3 -braids. We also discuss the limit distribution of random walks on locally free groups and express some conjectures about statistics of random walks on the group Bn .

S. Nechaev: Statistics of Knots and Entangled Random Walks

677

More extended discussion of the results concerning the statistics of Markov chains on the braid and locally free groups one can find in [52–54]. 3.1.1 Basic definitions and statistical model The braid group Bn of n strings has n − 1 generators {σ1 , σ2 , . . . , σn−1 } with the following relations: σi σi+1 σi = σi+1 σi σi+1

(1 ≤ i < n − 1)

σi σj = σj σi

(|i − j| ≥ 2)

σi σi−1

=

σi−1 σi

(3.1)

= e.

Any arbitrary word written in terms of “letters”–generators from the set −1 }–gives a particular braid. The geometrical {σ1 , . . . , σn−1 , σ1−1 , . . . , σn−1 interpretation of braid generators is shown below:

The length of the braid is the total number of the used letters, while the minimal irreducible length hereafter referred to as the “primitive word” is the shortest noncontractible length of a particular braid which remains after applying all possible group relations (3.1). Diagrammatically the braid can be represented as a set of crossed strings going from the top to the bottom appeared after subsequent gluing the braid generators. The closed braid appears after gluing the “upper” and the “lower” free ends of the braid on the cylinder. Any braid corresponds to some knot or link. So, it is feasible principal possibility to use the braid group representation for the construction of topological invariants of knots and links. However the correspondence between braids and knots is not mutually single valued and each knot or link can be represented by infinite series of different braids. This fact should be taken into account in course of knot invariant construction. Take a knot diagram K in general position on the plane. Let f [K] be the topological invariant of the knot K. One of the ways to construct the knot invariant using the braid group representation is as follows.

678

Topological Aspects of Low Dimensional Systems

[ [[ [ [ [[ [

f f

b'

=

. . . . .

b''

b

b''

f

. . . . .

b'

b

=f

. . . . .

. . . . .

Fig. 7. Geometric representation of equations (3.2).

1. Represent the knot by some braid b ∈ Bn . Take the function f f : Bn → C where C is a ring of polynomials. Demand f to take the same value for all braids b representing the given knot K. That condition is established in the well-known Markov-Birman theorem (see, for instance [55]): The function fK {b} defined on the braid b ∈ Bn is the topological invariant of a knot or link if and only if it satisfies the following “Markov condition”: fK {b0 b00 } = fK {b00 b0 } fK {b0 σn } = fK {σn b0 } = fK {b0 }

b0 , b00 ∈ Bn

(3.2)

where b0 and b00 are two subsequent sub-words in the braid – see Figure 7. 2. Now the invariant fK {b} can be constructed using the linear functional ϕ{b} defined on the braid group and called Markov trace. It has the following properties ϕ{b0 b00 } = ϕ{b00 b0 } ϕ{b0 σn } = τ ϕ{b0 } ϕ{b

0

σn−1 }

0

= τ¯ϕ{b }

(3.3)

S. Nechaev: Statistics of Knots and Entangled Random Walks

where

τ¯ = ϕ{σi−1 };

τ = ϕ{σi },

i ∈ [1, n − 1].

679

(3.4)

The invariant fK {b} of the knot K is connected with the linear functional ϕ{b} defined on the braid b as follows τ¯ 12 #(+)−#(−) ϕ{b} (3.5) fK {b} = (τ τ¯)−(n−1)/2 τ where #(+) and #(−) are the numbers of “positive” and “negative” crossings in the given braid correspondingly. The Alexander algebraic polynomials are the first well-known invariants of such type. In the beginning of 1980s Jones discovered the new knot invariants. He used the braid representation “passed through” the Hecke algebra relations, where the Hecke algebra, Hn (t), for Bn satisfies both braid group relations equation (3.1) and an additional “reduction” relation (see the works [55, 56]) (3.6) σi2 = (1 − t)σi + t. Now the trace ϕ{b} = ϕ(t){b} can be said to take the value in the ring C of polynomials of one complex variable t. Consider the functional ϕ(t) over the braid {b0 σi b00 }. Equation (3.6) allows us to get the recursion (skein) relations for ϕ(t) and for the invariant fK (t) (see for details [58]): ϕ(t){b0 σi b00 } = (1 − t)ϕ(t){b0 b00 } + tϕ(t){b0 σi−1 b00 } and + (t) − t fK

τ¯ τ

− fK (t) = (1 − t)

τ¯ 1/2 τ

0 fK (t)

(3.7)

(3.8)

+ − 0 ≡ f {b0 σi b00 }; fK ≡ f {b0 σi−1 b00 }; fK ≡ f {b0 b00 } and the fraction where fK τ¯ depends on the used representation. τ 3. The tensor representations of the braid generators can be written as follows X i+1 km i Rln (u)I (1) ⊗ · · · I (i−1) ⊗ Enk ⊗ Eml ⊗ I (i+1) ⊗ · · · I (n) σi (u) = lim u→∞

klmn

(3.9) where I (i) is the identity matrix acting in the position i; Enk is a matrix km is the matrix satisfying the Yang-Baxter with (Enk )pq = δnp δkq and Rln equation X ap X ap jb bq ia ia Rcr (v)Rkc (u + v)Rjb (u) = Rbq (u)Rcr (u + v)Rka (v). (3.10) abc

abc

In that scheme both known polynomial invariants (Jones and Alexander) ought to be considered. In particular, it has been discovered in [57, 58]

680

Topological Aspects of Low Dimensional Systems

that the solutions of equation (3.10) associated with the groups SUq (2) and GL(1, 1) are linked to Jones and Alexander invariants correspondingly. To be more specific: τ¯ (a) = t2 for Jones invariants, fK (t) ≡ V (t). The corresponding skein τ relations are (3.11) t−1 V + (t) − tV − (t) = (t−1/2 − t1/2 )V 0 (t) and τ¯ (b) = t−1 for Alexander invariants, fK (t) ≡ ∇(t). The corresponding τ skein relations3 are ∇+ (t) − ∇− (t) = (t−1/2 − t1/2 )∇0 (t).

(3.12)

To complete this brief review of construction of polynomial invariants from the representation of the braid groups it should be mentioned that the Alexander invariants allow also another useful description [59]. Write the generators of the braid group in the so-called Magnus representation   1 0 ···      0 ...  1 0 0    ..  ← jth row; A =  t −t 1  . ˆj =  ... σj ≡ σ A .    0 0 1   . .. 0   ··· 0 1 (3.13) Now the Alexander polynomial of the knot represented by the closed braid QN W = j=1 σαj of the length N one can write as follows   N Y σ ˆαj − e (3.14) (1 + t + t2 + . . . + tn−1 ) ∇(t){A} = det  j=1

where index j runs “along the braid”, i.e. labels the number of used generators, while the index α = {1, . . . , n−1, n, . . . , 2n−2} marks the set of braid −1 }. In generators (letters) ordered as follows {σ1 , . . . , σn−1 , σ1−1 , . . . , σn−1 our further investigations we repeatedly address to that representation. We are interested in the limit behavior of the knot or link invariants when the length of the corresponding braid tends to infinity, i.e. when the braid “grows”. In this case we can rigorously define some topological characteristics, simpler than the algebraic invariant, which we call the knot complexity. 3 Let us stress that the standard skein relations for Alexander polynomials one can obtain from equation (3.12) replacing t1/2 by −t1/2 .

S. Nechaev: Statistics of Knots and Entangled Random Walks

681

Call the knot complexity, η, the power of some algebraic invariant, fK (t) (Alexander, Jones, HOMFLY) (see also [26]) ln fK (t) · |t|→∞ ln |t|

η = lim

(3.15)

Remark. By definition, the “knot complexity” takes one and the same value for rather broad class of topologically different knots corresponding to algebraic invariants of one and the same power, being from this point of view weaker topological characteristics than complete algebraic polynomial. Let us summarize the advantages of knot complexity. (i) One and the same value of η characterizes a narrow class of “topologically similar” knots which is, however, much broader than the class represented by the polynomial invariant fK (t). This enables us to introduce the smoothed measures and distribution functions for η. (ii) The knot complexity η describes correctly (at least from the physical point of view) the limit cases: η = 0 corresponds to “weakly entangled” trajectories whereas η ∼ N matches the system of “strongly entangled” paths. (iii) The knot complexity keeps all nonabelian properties of the polynomial invariants. (iv) The polynomial invariant can give exhaustive information about the knot topology. However when dealing with statistics of randomly generated knots, we frequently look for rougher characteristics of “topologically different” knots. A similar problem arises in statistical mechanics when passing from the microcanonical ensemble to the Gibbs one: we lose some information about details of particular realization of the system but acquire smoothness of the measure and are able to apply standard thermodynamic methods to the system in question. The main purpose of the present section is the estimation of the limit probability distribution of η for the knots obtained by randomly generated closed Bn -braids of the length N . It should be emphasized that we essentially simplify the general problem “of knot entropy”. Namely, we introduce an additional requirement that the knot should be represented by a braid from the group Bn without fail. We begin the investigation of the probabilistic properties of algebraic knot invariants by analyzing statistics of the random loops (“Brownian bridges”) on simplest non-commutative groups. Most generally the problem can be formulated as follows. Take the discrete group Gn with a fixed finite number of generators {g1 , . . . , gn−1 }. Let ν be the uniform distribu−1 }. For convenience we suppose tion on the set {g1 , . . . , gn−1 , g1−1 , . . . , gn−1 −1 1 hj = gi for j = i and hj = gi for j = i + n − 1; ν(hj ) = 2n−2 for any j. We construct the (right-hand) side random walk (the random word) on Gn with a transition measure ν, i.e. the Markov chain {ξn }, ξ0 = e ∈ Gn and

682

Topological Aspects of Low Dimensional Systems

1 . It means that with the probabilProb(ξj = u|ξj−1 = v) = ν(v −1 u) = 2n−2 1 ity 2n−2 we add the element hαN to the given word hN −1 = hα1 hα2 . . . hαN −1 from the right-hand side4 . The random word W formed by N letters taken independently with the 1 from the set {g1 , . . . , gn−1 , uniform probability distribution ν = 2n−2 −1 −1 g1 , . . . , gn−1 } is called the Brownian bridge (BB) of length N on the group Gn if the shortest (primitive) word of W is identical to the unity. Two questions require most of our attention: 1. What is the probability distribution P (N ) of the Brownian bridge on the group Gn . 2. What is the conditional probability distribution P (k, m|N ) of the fact that the sub-word W 0 consisting of first m letters of the N –letter word W has the primitive path k under the condition that the whole word W is the Brownian bridge on the group Gn . (Hereafter P (k, m|N ) is referred to as the conditional distribution for BB.) It has been shown in the paper [41] that for the free group the corresponding problem can be mapped on the investigation of the random walks on the simply connected tree. Below we represent shortly some results concerning the limit behavior of the conditional probability distribution of BB on the Cayley tree. In the case of braids the more complicated group structure does not allow us to apply the same simple geometrical image directly. Nevertheless the problem of the limit distribution for the random walks on Bn can be reduced to the consideration of the random walk on some graph C(Γ). In case of the group B3 we are able to construct this graph evidently, whereas for the group Bn (n ≥ 4) we give upper estimations for the limit distribution of the random walks considering the statistics of Markov chains on so-called local groups.

3.1.2 Random process on P SL(2, ZZ), B3 and limit distribution of powers of Alexander invariant We begin with computing the distribution function for the conditional random process on the simplest nontrivial braid group B3 . The group B3 can be represented by 2 × 2 matrices. To be specific, the braid generators σ1 and σ2 in the Magnus representation [59] look as follows: σ1 =

−t 1 0 1

;

σ2 =

1 t

0 −t

,

(3.16)

where t is “the spectral parameter”. It is well known that for t = −1 the matrices σ1 and σ2 generate the group P SL(2, ZZ) in such a way that the 4 Analogously

we can construct the left-hand side Markov chain.

S. Nechaev: Statistics of Knots and Entangled Random Walks

whole group B3 is its central extension with the center 6λ t 4λ 4λ 6λ 6λ (σ1 σ2 σ1 ) = (σ2 σ1 σ2 ) = (σ1 σ2 ) = (σ2 σ1 ) = 0

0 t6λ

683

. (3.17)

First restrict ourselves with the examination of the group P SL(2, ZZ), for ˜2 = σ2 (at t = −1). which we define σ ˜1 = σ1 and σ The canonical representation of P SL(2, ZZ) is given by the unimodular matrices S, T : 0 1 1 1 S= ; T = . (3.18) −1 0 0 1 ˜2 σ ˜1 = σ ˜2 σ ˜1 σ ˜2 in the {S, T }-representation takes The braiding relation σ ˜1 σ the form (3.19) S 2 T S −2T −1 = 1 in addition we have S 4 = (ST )3 = 1

(3.20)

This representation is well known and signifies the fact that in terms of {S, T }-generators the group SL(2, ZZ) is a free product Z 2 ⊗ Z 3 of two cyclic groups of the 2nd and the 3rd orders correspondingly. ˜2 } is as follows The connection of {S, T } and {˜ σ1 , σ σ ˜1 = T σ ˜2 = T

(T = σ ˜1 ) −1

ST

−1

(S = σ ˜1 σ ˜2 σ ˜1 ).

(3.21)

The modular group P SL(2, ZZ) is a discrete subgroup of the group P SL(2, IR). The fundamental domain of P SL(2, ZZ) has the form of a cir cular triangle ABC with angles 0, π3 , π3 situated in the upper half-plane Imζ > 0 of the complex plane ζ = ξ + iη (see Fig. 8 for details). According to the definition of the fundamental domain, at least one element of each orbit of P SL(2, ZZ) lies inside ABC-domain and two elements lie on the same orbit if and only if they belong to the boundary of the ABC-domain. The group P SL(2, ZZ) is completely defined by its basic substitutions under the action of generators S and T : S:

ζ → −1/ζ

T :

ζ → ζ + 1.

(3.22)

Let us choose an arbitrary element ζ0 from the fundamental domain and construct a corresponding orbit. In other words, we raise a graph, C(Γ), which connects the neighboring images of the initial element ζ0 obtained under successive action of the generators from the set {S, T, S −1 , T −1 } to

684

Topological Aspects of Low Dimensional Systems

fundamental domain of the modular group

Fig. 8. The Riemann surface for the modular group The graph C(Γ) representing the topological structure of P SL(2, ZZ) is shown by the dashed line.

the element ζ0 . The corresponding graph is shown in Figure 8 by the broken line and its topological structure is clearly reproduced in Figure 9. It can be seen that although the graph C(Γ) does not correspond to the free group and has local cycles, its “backbone”, C(γ), has Cayley tree structure but with the reduced number of branches as compared to the free group C(Γ2 ). Turn to the problem of limit distribution of a random walk on the graph C(Γ). The walk is determined as follows: 1. Take an initial point (“root”) of the random walk on the graph C(Γ). Consider the discrete random jumps over the neighboring vertices of the graph with the transition probabilities induced by the uniform distribution ˜2 , σ ˜1−1 , σ ˜2−1 }. These probabilities are (see ν on the set of generators {˜ σ1 , σ Eq. (3.21)) Prob(ξn = T ζ0 | ξn−1 = ζ0 ) =

1 4

Prob(ξn = (T −1 ST −1 )ζ0 | ξn−1 = ζ0 ) = Prob(ξn = T −1 ζ0 | ξn−1 = ζ0 ) =

1 4

1 4

(3.23)

1 · 4 The following facts should be taken into account: the elements Sζ0 and S −1 ζ0 represent one and the same point, i.e. coincide (as it follows from Prob(ξn = (T S −1 T )ζ0 | ξn−1 = ζ0 ) =

S. Nechaev: Statistics of Knots and Entangled Random Walks

685

M

T O S

T

S T

graph C(Γ) backbone graph C(γ)

Fig. 9. The graph C(Γ) and its backbone graph C(γ) (see the explanations in the text).

Eq. (3.22)); the process is Markovian in terms of the alphabet {˜ σ1 , . . ., σ ˜2−1 } only; the total transition probability is conserved. 2. Define the shortest distance, k, along the graph between the root and terminal points of the random walk. According to its construction, this distance coincides with the length |W{S,T } | of the minimal irreducible word W{S,T } written in the alphabet {S, T, S −1 , T −1 }. The link of the distance, k, with the length |W{˜σ1 ,˜σ2 } | of the minimal irreducible word W{˜σ1 ,˜σ2 } written ˜2 , σ ˜1−1 , σ ˜2−1 } is as follows: (a) |W{˜σ1 ,˜σ2 } | = 0 in terms of the alphabet {˜ σ1 , σ if and only if k = 0; (b) for k 1 the length |W{˜σ1 ,˜σ2 } | has asymptotics: |W{˜σ1 ,˜σ2 } | = k + o(k). We define the “coordinates” of the graph vertices in the following way (see Fig. 9): (a) We apply the arrows to the bonds of the graph Γ corresponding to T generators. The step towards (backwards) the arrow means the application of T (T −1 ). (b) We characterize each elementary cell of the graph Γ by its distance, µ, along the graph backbone γ from the root cell. (c) We introduce the variable α = {1, 2} which numerates the vertices in each cell only. We assume that the walker stays in the cell M located at the distance µ along the backbone from the origin if and only if it visits one of two in-going vertices of M . Such labelling gives unique coding of the

686

Topological Aspects of Low Dimensional Systems

whole graph C(Γ). Define the probability Uα (µ, N ) of the fact that the N -step random walk along the graph C(Γ) starting from the root point is ends in α-vertex of the cell on the distance of µ steps along the backbone. It should be emphasized that Uα (µ, N ) is the probability to stay in any of Nγ (µ) = 3 · 2µ−1 cells situated at the distance µ along the backbone. It is possible to write the closed system of recursion relations for the functions Uα (µ, N ). However, here we attend to rougher characteristics of random walk. Namely, we calculate the “integral” probability distribution of the fact that the trajectory of the random walk starting from an arbitrary vertex of the root cell O has ended in an arbitrary vertex point of the cell M situated on the distance µ along the graph backbone. This probability, U (µ, N ), reads 1 X Uα (µ, N ). U (µ, N ) = 2 α={1,2}

The relation between the distances k, along the graph Γ, and µ along its backbone γ is such: k = µ + o(µ) for µ 1, what ultimately follows from the constructions of the graphs C(Γ) and C(γ). Suppose the walker stays in the vertex α of the cell M located at the distance µ > 1 from the origin along the graph backbone C(γ). The change ˜2 , σ ˜1−1 , σ ˜2−1 } is in µ after making of one arbitrary step from the set {˜ σ1 , σ summarized in the following table:

α=1 µ→µ+1

σ ˜1 = T σ ˜2 = T σ ˜1−1 σ ˜2−1

α=2

−1

=T

ST

−1

−1

= TS

−1

T

µ→µ

σ ˜2 = T

µ→µ−1

σ ˜1−1 σ ˜2−1

µ→µ+1

µ→ µ−1

σ ˜1 = T −1

=T

ST

−1

−1

= TS

−1

µ→ µ+1 µ→ µ+1

T

µ→µ

It is clear that for any value of α two steps increase the length of the backbone, µ, one step decreases it and one step leaves µ without changes. Let us introduce the effective probabilities: p1 – to jump to some specific cell among 3 neighboring ones of the graph C(Γ) and p2 – to stay in the given cell. Because of the symmetry of the graph, the conservation law has to be written as 3p1 + p2 = 1. By definition we have: p1 = ν = 14 . Thus we can write the following set of recursion relations for the integral probability

S. Nechaev: Statistics of Knots and Entangled Random Walks

687

U (µ, N ): 1 U (µ + 1, N ) + 4 1 U (µ, N + 1) = U (µ + 1, N ) + 4 U (µ, N = 0) = δµ,1 . U (µ, N + 1) =

1 1 U (µ, N ) + U (µ − 1, N ) (µ ≥ 2) 4 2 1 U (µ, N ) (µ = 1) 2

(3.24) The solution of equation (3.24) gives the limit distribution for the random walk on the group P LS(2, ZZ). The probability distribution U (k, N ) of the fact that the randomly generated N -letter word W{˜σ1 ,˜σ2 } with the uniform distribution ν = 14 over the ˜2 , σ ˜1−1 , σ ˜2−1 } can be contracted to the minimal irreducible generators {˜ σ1 , σ word of length k, has the following limit behavior  1 k=0  N   3/2 N h h 2 U (k, N ) ' √  k h k k/2 π(4 − h) 4(h − 2)   2 exp − 1k 4N N 3/2 (3.25) √ 2 . where h = 2 + 2 Corollary 1 The probability distribution U (k, m|N ) of the fact that in the ˜2 , σ ˜1−1 , σ ˜2−1 } randomly generated N -letter trivial word in the alphabet {˜ σ1 , σ the sub-word of first m letters has a minimal irreducible length k reads 2 k h 1 k2 1 h + exp · U (k, m|N ) = √ π(4 − h) (m(N − m))3/2 4 m N −m (3.26) Actually, the conditional probability distribution U (µ, m|N ) that the random walk on the backbone graph, C(γ), starting in the origin, visits after first m ( m N = const) steps some graph vertex situated at the distance µ and after N steps returns to the origin, is determined as follows U (µ, m|N ) =

U (µ, m)U (µ, N − m) U (µ = 0, N )Nγ (µ)

(3.27)

where Nγ = 3 · 2µ−1 ) and U (µ, N ) is given by (3.25). The problem considered above helps us in calculating the conditional distribution function for the powers of Alexander polynomial invariants of knots produced by randomly generated closed braids from the group B3 . Generally the closure of an arbitrary braid b ∈ Bn of the total length N gives the knot (link) K. Split the braid b in two parts b0 and b00 with

688

Topological Aspects of Low Dimensional Systems

b'

b''

Fig. 10. Construction of Brownian bridge for knots.

the corresponding lengths m and N − m and make the “phantom closure” of the sub-braids b0 and b00 as it is shown in Figure 10. The phantomly closed sub-braids b0 and b00 correspond to the set of phantomly closed parts (“sub-knots”) of the knot (link) K. The next question is what the conditional probability to find these sub-knots in the state characterized by the complexity η when the knot (link) K as a whole is characterized by the complexity η = 0 (i.e. the topological state of K “is close to trivial”). Returning the the group B3 , introduce normalized generators as follows ||σj±1 || = (det σj±1 )−1 σj±1 · To neglect the insignificant commutative factor dealing with norm of matrices σ1 and σ2 . Now we can rewrite the power of Alexander invariant (Eq. (3.14)) in the form η = [#(+) − #(−)] + η

(3.28)

in a given where #(+) and #(−) are numbers of generators σαj or σα−1 j QN braid and η is the power of the normalized matrix product j=1 ||σαj ||. The condition of Brownian bridge implies η = 0 (i.e. #(+) − #(−) = 0 and η = 0). Write ||σ2 || = T −1 (t)S(t)T −1 (t) (3.29) ||σ1 || = T (t);

S. Nechaev: Statistics of Knots and Entangled Random Walks

689

where T (t) and S(t) are the generators of the “t-deformed” group P SLt (2, ZZ) T (t) = S(t)

=

0 1 (−t)−1 (−t)1/2 ; 0 1 0 (−t)−1/2 0 0 1 (−t)−1/2 . −1 0 0 (−t)1/2

(3.30)

The group P SLt (2, ZZ) preserves the relations of the group P SL(2, ZZ) unchanged, i.e., (T (t)S(t))3 = S 4 (t) = T (t)S 2 (t)T −1 (t)S −2 (t) = 1 (compare to Eq. (3.19)). Hence, if we construct the graph C(Γt ) for the group P SLt (2, ZZ) connecting the neighboring images of an arbitrary element from the fundamental domain, we ultimately come to the conclusion that the graphs C(Γt ) and C(Γ) (Fig. 9) are topologically equivalent. This is the direct consequence of the fact that group B3 is the central extension of P SL(2, ZZ). It should be emphasized that the metric properties of the graphs C(Γt ) and C(Γ) differ because of different embeddings of groups P SLt (2, ZZ) and P SL(2, ZZ) into the complex plane. QN Thus, the matrix product j=1 ||σαj || for the uniform distribution of braid generators is in one-to-one correspondence with the N -step random walk along the graph C(Γ). Its power coincides with the respective geodesics length along the backbone graph C(γ). Thus we conclude that limit distribution of random walks on the group B3 in terms of normalized generators (3.29) is given by equation (3.25) where k should be regarded as the power Q of the product N α=1 ||σαj ||. Hence we come to the following statement. Take a set of knots obtained by closure of B3 -braids of length N with the uniform distribution over the generators. Then the conditional probability distribution U (η, m|N ) for the normalized complexity η of Alexander polynomial invariant (see Eq. (3.28)) has the Gaussian behavior and is given by equation (3.26) where k = η. 3.2 Random walks on locally free groups We aim at getting the asymptotics of conditional limit distributions of BB on the braid group Bn . For the case n > 3 it presents a problem which is unsolved yet. However we can estimate limit probability distributions of BB on Bn considering the limit distributions of random walks on the so-called “local groups” ([44, 48, 52–54]). The group LF n+1 (d) we call the locally free if the generators, {f1 , . . . , fn } obey the following commutation relations: (a) each pair (fj , fk ) generates the free subgroup of the group Fn if |j − k| < d; (b) fj fk = fk fj for |j − k| ≥ d.

690

Topological Aspects of Low Dimensional Systems

(Below we restrict ourselves to the case d = 2 and define LF n+1 (2) = LF n+1 .) The limit probability distribution for the N -step random walk (N 1) on the group Fn+1 to have the minimal irreducible length µ is 3µ2 const −N/6 µ sinh µ exp − (n = 3) P(µ, N ) ' 3/2 e 2N N ( (3.31) 2 ) 9 2 1 exp − µ− N (n 1). P(µ, N ) ' √ 10N 3 2 10πN We propose two independent approaches valid in two different cases: (1) for n = 3 and (2) for n 1. (1) The following geometrical image seems useful. Establish the one-toone correspondence between the random walk in some n-dimensional Hilbert space LHn (x1 , . . . , xn ) and the random walk on the group LF n+1 , written in terms of generators {f1 , . . . , fn−1 }. To be more specific, suppose that when a generator, say, fj , (or fj−1 ) is added to the given word in LF n , the walker makes one unit step towards (backwards for fj−1 ) the axis [0, xj [ in the space LHn (x1 , . . . , xn ). Now the relations (a)-(b) of the definition of the locally free group could be reformulated in terms of metric properties of the space LHn . Actually, the relation (b) indicates that successive steps along the axes [0, xj [ and [0, xk [ (|j − k| ≥ 2) commute, hence the section (xj , xk ) of the space LHn is flat and has the Euclidean metric dx2j + dx2k . Situation with the random trajectories in the sections (xj , xj±1 ) of the Hilbert space LHn appears to be completely different. Here the steps of the walk obey the free group relations (a) and the walk itself is mapped to the walk on the Cayley tree. It is well known that Cayley tree can be isometrically embedded (uniformly without gaps and selfintersections) into the 3-pseudosphere which gives the representation of the non-Euclidean plane with the constant negative curvature. Thus, sections (xj , xj+1 ) have the metric of Lobachevskii plane which can be written in the form x12 (dx2j + dx2j+1 ). j

For the group LF 4 these arguments result in the following metric of appropriate space LH(3) ds2 =

dx21 + dx22 + dx23 . x22

(3.32)

Actually, the space section (x1 , x3 ) is flat whereas the space sections (x1 , x2 ) and (x2 , x3 ) have Lobachevskii plane metric. The noneuclidean (hyperbolic) distance between two points M 0 and M 00 in the space H3 is defined as follows X 1 2 (xi (M 0 ) − xi (M 00 )) x2 (M 0 )x2 (M 00 ) i=1 3

cosh µ(M 0 M 00 ) = 1 +

(3.33)

S. Nechaev: Statistics of Knots and Entangled Random Walks

691

where {x1 , x2 , x3 } are the euclidean coordinates in the 3D-halfspace x2 > 0 and µ is regarded as geodesics on a 4-pseudosphere (Lobachevskii space). Some well known results concerning the limit behavior of random walks in spaces of constant negative curvature are reviewed in the next section where solutions of the diffusion equations in the Lobachevskii plane and space are given by equation (3.57) and equation (3.59) correspondingly. Thus we can conclude that the distribution function for random walk in Lobachevskii space Ps (µ, N ) defined by equations (3.59-3.62) gives also the probability for the N -letter random word (written in terms of uniformly distributed generators on F4 ) to have the primitive word of length µ (see Eq. (3.31)). (2) For the group LF n+1 (n 1) we extract the limit behavior of the distribution function evaluating the probabilities to increase and to decrease the length of the primitive word if we randomly add one extra letter to the given word. We follow below the line proposed by Desbois [52, 54]. First of all let us point out the main steps of our numerical computations: (a) We generate randomly (with uniform probability distribution) the words of lengths N ∈ [1000; 20000], while the number of generators, n, varies in the interval [3; 200]. The number of randomly generated words is of order of 1000. (b) We reduce the given word till the minimal irreducible (primitive) word. This can be done by using the braid (or locally free) group relations. The numerical procedure is as follows. First, we try to push each braid generator in the word as far as possible to the left. Some reductions can occur after that. Then, we play the same game but in the opposite direction, pushing each braid generator to the right performing possible reductions of the word, then–to the left again and so on... If no reductions occur during two consecutive steps, we stop the process. We compute the following quantities for braid and locally free groups: The mean length of the shortest (primitive) word hµi ∞ P

hµi =

µZ(µ, N )

µ=0 ∞ P

(3.34) Z(µ, N )

µ=0

and the variance Var(µ) ∞ P

2 Var(µ) ≡ µ2 − µ =

µ2 Z(µ, N )

µ=0 ∞ P µ=0

Z(µ, N )

2 − µ .

(3.35)

692

Topological Aspects of Low Dimensional Systems

The results of numerical simulations for the word statistics on braid (Bn ) and locally free (LF n (d)) groups are presented in the Table 1. Table 15 . Bn

Groups

LF n (2)

LF n (3)

LF n (4)

hµi N

Var(µ) N

hµi N

Var(µ) N

hµi N

Var(µ) N

hµi N

Var(µ)

n=3

0.29

0.85

0.50

0.76

0.50

0.76

0.50

0.75

n=5

0.49

0.77

0.60

0.63

0.71

0.48

0.75

0.46

n = 10

0.56

0.63

0.65

0.56

0.77

0.40

0.82

0.34

n = 20

0.59

0.63

0.66

0.54

0.79

0.39

0.84

0.29

n = 50

0.61

0.61

0.67

0.56

0.80

0.38

0.85

0.27

n = 100

0.61

0.61

0.67

0.52

0.80

0.36

0.86

0.26

n = 200

0.61

0.60

0.67

0.53

0.80

0.35

0.86

0.26

N

The maximal standard deviations in the Table 1 (and everywhere below) are: ±0.01 for the mean value hµi /N ±0.05 for the variance Var(µ)/N . 3.3 Analytic results for random walks on locally free groups Let us estimate now the quantities hµi /N and Var(µ)/N analytically. We present below two different approaches called “dynamical” and “statistical”. The “dynamical” approach is based on simple estimation of the probability to reduce the primitive word by random adding one extra letter. The estimate obtained by this method is in very good agreement with corresponding numerical simulations. However the “statistical” approach dealing with rigorous enumeration of all nonequivalent primitive words in the locally free group LF n (d) leads to another answer. The rest of this section is devoted to the explanation of the abovementioned discrepancy. Dynamical Consideration. Under the conditions n1 N n2

(3.36)

we can easily develop the dynamical arguments which are supported by results of numerical simulations presented above. The last inequality in (3.36) ensures the conditions, sufficient for finding the limit probability distribution of Markov chains on the groups of n generators. Actually, the number of letters in the word, N should be much larger that the number of all 5 The

groups LF n (d) are completely free when d ≥ n − 1.

S. Nechaev: Statistics of Knots and Entangled Random Walks

693

possible pairs in the set of 2n letters6 . Only in this case the corresponding Markov process has the reliable distribution function. The number of pairs is of order 4n2 , so we arrived at the inequality stated in (3.36). Take a randomly generated N –letter word W . This word is characterized by the length of the primitive word Wp (recall that Wp is the length of the word W obtained after all possible contractions allowed by the structure of the group LF n (d)7 ). Let us compute the probability π(d) of the fact that the primitive word Wp will be shortened in one letter after adding the letter fi (i ∈ [1, n]) to the word W from the right-hand side. It is easy to understand that the primitive word Wp can be reduced if: a) the last letter in the word Wp is just fi−1 . The probability of such 1 ; event is 2n b) the letter before the last in the word Wp is fi−1 and the last letter commutes with the letter fi . The probability of such event is 1 4d−2 2n 1 − 2n ; c) the third letter from the right end of the word Wp is fi−1 and two last letters commute with the letter fi . The probability of such event is 1 4d−2 2 ; 2n 1 − 2n d) ... and so on. Finally we arrive at the following expression for the probability π(d): l ∞ 1 4d − 2 1 X · = 1− π(d) = 2n 2n 4d − 2

(3.37)

l=0

The procedure described above assumes that the letters remaining in the word Wp are uniformly distributed–as in the initial (nonreduced word W ). The absence of “boundary effects” is ensured by the condition (3.36). Once having the probability π(d), we can write down the master equation for the probability P (µ, N ) of the fact that in randomly generated N –letter word the primitive path has the length µ P (µ, N + 1) = (1 − π(d)) P (µ − 1, N ) + π(d) P (µ + 1, N ) 6 The

total number of generators is 2n because each of n generators has the inverse

one.

7 Our

(µ ≥ 2) (3.38)

consideration is valid for any values of d.

694

Topological Aspects of Low Dimensional Systems

where the relation between P (µ, N ) and the partition function Z(µ, N ) introduced above is as follows Z(µ, N ) · P (µ, N ) = P ∞ Z(µ, N ) µ=0

The recursion relation (3.38) coincides with the equation describing the random walk on the halfline with the drift from the origin or, what is the same, with the equation describing the random walk on the simply Cayley tree with the coordinational number 1 = 4d − 2. (3.39) zeff = π(d) Taking into account the last analogy we can complete the equation (3.38) by the boundary conditions P (µ = 1, N + 1) = P (µ = 0, N ) + π(d) P (µ = 2, N ) P (µ = 0, N + 1) = πP (µ = 1, N ) P (µ, N = 0) = δµ,0 .

(3.40)

It is noteworthy that these equations are written just for the Cayley tree with zeff branches. The actual structure of the graph corresponding to the group LF n (d) is much more complex, thus equations (3.40) should be regarded as an aproximation. However the exact form of boundary conditions does not influence the asymptotic solution of equation (3.38) in vicinity of the maximum of the distribution function: 1 P (µ, N ) ' p 2 2π(zeff − 1)N ( 2 ) 2 zeff − 2 zeff N µ− · (3.41) exp − 8(zeff − 1)N zeff Thus, we find zeff − 2 2d − 2 hµ(d)i ' = N zeff 2d − 1 4(zeff − 1) 4d − 3 Var(µ, d) ' = · 2 N zeff (2d − 1)2 Substituting in equation values: hµ(d)i = N hµ(d)i = N hµ(d)i = N

(3.42)

(3.42) d = 2, 3, 4 we get the following numerical 2 ; 3 4 ; 5 6 ; 7

Var(µ, d) 5 = N 9 Var(µ, d) 9 = N 25 Var(µ, d) 13 = N 49

for d = 2 for d = 3 for d = 4

S. Nechaev: Statistics of Knots and Entangled Random Walks

695

what is in the excellent agreement with the asymptotic values (n 1) from the Table 1 for the same groups. The equation (3.41) gives the estimation from below for the limit distribution of the primitive words on the group Bn for n 1. Another statistical problem appears when we are interested in the consideration of the target space of the group LF n+1 , i.e. in the evaluation of the number of nonequivalent primitive words in the group LF n+1 (see for details [53]). Let Vn (µ) be the number of all nonequivalent primitive words of length µ on the group LF n+1 . When µ 1, Vn (µ) has the following asymptotic: µ 8π 2 ; n 1. (3.43) Vn (µ) = const 7 − 2 n To get equation (3.43) we write each primitive word Wp of length µ in the group LF n+1 in the so-called normal order (all fαi are different) similar to so-called “symbolic dynamics” used in consideration of chaotic systems m1

Wp = (fα1 ) where

s X

m2

(fα2 )

ms

. . . (fαs )

(3.44)

|mi | = µ (mi 6= 0 ∀ i; 1 ≤ s ≤ µ) and sequence of generators fαi

i=1

in equation (3.44) for all fαi satisfies the following local rules: (i) if fαi = f1 , then fαi+1 ∈ {f2 , f3 , . . . fn−1 }; (ii) if fαi = fk (1 < k ≤ n − 1), then fαi+1 ∈ {fk−1 , fk+1 , . . . fn−1 }; (iii) if fαi = fn , then fαi+1 = fn−1 . These local rules prescribe the enumeration of all distinct primitive words. If the sequence of generators in the primitive word Wp does not satisfy the rules (i)-(iii), we commute the generators in the word Wp up the normal order is restored. Hence, the normal order representation provides us with the unique coding of all nonequivalent primitive words in the group LF n+1 . The calculation of the number of distinct primitive words, Vn (µ), of the given length µ is rather straightforward: " s # µ X0 X X R(s) δ |mi | − µ (3.45) Vn (µ) = s=1

{m1 , ..., ms }

i=1

where R(s) is the number of all distinct sequences of s generators taken from the set {f1 , . . . , fn } and satisfying the local rules (i)-(iii) while the second sum gives the number of all possible representations of the primitive path of length µ for the fixed sequence of generators (“prime” means that the sum runs over all mi 6= 0 for 1 ≤ i ≤ s; δ is the Kronecker δ-function).

696

Topological Aspects of Low Dimensional Systems

It should be mentioned that the local rules (i)-(iii) define the generalized Markov chain with the states given by the n×n coincidence matrix Tˆn where the rows and columns correspond to the generators f1 , . . . , fn :

Tˆn (d) =

f1

f2

f3

f4

...

fn−1

fn

f1

0

1

1

1

...

1

1

f2

1

0

1

1

...

1

1

f3

1

1

0

1

...

1

1

f4 .. .

0 .. .

1 .. .

1 .. .

0 .. .

... .. .

1 .. .

1 .. .

fn−1

0

0

0

0

...

0

1

fn

0

0

0

0

...

1

0

(3.46)

The number of all distinct normally ordered sequences of words of length s with allowed commutation relations is given by the following partition function is h (3.47) Rn (s, d) = vin Tˆn (d) vout where  n

vin

z }| { = ( 1 1 1 ... 1 )

and

  vout =  

1 1 1 .. . 1

       n.    

(3.48)

Supposing that the main contribution in equation (3.45) results from s 1 we take for Rn (s) the following asymptotic expression Rn (s)

= s1

s (λmax n )

;

λmax n

4π 2 =3− 2 +O n

1 n3

(3.49)

is the highest eigenvalue of the matrix Tˆn (n 1). where λmax n The remaining sum in equation (3.45) is independent of R(s), so its calculation is trivial: " s # X X0 (µ − 1)! · (3.50) δ |mi | − µ = 2s (s − 1)!(µ − s)! i=1 {m1 , ..., ms }

S. Nechaev: Statistics of Knots and Entangled Random Walks

697

Collecting all terms in equation (3.45) and evaluating the sum over s we arrive at equation (3.43). The value Vn (µ, d) is growing exponentially fast with µ and the “speed” of this grows is clearly represented by the fraction Vn (µ + 1) 8π 2 '7− 2 (3.51) zeff − 1 = Vn (µ) µ1 n where zeff is the coordinational number of effective tree associated with the locally free group. Thus, the random walk on the group LF n+1 can be viewed as follows. Take the free group Γn with generators {f˜1 , . . . , f˜n } where all f˜i (1 ≤ i ≤ n) do not commute. The group Γn has a structure of 2n-branching Cayley tree, C(Γn ), where the number of distinct words of length µ is equal to V˜n (µ), V˜n (µ) = 2n(2n − 1)µ−1 .

(3.52)

The graph C(LF n+1 ) corresponding to the group LF n+1 can be constructed from the graph C(Γn ) in accordance with the following recursion procedure: (a) Take the root vertex of the graph C(Γn ) and consider all vertices on the distance µ = 2. Identify those vertices which correspond to the equivalent words in group LF n+1 ; (b) Repeat this procedure taking all vertices at the distance µ = (1, 2, . . .) and “gluing” them at the distance µ+2 according to the definition of the locally free group. By means of the described procedure we raise a graph which in average has zeff − 1 distinct branches leading to the “next coordinational sphere”. Thus this graph coincides (in average) with zeff -branching Cayley tree. We find further investigation of the random walks on the groups LF n+1 (d) for different values of d very perspective. It should give insight for consideration of random walk statistics on “partially commutative groups”. Moreover, the set of problems considered there has deep relation with the spectral theory of random matrices. 3.4 Brownian bridges on Lobachevskii plane and products of non-commutative random matrices The problem of word enumeration on locally non-commutative group has evident connection with the statistics of Markov chains on graphs having the Cayley tree–like structure and, hence, with random walk statistics on the surfaces of a constant negative curvature. (We have stressed already that the Cayley tree–like graphs are isometrically embedded in the surfaces of a constant negative curvature). Recall that the distribution function, P (r, t), for the free random walk in D-dimensional Euclidean space obeys the standard heat equation: ∂ P (r, t) = D∆P (r, t) ∂t

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1 and appropriate initial and normalwith the diffusion coefficient D = 2D ization conditions P (r, t = 0) = δ(r) Z P (r, t)dr = 1.

Correspondingly, the diffusion equation for the scalar density P (q, t) of the free random walk on a Riemann manifold reads (see [62] for instance) 1 ∂ √ ∂ ∂ P (q, t) = D √ g g −1 ik P (q, t) (3.53) ∂t g ∂qi ∂qk where P (q, t = 0) = δ(q) Z √ gP (q, t)dq = 1

(3.54)

and gik is the metric tensor of the manifold; g = det gik . Equation (3.53) has been subjected to thorough analysis for the manifolds of the constant negative curvature. Below we reproduce the corresponding solutions for the best known cases: for 2D– and 3D–Lobachevskii spaces (often referred to as 3– and 4–pseudospheres) labelling them by indices “p ” and “s ” for 2D– and 3D–cases correspondingly. For the Lobachevskii plane one has 1 0 (3.55) ||gik || = 0 sinh2 µ where µ stands for the geodesics length on 3-pseudosphere. The corresponding diffusion equation now reads 2 ∂ ∂ 1 ∂2 ∂ Pp (µ, ϕ, t) = D + + coth µ Pp (µ, ϕ, t). (3.56) ∂t ∂µ2 ∂µ sinh2 µ ∂ϕ2 The solution of equation (3.56) is believed to have the following form Z ∞ ξ exp − ξ2 − tD 4tD e 4 p √ dξ Pp (µ, t) = 3 cosh ξ − cosh µ 4π 2π(tD) µ (3.57) 1/2 − tD 2 4 µ µ e exp − . ' 4πtD sinh µ 4tD For the Lobachevskii space the corresponding metric tensor is 1 0 0 · 0 ||gik || = 0 sinh2 µ 2 2 0 0 sinh µ sin θ

(3.58)

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Substituting equation (3.53) for equation (3.58) we have Ps (µ, t) =

µ2 e−tD µ p exp − · 4tD 8π π(tD)3 sinh µ

(3.59)

For the first time this spherically symmetric solution of the heat equation (Eq. (3.53)) in the Lobachevskii space was received in [63]. In our opinion one fact must be given our attention. The distribution functions Pp (µ, t) and Ps (µ, t) give the probabilities to find the random walk (starting at the point µ = 0) after time t in some specific point located at the distance µ in corresponding noneuclidean space. The probability to find the terminal point of a random walk after time t somewhere at the distance µ is (3.60) Pp,s (µ, t) = Pp,s (µ, t)Np,s (µ) where Np (µ) = sinh µ

(3.61)

is the perimeter of circle of radius µ on the Lobachevskii plane and Ns (µ) = sinh2 µ

(3.62)

is the area of sphere of radius µ in the Lobachevskii space. The difference between Pp,s and Pp,s is insignificant in euclidean geometry, whereas in the noneuclidean space it becomes dramatic because of the consequences of the behavior of Brownian bridges in spaces of constant negative curvature. Using the definition of the Brownian bridge, let us calculate the probabilities to find the N -step random walk (starting at µ = 0) after first t steps at the distance µ in the Lobachevskii plane (space) under the condition that it returns to the origin on the last step. These probabilities are (N → ∞) Pp (µ, t)Pp (µ, N − t) Pp (0, t) µ2 1 1 N µ exp − + = 4πDt(N − t) 4D t N −t

Pp (µ, t|0, N ) =

Ps (µ, t)Ps (µ, N − t) Ps (µ, t|0, N ) = Ps (0, t) 3/2 1 µ2 1 N 2 + µ exp − · = 4D t N −t 8πt3/2 (N − t)3/2

(3.63)

Hence we come to the standard Gaussian distribution function with zero mean.

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Equations (3.63) describing the random walk on the Riemann surface of constant negative curvature have direct application to the conditional distributions of Lyapunov exponents for products of some non-commutative matrices. Let us consider the first of Eqs. (3.63). Changing the variϕ = arg z where z = x + iy; z¯ = x − iy we map the ables µ = ln 1+|z| 1−|z| ; 3–pseudosphere (µ, ϕ) onto the unit disk |z| < 1 known as the Poincare representation of the Lobachevskii plane. The corresponding conformal 4 dzd¯ z 1+iw Using the conformal transform z = 1−iw metric reads dl2 = (1−|z| 2 )2 . we recover the so-called Klein representation of Lobachevskii plane, where 4 dwdw ¯ and the model is defined in Imw > 0 (w = u+iv; w ¯ = u−iv). dl2 = − (w− w) ¯ 2 The following relations can be verified using conformal representations of the Lobachevskii plane metric (see, for instance [17]). The group of fractional transformations of Lobachevskii plane is isomorphic to: (i) the group SU (1, 1)/ ± 1 ≡ P SU (1, 1) in the Poincare model; (ii) the group SL(2, IR)/ ± 1 ≡ P SL(2, IR) in the Klein model. Moreover, it is known (see, for example [34]) that the Lobachevskii plane H can be identified with the group SL(2, IR)/SO(2). This relation enables us to resolve (at least qualitatively) the following problem. Take the Brownian bridge on the group H = SL(2, IR)/SO(2), i.e. demand the ck ∈ H (0 ≤ k ≤ N ) to products of N independent random matrices M be identical to the unit matrix. Consider the limit distribution of the ˆ for the first m matrices in that products. To have Lyapunov exponent, δ, a direct mapping of this problem on the random walk in the Lobachevskii plane, write the corresponding stochastic recursion equation for some vector uk Wk = vk 1 ck Wk ; W0 = (3.64) Wk+1 = M 1 where Mk ∈ H for all k ∈ [1, N ]. The BB–condition means that WN = W1

for N 1.

(3.65)

Let us consider the simplest case ck ; ck = 1 + M M

ck ] 1. norm[M

(3.66)

In this case the discrete dynamic equation (3.64) can be replaced by the differential one. Its stationary measure is determined by the corresponding Fokker-Plank equation (3.53). The Lyapunov exponent, δˆ of product of c coincides with the length of geodesics in the Klein reprandom matrices M resentation of the Lobachevskii plane. Hence, under the conditions (3.65), (3.66) we have for δˆ the usual Gaussian distribution coinciding with the first of equation (3.63). Without the BB–condition (i.e. for “open walks”) we reproduce the standard F¨ urstenberg behavior [39].

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Although this consideration seems rather crude, it clearly shows the origin of the main result: The “Brownian Bridge” condition for random walks in space of constant negative curvature makes the space “effectively flat” turning the corresponding limit probability distribution for random walks to the ordinary central limit distribution. The question whether this result is valid for the case of the random walk in noneuclidean spaces of non-constant negative curvature still remains. Finally we would like to introduce some conjectures which naturally generalize our consideration. The complexity η of any known algebraic invariants (Alexander, Jones, HOMFLY) for the knot represented by the Bn -braid of length N with the uniform distribution over generators has the following limit behavior: η2 const (3.67) P (η, N ) ∼ 3/2 η exp −α(n)N + β(n)η − δ(n)N N where α(n), β(n), δ(n) are numerical constants depending on n only. The knot complexity η in ensemble of Brownian Bridges from the group Bn shown in Figure 10 has Gaussian distribution, where hηi = 0;

2 1 η = δ(n)N. 2

(3.68)

These conjectures are not to be proven rigorously yet. The main idea is to employ the relation between the knot complexity η, the length of the shortest noncontractible word and the length of geodesics on some hyperbolic manifold. 4

Conformal methods in statistics of random walks with topological constraints

The last few years have been marked by considerable progress in understanding the relationship between Chern-Simons topological field theory, construction of algebraic knot and link invariants and conformal field theory (see, for review [64]). Although the general concepts have been well elaborated in the fieldtheoretic context, their application in the related areas of mathematics and physics, such as, for instance, probability theory and statistical physics of chain-like objects is highly limited. The present section is mainly concerned with the conformal methods in statistical analysis which allow us to correlate problems discussed in Sections 1 and 2 and the limit distributions of random walks on multiconnected Riemann surfaces. To be more specific, we show on the level

702

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of differential equations how simple geometrical methods can be applied to construction of non-commutative topological invariants. The latter might serve as nonabelian generalizations of the Gauss linking numbers for the random walks on multi-punctured Riemann surfaces. We also study the connection between the topological properties of random walks on the double punctured plane and behavior of four-point correlation functions in the conformal theory with central charge c = −2. The developed approach is applied to the investigation of statistics of 2D–random walks with multiple topological constraints. For instance, the methods presented here allow us to extract nontrivial critical exponents for the contractible (i.e., unentangled) random walks in the regular lattices of obstacles. Some of our findings support conjectures of Sections 2 and 3 and have direct application in statistics of strongly entangled polymer chains (see Sect. 5). 4.1 Construction of nonabelian connections for Γ2 and P SL(2, ZZ) from conformal methods We analyze the random walk of length L with the effective elementary step a (a ≡ 1) on the complex plane z = x + iy with two points removed. Suppose the coordinates of these points being A (z1 = (0, 0)) and B (z2 = (c, 0)) (c ≡ 1). Such choice does not indicate the loss of generality because by means of simultaneous rescaling of the length, L, of the random walk and of the distance, c, between the removed points we can always obtain of any arbitrary values of a and c. Consider the closed paths on z and attribute the generators g1 , g2 of some group G to the turns around the points A and B if we move along the path in the clockwise direction (we apply g1−1 , g2−1 for counter-clockwise move) (Fig. 11b). The question is: what is the probability P (µ, L) for the random walk of length L on the plane z to form a closed loop with the shortest noncontractible word written in terms of generators {g1 , g2 , g1−1 , g2−1 } to have the length µ (see also Sect. 2). Let the distribution function P (µ, L) be formally written as a path integral with a Wiener measure ( 2 ) Z Z Z L 1 dz(s) 1 ds . . . D{z} exp − 2 P (µ, L) = Z a 0 ds (4.1) −1 −1 ×δ W {g1 , g2 , g1 , g2 |z} − µ R where Z = P (µ, L)dµ and W {. . . |z} is the length of the shortest word on G as a functional of the path in the complex plane. Conformal methods enable us construct the connection and the topological invariant W for the given group as well as to rewrite equation (4.1) in a closed analytic form which is solvable at least in the limit L → ∞.

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Fig. 11. (a) The double punctured complex plane z with two basis loops C1 and C2 (b) the plane with two removed points; (c) the universal covering ζ with fundamental domain corresponding to free group Γ2 . The contours P1 and P2 are the images of the loops C1 and C2 .

Let ζ(z) be the conformal mapping of the double punctured plane z = x + iy on the universal covering ζ = ξ + iλ. The Riemann surface ζ is constructed in the following way. Make three cuts on the complex plane z between the points A and B, between B and (∞) and between (∞) and A along the line Imz = 0 (see Fig. 11b). These cuts separate the upper (Imz > 0) and lower (Imz < 0) half-planes of z. Now perform the conformal transform of the half-plane Imz > 0 to the fundamental domain of the group G{g1 , g2 }–the curvilinear triangle lying in the half-plane Imζ > 0 of the plane ζ (Fig. 11c). Each fundamental domain represents the Riemann sheet corresponding to the fibre bundle above z. The whole covering space ζ is the unification of all such Riemann sheets. The coordinates of initial and final points of any trajectory on universal covering ζ determine [66]: (a) The coordinates of corresponding points on z; (b) The homotopy class of any path on z. In particular, the contours on ζ are closed if and only if W {g1 , g2 |z} ≡ 1, i.e. they belong to the trivial

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Topological Aspects of Low Dimensional Systems

homotopy class. Coordinates of ends of the trajectory on universal covering ζ can be used as the topological invariant for the path on double punctured plane z with respect to the action of the group G. Thus, we characterize the topological invariant, Inv(C), of some closed directed path C starting and ending in an arbitrary point z0 6= {z1 , z2 , ∞} on the plane z by the coordinates of the initial, ζin (z0 ), and final, ζfin (z0 ), points of the corresponding contour P in the covering space ζ. The contour P connects the images of the point z0 on the different Riemann sheets. Write Inv(z) (C) as a full derivative along the contour C: I dζ(z) def dz. (4.2) Inv(z) (C) = ζin − ζfin = C dz dζ(z) is very straightforward. dz Actually, the invariant, Inv(C), can be associated with the flux through the contour C on the plane (x, y): I I ∇ζ(x, y)ndr = ν × ∇ζ(x, y)v(s)ds (4.3) Inv(C) ≡ Inv(x,y) (C) =

The physical interpretation of the derivative

C

C

where: n is the unit vector normal to the curve C, dr = ex dx + ey dy on the plane (x, y); v(s) = dr ds denotes the “velocity” along the trajectory; and ds stands for the differential path length. Simple transformations used in equation (4.3) are: (a) ndr = ex dy − ey dx = dr × ν; (b) ∇ζ(x, y)(dr × ν) = (ν × ∇ζ(x, y))dr, where ν = (0, 0, 1) is the unit vector normal to the plane (x, y). The vector product A(x, y) = ν × ∇ζ(x, y)

(4.4)

can be considered a non-abelian generalization of the vector potential of a solenoidal “magnetic field” normal to the plane (x, y) and crossing it in the points (x1 , y1 ) and (x2 , y2 ). Thus, A defines the flat connection of the double punctured plane z with respect to the action of the group G. It is easy to show how the basic formulae (4.2) and (4.3) transform in case of commutative group Gcomm {g1 , g2 } which distinguishes only the classes of homology of the contour C with respect to the removed points on the plane. The corresponding conformal transform is performed by the function ζ(z) = ln(z − z1 ) + ln(z − z2 ). This immediately gives the abelian connection and the Gauss linking number as a topological invariant: X r − rj ; A(r) = ν × |r − rj |2 j={1,2} I X I (y − yj )dx − (x − xj )dy A(r)dr = = 2π(n1 + n2 ) Inv(C) = 2 2 C C (x − xj ) + (y − yj ) j={1,2}

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where n1 and n2 are the winding numbers of the path C around the points A and B of the plane (x, y). Substituting equation (4.1) written in the Euclidean coordinates (x, y) for equation (4.3) and using the Fourier transform for the δ-function, we can rewrite equation (4.1) as follows Z ∞ 1 e−iqµ P (q, L)dq (4.5) P (µ, L) = 2π −∞ where 1 P (q, L) = Z

Z

Z ...

 

1 D{r} exp − 2  a

ZL

dr(s) ds

2

dr(s) − iqA(r) ds

0

 

! ds



(4.6) The function P (q, L) coincides with the Green function P (r0 , r = r0 , q, L) of the non-stationary Schr¨ odinger-like equation for the free particle motion in a “magnetic field” with the vector potential (4.4): 2 1 ∂ P (r0 , r, q, L) − ∇ − iqA(r) P (r0 , r, q, L) = δ(L)δ(r − r0 ) (4.7) ∂L 2a where q plays a role of a “charge” and the magnetic field is considered transversal, i.e. rot A(r) = 0. Describe now the constructive way of getting the desired conformal transform. The single-valued inverse function z(ζ) ≡ ζ −1 (z) is defined in the fundamental domain of ζ–the triangle ABC. The multivalued function φ(ζ) is determined as follows: – the function φ(ζ) coincides with z(ζ) in the basic fundamental domain; – in all other domains of the covering space ζ the function φ(ζ) is analytically continued through the boundaries of these domains by means of fractional transformations consistent with the action of the group G. Consider two basic contours P1 and P2 on ζ (see Fig. 11c) being the conformal images of the loops enclosing points A and B in Figure 11b. The function φ(z) (z 6= {z1 , z2 , ∞}) obeys the following transformations: i h a1 φ(z) + b1 C ; φ z →1 z → φ˜1 (z) = c1 φ(z) + d1 i h a2 φ(z) + b2 C φ z →2 z → φ˜2 (z) = c2 φ(z) + d2 where

a1 c1

b1 d1

= g1 ;

a2 c2

b2 d2

(4.8)

= g2

are the matrices of basic substitutions of the group G{g1 , g2 }.

(4.9)

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Topological Aspects of Low Dimensional Systems

We assume ζ(z) to be a ratio of two fundamental solutions, u1 (z), and, u2 (z), of some second order differential equation with peculiar points {z1 = (0, 0), z2 = (0, 1), z3 = (∞)}. As it follows from the analytic theory of differential equations [68], the solutions u1 (z) and u2 (z) undergo the linear transformations when the variable z moves along the contours CA and CB corresponding to the turns around points A and B in Figure 11b: CA : CB :

u ˜1 (z) u ˜2 (z) u˜1 (z) u˜2 (z)

= g1

= g2

u1 (z) u2 (z) u1 (z) u2 (z)

; ·

(4.10)

The problem of restoring the form of differential equation knowing the monodromy matrices g1 and g2 of the group G known as Riemann-Hilbert problem has an old history [68]. In our particular case we restrict ourselves with the well investigated groups Γ2 (the free group) and P SL(2, ZZ) (the modular group) (3.16). Thus, we have the following second-order differential equations: z(z − 1)

d 1 d2 (f ) u (z) + (2z − 1) u(f ) (z) + u(f ) (z) = 0 dz 2 dz 4

(4.11)

for the free group and d2 z(z − 1) 2 u(m) (z) + dz

5 z−1 3

d (m) 1 u (z) + u(m) (z) = 0 dz 12

(4.12)

for the modular group. The function which performs the conformal mapping of the upper halfplane Imz > 0 on the fundamental domain (the curvilinear triangle ABC) of the universal covering ζ now reads (f,m)

ζ(z) = (f,m)

(f,m)

u1

(z)

(f,m) u2 (z)

(4.13)

where u1,2 (z) and u1,2 (z) are the basic solutions of (4.11) and (4.12) for Γ2 and P SL(2, ZZ) respectively. As an example we give an explicit form of the complex potential A(z) for the free group Γ2 . Substituting equation (4.2) for equation (4.13), we get 1 F1 (z)F4 (z) F3 (z) dζ(z) = − (4.14) A(z) = dz 2(z − 1) F22 (z) F2 (z)

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where √ 1/ Z z

F1 (z) = 1

Z1

dκ

p ; F2 (z) = (1 − κ2 )(1 − zκ2 ) √ 1/ Z zr

F3 (z) = 1

1 − κ2 dκ; F4 (z) = 1 − zκ2

The asymptotic of (4.14) is as follows  1   dζ(z)  z ∼  1 dz   z−1

0

dκ p 2 (1 − κ )(1 − zκ2 )

Z1 r 0

1 − κ2 dκ. 1 − zκ2

z→0 z→1

(compare to the abelian case). 4.2 Random walk on double punctured plane and conformal field theory The geometrical construction described in the previous section is evidently related to the conformal field theory. In the most direct way this relation could be understood as follows. The ordinary differential equations equation (4.11) and equation (4.12) can be associated with equations on the four-point correlation function of some (still not defined) conformal field theory. The question remains whether it is always possible to adjust the central charge c of the corresponding Virosoro algebra and the conformal dimension ∆ of the critical theory to the coefficients in equations like (4.11, 4.12). The question has positive answer and we show that on the example of the random walk on the double punctured plane with the monodromy of the free group. We restrict ourselves to the “critical” case of infinite long trajectories, i.e. we suppose L → ∞. In the field-theoretic language that means the consideration of the massless free field theory on z. Actually, the partition function of the selfintersecting random walk on z written in the field representation is generated by the scalar Hamiltonian H = 12 (∇ϕ)2 + mϕ2 where the mass m functions as the “chemical potential” conjugated to the length of the path (m ∼ 1/L). Thus, for L → ∞ we have mc = 0 which corresponds to the critical point in conformal theory [65]. We introduce the conformal operator, ϕ(z), on the complex plane z. The dimension, ∆, of this operator is defined by the conformal correlator hϕ(z)ϕ(z 0 )i ∼

1 2∆

|z − z 0 |

·

(4.15)

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Topological Aspects of Low Dimensional Systems

Let us suppose ϕ(z) to be a primary field, then the four-point correlation function hϕ(z1 )ϕ(z2 )ϕ(z3 )ϕ(z4 )i satisfies the equation following from the conformal Ward identity [65, 69, 70]. In form of ordinary Riemann differential equation, equation on the conformal correlator ψ(z|z1 , z2 , z3 ) = hϕ(z)ϕ(z1 )ϕ(z2 )ϕ(z3 )i with the fixed points {z1 = (0, 0), z2 = (1, 0), z3 = ∞} reads [65, 69] d2 1 d ∆ 1 d ∆ 2∆ 3 + − 2− + + × 2(2∆ + 1) dz 2 z dz z − 1 dz z (z − 1)2 z(z − 1) ψ(z|z1 , z2 , z3 ) = 0. Performing the substitution ψ(z|z1 , z2 , z3 ) = [z(z − 1)]−2∆ u(z) we get the equation 2 2 z(z − 1)u00 (z) − (1 − 4∆)(1 − 2z)u0 (z) − (2∆ − 8∆2 )u(z) = 0 3 3

(4.16)

which coincides with equation (4.11) for one single value of ∆ 1 ∆=− · 8

(4.17)

The conformal properties of the stress-energy tensor, T (z), are defined by the coefficients, Ln , in its Laurent expansion, T (z) =

∞ X

Ln · n+2 z n=−∞

These coefficients form the Virosoro algebra [65] [Ln , Lm ] = (n − m)Ln+m +

1 C(n3 − n)δn+m,0 12

where the parameter, c, is the central charge of the theory. Using the relation c = 2∆(5−8∆) (2∆+1) established in [69] and equation (4.17) we obtain c = −2.

(4.18)

We find the following fact, mentioned by B. Duplantier, very intriguing. As he has pointed out, the value ∆ = − 81 (Eq. (4.17)) coincides with the surface exponent (i.e. with the conformal dimension of the two point correlator near the surface) for the dense phase of the O(n = 0) lattice model (or, what is the same, for the Potts model with q = 0) describing statistics of the so-called “Manhattan random walks” (known also as “dense polymers”–see

S. Nechaev: Statistics of Knots and Entangled Random Walks

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the paper [27]). Recall that Potts model has been already mentioned in the Section 1 in connection with construction of algebraic knot invariants. It is hard to believe that such coincidence is occasional and we hope that the relation between these problems will be elucidated in the near future. The conformal invariance of the random walk [66, 67] together with the geometrical interpretation of the monodromy properties of the four-point conformal correlator established above enable us to express the following assertion: The critical conformal field theory characterized by the values c = −2 and ∆ = − 18 gives the field representation for the infinitely long random walk on the double punctured complex plane. With respect to the four-point correlation function, we could ask what happens with the gauge connection Aj (z) if the argument zj of the primary field ψ(zj ) moves along the closed contour C around three punctures on the plane. From the general theory it is known that Aj (z) can be written as Aj (z) =

2 X Ri Rj k z − zi

(4.19)

i6=j

where k is the level of the corresponding representation of the Kac-Moody algebra and Ri , Rj are the generators of representation of the primary fields ψ(zi ), ψ(zj ) in the given group [71]. The holonomy operator χ(C) associated with Aj (z) reads I Aj (z)dz . (4.20) χ(C) = P exp − C

It would be interesting to compare equation (4.14) (with one puncture at infinity) to equation (4.19). Besides we could also expect that equation (4.2) would allow us to rewrite the holonomy operator (4.20) as follows χ(C) = exp (ζin − ζfin ) . At this point we finish the brief discussion of the field-theoretical aspects of the geometrical approach presented above. 4.3 Statistics of random walks with topological constraints in the two–dimensional lattices of obstacles The conformal methods can be applied to the problem of calculating the distribution function for random walks in regular lattices of topological obstacles on the complex plane w = u + iv. Let the elementary cell of the lattice be the equal-sided triangle with the side length c. Introduce the distribution function P (w0 , w, L|hom) defining the probability of the fact that the trajectory of random walk starting at the point

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Topological Aspects of Low Dimensional Systems

w0 comes after “time” L to the point w and all paths going from w0 to w belong to the same homotopy class with respect to the lattice of obstacles. Formally we can write the diffusion equation ∂ a ∆w P (w, L|hom) = P (w, L|hom) 4 ∂L

(4.21)

with initial and normalization conditions: P (w, L = 0|hom) = δ(z0 ); X |w − w0 |2 1 exp − P (w0 , w, L|hom) = · πaL aL

{hom}

The conformal methods can be used to find the asymptotic solution of equation (4.21) when L a. Due to the conformal invariance of the Brownian motion, the new random process in the covering space will be again random but in the metric-dependent “new time”. In particular, we are interested in the probability to find the closed path of length L to be unentangled in the lattice of obstacles. The construction of the conformal transformation ζ(w) (explicitly described in [66]) can be performed in two steps–see Figure 11: 1. First, by means of auxiliary reflection w(z) we transfer the elementary cell of the w-plane to the upper half-plane of the Im(z) > 0 of the double punctured plane z. The function w(z) is determined by the ChristoffelSchwarts integral Z z d˜ z c (4.22) w(z) = B 13 , 13 0 z˜2/3 (1 − z˜)2/3 where B 13 , 13 is the Beta-function. The correspondence of the branching points is as follows: A(w = 0)

˜ = 0) → A(z

B(w = c)

˜ = 1) → B(z

π

C w = c e−i 3

˜ = ∞). → C(z

2. The construction of the universal covering ζ for the double punctured complex plane z is realized by means of automorphic functions. If the covering space is free of obstacles, the corresponding conformal transform should be as follows −

z2 − z + 1 2 z(ζ) = 2 2z (z − 1)2 z 0 (ζ) 1

(4.23)

S. Nechaev: Statistics of Knots and Entangled Random Walks

711

where z(ζ) is the so-called Schwartz’s derivative z 000 (ζ) 3 − z(ζ) = 0 z (ζ) 2

z 00 (ζ) z 0 (ζ)

2 ,

z 0 (ζ) =

dz · dζ

It is well known in the analytic theory of differential equations [68] that the solution of equation (4.23) can be represented as ratio of two fundamental solutions of some second order differential equation with two branching points, namely, of equation (4.11). The final answer reads z(ζ) ≡ k 2 (ζ) =

θ24 (0, eiπζ ) θ34 (0, eiπζ )

(4.24)

where θ2 (0, ζ) and θ3 (0, ζ) are the elliptic Jacobi Theta-functions. We recall their definitions ∞ X π eiπζn(n+1) cos(2n + 1)χ θ2 χ, eiπζ = 2ei 4 ζ n=0

θ3 χ, e

iπζ

=1+2

∞ X

(4.25) e

iπζn2

cos 2nχ.

n=0

˜ B, ˜ C˜ have the images in the vertex points of zeroThe branching points A, angled triangle lying in the upper half-plane of the plane ζ. We have from equation (4.24): ˜ = 0) ¯ = ∞) A(z → A(ζ ˜ = 1) B(z

¯ = 0) → B(ζ

˜ = ∞) C(z

¯ = −1). → C(ζ

The half-plane Im(ζ) > 0 functions as a covering space for the plane w with the regular array of topological obstacles. It does not contain any branching point and consists of the infinite set of Riemann sheets, each of them having form of zero-angled triangle. These Riemann sheets correspond to the fibre bundle of w. The conformal approach gives us a well defined nonabelian topological invariant for the problem–the difference between the initial and final points of the trajectory in the covering space (see Sect. 3.1). Thus, the diffusion equation for the distribution function P (ζ, L) in the covering space ζ with given initial point ζ0 yields a ∂2 2 ∂ P (ζ, ζ0 , L) P (ζ, ζ0 , L) = |w0 (ζ)| 4 ∂ζ∂ζ ∂L

(4.26)

712

Topological Aspects of Low Dimensional Systems

where we took into account that under the conformal transform the Laplace operator is transformed in the following way dζ 2 ∆ζ ∆w = dw

dζ 2 1 dw = |w0 (ζ)|2 ·

and

In particular, the value P (ζ = ζ0 , ζ0 , L) gives the probability for the path of length L to be unentangled (i.e. to be contractible to the point) in the lattice of obstacles. 2 The expression for the Jacobian |w0 (ζ)| one can find using the properties 0 of Jacobi Theta-functions [72]. Write w (ζ) = w0 (z) z 0 (ζ), where w0 (z) =

c

16/3

1

1 3, 3

B

θ3

8/3

θ2

8/3

θ0

and z 0 (ζ) = iπ

θ24 θ04 ; θ34

π d ln i θ04 = 4 dζ

θ2 θ3

(we omit the arguments for compactness). The identity dθ1 (χ, eiπζ ) 0 iπζ = πθ0 (χ, eiπζ ) θ2 (χ, eiπζ ) θ3 (χ, eiπζ ) θ1 (0, e ) ≡ dχ χ=0 enables us to get the final expression 8/3 2 |w0 (ζ)| = c2 h2 θ10 0, eiπζ ,

h=

1 π 1/3 B

1 1 3, 3

' 0.129

(4.27)

where π

θ1 (χ, eiπζ ) = 2ei 4 ζ

∞ X

(−1)n eiπn(n+1)ζ sin(2n + 1)χ.

(4.28)

n=0

Return to equation (4.26) and perform the conformal transform of the upper half-plane Imζ > 0 to the interior of the unit circle on the complex plane τ in order to use the symmetry properties of the system. It is convenient to choose the following mapping of the vertices of the fundamental triangle ¯ C¯ A¯B ¯ = ∞) → A0 (ζ˜ = 1) A(ζ ¯ = 0) B(ζ

→ B 0 (ζ˜ = e−i

2π 3

)

¯ = −1) → C 0 (ζ˜ = ei 2π 3 ). C(ζ

S. Nechaev: Statistics of Knots and Entangled Random Walks

713

0.8 0.6 f 0.4 0.5

0.2 0 0

y

-0.5 0

-0.5

x

0.5

Fig. 12. Relief of the function g(r, ψ)—see explanations in the text.

The corresponding transform reads π

ζ(τ ) = e−i 3

2π

τ − ei 3 −1 τ −1

(4.29)

and the Jacobian |w0 (τ )|2 takes the form 3c2 h2 0 iπζ(τ ) 8/3 0 2 |w (τ )| = θ 0, e . |1 − τ |4 1 In Figure 12 we plot the function g(r, ψ) =

(4.30)

1 0 |w (τ )|2 where τ = r eiψ . c2

The gain of such representation becomes clear if we average the function g(r, ψ) with respect to ψ. The numerical calculations give us: 1 r→1 2π

lim hg(r, ψ)iψ ≡ lim

r→1

Z

2π

g(r, ψ)dψ = 0

$ (1 − r2 )2

(4.31)

where $ ' 0.0309 (see Fig. 13). Thus it is clear that for r rather close to 1 the diffusion is governed by the Laplacian on the surface of the constant negative curvature (the Lobachevskii plane). Representation of the Lobachevskii plane in the unit circle and in the upper half-plane (i.e. Poincare and Klein models) has

714

Topological Aspects of Low Dimensional Systems

0,5

2π ψ ( 1-r 2 ) 2

0,4

0,3

0,2

0,1

0,0 0,0

0,2

0,4

0,6

0,8

1,0

r

Fig. 13. Plot of product 2π hg(r, ψ)iψ × (1 − r 2 )2 as a function of r.

been discussed in Section 2.4. Finally the diffusion equation (4.26) takes the following form: ∂ P (r, ψ, N ) = D(1 − r2 )2 ∆r,ψ P (r, ψ, N ) ∂N

(4.32)

2

a where D = 4$c 2 is the “diffusion coefficient” in the Lobachevskii plane and N = L/a is the dimensionless chain length (i.e. effective number of steps). Changing the variables (r, ψ) → (µ, ψ), where µ = ln 1+r 1−r , we get the unrestricted random walk on the 3-pseudosphere (see Eq. (3.57)). Correspondingly the distribution function P (µ, N ) reads Z ∞ ξ exp − ξ2 ND 4N D e− 4 p √ dξ. (4.33) P (µ, N ) = cosh ξ − cosh µ 4π 2π(N D)3 µ

The physical meaning of the geodesics length on 3-pseudosphere, µ, is straightforward: µ is the length of so-called “primitive path” in the lattice of obstacles, i.e. length of the shortest path remaining after all topologically allowed contractions of the random trajectory in the lattice of obstacles. Hence, µ can be considered a nonabelian topological invariant, much more powerful than the Gauss linking number. This invariant is not complete except one point µ = 0 where it precisely classifies the trajectories belonging

S. Nechaev: Statistics of Knots and Entangled Random Walks

715

to the trivial homotopic class. Let us note that the length η is proportional to the length of the primitive (irreducible) word written in terms of generators of the free group Γ2 . 5

Physical applications. Polymer language in statistics of entangled chain–like objects

Topological constraints essentially modify the physical properties of statistical systems consisting of chain-like objects of completely different nature. It should be said that topological problems are widely investigated in connection with quantum field and string theories, 2D-gravitation, statistics of vortices in superconductors and world lines of anyons, quantum Hall effect, thermodynamic properties of entangled polymers etc. Modern methods of theoretical physics allow us to describe rather comprehensively the effects of nonabelian statistics on physical behavior for each particular referred system; however, in our opinion, the following general questions remain obscure: How does the changes in topological state of the system of entangled chain-like objects effect their physical properties? How can the knowledge accrued in statistical topology be applied to the construction of the Ginzburg-Landau-type theory of fluctuating entangled chain-like objects? In order to have representative and physically clear image for the system of fluctuating chains with the full range of nonabelian topological properties it appears quite natural to formulate general topological problems in terms of polymer physics. It allows us: to use a geometrically clear image of polymer with topological constraints as a model corresponding to the path integral formalism in the field theory; to advance in investigation of specific physical properties of biological and synthetical polymer systems where the topological constraints play a significant role. For physicists the polymer objects are attractive due to many reasons. First of all, the adjoining of monomer units in chains essentially reduces all equilibrium and dynamic properties of the system under consideration. Moreover, due to that adjoining the behavior of polymers is determined by the space-time scales larger than for low-molecular-weight substances. The chain-like structure of macromolecules causes the following peculiarities (see, for instance [73]): the so-called “linear memory” (i.e. fixed position of each monomer unit along the chain); the low translational entropy (i.e. the restrictions on independent motion of monomer units due to the presence of bonds); large space fluctuations (i.e. just a single macromolecule can be regarded as a statistical system with many degrees of freedom). It should be emphasized that the above mentioned “linear memory” leads to the fact that different parts of polymer molecules fluctuating in

716

Topological Aspects of Low Dimensional Systems

space can not pass one through the others without the chain rupture. For the system of non-phantom closed chains this means that only those chain conformations are available which can be transformed continuously into one another, which inevitably give rise to the problem of knot entropy determination (see Sect. 2 for details). 5.1 Polymer chain in 3D-array of obstacles The 3D-model “polymer chain in an array of obstacles” (PCAO) can be defined as follows ([43, 49, 66]). Suppose a polymer chain of length L = N a is placed between the edges of the simple cubic lattice with the spacing c, where N and a are the number of monomer units in the chain and the length of the unit correspondingly. We assume that the chain cannot cross (“pass through”) any edges of the lattice. The PCAO-model can be considered as the basis for a mean-field-like self-consistent approach to the major problem of entropy calculation of ensembles of strongly entangled fluctuating chains. Namely, choose the test chain, specify its topological state and assume that the lattice of obstacles models the effect of entanglements with the surrounding chains (“background”). Neglecting the fluctuations of the background and the topological constraints which the test chain produces for itself, we lose information about the correlations between the test chain and the background. However even in the simplest case we arrive at some nontrivial results concerning statistics of the test chain caused by topological interactions with the background. This means that for the investigation of properties of real polymer systems with topological constraints it is not sufficient to be able to calculate the statistical characteristics of chains in lattices of obstacles, but it is also necessary to be able to adjust any specific physical system to the unique lattice of obstacles, which is much more complicated task. So, let us take a closed polymer chain without volume interactions (i.e. a chain with selfintersections) in the trivial topological state with respect to the 3D lattice of obstacles. It means that the chain trajectory can be continuously contracted to the point. It is clear that because of the obstacles, the macromolecule will adopt more compact conformation than the standard random walk without any topological constraints. It is convenient to begin with the lattice realization of the problem. In this case the polymer chain can be represented as a closed N -step random walk on a cubic lattice with the length of elementary step a being equal to the spacing of the array of obstacles, c. The general case a 6= c will be considered later. The random walk on a 3D-cubic lattice in the presence of the regular array of topological constraints produced by uncrossible strings on the dual lattice is equivalent to the free random walk on the graph–the Cayley tree with the branching number z = 6.

S. Nechaev: Statistics of Knots and Entangled Random Walks The the average space dimension R(N ) ≡ tangled N -step random walk is ([43]):

717

p R2 (N ) of the closed unen-

R ∼ aN 1/4 .

(5.1)

The outline of the derivation of the result (5.1) is as follows. First of all note that the Cayley tree with z branches (called latter as z–tree with z = 2D branches) plays a role of the universal covering and is just a visualization of the free group Γ∞ with the infinite number of generators. At the same time Γ∞ / ZZD = Γz/2 , where Γz/2 is the free group with z generators. Writing down the recursion relations for the probability P (k, N ) for the N -step random walk on the z-tree (compare to (3.24)– (3.40)), we can easily find the conditional limiting distribution for the func(k,N −m) . Recall that P (k, m|N ) gives the condition P (k, m|N ) = P (k,m)P z(z−1)k−1 tional probability distribution of the fact that two sub-chains C1 and C2 of lengths m and N − m have the common primitive path k under the condition that the composite chain C1 C2 of length N is closed and unentangled in regard to the obstacles; 3/2 k2 N N 2 k exp − · (5.2) P (k, m|N ) ' 2m(N − m) 2m(N − m) This equation enables us to get the following expressions for the mean length of the primitive path, hk(m)i of closed unentangled N -link chain divided into two parts of the lengths m and N − m correspondingly r N X 2 2m(N − m) (N 1). (5.3) kP (k, m|N ) ' √ hk(m)i = N π k=0

The primitive path itself can be considered a random walk in a 3D space with restriction that any step of the primitive path should not be strictly opposite one. Therefore the mean-square distance in the

to the previous space (r0 − rm )2 between the ends of the primitive path of k(m) steps is equal to

z hk(m)ia2 (5.4) (r0 − rm )2 = z−2 where rm is the radius-vector of a link with the number m and the boundary

conditions are: rN = r0 = 0. The mean-square gyration radius, Rg2 of N -step closed unentangled random walk in the regular lattice of obstacles reads

Rg2

=

1 X

(rn − rm )2 = 2 2N n6=m

=

N 1 X

(r0 − rm )2 2N m=1 √ 2π 2 √ z a N. z−2 8

(5.5)

718

Topological Aspects of Low Dimensional Systems

This result should be compared to the mean-square radius of the

2gyration 1 2 a N. = 12 closed chain without any topological constraints, Rg,0 The relation R ∼ N 1/4 is reminiscent of the well-known expression for the dimension of randomly branched ideal macromolecule. The gyration radius of an ideal “lattice animal” containing N links is proportional to N 1/4 . It means that both systems belong to the same universality class. Now we turn to the mean-field calculation of the critical exponent ν of nonselfintersecting random walk in the regular lattice of obstacles [75]. Within the framework of Flory-type mean-field theory the nonequilibrium free energy, F (R), of the polymer chain of size R with volume interactions can be written as follows F (R) = Fint (R) + Fel (R)

(5.6)

where Fint (R) is the energy of the chain self-interactions and Fel (R) is the “elastic”, (i.e. pure entropic) contribution to the total free energy of the system. Minimizing F (R) with respect to R for fixed chain length, L = N a, we get the desired relation R ∼ N ν . Write the interacting part of the chain free energy written in the virial expansion (5.7) Fint (R) = V Bρ2 + Cρ3 where V ∼ Rd is the volume occupied by the chain in d-dimensional space; T −θ and C = const > 0 are the two– ρ= N V is the chain density; B = b θ and three– body interaction constants respectively. In the case B > 0 third virial coefficient contribution to equation (5.7) can be neglected [73]. The “elastic” part of the free energy Fel (R) of an unentangled closed chain of size R and length N a in the lattice of obstacles can be estimated as follows Z Fel (R) = const + ln P (R, m = N/2, N ) = const + ln dk P (k, N ) P (R, k) (5.8) where the distribution function P (k, m, N ) is the same as in equation (5.2) and P (R, k) gives the probability for the primitive path of length k to have the space distance between the ends equal to R: P (R, k) =

D 2πack

d/2

DR2 exp − · 2ack

(5.9)

Substituting equation (5.8) for equations (5.2) and (5.9) we get the following estimate 1/3 R4 4/3 + o R . (5.10) Fel (R) = − a 2 c2 N

S. Nechaev: Statistics of Knots and Entangled Random Walks

719

Equations (5.7) and (5.10) allow us to rewrite equation (5.6) in the form N2 F (R) ' B D − R

R4 a 2 c2 N

1/3 ·

(5.11)

Minimization of equation (5.11) with respect to R for fixed N yields R ∼ B 3/(4+3D) (ac)2/(4+3D) N ν ;

ν=

7 · 4 + 3D

(5.12)

The upper critical dimension for that system is Dcr = 8. For D = 3 equation (5.12) gives (5.13) R ∼ N 7/13 . It is interesting to compare equation (5.12) to the critical exponent νan of the lattice animal with excluded volume in the D–dimensional space, 3 , which gives νan = 37 for D = 3. The difference in exponents sigνan = 4+D nifies that the unentangled ring with volume interactions and the nonselfintersecting “lattice animal” belong to different universality classes (despite in the absence of volume interactions they belong to the same class). 5.2 Collapsed phase of unknotted polymer In this Section we show which predictions about the fractal structure of a strongly collapsed phase of unknotted ring polymer can be made using the concept of “polymer chain in array of obstacles”. 5.2.1 “Crumpled globule” concept in statistics of strongly collapsed unknotted polymer loops Take closed nonselfintersecting polymer chain of length N in the trivial topological state8 . After a temperature decrease the formation of the collapsed globular structure becomes thermodynamically favorable [78]. Supposing that the globular state can be described in the virial expansion we intro<0 duce as usual two– and three–body interaction constants: B = b T −θ θ and C = const > 0. But in addition to the standard volume interactions we would like to take into account the non-local topological constraints which obviously have a repulsive character. In this connection we express our main assertion [79]. The condition to form a trivial knot in a closed polymer changes significantly all thermodynamic properties of a macromolecule and leads to specific 8 The fact that the closed chain cannot intersect itself causes two types of interactions: a) volume interactions which vanish for infinitely thin chains and b) topological constraints which remain even for chain of zero thickness.

720

Topological Aspects of Low Dimensional Systems

non-trivial fractal properties of a line representing the chain trajectory in a globule. We call such structure crumpled (fractal) globule. We prove this statement consistently describing the given crumpled structure and showing its stability. It is well-known that in a poor solvent there exists some critical chain length, g ∗ , depending on the temperature and energy of volume interactions, so that chains which have length bigger than g ∗ collapse. Taking long enough chain, we define these g ∗ -link parts as new block monomer units (crumples of minimal scale). Consider now the part of a chain with several block monomers, i.e. crumples of smallest scale. This new part should again collapse in itself, i.e. should form the crumple of the next scale if other chain parts do not interfere with it. The chain of such new sub-blocks (crumples of new scale) collapses again and so on until the chain as a whole (see Fig. 14) forms the largest final crumple. Thus the procedure is completed when all initial links are united into one crumple of the largest scale. It should be noted that the line representing the chain trajectory obtained through the procedure described above resembles the 3D-analogue of the well known self-similar Peano curve. The specific feature of the crumpled globule is in the fact that different chain parts are not entangled with each others, completely fill the allowed volume of space and are “collapsed in themselves” starting from the characteristic scale g ∗ . It may seem that due to space fluctuations of the chain parts all that crumples could penetrate each others with the loops, destroying the self-similar scale-invariant structure described above. However it can be shown on the basis of PCAO–model that if the chain length in a crumple of an arbitrary scale exceeds Ne then the crumples coming in contact do not mix with each other and remain segregated in space. Recall that Ne is the characteristic distance between neighboring entanglements along the chain expressed in number of segments and, as a rule, the values of Ne lie in the range 30 ÷ 300 [73]. Since the topological state of the chain part in each crumple is fixed and coincides with the state of the whole chain (which is trivial) this chain part can be regarded as an unknotted ring. Other chain parts (other crumples) function as effective lattice of obstacles surrounding the “test” ring–see Figure 15. Using the results of the Section 5.1 (see Eq. (5.5)) we conclude that any M -link ring subchain without volume interactions not entangled with any of obstacles has the size R(0) (M ) ∼ aM 1/4 . If R(0) is the size of an equilibrium chain part in the lattice of obstacles, the entropy loss for ring chain, S, as a function of its size, R, reaches its maximum for R ' R(0) (see Eq. (5.10)) and the chain swelling for values of R exceeding R(0) (M ) is entropically unfavorable. At the same time in the presence of excluded volume the following obvious inequality must be fulfilled

S. Nechaev: Statistics of Knots and Entangled Random Walks

(a)

(b)

(c)

(d)

721

Fig. 14. (a)–(c) Subsequent stages of collapse; (d) Self-similar structure of crumpled globule segregated on all scales.

R(M ) ∼ aM 1/3 , which follows from the fact that density of the chain in the globular phase ρ ∼ R3 /N is constant. In connection with the obvious relation R(M ) > R(0) (M ) we conclude that the swelling of chains in crumples due to their mutual inter-penetration with the loops does not result in the entropy gain and, therefore, does not occur in the system with finite density. It means that the size of crumple on each scale is of order of its size in dense packing state and the crumples are mutually segregated in space. These questions are discussed in details in the work [79]. The system of densely packed globulized crumples corresponds to the chain with the fractal dimension Df = 3 (Df = 3 is realized from the

722

Topological Aspects of Low Dimensional Systems

Fig. 15. (a) Part of the closed unknotted chain surrounded by other parts of the same chain; (b) unentangled ring in lattice of obstacles. The obstacles replace the effect of topological constraints produced by other part of the same chain.

minimal scale, g ∗ , up to the whole globule size). The value g ∗ is of order g ∗ = Ne (ρa3 )−2 ,

(5.14)

where ρ is the globule density. This estimation was obtained in [79] using the following arguments: g = (ρa3 )−2 is the mean length of the chain part between two neighboring (along the chain) contacts with other parts; consequently Ne g is the mean length of the chain part between topological contacts (entanglements). Of course, as to the phantom chains, Gaussian blobs of size g are strongly overlapped with others because pair contacts between monomers are screened (because of so-called θ-conditions [78]). However for nonphantom chains these pair contacts are topologically essential because chain crossings are prohibited for any value and sign of the virial coefficient. The entropy loss connected with the crumpled state formation can be estimated as follows: N (5.15) S ' − ∗· g Using equation (5.15) the corresponding crumpled globule density, ρ, can be obtained in the mean-field approximation via minimization of its free energy. The density of the crumpled state is less than that of usual equilibrium state what is connected with additional topological repulsive-type interactions

S. Nechaev: Statistics of Knots and Entangled Random Walks

723

between crumples: ρcrump =

ρeq < ρeq 1 + const(a6 /CNe )

(5.16)

where ρeq is the density of the usual (Lifshits’) globule. The direct experimental verification of the proposed self-similar fractal structure of the unknotted ring polymer in the collapsed phase meets some technical difficulties. One of the ways to justify the “crumpled globule” (CG) concept comes from its indirect manifestations in dynamic and static properties of different polymer systems. The following works should be mentioned in that context: 1. The two-stage dynamics of collapse of the macromolecule after abrupt changing of the solvent quality, found in recent light scattering experiments by Chu and Ying (Stony Brook) [80]. 2. The notion about the crumpled structure of the collapsed ring polymer allowed to explain [84] the experiments on compatibility enhancement in mixtures of ring and linear chains [85], as well as to construct the quantitative theory of a collapse of N –isopropilacrylamide gel in a poor water [83]. 3. The paper [82] where the authors claim the observation of the crumpled globule in numerical simulations. 5.2.2 Knot formation probability We can also utilize the CG-concept to estimate the trivial knot formation probability for dense phase of the polymer chain. Let us repeat that the main part of our modern knowledge about knot and link statistics has been obtained with the help of numerical simulations based on the exploiting of the algebraic knot invariants (Alexander, as a rule). Among the most important results we should mention the following ones: – The probability of the chain self-knotting, p(N ), is determined as a function of chain length N under the random chain closure [1, 86]. In the work [87] (see also the recent paper [88]) the simulation procedure was extended up to chains of order N ' 2000, where the exponential asymptotic of the type p0 ∼ exp(−N/N0 (T )) has been found for trivial knot formation probability for chains in good and θ-solvents. A statistical study of random knotting probability using the Vassiliev invariants has been undertaken in recent work [88]. – The knot formation probability p is investigated of q as a function 2 Rg2 / Rg,0 , Rg2 is the swelling ratio α (α < 1) where α =

724

Topological Aspects of Low Dimensional Systems

nontrivial knotting probability, p

1,0

p(α)=0.925 exp(-0.03/α6) p(α)=1.2 exp(-0.25/α2)

0,8

0,6

0,4

0,2

0,0

0,0

0,2

0,4

0,6

0,8

1,0

swelling ratio, α

Fig. 16. Dependence of non-trivial knot formation probability, p on swelling parameter, α, in globular state.

2 1 mean-square gyration radius of the closed chain and Rg,0 N a2 = 12 is the same for unperturbed (α = 1) chain–see Figure 16, where points correspond to the data of reference [1]; dashed line gives approximation in weak compression regime and solid line–the approximation based on the concept of crumpled globule. It has been shown that this probability decreases sharply when a coil contracts from swollen state with α > 1 to the Gaussian one with α = 1 [89] and especially when it collapses to the globular state [1, 86]. – It has been established that in region α > 1 the topological constraints are screened by volume interactions almost completely [89]. – It has been shown that two unentangled chains (of the same length) even without volume interactions in the q coil state repulse each other as

2 Rg,0 [1, 90]. impenetrable spheres with radius of order Return to Figure 16, where the knot formation probability p is plotted as a function of swelling ratio, α, in the globular region (α < 1). It can be seen that in compression region, especially for α < 0.6 data of numerical

S. Nechaev: Statistics of Knots and Entangled Random Walks

725

experiment are absent. It is difficult to discriminate between different knots in strongly compressed regime because it is necessary to calculate Alexander polynomial for each generated closed contour. It takes of order O(l3 ) operations (l is the number of self-interactions in the projection). This value becomes as larger as denser the system. Let us present the theoretical estimations of the non-trivial knot formation probability p(α) in dense globular state (α < 0.6) based on the CG-concept. The trivial knot formation probability under random linear chain closure, can be defined by the relation: q(α) =

Z(α) , Z0 (α)

q(α) = 1 − p(α)

(5.17)

where Z(α) is the partition function of unknotted closed chain with volume interactions for fixed value of swelling parameter, α, and Z0 (α) is that of “shadow” chain without topological constraints but with the same volume interactions. Both partition functions can be estimated within the framework of the mean field theory. To do so, let us write down the classic Flory-type representation for the free energy of the chain with given α (in equations below we suppose for the temperature T ≡ 1): F (α) = − ln Z(α) = Fint (α) + Fel (α)

(5.18)

Fel (α) = −S(α).

(5.19)

where Here the contributions Fint (α) from the volume interactions to the free energies of unknotted and shadow chain of the same density (i.e. of the same α) are equivalent. Therefore, the only difference concerns the elastic part of free energy, Fel , or, in other words, the conformational entropy. Thus, the equation (5.17) can be represented in the form: (5.20) q(α) = exp −F (α) − F0 (α) = exp S(α) − S0 (α) . According to Fixmann’s calculations [91] the entropy of phantom chain S0 (α) (S0 (α) = ln Z0 (α)) in region α < 1 can be written in the following form: (5.21) S0 (α) ' −α−2 . In the weak compression region 0.6 < α ≤ 1 the probability of nontrivial knotting, p(α), can be estimated from the expression of the phantom ring entropy (Eq. (5.21)). The best fit of numerical data [1] gives us (0.6 < α ≤ 1) (5.22) p(α) = 1 − A1 exp −B1 α−2 where A1 and B1 are the numerical constant.

726

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The nontrivial part of our problem is reduced to the estimation of the entropy of strongly contracted closed unknotted ring (α 1). Using equations (5.14) and (5.15) and the definition of α we find S(α) ' −

1 −6 α . Ne

(5.23)

In the region of our interest (α < 0.6) the α−2 -term can be neglected in comparison with α−6 . Therefore, we the final probability estimate has the form: 1 (α < 0.6) (5.24) p(α) = 1 − A2 exp − α−6 Ne where A2 and Ne are the numerical constants (their values are given below). The probabilities of the nontrivial knot formation, p(α), in weak and strong compression regions are shown in Figure 16 by the dotted and solid lines respectively. The values of the constants are: A1 = 1.2, B1 = 0.25, A2 = 0.925, Ne = 34; they are chosen by comparing equations (5.22) and (5.24) with numerical data of reference [1]. 5.2.3 Quasi-knot concept in collapsed phase of unknotted polymers Speculations about the crumpled structure of strongly contracted closed polymer chains in the trivial topological state could be partially confirmed by the results of Sections 1 and 2. The crucial question is: why the crumples remain segregated in a weakly knotted topological state on all scales in course of chain fluctuations. To clarify the point we begin by defining the topological state of a crumple, i.e. the unclosed part of the chain. Of course, mathematically strict definition of a knot can be formulated for closed (or infinite) contours exclusively. However the everyday experience tells us that even unclosed rope can be knotted. Thus, it seems attractive to construct a non-rigorous notion of a quasi-knot for description of long linear chains with free ends. Such ideas were expressed first in 1973 by Lifshits and Grosberg [92] for the globular state of the chain. The main conjecture was rather simple: in the globular state the distance between the ends of the chain is of order R ∼ aN 1/3 , being much smaller than the chain contour length L ∼ N a. Therefore, the topological state of closed loop, consisting of the chain backbone and the straight end-to-end segment, might roughly characterize the topological state of the chain on the whole. The composite loop should be regarded as a quasi-knot of the linear chain. The topological state of a quasi-knot can be characterized by the knot complexity, η, introduced in Section 3 (see Eq. (3.15)). It should be noted that the quasi-knot concept failed for Gaussian chains where the large space

S. Nechaev: Statistics of Knots and Entangled Random Walks

727

fluctuations of the end-to-end distance lead to the indefiniteness of the quasitopological state. Our model of crumpled globule can be reformulated now in terms of quasi-knots. Consider the ensemble of all closed loops of length L generated with the right measure in the globular phase. Let us extract from this ensemble the loops with η(L) = 0 and find the mean quasi-knot complexity, hη(l)i, of an arbitrary subpart of length l (l/L = h = const; 0 < h < 1) of the given loop. In the globular state the probability π(r) to find the end of the chain of length L in some point r inside the globule of volume R3 is of order π(r) ∼ R13 being independent on r (this relation is valid when La R2 ). So, for the globular phase we could roughly suppose that the loops in the ensemble are generated with the uniform distribution. Thus our system satisfies the “Brownian Bridge” condition and according to conjecture of the Section 3 (Eq. (3.67)) we can apply thepfollowing scaleinvariant estimate for the averaged quasi-knot complexity hη 2 (l)i p

hη 2 (l)i ∼ l1/2 = h1/2 L1/2 .

(5.25) p This value should be compared to averaged complexity hη 2 (l)i of the part of the same length l in the equilibrium globule created by an open chain of length L, i.e. without the Brownian Bridge condition p hη 2 (l)i ∼ l = hL. (5.26) Comparing equations (5.25) and (5.26) we conclude that any part of an unknotted chain in the globular state is far less knotted than the same part of an open chain in the equilibrium globule, which supports our mean-field consideration presented above. Let us stress that our statement is thermodynamically reliable and is independent of kinetics of crumpled globule formation. 6

Some “tight” problems of the probability theory and statistical physics

Usually, in the conclusion it is accepted to overview the main results and imperceptibly prepare the audience to an idea how important the said is... We would not like go not by a usual way and to make a formal conclusion, because the summary of received results together with brief exposition of ideas and methods were indicated in the introduction and some incompleteness of account could only stimulate the fantasy. On the contrary, we will try to pay attention to some hidden difficulties, which we permanently met on our way, as well as to formulate possible, yet unsolved problems, logically following from our consideration. Thus, we shall schematically designate borders of given research and shall allow

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the audience most to decide, whether a given subject deserves of further attention or not. 6.1 Remarks and comments to Section 2 1. The derivation of equations (2.59)–(2.60) assumed the passage from model with short–range interactions to the mean–field–theory, in which all spins are supposed to interact with each others. From the topological point of view such approximation is unphysical and requires the additional verification. We believe, that the considered model could be investigated with help of conformal theories and renormalisation group technique in the case of “weak disorder”, i.e. when exists the strong asymmetry in choice of vertex crossings on a lattice. 2. As it was shown above, utilization of Jones topological invariant with necessity results in the study of thermodynamic properties of a Potts model. In the work [57] was mentioned, that Alexander polynomials are naturally connected to a partition function of a free fermion model and, hence, to an Ising model. Probably, the use of similar functional representation of Alexander polynomials in the frameworks of our disordered model would result in more simple equations, concerned with statistical properties of the Ising spin glasses. 3. All results received in this work are sticked to a model, which is effectively two–dimensional, since we are interested in statistical properties of a planar projection of knot in which all space degrees of freedom are thrown away and the topological disorder is kept only. Thus, physically, the model corresponds to the situation of a globular polymer chain located in a narrow two–dimensional slit. In connection with that the following question is of significant interest: how the fluctuations of a trajectory in a three–dimensional space modify our consideration and, in particular, the answer (2.67)? 6.2 Remarks and comments to Sections 3 and 4 1. The investigation of topological properties of trajectories on multiconnected manifolds (in planes with sets of removed points) from the point of view of the conformal field theory assumes a construction of topological invariants on the basis of monodromy properties of correlation functions of appropriate conformal theories. In connection with that there is a question concerning the possibility of construction of conformal theory with the monodromies of the locally–free group considered in work. 2. Without any doubts the question about the relation between topological invariants design and spectral properties of dynamic systems on hyperbolic manifolds is of extreme importance. The nature of mentioned connection consists in prospective dependence between knot invariants

S. Nechaev: Statistics of Knots and Entangled Random Walks

729

(in the simplest case, Alexander polynomials), recorded in the terms of a trace of products of elements of some hyperbolic group (see expression (3.14)) and trace formulae for some dynamic system on the same group. 3. Comparing distribution function of primitive paths µ (3.57) with the distribution function of a knot complexity of η (3.15)), we can conclude that both these invariants have the same physical sense: a random walk in a covering space, constructed for lattice of obstacles, is equivalent from the topological point of view to a random walk on a Cayley tree. Thus, the knot complexity is proportional to a length of the primitive (irreducible) word, written in terms of group generators, i.e. is proportional to a geodesic length on some surface of constant negative curvature. We believe, that the detailed study of this interrelation will appear rather useful for utilization of algebraic invariants in the problems concerning statistics of ensembles of fluctuating molecules with a fixed topological state of each separate polymer chain. 3. Questions, considered in Section 3 admit an interpretation in spirit of spin glass theories, discussed at length of the Section 2. Let us assume, that there is a closed trajectory of length L, which we randomly drop on a plane with regular set of removed points. Let one point of a trajectory is fixed. The following question appears: what is a probability to find a random trajectory in a given topological state with respect to the set of removed points? The topological state of a trajectory is a typical example of the quenched disorder. According to the general concept, in order to find an appropriate distribution function (statistical sum), it is necessary to average the moments of topological invariant over a Gaussian distribution (i.e. with the measure of trajectories on a plane). The same assumptions are permitted us to assume, that the function g(r, ψ) (Jacobian of conformal transformation)– see Figure 12 has a sense of an ultrametric “potential”, in which the random walk takes place and where each valley corresponds to some given topological state of the path. The closer r is to 1, the higher are the barriers between neighboring valleys. Thus, all reasonably long (La c2 ) random trajectories in such potential will become “localized” in some strongly entangled state, in the sense that the probability of spontaneous disentanglement of a trajectory of length La is of order of exp −const La c2 . Probably this analogy could be useful in a usual theory of spin glasses because of the presence of explicit expression for the ultrametric Parisi phase space [20] in terms of a double-periodic analytic functions. 6.3 Remarks and comments to Section 5 1. We would like to express the conjecture (see also [93]) concerning the possibility of reformulation of some topological problems for strongly collapsed chains (see Sect. 5.4) in terms of integration over the set of trajectories with fixed fractal dimension but without any topological constraints.

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We have argued that in ensemble of strongly contracted unknotted chains (paths) most of them have the fractal dimension Df = 3; We believe that almost all paths in the ensemble of lines with fractal dimension Df = 3 are topologically isomorphic to simple enough (i.e. close to the trivial ) knot. Let us remind, that the problem of the calculation of the partition function for closed polymer chain with topological constraints can be written as an integral over the set Ω of closed paths with fixed value of topological invariant (see Sect. 1): Z Z Z (6.1) Z = Dw {r} e−H = . . . Dw {r}e−H δ[I − I0 ], Ω

where Dw {r} means integration with the usual Wiener measure and δ[I −I0 ] cuts the paths with fixed value of topological invariant (I0 corresponding to the trivial knots). If our conjecture is true, then the integration over Ω in equation (6.1) for the chains in the globular phase (i.e. when La R2 ) can be replaced by the integration over all paths without any topological constraints, but with special new measure, Df {r}: Z Z (6.2) Z = . . . Df {r} e−H . The usual Wiener measure Dw {r} is concentrated on trajectories with the fractal dimension Df = 2. Instead of that, the measure Df {r} with the fractal dimension Df = 3 for description of statistics of unknotted rings should be used. 2. We believe, that the distribution of knot complexity found for some model systems can serve as a starting point in construction of a mean–field Ginsburg–Landau–type theory of fluctuating polymer chains with a fixed topology. From a physical point of view it seems to be important to rise the mean–field theory which takes into account the influence of topological restrictions on phase transitions in bunches of entangled directed polymers. 3. Let us note, that despite a number of experimental works, indirectly testifying for the existence of a fractal globule (see Sect. 5 and references), the direct observation of this structure in real experiments is connected to significant technical difficulties and is so far not carried out. We believe, that the organization of an experiment on determination of a microstructure of an entangled ring molecule in a globular phase could introduce final clarity to a question on a crumpled globule existence. References [1] Vologodskii A.V. and Frank-Kamenetskii M.D., Usp. Fiz. Nauk 134 (1981) 641 (in Russian); Vologodskii A.V., Lukashin A.V., Frank-Kamenetskii M.D. and

S. Nechaev: Statistics of Knots and Entangled Random Walks

[2] [3] [4] [5]

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Anshelevich V.V., Zh. Exp. Teor. Fiz. 66 (1974) 2153; Vologodskii A.V., Lukashin A.V. and Frank-Kamenetskii M.D., Zh. Exp. Teor. Fiz. 67 (1974) 1875; FrankKamenetskii M.D., Lukashin A.V. and Vologodskii A.V., Nature 258 (1975) 398. Wilsceck F., Fractional Statistics and Anyon Superconductivity (Singapore: WSPC, 1990). Majid S., Int. J. Mod. Phys. A 8 (1993) 4521. G´ omez C. and Sierra G. J. Math. Phys. 34 (1993) 2119. Jones V.F.R., Pacific J. Math. 137 (1989) 311; Likorish W.B.R., Bull. London Math. Soc. 20 (1988) 558; Wadati M., Deguchi T.K. and Akutsu Y., Phys. Rep. 180 (1989) 247. Integrable Models and Strings Lect. Not. Phys. 436 (Springer: Heidelberg, 1994); Kauffman L.H., AMS Contemp. Math. Ser. 78 (1989) 263. Kauffman L.H., Topology 26 (1987) 395. Vassiliev V.A., Complements of Discriminants of Smooth Maps: Topology and Applications Math. Monographs 98 (Trans. AMS, 1992); Bar-Natan D., preprint Harvard, 1992. Edwards S.F., Proc. Roy. Soc. 91 (1967) 513. Yor M., J. Appl. Prob. 29 (1992) 202; Math. Finance 3 (1993) 231. Comtet A., Desbois J. and Monthus C., J. Stat. Phys. 73 (1993) 433; Desbois J., J. Phys. A 25 (1992) L-195, L-755; Desbois J. and Comtet A., J. Phys. A 25 (1992) 3097. Wiegel F., Introduction to Path-Integrals Methods in Physics and Polymer Science (Singapore: WSPC, 1986). Khandekar D.S. and Wiegel F., J. Phys. A 21 (1988) L-563; J. Phys. France 50 (1989) 2205. Hofstadter D., Phys. Rev. B 14 2239 (1976); Guillement J., Helffer B. and Treton P., J. Phys. France 50 (1989) 2019. Wiegmann P.B. and Zabrodin A.V., Phys. Rev. Lett. 72 (1994) 1890; Nucl. Phys. B 422 [FS] (1994) 495. Bellissard J., Lect. Notes Phys. 257 (1986) 99. Dubrovin B.A., Novikov S.P. and Fomenko A.T., Modern Geometry (Moscow: Nauka, 1979). Wu F.Y., Rev. Mod. Phys. 54 (1982) 235. Baxter R.J., Exactly Solvable Models in Statistical Mechanics (London: Academic Press, 1982). Mezard M., Parisi G. and Virasoro M., Spin Glass Theory and Beyond (Singapore: WSPC, 1987). Lifshits I.M., Gredeskul S.A. and Pastur L.A., Introduction to the theory of disordered systems (Moscow: Nauka, 1982). Reidemeister K., Knotentheorie (Berlin: Springer, 1932). Kauffman L.H., Knots and Physics (Singapore: WSPC, 1991). Grosberg A. Yu and Nechaev S., J. Phys. A 25 (1992) 4659. Wu F.Y., J. Knot Theory Ramific. 1 (1992) 32. Grosberg A. Yu and Nechaev S., Europhys. Lett. 20 (1992) 613. Duplantier B. and David F., J. Stat. Phys. 51 (1988) 327. Edwards S.F. and Anderson P.W., J. Phys. F 5 (1975) 965. Cwilich G. and Kirkpatrick T.R., J. Phys. A 22 (1989) 4971. De Santis E., Parisi G. and Ritort F., preprint: cond-mat/9410093. Gross D.J., Kanter I. and Sompolinsky H., Phys. Rev. Lett. 55 (1985) 304.

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[32] Bellissard J. and Simon B., J. Funct. Anal. 48 (1982) 408. [33] Gutzwiller M., Chaos in Classical and Quantum Mechanics (Heidelberg: Springer, 1990). [34] Terras A., Harmonic Analysis on Symmetric Spaces and Applications (Berlin: Springer, 1979). [35] Hejhal D., The Selberg Trace Formula for P SL(2, R) Lect. Notes Math. 548 (Berlin: Springer, 1976); 1001 (Berlin: Springer, 1983). [36] Chassaing P., Letac G. and Mora M., Prob. Measures Groups Lect. Not. Math. 1064 (1983). [37] Nechaev S.K., Statistics of Knots and Entangled Random Walks (Singapore: WSPC, 1996). [38] Lerda A., Anyons Lect. Notes Phys. 14 (Berlin: Springer, 1992). [39] F¨ urstenberg H., Trans. Am. Math. Soc 198 (1963) 377; Tutubalin V., Prob. Theor. Appl. (in Russian) 10 (1965) 15. [40] Nechaev S. and Sinai Ya, Bol. Sci. Bras. Mat. 21 (1991) 121. [41] Koralov L., Nechaev S. and Sinai Ya, Prob. Theor. Appl. 38 (1993) 331. [42] Mirlin A. and Fyodorov Y., J. Phys. A 24 (1991) 227; Fyodorov Y., Basic Features of Efetov’s supersymmetry approach Mesoscopic Quantum Physics, Les Houches, session LXI, 1994 (Amsterdam: Elsevier, 1995). [43] Khokhlov A. and Nechaev S., Phys. Lett. A 112 (1985) 156; Nechaev S., Semenov A. and Koleva M., Physica A 140 (1987) 506. [44] Nechaev S. and Vershik A., J. Phys. A 27 (1994) 2289. [45] Monastyrsky M. and Nechaev S., Mod. Phys. Lett. A (1997). [46] Nechaev S., Grosberg A. Yu and Vershik A.M., J. Phys. A 29 (1996) 2411. [47] Kesten H., Trans. Amer. Math. Soc. 92 (1959) 336. [48] Vershik A.M., Topics in Algebra 26, pt. 2 (1990) 467 (Warszawa: Banach Center Publication, PWN Publ.); Proc. Am. Math. Soc. 148 (1991) 1. [49] Helfand E. and Pearson D.S., J. Chem. Phys. 79 (1983) 2054; Rubinstein M. and Helfand E., J. Chem. Phys. 82 (1985) 2477. [50] Cates M.E. and Deutsch J.M., J. Phys. France 47 (1986) 2121. [51] Vershik A.M., private communication. [52] Desbois J. and Nechaev S., J. Stat. Phys. 88 (1997) 201. [53] Comtet A. and Nechaev S., J. Phys. A 31 (1998) 2767. [54] Desbois J. and Nechaev S., J. Phys. A 31 (1998) 5609. [55] Jones W.F.R., Bull. Am. Math. Soc. 12 (1985) 103. [56] Vershik A.M. and Kerov S., Dokl. Ac. Nauk SSSR 301 (1988) 777 (in Russian). [57] Kauffman L.H. and Saleur H., Comm. Math. Phys. 141 (1991) 293. [58] Akutsu Y. and Deguchi T.K., Phys. Rev. Lett. 67 (1991) 777; Akutsu Y. and Wadati M., J. Phys. Soc. Jpn. 56 (1987). [59] Birman J., Knots, Links and Mapping Class Groups Ann. Math. Stud. 82 (Princeton: Princeton Univ. Press, NJ, 1976). [60] Mumford M., Tata Lectures on Theta, I, II Progress in Mathematics 28, 34 (Boston, MA: Birkhauser, 1983). [61] Letchikov, Russian Math. Surv. 51 (1996) 49. [62] Zinn-Justin J., Quantum Field Theory and Critical Phenomena (Oxford: Clarendon, 1989). [63] Gerzenshtein M.E. and Vasiljev V.B., Prob. Theor. Appl. 4 (1959) 424 (in Russian); Karpelevich F.I. and Tutubalin V.N. and Shour M.G., Prob. Theor. Appl. 4 (1959) 432 (in Russian).

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SEMINAR 2

TWISTING A SINGLE DNA MOLECULE: EXPERIMENTS AND MODELS

V. CROQUETTE LPS, ENS, UMR 8550 CNRS, associ´ e aux Universit´es Paris VI et Paris VII, 24 rue Lhomond, 75231 Paris Cedex 05, France

Contents 1 Introduction

737

2 Single molecule micromanipulation 739 2.1 Forces at the molecular scale . . . . . . . . . . . . . . . . . . . . . 739 2.2 Brownian motion: A sensitive tool for measuring forces . . . . . . 740 3 Stretching B-DNA is well described by the worm like chain model 740 3.1 The Freely Jointed Chain Elasticity model . . . . . . . . . . . . . . 740 3.2 The overstretching transition . . . . . . . . . . . . . . . . . . . . . 743 4 The 4.1 4.2 4.3 4.4

torsional buckling instability The buckling instability at T = 0 . . . . . . . . . . . . . . . . The buckling instability in the Rod Like Chain (RLC) model Elastic Rod Model of Supercoiled DNA . . . . . . . . . . . . Theoretical analysis of the extension versus supercoiling experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Critical torques are associated to phase changes . . . . . . . .

744 . . . 744 . . . 746 . . . 746 . . . 751 . . . 754

5 Unwinding DNA leads to denaturation 754 5.1 Twisting rigidity measured through the critical torque of denaturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 5.2 Phase coexistence in the large torsional stress regime . . . . . . . . 758 6 Overtwisting DNA leads to P-DNA 760 6.1 Phase coexistence of B-DNA and P-DNA in the large torsional stress regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 6.2 Chemical evidence of exposed bases . . . . . . . . . . . . . . . . . . 762 7 Conclusions

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T. Strick1 , J.-F. Allemand1 , D. Bensimon1 , V. Croquette1 , C. Bouchiat2 , M. M´ezard2 and R. Lavery3

Abstract We present experiments and models investigating the twisting of a single DNA molecule stretched by a constant force. We first describe the low stress regime where interwound structures appear. We find that these structures reduce the molecule’s extension and may be described by a discretized Rod-Like Chain Model requiring two parameters: the bending and the twisting rigidity of the molecule. We discuss the experimental evaluation of the ratio of these two elasticity coefficients. Finally, we show that increasing both the torsional and stretching stresses induces a cooperative phase transition to new structures (which are different for positive and negative twisting). Once these structures appear, they coexist with the canonical Watson-Crick B-DNA over a large range of forces and degrees of coiling. We shall describe these stress-induced transitions and discuss the new structures and their possible biological relevance.

1

Introduction

For the past few years, micromanipulation techniques have given physicists and biologists the opportunity to study a single biopolymer, such as a DNA molecule or a protein. Physically-inspired questions such as the elastic behavior of a single polymer are no longer gedanken experiments but routine measurements thanks to the pioneering work of Smith et al. [1]. In a more biophysical spirit, the lock-key model of binding between ligand 1 LPS, ENS, UMR 8550 CNRS, associ´ e aux Universit´ es Paris VI et Paris VII, 24 rue Lhomond, 75231 Paris Cedex 05, France. 2 LPT, ENS, UMR 8550 CNRS, associ´ e aux universit´ es Paris VI et Paris VII, 24 rue Lhomond, 75231 Paris Cedex 05, France. 3 Laboratoire de Biochimie Th´ eorique, UPR 9080 du CNRS, Institut de Biologie Physico-Chimique, 13 rue Pierre et Marie Curie, 75005 Paris, France.

c EDP Sciences, Springer-Verlag 1999

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and receptor has been tested directly by measuring the system’s effective tensile strength [2]. The force developed by a single motor protein such as a myosin molecule advancing along an actin filament [3] or a kinesin group stepping on a tubulin fiber [4] is studied quantitatively and furthers our understanding of muscle contraction. These experiments bring new insights to the study of protein activity and emphasize the mechanical aspects of enzymatic behavior. A beautiful demonstration was given by Kinoshita et al. [5] who showed experimentally that the F1 -ATPase works like a small rotary motor with a stator, a rotor and a cam-shaft. This enzyme, present in all our cells, actually synthesizes adenosine triphosphate (ATP), the fuel which drives most enzymatic reactions. Micromanipulation techniques take advantage of a variety of tools ranging from optical tweezers, magnetic tweezers and microneedles to single molecule fluorescence. We shall focus here on one such technique and two types of experiments on a single polymer: stretching and twisting a DNA molecule. The first polymer stretching experiments were carried out on torsionally unconstrained DNA at low forces, so that the work of stretching went mostly into a reduction of the configurational entropy of the DNA polymer. The resulting force vs. extension curves fit very nicely the predictions of the Worm-Like Chain (WLC) model appropriate for semi-flexible polymers [6]. More recently, a torsional stress [7] was applied to a single DNA molecule. Whereas force stress is not very common in vivo, DNA is always under torsional stress in all living cells. Altering this constraint is lethal for the cell. One of the reasons explaining the importance of DNA twisting is the existence of interwound structures. We show here that this mechanism may now be studied with great accuracy experimentally, theoretically and with computer modelization. The recent analytical treatment of the Rod-Like-Chain Model has greatly improved our understanding of twisted DNA. However, this model ceases to correctly describe the DNA molecule when large stretching or twisting stresses are applied to it. Experiments performed at high stretching forces have shown that DNA overstretches [8,9], leading to a new molecular structure. We shall also describe how a large torsional stress [7, 10] induce the molecule to undergo a transition from the classical Watson-Crick double-helical B form [11] to new structural forms. In this article, these three approaches have been regrouped: the experimental work was performed by T. Strick, J.-F. Allemand, D. Bensimon and V. Croquette, the theoretical analysis based on the Rod-Like Chain Model is discussed by C. Bouchiat and M. M´ezard while R. Lavery performed the molecular modeling.

V. Croquette et al.: Twisting a Single DNA Molecule

Translation and Rotation Motions

µ

S

N

BIOTIN

µ

Magnets Flows (beads, buffers, enzymes, etc...)

739

Capillary DNA

Objective x 63 ANTIDIGOXIGENIN

DIGOXIGENIN

Beads Glass slide

Inverted Microscope Computer

CCD camera

Fig. 1. Schematic view of the apparatus used to twist and stretch single DNA molecules. DNA molecules were first prepared with biotin attached to one end and digoxigenin (dig) bound to the other. These end-labeled DNA molecules are incubated with streptavidin-coated magnetic beads and then flowed into a square glass capillary coated with an antibody to dig, antidig. The DNA molecules bind specifically to the bead via biotin/streptavidin coupling and to the glass via dig/antidig coupling. The capillary is placed above an inverted microscope. Magnets are placed above the capillary. By approaching the magnets we increase the stretching force on the bead and thus on the molecule. By rotating the magnets the molecule is twisted at constant force. A frame grabber installed in a PC allows for tracking of the Brownian fluctuations hδx2 i of the bead. The determination of hδx2 i and of the molecule’s extension l leads to a measure of the stretching force F = kB T l/hδx2 i.

2

Single molecule micromanipulation

2.1 Forces at the molecular scale The upper bound for forces in micromanipulation experiments is the tensile strength of a covalent bound, on the order of eV/˚ A or about 1 nN. The smallest measurable force is set by the Langevin force which is responsible for the Brownian motion of the sensor. Because of its random nature, the Langevin force is a noise density in force which is simply written as √ fn = 4kB T 6πηr (η is the viscosity of the medium, r is the typical size of the sensor). Such a sensor may be √ modelized by a one-micron-diameter object in water, leading to fn ∼ 10fN/ Hz. In between those two extremes lies the forces typical of interactions at the molecular scale, which are of order kB T /nm ∼ 4 pN. This is roughly the stall force of a single molecular motor such as myosin (4 pN [12]) or RNA-polymerase (15 pN to 30 pN [13,14]). It is also the typical force needed to unpair the DNA bases (about 15 pN [15]). At a force of 75 pN DNA overstretches [8, 9].

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Topological Aspects of Low Dimensional Systems

2.2 Brownian motion: A sensitive tool for measuring forces To produce and measure such forces on a DNA molecule we use a single molecule manipulation technique [7]. Briefly, it consists in stretching a single DNA molecule (λ-DNA ∼ 50 000 base-pairs ∼ 16 µm is often used) bound at one end to a surface and at the other to our sensor: a magnetic bead (see Fig. 1). The specific binding of the molecule’s ends is achieved using two ligand/receptor pairs. Small magnets, whose position and rotation can be controlled, are used to pull on and rotate the bead and thus stretch and twist the molecule. The tethered bead (1 − 4.5 µm in diameter) exhibits Brownian motion whose amplitude gives access to the force applied to the bead: the stronger the force, the smaller its fluctuations. This system allows us to apply and measure forces ranging from a few femtoNewtons to nearly 100 picoNewtons (see [16]). 3

Stretching B-DNA is well described by the worm like chain model

Figure 2 shows the force vs. extension curve of single DNA molecules in the range 0.06 pN < F < 100 pN. At low force this curve may be understood using theoretical models of polymer elasticity. At high forces these theories break down due to the complex internal structure of DNA which remodels itself under strong mechanical constraints. 3.1 The Freely Jointed Chain Elasticity model The simplest elasticity model used to describe polymers is the Freely Jointed Chain (FJC) Model, where each monomer corresponds to a unitary segment b whose orientation is completely independent of its neighbors’. Under a stretching force F, such a segment is equivalent to a spin in a magnetic field, the entire polymer length being equivalent to the magnetization of a paramagnetic substance in a magnetic field. On one hand, the system likes to align itself with the force thus gaining an energy F b. On the other hand, it wants to adopt a random orientation in order to maximize its entropy. As expected, the competition between the two regimes will depend upon the ratio F b/kB T . For small forces (F < kB T /b), the polymer adopts a random coil configuration, and its end-to-end extension is small. For large forces, the polymer is nearly completely stretched. The extension versus force curve is described by the Langevin function: l = coth l0

Fb kB T

−

Fb · kB T

(1)

If we want to apply this model to DNA, we must take b = 3.4 ˚ A which is the distance between two successive bases. However, two consecutive bases are

V. Croquette et al.: Twisting a Single DNA Molecule

741

Fig. 2. Force versus relative extension curves of single DNA molecules. The dots correspond to several experiments performed over a wide range of forces. The force was measured using the Brownian fluctuation technique [7]. The full line curve is a best fit to the WLC model for forces smaller than 5 picoNewtons. At high forces, the molecule first elongates slightly, as would any material in the elastic regime. Above 70 pN, the length abruptly increases, corresponding to the appearance of the new S-DNA phase.

stacked on each other and may not be considered as independent in their respective orientation. Thus, the FJC model does not describe the force vs. extension curve with the simple hypothesis that b = 3.4 ˚ A. There is however, some flexibility in DNA and two consecutive bases do not point in exactly the same direction. A first attempt to improve the model is to assume that the chain segment length is not the distance between two bases but an effective length b which is chosen in such way that the bending elastic energy due to thermal fluctuations in a length b is precisely kB T /2. The length b is called the Khun length and is two times ξ, the persistence length (the distance over which the orientational correlation decreases by a factor e). For DNA, this persistence length is large compared with the distance between two bases: ξ ∼ 50 nm. Moreover, it appears that the elastic model describing this effective chain is no longer the FJC model but the Worm Like Chain model. The WLC model has the same qualitative behavior as the FJC: small extension for F < kB T /b and large extension otherwise, but its precise behavior is different.

742

Topological Aspects of Low Dimensional Systems

As can be seen at low forces (F < 6 pN) the WLC model (continuous curve) fits the data very well over three decades in force. This model describes a DNA molecule as a semi-flexible polymer chain of length l0 and bending modulus B (or persistence length ξ = B/kB T ). The energy EWLC of a given configuration ˆt(s) of a molecule stretched by a force F along the z−axis is:

EWLC =

B 2

Z 0

l0

∂ˆt(s) ∂s

2

Z ds − F

l0

ˆt(s) · ˆzds

(2)

0

where ˆt(s) is the local tangential vector at curvilinear coordinate s along the molecule. The first term is the bending energy, while the second sets the extension l of the molecule: 0 < l < l0 . The partition function Z(L, F, ˆt0 , ˆt1 ) is given by the path integral: Z Z(L, F, ˆt0 , ˆt1 ) =

D(ˆt) exp (−β EWLC )

(3)

D(ˆt) is the functional integration measure relative to the paths drawn on the unit sphere starting at the point ˆt(s = 0) = ˆt0 and ending at ˆt(s = l0 ) = ˆt1 . By performing an analytic continuation of the s integral in EWLC towards the imaginary axis, one gets the action integral associated with the precession of a dumbbell molecule, having a transverse moment of inertia proportional to ξ and an electric dipole moment along ˆt; the applied electric field is, up to a constant, the stretching force F. The partition function Z(L, F, ˆt0 , ˆt1 ) goes over to the transition probability amplitude hˆt1 tf | t = 0 ˆt0 i ( t ∝ Im(s)) for a dumbbell, with its axis initially oriented along ˆt0 , and at the final time tf with it axis oriented along ˆt1 . By a simple adaptation of the usual rules of quantum mechanics, one gets the partition function: l0 ˆ ˆ ˆ ˆ Z(L, F, t0 , t1 ) = ht1 | exp − HWLC | ˆt0 i ξ

(4)

ˆ WLC is obtained from the dumbbell where the dimensionless operator H molecule Hamiltonian by appropriate changes of units. This expression of ˆ WLC eigenstate the partition function can be written as a sum over the H l0 contributions, each term being proportional to exp(− ξ n ), where n is the associated eigenvalue. Since l0 ξ the sum is dominated by the groundstate contribution. The ground state energy gWLC = 0 is a function of a single dimensionless parameter: u = F ξ/kB T . Hence the free energy of a stretched DNA chain is:

V. Croquette et al.: Twisting a Single DNA Molecule

l0 F = − log Z = gWLC (F ξ/kB T ). kB T ξ

743

(5)

The relative extension of the molecule at a given force is: x = l/l0 = −∂gWLC /∂u. Inverting this relation, the exact solution can easily be computed [17]. A convenient and very accurate approximation (within 0.1%) is: X 1 1 Fξ =x− + + ai xi 2 kB T 4 4(1 − x) i=2 7

u=

(6)

with a2 = −0.5164228, a3 = −2.737418, a4 = 16.07497, a5 = −38.87607, a6 = 39.49944, a7 = −14.17718 [17]. Fitting the force vs. extension data to this theoretical prediction yields the most accurate estimate of the DNA’s persistence length ξ and allows the study of its dependence on ionic conditions [18]. Beyond this entropic regime, i.e. from ∼ 6 pN to about 70 pN, DNA behaves like an elastic rod with stiffness EA ∼ 1000 pN [9] (where E is the Young modulus of DNA and A its effective cross-sectional area [19]). Neglecting entropic contributions, the force vs. extension curve follows a simple Hookean law: F = EA(x − 1) (with x > 1). Notice that there exist some ad-hoc formula interpolating between the entropic and Hookean regimes, e.g. replacing the term (1 − x)2 in equation (6) by (1 − x + F/EA)2 [20]. 3.2 The overstretching transition At F ∼ 70 pN, a very interesting cooperative (i.e. quasi-first order) transition is observed from a slightly stretched (∼ 10%) B-DNA phase to a highly stretched (∼ 80%) phase, called S-DNA [8, 9, 21, 22]. Although there are, as yet, no crystallographic data on the structure of S-DNA, very old experiments [23] (pre-dating Watson and Crick’s discovery of the double helical structure of DNA [11]) have shown that stretched DNA fibers (consisting of many aligned molecules) undergo a birefringent transition apparently due to a tilt of the DNA bases upon stretching. Recent numerical calculations indeed suggest that a tilted structure for S-DNA is one of two possibilities (the other being a straight ladder) depending on precisely how the strands are pulled [8, 24]. A phenomenological Ising-like description of the B-DNA → S-DNA transition has been proposed [8, 21], where the force plays the same role as the magnetic field in a ferromagnetic context. In this model, the observed sharpness of the transition (its high cooperativity) is associated with a large

744

Topological Aspects of Low Dimensional Systems

interfacial energy between the B and S phases, suggesting that the typical domain size is about 100 bases long [21]. 4

The torsional buckling instability

To describe DNA under torsional stress it is first necessary to introduce the relevant experimental parameters. The number of times the two strands of the DNA double-helix are intertwined – the linking number of the molecule (Lk) – is one of these parameters. It corresponds to the the sum of two geometrical components of the system, its writhe (W r) and its twist (T w): Lk = W r +T w. W r is a measure of the coiling of the DNA axis about itself, similar to the way in which a twisted phone cord forms interwound structures in order to relieve torque. T w represents the helical winding of the two strands around each other. For unconstrained linear DNA molecules, assuming the absence of any spontaneous local curvature, Lk = Lk0 = T w0 (= the number of helical turns in B-DNA) [25]. One defines the relative change in linking number, or degree of supercoiling, σ = (Lk − Lk0 )/Lk0 = ∆Lk/Lk0 . The value of σ for most circular molecules isolated from cells or virions is roughly −0.06. At fixed Lk the ratio T w/W r depends on the force stretching the molecule, the writhe being suppressed by high forces. As a consequence, pulling on a supercoiled molecule increases its effective torque. In our experiment, provided that the DNA binding is achieved at multiple points at both ends, a torsional constraint may be applied to the molecule by simply rotating the magnets. As one turn of the magnets implies a change of one turn of the molecule, we have simply ∆Lk = ±n, where ±n is the number of turns by which the magnets are made to rotate. In this paper we also use sometimes the scaled supercoiling variable η = 2πnξ/L. 4.1 The buckling instability at T = 0 Twisting DNA leads to a torsional buckling instability analogous to that observed on telephone cords or rubber tubes. This instability leads to the formation of interwound structures known as plectonemes. Of course, a DNA molecule is also animated by very strong thermal fluctuations which play an important role [25a,25b]. However it is instructive to first consider the purely mechanical (zero temperature) instability of a rubber tube of length l and elastic twist rigidity normalized by kB T . If we firmly hold one end of the tube while simultaneously rotating and pulling on the second end with a force F , we observe the following phenomenon (see Fig. 3): when the twist constraint is small, the associated torque Γ increases linearly with the twist angle θ, Γ = kB T Cθ/l and the tube remains straight. As the tube is further twisted, a critical twist angle θc,b and torque Γc,b are reached and the tube ceases to be straight: it locally buckles and forms a small loop of radius Rc,b . The torsional energy

V. Croquette et al.: Twisting a Single DNA Molecule

745

F=.33pN Torque

3

Writhe Twist

Plecto

l 2

Magnetic Bead

Magnetic Bead DNA

l

1

DNA Glass surface

n c,bor σ c,b

n or σ

1

2

3

4

Η

Fig. 3. Left: schematic view of the buckling transition for a twisted rubber tube (dotted line) or a DNA molecule (solid line). Below a critical number of turns nc,b the rubber tube’s torque increases linearly as it stores twisting energy. When nc,b turns have been added the system abruptly exchanges twisting energy for bending energy and plectonemes begin to form. The plectonemes grow linearly with subsequent twisting and the torque remains constant thereafter. In the case of DNA, the same picture is basically true, except for the fact that thermal fluctuations round off the transition which takes place at nc,b . Right: Results from the RLC model corresponding to a stretching force of F = 0.33 pN. The x-axis represents the reduced supercoiling variable η (see Eq. (15)), and the y-axis is in arbitrary units. The long-dash curve represents the torque acting on the DNA: as described above, it increases linearly until ηc,b ∼ 1 and remains essentially constant thereafter. The short-dash curve represents the ratio of writhe to twist: note that it is never zero and increases rapidly as η > 1. Finally, the full line give a measure of, the fraction of stable plectonemic structures which becomes noticeable after the torsional buckling transition has been passed.

thus gained is 2πΓc,b , whereas the energy cost (due to bending and work against F ) is (see Eq. (2)): p πB/R + 2πRF . This energy is minimized for a loop of radius Rc,b = B/2F . The critical torque for the formation of plectonemes is controlled by the√balance between energy gain and cost, i.e. by the stretching force: Γc,b = 2BF . As we twist the tube further, we increase the length of the plectonemes but the torque in the tube remains basically fixed at its critical value Γc,b .

746

Topological Aspects of Low Dimensional Systems

Fig. 4. Relative extension of a DNA molecule versus the degree of torsion η for various stretching forces. For the three low force curves, the behavior is symmetrical versus η (or σ), the shortening corresponds to the formation of plectonemes. For those low forces the comparison between experimental data (points) and theory for the Rod-Like-Chain model (full-line) is made. When the force is increased above 0.5 pN, the curve becomes asymmetric: plectonemes still form for positive η(or σ) while denaturation prevent there occurrence for negative η(or σ). For forces, larger than 3 pN no more plectonemes are observed.

4.2 The buckling instability in the Rod Like Chain (RLC) model For DNA, the picture is pretty much the same [26]. The thermal fluctuations which will be most important near the mechanical instability at θc,b will tend to round it off. Hence as one is coiling a DNA molecule under fixed force F , one observes the following behavior (see Fig. 4): at low degrees of supercoiling |σ| the molecule’s extension varies little. Beyond a critical value σc,b (which depends on the force), the molecule shortens continuously, forming plectonemes as it is further twisted. An accurate theoretical treatment of this behavior is given by the RLC model developed by Bouchiat and M´ezard [17] and described lelow. 4.3 Elastic Rod Model of Supercoiled DNA In an attempt to describe the entropic elasticity of a single supercoiled DNA molecule, Bouchiat and M´ezard have performed an exact treatment of the thermal fluctuations of a twisted elastic rod [17, 27]. The Rod-Like

V. Croquette et al.: Twisting a Single DNA Molecule

747

Chain (RLC) model is a non-trivial extension of the WLC model previously described. The deformation of the molecular chain under the stretching and winding constraints is specified by the local trihedron {ˆ ei (s)} = ˆ (s), ˆt(s)} where s is the arc length along the molecule, ˆt is the {ˆ u(s), n ˆ (s) is along the basis line and n ˆ (s) = unit vector tangent to the chain, u ˆt(s) ∧ u ˆ (s). The evolution of the triedron {ˆ ei (s)} along the chain is obtained by applying a rotation R(s), parametrized by the usual Euler angles θ(s),φ(s) and ψ(s), to a reference triedron attached to a rectilinear relaxed molecule. The RLC elastic energy ERLC is obtained by adding to the WLC energy given previously in equation (2) a twist energy ET : Z ERLC

=

Ω(s) =

EWLC + ET = EWLC + ∂φ ∂ψ + cos θ(s) ∂s ∂s

l0

ds 0

C 2 (Ω(s)) 2

(7) (8)

where Ω(s) is the twist per unit length and represents the deviation from the DNA’s natural helicity. It is easily seen that the linear energy density associated with the RLC model defined above involves an elastic rigidity tensor having a cylindrical symmetry relative to the molecular axis ˆt(s). It is therefore legitimate to question the validity of a model which ignores from the start the DNA helical structure. However, one can argue that cylindrical symmetry remains practically unbroken if the RLC model is used with data taken with a finite resolution ∆ l in the length measurements, typically of the order of three times the double helix pitch p in the experiments analyzed in the present paper. As shown in reference [28], all cylindrical asymmetric terms in the elastic energy are washed out by the finite length resolution of the experiments. As in the case of the WLC model the computation of the molecule’s partition function can be mapped into a well-defined Quantum Mechanics problem: instead of a dumbbell molecule one has to deal with a symmetric top molecule whose longitudinal moment of inertia is proportional to the elastic twist rigidity C. At this point one may get the impression that the solution of the RLC model can be obtained by a relatively straightforward extension of the method described previously for the WLC model. The mapping of the elastic rod thermal fluctuations problem onto the quantum symmetric top problem is in fact a somewhat formal procedure in the sense ˆ RLC as a differthat the path integral formalism yields the Hamiltonian H ential operator which is not completely defined, unless the functional space on which it is acting is properly specified. Physical considerations will lead us to choose different functional spaces for the two problems at hand. In the symmetric top problem the space is that of 2π periodic functions of

748

Topological Aspects of Low Dimensional Systems

the Euler angle φ and ψ, but in the study of RLC thermal fluctuations the space is that of general functions of these angles, without any constraint of periodicity. The above considerations, which may seem somewhat technical, have important physical consequences. In order to solve the RLC model, ones has to to deal with a quantum spherical top when its angular momentum along the top axis is not quantized. Such a problem is mathematically singular and as a consequence the continuous limit of the RLC model considered so far does not give an adequate description of supercoiled DNA. The remedy is to introduce a discretized version of the RLC model, involving an elementary length scale b about twice the double helix pitch p. This is consistent with the previous considerations where the empirical length resolution ∆ l ≈ 3p was invoked in order to justify the cylindrical symmetry of the tensor of elastic rigidities. The experimental supercoiling constraint, applied to the free end of the DNA chain, is implemented in the RLC model by requiring that: Z

l0

ψ(l0 ) + φ(l0 ) =

ds 0

∂ψ ∂φ + ∂s ∂s

= χ = 2 πn

(9)

where n is the number of turns of the magnetic bead. It is of interest to introduce the total twist Tw along the chain: Z Tw =

Z

l0

0

l0

ds Ω(s) =

ds 0

∂ψ ∂φ + cos θ(s) ∂s ∂s

·

(10)

Then one defines by subtraction a “local writhe” χW contribution to the supercoiling angle: Z χW = χ − T w =

L

ds 0

∂φ (1 − cos θ). ∂s

(11)

The above decomposition, which results from a trivial manipulation, is reminiscent of the decomposition of the linking number into twist and writhe for closed chains. There is however a crucial difference: here the supercoiling angle χ is not a topological quantum number but a continuous angular variable. Let us call Z(χ1 , χ2 , F ) the partition function relative to a situation where both the twist and the writhe take specified values: Tw = χ1 and χW = χ2 . Using simple path integral manipulations, one proves the factorization property: Z(χ1 , χ2 , F ) = ZT (χ1 ) ZW (χ2 ). Physically it implies that, in the absence of a supercoiling constraint, the twist and the writhe fluctuate independently. The supercoiling-constrained partition function Z(χ, F ) is then written as a convolution product:

V. Croquette et al.: Twisting a Single DNA Molecule

749

Z Z(χ, F ) =

dχ1 dχ2 ZT (χ1 ) ZW (χ2 ) δ ((χ1 + χ2 − χ)).

(12)

The twist partition function ZT (χ1 ) coincides with that of the inflexible C χ1 2 rod: ZT (χ1 ) = exp − 2 l0 . The writhe partition function is written as a R ˜ ˜ Fourier integral: ZW (χ) = dk 2π exp(−ik χ)ZW (k), where ZW (k) is given by the path integral: Z Z˜W (k) =

EWLC D(θ, φ) exp i k χW − · kB T

(13)

The Quantum Mechanics problem associated with Z˜W (k) is shown to be that of an electrically charged particle moving on a sphere under the joint action of an electric field and the field of a magnetic monopole of charge k. Following standard Quantum Mechanics rules, the corresponding ˆ RLC (k) is obtained by adding the singular writhe potential Hamiltonian H ˆ WLC : to H VW (k 2 , θ) =

k 2 1 − cos θ · 2 1 + cos θ

(14)

The singularity of VW (k 2 , θ) at θ = π is a clear manifestation of the singular nature of the continuous RLC model. Among the pathological features of the model, the simplest concerns the spontaneous fluctuations of writhe in the absence of supercoiling. The second moment is given by: 2 l0 1 ∂ 2 Z˜W (k) ∂0 (u, k 2 ) lim = ˜W (k) ∂ k 2 ξ k2 →0 ∂ k2 →0 Z

hχW 2 i = lim 2 k

ˆ RLC (k) ground state energy. Using standard perturwhere 0 (u, k 2 ) is the H bation theory at small k 2 , one finds hχW 2 i = (L/A)h (1−cos θ)/(1+cos θ) i0 ˆ WLC The symbol h i0 stands for the quantum average taken over the H ground state wave function. The result diverges since for a finite stretching force the wave function is finite at θ = π. Furthermore, one can prove that at small forces, say F ≤ 0.1pN , the relative extension is not modified when χ varies in the range of zero to a few l0 /ξ, in striking contradiction with experiments. This problem is clearly associated with the pole at θ = π in the writhe potential. Introducing by hand a cut-off near θ = π appears as an ad hoc procedure but it turns out that a small-distance cutoff b generates an angular cutoff: sin2 θ ≥ ξb . As pointed out previously, the RLC model with

750

Topological Aspects of Low Dimensional Systems

its cylindrical symmetric rigidities tensor is realistic only in the presence of a finite resolution ∆ l in the length measurement. In this context, the existence a length cutoff b ∼ ∆ l appears to be rather natural. It should be stressed that the effects of this length cutoff, for the values of b to be considered below, are reduced to the level of a few percent in the absence of a supercoiling constraint. As a consequence, the continuous version of the WLC model remains a remarkably good description of the DNA force extension measurements in the χ = 0 limit. The EWLC integral discretization prescription is rather straightforward and has the nice property of having a periodicity of 2π with respect to the finite difference φn − φn−1 . The discretization rule for the writhe integral χW is obtained by requiring the same periodicity property. To solve the discretized model RLC we have followed three methods. The first one a) is based upon a regularized continuous RLC model obtained by multiplying the singular writhe potential by a cut-off function R( ξb sin2 θ). The function R(x) is not the result of an educated guess but is actually derived from the discretized RLC model transfer matrix theory [28]. It is given as the ratio of two Bessel functions: R(x) = I1 (x)/I0 (x). The ground state energy 0 (u, k 2 ) and its partial derivatives are then obtained from the techniques used to get high-precision force versus extension curves in the WLC model. The second method b) directly uses transfer matrix techniques and is numerically very efficient; for details see [28]. The methods a) and b) give results which agree to within a few percent. The third method c) is based on Monte Carlo simulations and was initially used to validate the discretized RLC model. Nevertheless, it still appears to be the only realistic way to incorporate in a quantitative fashion the self-avoidance effects which are neglected in the present RLC model. The experiments suggest that the variations of the relative elongation hz(l0 )/l0 i versus the supercoiling angle χ scale as a function of χ/l0 . It is convenient to introduce an intensive supercoiling variable η = χξ/l0 which will be at most of the order of a few units in the validity domain of the RLC model. It is related to the more familiar variable σ by: η = 2πnξ/L = 2πξσ/p ' 95 σ.

(15)

The partition function Z(χ, F ) defined by equation (12) is written as an ordinary product in Fourier space. The k-integral is computed by the saddle point method in the limit l0 /ξ 1 with η fixed. The saddle point is imaginary, kc = iκ(u) and is given by the equation: B ∂0 η = + 2 2 (u, −κ2 ). κ C ∂k

(16)

V. Croquette et al.: Twisting a Single DNA Molecule

751

The saddle point contribution to the partition function Z(η, u) reads as follows: l0 ln (Z(η, u))) = − ξ

κ2 B 2 + η κ + O(1). 0 (u, −κ ) − 2C

(17)

One immediately gets the molecule’s relative extension h z(ll00 ) i: ∂0 (u, −κ2 ) 1 ∂ ln Z h z(l0 ) i =− · = l0 kB T ∂F ∂α

(18)

A similar evaluation of the torque Γ acting upon the free end of the molecule leads to the the remarkable result: ∂ ln Z ξ ∂ ln Z Γ =− =− = κ(u). kB T ∂χ l0 ∂η

(19)

One should note that in both derivations it is necessary to explicitly use the saddle point equation. Then, introducing the thermodynamic potential G/kB T = − ln (Z(η, u))−κχ, one verifies that the supercoiling angle χ given ∂ by equation (16) satisfies the thermodynamic relation χ = − ∂κ

G kB T

.

From the knowledge of 0 (u, −κ2 ), the combination of equations (16) and (18) yields a parametric representation of the extension versus supercoiling curves, the parameter being the torque in units of kB T . 4.4 Theoretical analysis of the extension versus supercoiling experiments The buckling instability is a competition between the bending and the twisting energy of the molecule. At low stretching and twisting stress, far from where the molecule undergoes structural transitions, the purely elastic RLC model is expected to accurately describe the experiment. We have thus chosen to measure the extension versus supercoiling curves at low forces, namely F = 0.116 pN, 0.197 pN and 0.328 pN in order to compare theory and experiment. These data still have to be considered as preliminary since a point-by-point evaluation of the systematic uncertainties has not yet been performed. Only the statistical uncertainties involved in the measurements have been considered in this preliminary analysis. In the theoretical approach, it is convenient to use the scaled supercoiling variable η = 2πnξ/L, which turns out to be of the order of unity in the domain of interest. (For ξ = 51.3 nm and p = 3.4 nm, η = 94.8σ.) We have also excluded from the analysis regions where the validity of the RLC model may be questioned.

752

Topological Aspects of Low Dimensional Systems C B

2

1.8

1.6

1.4

1.2 0.06

0.08

0.1

0.12

0.14

0.16

0.18

b 0.2 B

Fig. 5. Empirical determination of the cut off length b and the twist to bending rigidity ratio C/B from the extension versus supercoiling curves analysis. Note when b/B takes values below 0.1 the RCL model fit to the data becomes poorer. The variance increases by a factor 5, suggesting the presence a singularity near b = 0.

First, we exclude values of the relative elongation such that hz(l0 )i/l0 ≤ 0.1. The RLC model predicts that when the reduced supercoiling parameter η increases from 0 to few units the probability distribution of θ develops a peak near θ = π. In particular, when 0 ≤ hz(l0 )i/l0 ≤ 0.1, the RLC model is likely to generate configurations with z(s) ≤ 0, which are excluded by the presence of a wall. Second, we exclude high values of η by imposing the condition η ≤ 1.5 for F = 0.197 pN and F = 0.328 pN. There is theoretical evidence that at F = 0.328 pN the RLC model generates plectonemic-like configurations above a critical value ηc ≈ 1. Because the RLC model ignores self-avoiding effects which are of course present in the actual experiments this region may not be well described by this model. For example, in the experiments, plectonemes must have a radius larger than the DNA’s Debye radius. Whereas, the RLC model can generate plectonemes with an arbitrary small radius.

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Using equation (16) the ratio C/B can be written as a function of the reduced supercoiling variable η and the torque κ: η ∂0 B = − 2 2 (u, −κ2 ). C κ ∂k

(20)

With the help of interpolation techniques, one can invert equation (18) in order to obtain κ as a function of hz(l0 )i/l0 . In this way, every point of the extension versus η curve (with coordinates (η, hz(l0 )i/l0 )) is associated with an empirical value of the twist to bending rigidity ratio C/B, once a choice of the cut-off length b has been made. If the RLC model is to give a good representation of the data there must exist a value of b such that the set of empirical value C/B obtained from an extension versus supercoiling curve clusters nicely around the average value hC/Bi. For F = 0.116 pN, we have plotted in Figure 5 the average ratio hC/Bi 2 and the variance σr = (h (C/B)2 i − h C/Bi )1/2 versus the cut-off length b. The RLC model with b = 0.14 B/kB T (this value is approximately equal to twice that of the double helix pitch p), leads to a remarkably good agreement σr = 0.03. It is interesting to note that the average value with the data: hC/Bi hC/Bi varies slowly with b while the variance σr increases rapidly if one goes to small values of the cut-off; this is consistent with the fact that the RLC model becomes singular in the limit b → 0. Analogous results, somewhat less precise, have been obtained for the two other values of F considered in this section. They favor the same value of b. Performing a weighted average upon the whole set of C/B empirical values, taking b = 0.14ξ, one obtains the following empirical determination of the ratio of the two elastic rigidities involved in the RLC model: C = 1.64 ± 0.04 leading to C/(kB T ) = 82 ± 10 nm. B

(21)

The error quoted for C/B is the statistical one. Thus the statistical error on C/(kB T ) is of the order of 2 nm, but since we expect systematic errors to be non negligible we believe a 10 nm error to be reasonable. Since the data used in this analysis does not incorporate systematic uncertainties in a quantitative way, the above number should be considered as somewhat preliminary, but as its stands, it constitutes a significative improvement upon the present empirical situation: 0.8 ≤ C/B ≤ 2.0 [28a]. The theoretical extension versus η curves have been computed using the ratio C/B taken from experiment and the favored value b = 0.14ξ. The comparison with the experimental data is shown in Figure 4. The agreement looks very satisfactory and indicates an overall consistency of the procedure.

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4.5 Critical torques are associated to phase changes The torsional buckling instability just described treats the DNA molecule as a continuous elastic tube. It ignores the underlying double-helical structure of the molecule, and its relevance is therefore limited to very low forces (F < 0.4 pN) or low degrees of supercoiling (−0.015 < σ < 0.037). For higher forces and degrees of supercoiling, the buildup of torque in the molecule can be large enough to actually modify its internal structure. This is evidenced by breaking of the σ → −σ symmetry in the extension vs. supercoiling curves Figure 4. As a critical force is reached (ipso facto a critical torque), the molecule undergoes a transition from a contracted state (plectonemic B-DNA) to a highly extended one. As we shall see below, this state is characterized by the coexistence of B-DNA with denatured DNA (dDNA, for σ < 0) and with a new phase called P-DNA (for σ > 0). In 10 mM phosphate buffer (PB), the critical force Fc− required to induce the formation of localized regions of denatured DNA is Fc− ∼ 0.5 pN. The critical degree of unwinding beyond which DNA can be induced to denature is σc− = −0.015 and the associated critical torque is Γc− ∼ 8 pN nm. For overwound molecules the relevant values are: Fp c+ = 3 pN, σc+ = 0.037 and Γc+ ∼ 20 pN nm. Notice that Γc− /Γc+ ∼ Fc− /Fc+ , as our simple calculation above showed. These torque-induced transitions are reminiscent of the B → S transition observed in overstretched DNA, although there are some major differences. First, they occur at forces two orders of magnitude smaller than the 70 pN necessary for the generation of S-DNA. Second, whereas B-DNA is completely transformed to S-DNA upon increase of the force, this is not the case here. Due to the topological constraint of a fixed linking number, the proportion of the new phase in B-DNA is determined by the degree of supercoiling σ and the intrinsic twist of the new phase (this is a rather unusual statistical mechanics situation). We shall now present our evidence for the existence of these twist-induced phases and the possibility to measure the twist rigidity of DNA offered by the measurement of the threshold of denaturation. 5

Unwinding DNA leads to denaturation

Let us first consider negative supercoiling (σ < 0). It is known [29, 30] that for values of σ < −0.07, unstretched DNA undergoes localized denaturation. When stretched, similar local denaturation is observed at smaller values of |σ|, due to the increased torsional stress resulting from the inhibition of writhe (see Figs. 3 and [7, 16]). The denaturation is detected mechanically in the F vs. l curves by a sharp increase in the extension of the molecule at a force Fc− ∼ 0.5 pN. It is also seen by the breaking of the (σ → −σ) symmetry in the l vs. σ curves (see Fig. 4 right). If, as suggested by these

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Γ ∆

Γc

Ed nc

n

Fig. 6. Left: The extra work performed while stretching an overwound DNA. The molecule is overwound from point A to point A+ and then stretched along the σ > 0 curve to point B + . The extra work performed while stretching is the shaded area between the σ > 0 and the σ = 0 curves. RIGHT Dependence of the torque on the twist (number of turns). In a DNA molecule as in a twisted rod the torque increases linearly with the twist angle (number of turns). If the molecule melts because of torsional yield as expected when underwound, the torque stabilizes at a value Γc as it does in a rod which undergoes a torsional buckling instability. The difference ∆ in the work of over and under-twisting is the shaded area shown here and in Figure 7 left.

results, the twisted molecule separates into a pure B-DNA phase with a critical degree of supercoiling σc− = −0.015 and denatured regions with σd ∼ −1, then every extra turn applied to the molecule should increase the fraction of dDNA by 10.5 base-pairs (bp) [11].

5.1 Twisting rigidity measured through the critical torque of denaturation The denaturation transition offers a second way to evaluate the elastic torsional persistence length C/kB T using a very simple model. It consists in measuring the difference in work done while stretching a single DNA molecule wound either positively or negatively by the same number of turns. In the following we shall use the force versus extension measurements on DNA supercoiled by ±n turns, i.e. with the same |σ|, to estimate the elastic twist rigidity, C, the critical torque at denaturation Γc and the energy of denaturation per base pair (bp), d .

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Consider the case of a DNA of crystallographic length l0 at an initial extension lA+ (< l0 ). Let us coil the molecule by n > 0 turns to state A+ , (Fig. 6), requiring a torsional energy TA+ (F < 0.2 pN) and then extend it to state B + (F = 4 pN), so as to pull out its plectonemes and eliminate its writhe. Alternatively state B + could be reached by first stretching the torsionally relaxed DNA and then twisting it. In that case its torsional energy TB + is purely twist and by energy conservation we must have: TA+ + ∆WAB + = TB + =

C (2πn)2 . 2l0

(22)

Here ∆WAB + is the extra work performed in stretching a coiled molecule from A+ to B + , the shaded area in Figure 6. For the sake of simplicity we neglect the correction to the bare torsional constant C0 due to the thermal fluctuations [26, 27]. We shall see later that this approximation (C ≈ C0 ) is justified. Consider now the case in which DNA is underwound by −n turns to state A− and then stretched to state B − . By the same reasoning as above we may write: TA− + ∆WAB − = TB − .

(23)

Since when underwound the molecule partially denatures as it is pulled from A− to B − , the torsional energy TB − will consist of twist energy and energy of denaturation. We can nevertheless estimate TB − by considering the alternative pathway for reaching B − by first stretching the molecule and then twisting it. In this case as the molecule is underwound, the torque Γ initially rises as in a twisted rod:

Γ=

C 2πn. l0

(24)

When Γ reaches a critical value Γc after −nc turns, the molecule starts to denature. Any further increase in n enlarges the denaturation region, without affecting the torque in the molecule which stabilizes at Γ = Γc . The energy of denaturation is thus simply, see Figure 6:

Ed = 2π(n − nc )Γc while TB − =

C (2πnc )2 + Ed . 2l0

(25)

Since at low force the elastic behavior of a DNA molecule is symmetric under n → −n: TA+ = TA− . Thus subtracting equation (23) from equation (22) yields:

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Fig. 7. Left: Difference in the work of stretching over and underwound DNA. ◦: DNA unwound by n = −150 turns. : DNA overwound by n = 150 turns. The solid curves are polynomial fits to the force-extension data. The bottom curve is the theoretical (worm-like chain) fit to the data obtained for this molecule at σ = 0: l0 ∼ 15.7 µm and ξp = A/kB T ∼ 48 nm (Bustamante et al. 1994). The shaded surface between the σ > 0 and σ < 0 curves represents the work difference ∆. In both cases, point A+ (respectively, A− ) is reached by overwinding (underwinding) the DNA which is initially at low extension (point A, not shown). Point B + (B − ) is reached by stretching the molecule along the appropriate σ > 0 (σ < 0) curve. Right: Plot of the square root of the work difference ∆ vs. the number of turns n the molecule is over or underwound. The straight line is a best fit through the experimental points. From its slope we extract the value C/kB T = 86 ± 10 nm and from its intercept with the n-axis, nc = 66 turns, we infer Γc = 2πnc C/l0 = 9 pN nm.

∆ ≡ ∆WAB + − ∆WAB − = TB + − TB − =

2π 2 C (n − nc )2 . l0

∆ is the measured difference between the work performed while stretching an overwound molecule and the work done while pulling on an √ underwound one, see shaded area in Figure 7 left. Plotting the value of ∆ vs. n, one obtains a straight line (see Fig. 7 right), the slope of which allows one to determine the value of the torsional constant: C/kB T = 86 ± 10 nm. The intercept of that line with the n-axis yields nc = 66 turns, from which one can estimate the critical torque Γc = 9 pN nm and denaturation energy per bp d = Ed /10.5(n − nc ) = 1.35kB T . Although the error bar on the

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measurement of C is still rather large, this method can be improved to yield a more precise value of C. It is nevertheless at least consistent with the current very imprecise estimate of C/kB T = 75 ± 25 nm. We may now estimate the validity of our approximation neglecting the correction to C0 due to the thermal fluctuations [26]. At high force these renormalize C as: 1 kB T 1 √ · = + C C0 4A AF

(26)

The last term on the right implies a correction of only 5% to the value of C at F = 4 pN, smaller than our experimental uncertainty. A similar result for the C renormalization has been obtained within the RLC model [27,28]. It is interesting to note that the value of C determined here is in good agreement with the one obtained previously (see Sect. 4.4) from the measurement of the molecular extension versus σ at constant force [27], a totally independent measurement based on the model of a Rod Like Chain polymer. 5.2 Phase coexistence in the large torsional stress regime Now let us consider that we untwist DNA far more that what was needed to reach the denaturation threshold. Our results indicate that we thus induce a large denaturation bubble and that two DNA phases, B-DNA and denatured DNA, coexist along the same molecule. Physical evidence for this coexistence can be deduced from the force versus extension curves. Applying a constant force F on the molecule, its extension is the sum of the extension of the B-DNA component (1 − α)lB (F ) and of the denatured DNA αld (F ). Here α is the proportion of denatured DNA in the molecule, and lB (F ) and ld (F ) are respectively the extension (at a given force F ) of the pure B-DNA and dDNA structures. Since in general the normalized extensions of these two components (again, at a given force) are different, the extension of the entire molecule versus the degree of twisting will display a simple linear behavior. This is precisely what we observe in the experiment: the force/extension curve evolves linearly with σ. The slope of this linear behavior is given by the linking number of both phases: σ = (1−α)σB +ασd . Assuming that the portion of B-DNA is torsionally relaxed (we neglect its sub-threshold twist value: σB = σc− ∼ −0.015) and that dDNA is topologically equivalent to two parallel strands (σd = −1), one has: ασd ≈ σ. Combining these results leads to a linear relation between the extension of the molecule and the degree of supercoiling: l(F, σ) = (1 − σ/σd ) lB (F ) + (σ/σd ) ld (F ).

(27)

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Fig. 8. Experimental evidence for the coexistence of B-DNA and denatured DNA at negative supercoiling in 10 mM PB. (A) Force (F ) vs. extension curves for a single DNA molecule obtained at different degrees of supercoiling (−1 < σ < 0) above the transition to dDNA. (B) Fraction of denatured DNA measured as a function of σ.

The force versus extension curves obtained at different values of −1 < σ < 0 can all be collapsed onto a single ld (F ) curve (independent of σ), in agreement with equation (27) (data not shown). To further corroborate the physical picture previously described we have performed two kinds of experiments:

• A biological experiment where we have hybridized single-stranded DNA fragments homologous to different regions of our DNA [31]. These experiments show that the stretched unwound DNA forms multiple denaturation bubbles in a sequence-dependent manner: denaturation begins in A+T rich regions before progressing to G+C rich regions. • A chemical experiment where we have chemically modified the bases which were not involved in Waston-Crick base pairing [10]. This was done by incubating an undertwisted and stretched DNA molecule with glyoxal, a reagent specific for unpaired bases.

Both experiments confirm the presence of denatured regions in stretched underwound DNA in the proportions described in equation (27), i.e. about −σ.

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Fig. 9. Mechanical characterization of P-DNA. A) Elasticity curves showing two sharp transitions at 3 pN and 25 pN. The first transition (not shown for all curves) is associated with the disappearance of plectonemes in B-DNA and the formation of P-DNA. The second transition, showing hysteresis, is attributed to the disappearance of plectonemes in the P-DNA sub-phase. The existence of such plectonemic structures explains the shortening of the molecule at relatively low forces 3 pN < F < 10 pN. At higher forces, these curves show that P-DNA is actually longer than B-DNA. B) Complete set of decreasing force scan curves as in A) with σi = i × 0.343. C) Rescaling of the curves in B).

6

Overtwisting DNA leads to P-DNA

6.1 Phase coexistence of B-DNA and P-DNA in the large torsional stress regime Let us now consider positive supercoiling (n > 0). The elastic behavior of a stretched, overwound DNA reveals the existence of a sharp transition at

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Fig. 10. Structure of P-DNA deduced from numerical energy minimization of a molecular model of DNA at T = 0. Space filling models of a (dG)18 .(dC)18 fragment in B-DNA (top) and P-DNA (bottom) conformations. The backbones are colored red and the bases blue (guanine) and yellow (cytosine). These models were created with the JUMNA program [32, 33], by imposing twisting constraints on helically symmetric DNA’s with regular repeating base sequences.

Fc+ ∼ 3 pN (see Fig. 9A). By analogy with negative supercoiling, we suggest that stretched, overwound DNA undergoes a phase separation between a fraction of pure B-DNA and a fraction with a new molecular structure, which we term P-DNA. As shown previously, see equation (27), the coexistence of two phases implies a linear dependence between the extension of the molecule, l(F, σ), and σ. This linearity is indeed observed up to σ ≈ +3, where the extension goes to zero for forces < 25 pN. The natural twist of the new P-DNA phase, σp , is thus +3 (Lk = 4Lk0 ), which corresponds to ∼ 2.6 base-pairs per turn. Using this value of σp in equation (27) (σp replacing σd ), we find that the experimental force versus extension curves for 0.037 < σ < 3, do indeed collapse to a single curve lp (F ), the extension at given force F of the pure P-DNA phase (see Fig. 9). Molecular modeling has been used to investigate possible structures for this highly twisted DNA using the JUMNA program [32, 33] to minimize the energy of a DNA molecule at T = 0 and under twist constraints with −6 < σ < +4. As σ increases, the phosphate backbones move to the center of the structure and the bases are expelled. The fact that extreme twisting leads to base-pair disruption can be understood by noting that the distance between successive phosphates within one phosphodiester strand cannot exceed roughly 7.5 ˚ A. For a rise of 3 ˚ A,

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the maximum length of the phosphate-phosphate (P-P) vector projected into the plane perpendicular to the helical axis is therefore roughly 6.9 ˚ A. For a helix radius of roughly 10 ˚ A (as in B-DNA), simple geometry gives the angle subtended by the projected P-P vector (that is, the maximum twist) as roughly 40◦ . To increase this angle, it is necessary to decrease the radius A, 70◦ at 6 ˚ A and, finally, 180◦ at 3.45 ˚ A) – which of the helix (50◦ at 8 ˚ implies bringing the backbones to the center of the helix, and, consequently, pushing the bases out. Note that at intermediate forces (3 pN < F < 25 pN), the extension of a strongly overwound DNA (σ → σp ∼ 3) decreases to zero (Fig. 9A). We propose that this shortening is due to the formation of plectonemic conformations of P-DNA stabilized by interactions between their exposed unpaired bases, a possibility suggested by the numerical simulations. When the stretching force, F , exceeds about 25 pN, the molecule extends by destroying these plectonemes (as discussed in [34,35]). The hysteresis observed upon increasing the force could then be due to sporadic and cooperative base unpairing in these plectonemic structures. A simple theoretical model incorporating plectonemes in P-DNA nicely fits our measurements [10]. 6.2 Chemical evidence of exposed bases To show that the bases in P-DNA are really exposed, we have followed a protocol similar to the one mentioned previously to demonstrate the existence of localized unpaired regions in underwound DNA by reacting the exposed bases with glyoxal. We indeed find that P-DNA, like denatured DNA, presents bases exposed to the solution and which therefore react with glyoxal. However the number of base pairs for the P-DNA reaction is one third that of denatured DNA. This further confirms the value of σp = 3 for the P-DNA phase whose proportion in the molecule incubated in glyoxal was as expected, (σinc − σc+ )/(σp − σc+ ) ≈ (σinc − σc+ )/3. Note that an earlier chemical detection of structural alterations within positively supercoiled DNA is described in [36]. 7

Conclusions

Using simple micromanipulation techniques, we have investigated the elastic response of a twisted and stretched DNA molecule. For small stress levels, we observe a buckling instability which leads to the formation of interwound structures which shorten the molecule. The analytical treatment of this instability in the presence of Brownian fluctuations allow an accurate description of the experimental results. In particular, the ratio between the twisting and the bending elasticity coefficients can be determined with a reasonable accuracy. The theoretical model allows us to show that the relevant stress parameter for the DNA molecule is the applied torque.

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In particular, by converting writhe into twist the stretching force simply increases the torque on the molecule. When this torque becomes too large, changes in DNA structure occur locally along the molecule. This leads to a partition between two phases such as denaturation bubbles at the entropic force-scale (F ∼ 0.5 pN) when one untwists the molecule, or hypertwisted P-DNA at higher forces (3 pN) when one overtwists it. This new P-DNA structure detected by micromanipulation was probably already observed by X-ray crystallography: the genome of the Pf1 virus (a circular, single-stranded DNA) was observed to be packaged in a structure [37] which is very similar to P-DNA. However the DNA structure of this virus was not believed to represent a canonical structure but rather a very particular one induced by its capside protein. Micromanipulation experiments that show, these proteins just lower the energy barrier between B-DNA and this hypertwisted P-DNA where the sugar-phosphate backbone is on the inside of the molecule and the base-pairs on the outside. The structure is apparently stabilized in vivo by the mechanical constraints applied by the specialized packaging proteins. The helical pitch of the structure observed by X-ray crystallography is very close to the experimentallymeasured winding of P-DNA and to the structures obtained by molecular modeling. Finally, cryo-electron microscopy has shown that the plectonemic structures which are stabilized at low forces in supercoiled DNA are important for the activity of topoisomerases, the enzymes whose role in vivo is to regulate DNA supercoiling. Topoisomerase II, responsible for resolving DNA knots and entanglement by passing one strand through another, has been seen by electron microscopy to preferentially interact with DNA crossovers. We have therefore begun to use our mechanical control of plectonemic supercoils to study in real-time the activity of topoisomerases on DNA. It is our hope that the mechanical control of DNA structure will help advance the study of DNA-protein interactions from the perspective of an underlying mechanism for protein action, i.e. deformation-induced metastable states of DNA which can facilitate complexation and play a role in specific recognition. References [1] Smith S.B., Finzi L. and Bustamante C., Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads. Science 258 (1992) 1122–1126. [2] Florin E.L., Moy V.T. and Gaub H.E., Adhesion force between individual ligandreceptor pairs. Science 264 (1994) 415–417. [3] Spudich J.A., How molecular motors work. Nature 372 (1994) 515–518. [4] Svoboda K., Schmidt C.F., Schnapp B.J. and Block S.M., Direct observation of kinesin stepping by ical trapping interferometry. Nature 365 (1993) 721–727.

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[5] Noji H., Yasuda R., Yoshida M. and Kinosita K., Direct observation of the rotation of F1 -ATPase. Nature 386 (1997) 299–302. [6] Marko J.F. and Siggia E., Stretching DNA. Macromolecules 28 (1995) 8759–8770. [7] Strick T., Allemand J.F., Bensimon D., Bensimon A. and Croquette V., The elasticity of a single supercoiled DNA molecule. Science 271 (1996) 1835–1837. [8] Cluzel P., Lebrun A., Heller C., Lavery R., Viovy J.-L., Chatenay D. and Caron F., DNA: an extensible molecule. Science 271 (1996) 792–794. [9] Smith S.B., Cui Y. and Bustamante C., Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules. Science 271 (1996) 795–799. [10] Allemand J.-F., Bensimon D., Lavery R. and Croquette V., Stretched and overwound DNA form a Pauling-like structure with exposed bases. Proc. Natl. Acad. Sci. USA (1998) (in press). [11] Watson J.D. and Crick F.H.C., Molecular structure of nucleic acids. Nature 171 (1953) 737–738. [12] Finer J.T., Simmons R.M. and Spudich J.A., Single myosin molecule mechanics: piconewton forces and nanometre steps. Nature 368, (1994) 113–119. [13] Yin H., Wang M.D., Svoboda K., Landick R., Block S. and Gelles J., Transcription against a applied force. Science 270, (1995) 1653–1657. [14] Wang M.D., Schnitzer M.J., Yin H., Landick R., Gelles J. and Block S., Force and velocity measured for single molecules of RNA polymerase. Science 282 (1998) 902–907. [15] Essevaz B.-Roulet, Bockelmann U. and Heslot F., Mechanical separation of the complementary strands of DNA. Proc. Nat. Acad. Sci 94 (1997) 11935–11940. [16] Strick T., Allemand J.-F., Bensimon D. and Croquette V., The behavior of supercoiled DNA. Biophys. J. 74 (1998) 2016–2028. [17] Bouchiat C., Wang M.D., Block S.M., Allemand J.-F. and Croquette V., Estimating the persitence length of a worm-like chain molecule from force-extension measrements. Biophys. J. 76, (1999) 409–413. [18] Baumann C., Smith S., Bloomfield V. and Bustamante C., Ionic effects on the elasticity of single DNA molecules. Proc. Natl. Acad. Sci. (USA) 94 (1997) 6185– 6190. [19] Hogan M.E. and Austin R.H., Importance of DNA stiffness in protein-DNA binding specificity. Nature 329 (1987) 263–266. [20] Wang M.D., Yin H., Landick R., Gelles J. and Block S., Stretching DNA with optical tweezers. Biophys. J. 72 (1997) 1335–1346. [21] Cizeau P. and Viovy J.L., Modeling extreme extension of DNA. Biopolymers 42 (1997) 383–385. [22] Marko J.F., Dna under high tension: Overstreching undertwisting and relaxation dynamics. Phys. Rev. E 57 (1998) 2134–2149. [23] Wilkins M.H.F., Gosling R.G. and Seeds W.E., Nucleic acid: an extensible molecule? Nature 167 (1951) 759–760. [24] Lebrun A. and Lavery R., Unusual DNA conformations. Curr. Op. Struct. Biol. 7 (1997) 348–354. [25] White J.H., Self linking and the gauss integral in higher dimensions. Am. J. Math. 91 (1969) 693–728. [25a] Marko J.F. and Siggia E.D., Fluctuations and supercoiling of DNA, Science 265 (1995) 506-508; Marko J.F. and Siggia E.D., Statistical mechanics of supercoiled DNA, Phys. Rev. E 52 (3) (1995) 2912-2938. [25b] Vologodskii A.V., Leverne S.D., Kelnin K.V., Frank-Kamenetski H. and Cozzarelli N.R., J. Mol. Biol. 227 (1992) 1224; Marko J.F. and Vologodskii A.V., Biophys. J. 73 (1997) 123.

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[26] Moroz J.D. and Nelson P., Torsional directed walks, entropic elasticity and DNA twist stiffness. Proc. Nat. Acad. Sci. 94 (1997) 14418–14422. [27] Bouchiat C. and M´ezard M., Elasticity theory of a supercoiled DNA molecules. Phys. Rev. Lett. 80 (1997) 1556–1559. [28] Bouchiat C. and M´ezard M., Elasticity rod model of supercoiled DNA molecules. Preprint, LPTENS:99/4 (1999). [28a] Hagerman P.J., Ann. Rev. Biophys. Chem. 17 (1988) 265-268. [29] Kowalski D., Natale D.A. and Eddy M.J., Stable DNA unwinding, not breathing, accounts for the nuclease hypersensitivity of A+T rich regions. Proc. Natl. Acad. Sci. (USA) 85 (1988) 9464–9468. [30] Palecek E., Local supercoil-stabilized structures. Crit. Rev. Biochem. Mol. Biol. 26 (1991) 151–226. [31] Strick T., Croquette V. and Bensimon D., Homologous pairing in streched supercoiled DNA. Proc. Nat. Acad. Sci. (USA) 95 (1998) 10579–10583. [32] Lavery R., Adv. Comput. Biol. 1 (1994) 69–145. [33] Lavery R., Zakrzewska K. and Sklenar H., Comput. Phys. Commun. 91 (1995) 135–158. [34] Marko J.F. and Siggia E., Statistical mechanics of supercoiled DNA. Phys. Rev. E 52 (1995) 2912–2938. [35] Marko J.F. and Siggia E.D., Fluctuations and supercoiling of DNA. Science 265 (1994) 506–508. [36] Mc J.Clellan A. and Lilley D., Structural alteration in alterning adenine-thymine sequences in positively supercoiled DNA. J. Mol. Biol. 219 (1991) 145–149. [37] Liu D.J. and Day L.A., Pf1 virus structure: helical coat protein and DNA with paraxial phosphates. Science 265 (1994) 671–674.

COURSE 9

INTRODUCTION TO TOPOLOGICAL QUANTUM NUMBERS

D.J. THOULESS Dept. of Physics, Box 351560, University of Washington, Seattle, WA 98195, U.S.A.

Contents Preface

769

1 Winding numbers and topological classiﬁcation 769 1.1 Precision and topological invariants . . . . . . . . . . . . . . . . . . 769 1.2 Winding numbers and line defects . . . . . . . . . . . . . . . . . . 770 1.3 Homotopy groups and defect classiﬁcation . . . . . . . . . . . . . 772 2 Superﬂuids and superconductors 2.1 Quantized vortices and ﬂux lines . . . . . . . . . . . . . . . . . . . 2.2 Detection of quantized circulation and ﬂux . . . . . . . . . . . . . 2.3 Precision of circulation and ﬂux quantization measurements . . . 3 The 3.1 3.2 3.3

775 775 781 784

Magnus force 786 Magnus force and two-ﬂuid model . . . . . . . . . . . . . . . . . . 786 Vortex moving in a neutral superﬂuid . . . . . . . . . . . . . . . . 788 Transverse force in superconductors . . . . . . . . . . . . . . . . . 792

4 Quantum Hall eﬀect 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Proportionality of current density and electric 4.3 Bloch’s theorem and the Laughlin argument . 4.4 Chern numbers . . . . . . . . . . . . . . . . . 4.5 Fractional quantum Hall eﬀect . . . . . . . . 4.6 Skyrmions . . . . . . . . . . . . . . . . . . . .

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794 794 795 796 799 803 806

5 Topological phase transitions 5.1 The vortex induced transition in superﬂuid helium ﬁlms 5.2 Two-dimensional magnetic systems . . . . . . . . . . . . 5.3 Topological order in solids . . . . . . . . . . . . . . . . . 5.4 Superconducting ﬁlms and layered materials . . . . . . .

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807 807 813 814 817

. . . ﬁeld . . . . . . . . . . . .

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phase of superﬂuid 3 He 819 6 The 6.1 Vortices in the A phase . . . . . . . . . . . . . . . . . . . . . . . . 819 6.2 Other defects and textures . . . . . . . . . . . . . . . . . . . . . . . 823 7 Liquid crystals 826 7.1 Order in liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . 826 7.2 Defects and textures . . . . . . . . . . . . . . . . . . . . . . . . . . 828

INTRODUCTION TO TOPOLOGICAL QUANTUM NUMBERS

D.J. Thouless

Preface These lecture notes were prepared rather soon after I completed my book on Topological quantum numbers in nonrelativistic physics, which was published by World Scientiﬁc Publishing Co. Pte. Ltd., Singapore, in early 1998. I have not attempted to make a completely fresh presentation, but have cannibalized the text of my book to produce something shorter, with a diﬀerent ordering of topics. I wish to thank the publishers for allowing me to do this self-plagiarization. 1

Winding numbers and topological classiﬁcation

1.1 Precision and topological invariants High precision work generally depends on two ingredients. These are reproducibility, and the reduction of a measurement to a counting procedure. A ruler is a device for comparing a length with the number of marks along the ruler, and a vernier allows interpolation between marks on the main scale also to be done by counting. A pendulum clock and its successors are devices for comparing a time interval with the number of ticks that occur in the interval. Such devices are not completely reproducible, and may vary when conditions change. The earth’s rotational and orbital motion provide time standards that can be used for calibration, but they are diﬃcult to measure with very high precision, and we know that the rotational motion is subject to random as well as to systematic changes. Cesium atoms and ammonia molecules are reproducible, and they can form the basis for length measurements in which interference fringes are counted, or as time standards by driving the system in resonanance with with a standard atomic or molecular transition and counting beats against some uncalibrated frequency. This work was supported in part by the U.S. National Science Foundation, grant number DMR–9528345. c EDP Sciences, Springer-Verlag 1999

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The measurement of g for the electron to parts in 1011 is achieved, in part, by measuring the frequency diﬀerence between the spin and twice the orbital resonance frequency in a magnetic ﬁeld. Counting can be made very precise because, although any particular counter may make mistakes, comparison between the outputs of several independent counters can reduce the error rate to an extremely low value. Over the past 30 years several devices have been developed for use in high precision work where the devices themselves are manifestly not reproducible, but nevertheless give fantastically reproducible results. Among such devices are the SQUID magnetometer, which compares magnetic ﬂux in a superconducting ring with the quantum of ﬂux h/2e for a superconductor, the Josephson voltmeter, which compares the frequency of a microwave device with the frequency 2 eV /h generated by a voltage V across a superconducting weak link, and a quantum Hall conductance standard, which compares electrical conductance with the quanta e2 /h for conductance in a quantum Hall device. In none of these cases does the fabrication of the device have to be very tightly controlled, but there are good theoretical reasons and very strong experiments to show that diﬀerent devices, even those made of diﬀerent materials, give measurements that are essentially identical. This insensitivity to details is a characteristic of topological quantum numbers that is one of the themes of my next three lectures. We are used to thinking of quantum numbers like angular momentum which are related to invariance principles, and which can be studied from the algebra of the generators of the symmetry group. Such quantum numbers are sensitive to breaking of the symmetry, and are generally not useful in environments that are poorly controlled, such as interfaces between solids. 1.2 Winding numbers and line defects The simplest type of topological quantum number that I discuss is the winding number of an angle such as the phase of a condensate wave function in a superﬂuid or superconductor. If a neutral superﬂuid with a complex scalar order parameter, such as superﬂuid 4 He, is contained in the interior of a torus, or if a superconductor is made in the form of a ring, the condensate wave function has the form Ψ(r) = |Ψ| exp(iS) .

(1.1)

Single-valuedness of the condensate wave function Ψ implies that the phase S is locally single valued, but it may change by a multiple of 2π on a closed path that goes round the hole in the middle of the ring. The winding number W =

1 2π

gradS · dr,

(1.2)

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where the path of the integral is a simple loop round the hole, is an integer. This just depends on the topology of the region containing the superﬂuid or superconductor, and on the nature of the order parameter, and is independent of detailed geometrical features such as symmetry, or of detailed material properties such as homogeneity. I have given a description of the mathematical deﬁnition of a winding number, but have not yet said why it is relevant to physics. In the case of a neutral superﬂuid equation (1.2) deﬁnes the quantum number associated with superﬂuid circulation, whose quantization was argued by Onsager [1]. This is a quantity that can be measured, and this was ﬁrst done by Vinen [2]. Measurement of the circulation of a neutral superﬂuid is diﬃcult, and there are some real problems with its deﬁnition. The winding number for a superconductor counts the number of quanta of magnetic ﬂux, and this can be measured with very high precision, thanks to the Josephson eﬀects. Winding numbers are not only of importance in nonsimply connected systems, such as the interior of a torus. In superﬂuids and superconductors the order parameter can go to zero along curves that either run across the system or form closed loops within the system. Around such line singularities there may be nonzero integer winding numbers. Quantized vortex loops were detected in superﬂuid 4 He by Rayﬁeld and Reif [3]. In Type II superconductors ﬂux lines are line singularities which carry one quantum of magnetic ﬂux. A nonzero winding number assures the topological stability of the line singularity. Consider some small cylindrical region with its axis close to part of a curve along which the order parameter goes to zero. If the winding number is nonzero around some loop on the boundary of the region, there is no continuous change of the order parameter which can be made that will remove the singularity, since the winding number, with integer values, cannot be changed continuously, and must remain nonzero. If the winding number is zero the interior of the cylinder can be ﬁlled in with a continuous order parameter which is nonzero everywhere inside, so that the line singularity has two disconnected ends. This cylinder can then be expanded in the perpendicular along the line defect in such a way that the whole defect is replaced by a continuous nonzero value of the order parameter. In 1931, Dirac [4, 5] gave an argument for the quantization of electric charge in which a similar winding number appears, essentially the integral of the magnetic vector potential round a closed circuit. This winding number appears in the phase of the wave function when you try to write a quantum theory for an electrically charged particle in the presence of a magnetic monopole. The singularitiy that the circuit encloses is not a physical singularity, but a line singularity of the vector potential in the chosen gauge. Although magnetic monopoles have not been experimentally detected, and may not exist, there is no doubt that electric charge is quantized with a

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very high precision. Already in 1925 Piccard and Kessler [6] had shown that the charge of CO2 molecules was suﬃciently low that electrons and protons must have charges whose magnitudes diﬀer by less than one part in 1020 , and later work has improved on this bound by more than an order of magnitude [7]. The quantum Hall eﬀect provides another example of a topological quantum number which corresponds to a physical variable that can be measured with high precision – the Hall conductance. In this case the simplest system on which a measurement can be made is a two-dimensional electron gas with two pairs of leads – one pair to pass the current through the system, and another pair to measure the voltage. Such a circuit has the topology of a torus with a single hole in it, as I discuss later. It turns out that the Hall conductance can be related to a Chern number, a number associated with a torus, rather than a winding number associated with a loop. This is another example of a topological quantum number which is quantized with high precision, although the precision has not been established as accurately as ﬂux quantization and charge quantization have been.

1.3 Homotopy groups and defect classiﬁcation For systems in which the line defects are characterized by winding numbers, such as superﬂuids characterized by a complex scalar order parameter, or magnets with a preferred plane of magnetization, there is an obvious way of combining defects. The winding number round a path that encloses several defects is the algebraic sum of the winding numbers of associated with each of the defects that goes through a simply connected surface that is bounded by this path. One can easily show this by continuously deforming the path until it is broken up into a sum of loops which each contain only one singularity. Conversely one can combine paths round individual defects, expanding them in a continuous way, and gluing together the separate paths to form a big path surrounding more than one defect. The algebra of this process is just the algebra of addition of signed integers. The assignment of a phase angle to each point on a path round one or more singularities can be regarded as a mapping of the loop onto a circle. Such a mapping has a single topological invariant, the winding number. The law of combination of mappings of diﬀerent loops, by the process of continuous deformation and gluing together, deﬁnes a group, the ﬁrst homotopy group π1 , which in this case is just the group Z of integers. Other types of order parameter can give diﬀerent homotopy groups. For the Heisenberg ferromagnet, in which the magnetization can be oriented in any direction, the mapping of a loop surrounding a line defect will be onto the surface of the 2-sphere corresponding to the possible directions of magnetization. The mapping of a loop onto the surface of a sphere can

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always be contracted to a single point, so the homotopy group π1 in this case is just the trivial group which has only the identity element. In the A phase of superﬂuid 3 He and in some of the liquid crystal phases the homotopy groups are ﬁnite, so there are only a ﬁnite number of diﬀerent topological states of such systems. For example, for a uniaxial nematic liquid crystal, with the order parameter speciﬁed by a director on the surface of a projective sphere (a sphere in which opposite points are equivalent) rather than by a vector on the surface of a sphere, there is a topological invariant for a system conﬁned to the interior of a torus. This invariant takes the value zero if the director passes through the equator of the projective sphere an even number of times on a path round the system, and unity if the director passes through the equator of the projective sphere an odd number of times. The homotopy group π1 in this case is just the group Z2 with two members. For biaxial nematics and for cholesterics the symmetry group of the order parameter is noncommutative, and this gives some extra complications to the theory of defects in such materials. For these systems in which the homotopy group is ﬁnite we cannot expect a measurable physical variable to be quantized in the way that circulation in 4 He and ﬂux in superconductors are quantized. The superﬂuid phases of liquid 3 He illustrate this point. Whereas circulation in the B phase is quantized in much the same way as the circulation in 4 He, there is no quantized circulation in the A phase. There is a topological quantum number, but it has only the values 0 and 1, and does not correspond in any direct way to the circulation of the ﬂuid. Such quantum numbers are useful for classifying defects, and for determining whether two apparently diﬀerent states of the system can actually be continuously changed from one to the other. I use the term topological quantum numbers regardless of whether the topological invariant actually has anything to do with quantum mechanics. I have already mentioned the examples of magnets and liquid crystals, where the order parameters have little connection with Planck’s constant. A crystalline solid is another case of a system with topological quantum numbers. There are two important order parameters in a solid, which are the position of the actual unit cell with respect to an ideal unit cell, and the orientation of the unit cell. Long range correlation of the positional order is required for the observation of the sharp Bragg peaks which are measured in an X-ray diﬀraction experiment – the Debye-Waller factor gives the reduction in magnitude of this long range order. Elasticity theory deals with the eﬀects of slow modulation of positional and orientational order. Crystal dislocations are the topological defects associated with the positional order. Orientational changes are less important in solids than positional changes, because they are costly in the elastic energy associated with the accompanying changes in positional order, but disclinations are the defects associated with orientational order.

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Topological Aspects of Low Dimensional Systems

For defect surfaces and for point defects there are other types of homotopy groups π0 and π2 that determine the topological stability of the defects. If the order parameter is singular everywhere on a surface, then it is singular at some point on any short line segment that crosses the surface. Such a singularity on a line segment can only be topologically stable if the order parameter has two disconnected values on the two sides of the surface. The homotopy group π0 is therefore nontrivial only when the order parameter has values lying in distinct regions, as it does in the Ising model of magnetism, or in the Potts model. In some of Onsager’s early discussions of the circulation of a superﬂuid [8] the idea was suggested that regions of different quantized circulation should be nested inside one another, separated by singular surfaces — vortex sheets. Such vortex sheets are topologically unstable, as the order parameter can be made continuous across the sheet in small regions, which then can be expanded until the vortex sheet has broken up into an array of vortex lines. Only when the order parameter space breaks up into distinct regions, which it does in the A phase of 3 He [10, 11], can a vortex sheet be stable. A point defect can be surrounded by a spherical surface, and the behavior of the order parameter on this sphere deﬁnes the homotopy group π2 which describes the topological properties of a point defect. If the order parameter is an angle the order parameter space is a circle. All continuous maps of a 2sphere onto a circle are trivial, and can be shrunk to a single point, so π2 for superﬂuid 4 He or for a planar magnet is trivial. For the Heisenberg model the direction of the order parameter lies on a sphere, and the mapping of one sphere onto another can be characterized by the topological invariant known to Euler

Nw

= =

1 4π

2π

π

∂(θd , φd ) ∂(θs , φs ) 0 0 π 2π ˆ ˆ 1 ˆ · ∂d × ∂d , dφs dθs d 4π 0 ∂θs ∂φs 0 dφs

dθs sin θd

(1.3)

ˆ is the direction of the order parameter and θd , φd its polar angles, where d while θs , φs are the polar angles of points on the sphere relative to the point defect. This quantity has integer values, and an example of a defect whose quantum number is +1 is shown in Figure 1. Point defects of this sort can be combined according to the rules of integer addition. A defect with quantum number −1 would be obtained by reversing the direction of the magnetization everywhere. In addition to these topologically stable line and point defects there is considerable interest in extended defects, domain walls, textures or solitons, which can also be characterized by a topological quantum number. A domain wall between two oppositely aligned domains of a ferromagnet is

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Fig. 1. Magnetization pointing outwards in the space between two spherical enclosing surfaces. This is known as a hedgehog.

an example of such a structure, and a number of other cases occur in the context of superﬂuid 3 He and liquid crystals. Such textures may be stabilized by the eﬀect of boundary conditions, although they are not topologically stable defects. For example, if the magnetization at two ends of an ideal isotropic magnetic bar are constrained to point in diﬀerent directions there will be a domain wall dividing the two diﬀerent directions of magnetization. 2

Superﬂuids and superconductors

2.1 Quantized vortices and ﬂux lines The superﬂuid component of a neutral superﬂuid is supposed to ﬂow with a velocity determined by the phase S of the condensate wave function, so that for 4 He the superﬂuid velocity is ¯ h gradS , (2.1) m4 where, if the ﬂow is incompressible, S satisﬁes the Laplace equation. The ﬂow is therefore potential ﬂow, except where there are singularities. The normal component can be envisaged as a gas of excitations moving in the medium determined by the condensate wave function. On a large scale the motion of superﬂuid helium is not really irrotational even when the density of the normal component is very small. Rotation of a beaker of superﬂuid brings the surface into much the same parabolic shape that rotation of a normal ﬂuid would produce. This led Onsager [1] to argue that the curl of the superﬂuid velocity should be concentrated into singular lines – quantized vortices. Since the phase S of the order parameter has to be single-valued modulo 2π, the circulation round one or more of these singularities must be a vs =

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Topological Aspects of Low Dimensional Systems

multiple of κ0 = h/m4 : ¯ h 2πn¯h = nκ0 , grad S · dr = vs · dr = m4 m4

(2.2)

which is just the form given in equation (1.2). Rigid body rotation with angular velocity ω is simulated by a density 2ω/κ0 per unit area of these quantized vortices. At a speed of one revolution per second this leads to a concentration of vortex lines of about 1.25 mm−2 . These vortex lines behave in many ways like the vortices of classical hydrodynamic theory, and much of the relevant theory can be found in Lamb’s book on Hydrodynamics. Vortices are carried along by the local potential ﬂow velocity, produced generally by a combination of externally imposed ﬂow and the ﬂow produced by vortex lines themselves. Vortex lines have an energy per unit length which is ρκ2 times a logarithmic factor, where ρ is the mass density of the ﬂuid and κ is the circulation round the vortex. The argument of the logarithm is the ratio of two length scales, a large length which is the size of the container or the distance between vortices, and a small distance cut-oﬀ a0 which gives the size of the vortex core, the distance at which potential ﬂow no longer occurs. Straight vortices in an inﬁnite container, or along the axis of a cylindrical container, are stable, and have normal modes of circularly polarized vibration. Circular vortex rings of radius R are also stable, and propagate in a direction perpendicular to their planes at a speed proportional to κ/R, also multiplied by a logarithm of R/a0 . In Feynman’s description the wave function ΨV for a system with a vortex line centered on the axis of cylindrical coordinates r, φ, z has the form   φj − α(rj ) Ψ0 , (2.3) ΨV ≈ exp in j

j

where Ψ0 is the ground state. The velocity is n¯h/m4 r in the azimuthal direction. The factor exp[−α(r)] is designed to reduce the density near the vortex core where the velocity is high. Similar results are obtained in the work of Pitaevskii [12] and of Gross [13], who studied a nonlinear Schr¨ odinger equation in detail. In most models the energy is lowest if the vortices carry a single quantum of circulation, so n is ±1. In general the vortex will not be straight, but will follow some curving, time-dependent path, but in principle the theories can be modiﬁed to allow for this, although the practical diﬃculties are great. The phase S of the order parameter will change by a multiple of 2π when it goes round a closed path that surrounds vortex lines. The order

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parameter goes to zero on the vortex lines themselves, where the topological singularity resides. As I have argued in Section 1, such a singularity is topologically stable. This quantization of circulation is sometimes regarded as a manifestation of a quantization of angular momentum of h ¯ per helium atom, but this is not appropriate, as the quantization of circulation is much more robust than quantization of angular momentum. Quantized vortex lines not only solve the paradox of apparent rigid body rotation in terms of a uniform density of vortex lines, but they play many other important roles. Phonons and rotons do not transfer energy directly to or from the superﬂuid, but they can scatter oﬀ the vortex lines, giving up energy and momentum to vortex waves, so that there is mutual friction between the normal ﬂuid and the vortices. This was used by Hall and Vinen [14, 15] to detect the vortex array. Vortices provide one plausible mechanism for relaxation of superﬂuid ﬂow around a ring. A direct transition between two states with diﬀerent circulation would have an incredibly small matrix element, but a unit of circulation can be lost in a continuous manner by a vortex moving from one side to the other of the ring, in the manner shown in Figure 2. The energy and the average circulation are both reduced as the vortex moves in such a direction that its own circulation enhances the ﬂow velocity ahead of it and reduces the ﬂow velocity behind it. In general there will be a barrier to the initial formation of a vortex, which must be overcome by thermal activation or by quantum tunneling, but then the vortex moves across the circulating superﬂuid, losing energy to the normal ﬂuid. This is discussed in work by Vinen [16], Langer and Fisher [17], Muirhead et al. [18], and in the book by Donnelly [19]. The other plausible method for decay of circulation is by phase slip, in which the magnitude of the order parameter temporarily goes to zero on a cross-section of the ring, and the phase slips by a multiple of 2π across that cross-section. It was suggested by Onsager [1], and later by Feynman [20], that the λ-transition from superﬂuid to normal ﬂuid might be due to the thermal nucleation of indeﬁnitely long vortices rather than to the complete destruction of the order parameter. This is not widely believed to be correct for bulk helium, because the superﬂuid transition seems to have critical exponents close to those expected for a standard planar spin model in which the order parameter goes continuously to zero at the critical point. However, a phase transition analogous to the one suggested by Onsager and Feynman occurs in helium ﬁlms, as is discussed in Section 5. Finally, vortex cores are regions of low pressure and density, and they act as sites for the trapping of impurities such as ions and 3 He atoms. Ions trapped on vortices provide some of the best tools for studying vortices. The trapping of 3 He atoms has important consequences for the energetics of vortex nucleation, since such impurities usually occur in helium unless care

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Fig. 2. A vortex moving across a persistent current. When it crosses the ring (from the outside to the inside in the case shown) it reduces the circulation round the ring by κ0 .

is taken to remove them, and the trapped impurities lower the nucleation energy for vortices. The B phase of superﬂuid 3 He is similar to superﬂuid 4 He in many ways. Like the superconductors which are discussed below, the condensate is made up of weakly bound pairs of fermions, so the transition temperature is much lower, and there is a long correlation length. The pairs are in a triplet P state, but in the B phase spin and orbital angular momenta are combined in such a way that the square modulus of the gap parameter is isotropic. One possible way of doing this is to form a J = 0 combination of the orbital and spin angular momentum, but relative rotation of spin and orbital space does not aﬀect the pairing energy, and only changes the very small hyperﬁne energy. If this variable orientation between spin and orbital axes is ignored the order parameter is essentially a complex scalar, and the superﬂuid velocity is proportional to the gradient of its phase. Circulation is therefore quantized, with a quantum of circulation h/2m3 , since the basic units are pairs of atoms of mass m3 . There are many parallels between superﬂuidity in liquid 4 He and superconductivity in metals, but there are also some important diﬀerences. The order parameter is a complex scalar in both cases, so there are analogs to

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quantization of circulation and to vortex lines. The diﬀerences come under three heads: 1. Electrons are fermions while 4 He atoms are bosons, so that the Bose condensation that occurs in superconducting metals is of very weakly bound electron pairs, rather than of the well-separated bosons of liquid 4 He. 2. Electron pairs are charged, so there is an important coupling with the electromagnetic ﬁeld. 3. Electron pairs are in intimate contact with the periodic potential formed by the lattice of positive ions, and also with the impurities in that lattice. For some other systems these diﬀerences can come in diﬀerent combinations. For example, liquid 3 He and neutron stars give fermion pairing without electric charge or background eﬀects (lattice and ﬁxed impurities). In thin helium ﬁlms the disorder of the substrate potential is important. In thin ﬁlm superconductors the background potential is important, but the electric charge is much less important. Understanding of superconductors is based on the BCS (Bardeen, Cooper and Schrieﬀer) theory of superconductivity [21], which involves a sort of Bose condensation of pairs of electrons whose binding energy is much less than the Fermi energy. In the standard BCS superconductor the pair is in a rather large (≈100 nm) singlet state, and the pairing energy is a very small fraction of the Fermi energy, typically a millivolt or less. The core of a vortex behaves in some ways like a normal metal, although there is an energy gap of order ∆2 /EF in the core, where ∆ is the energy gap of the superconductor and EF is the Fermi energy. This is very small for conventional superconductors. The electric charge is very important, and leads to a number of consequences. Firstly, currents in superconductors are easy to detect, because they produce magnetic ﬁelds, whereas supercurrents are hard to detect in superﬂuid 4 He. As a result of this, very accurate measurements can be made. It was known very early to Kammerlingh-Onnes that supercurrents in a metal ring can have a negligible decay rate, whereas neutral superﬂuid ﬂow escaped detection for twenty ﬁve years. The most important result of the coupling of the electromagnetic ﬁeld to the order parameter from the theoretical point of view is the Meissner eﬀect – magnetic ﬂux is expelled from superconductors. The Meissner eﬀect was explained by London in terms of the rigidity of the superconducting wave function. Since the current density operator is (e/m)(i¯ hgrad − 2eA),

(2.4)

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Topological Aspects of Low Dimensional Systems

a rigid wave function that does not respond to changes in the vector potential gives a current density equal to e¯ h e2 grad S + A ns , (2.5) j=− 2m m where ns is the superconducting electron density, S is the phase of the superconducting wave function, and ns /2 is its square modulus. The curl of this equation gives e 2 ns B. (2.6) curl j = − m The curl of this equation, combined with Amp`ere’s law curl B = µ0 j and the charge conservation law div j = 0, gives the London equation ∇2 j =

e2 µ0 ns j. m

(2.7)

This shows that the current density j has exponential decay over a distance

m λL = , (2.8) µ0 e2 ns the London penetration depth. The conclusion of this argument is that all supercurrents are concentrated into the surface of a superconductor, and the current density in the interior of a sample falls oﬀ exponentially with the ratio of the distance from the surface to the London penetration depth. From equation (2.6) it is clear that the ﬂux density is also zero in the interior of the superconductor, and the Meissner eﬀect is obtained. This gives a good description of a strongly Type II superconductor. For a Type I superconductor the same qualitative eﬀects occur, but there is more adjustment of the condensate wave function to the magnetic ﬁeld, so that the expression (2.8) for the penetration depth is altered. Multiplication of equation (2.5) by m/e2 ns and integration round a closed loop inside the superconductor gives h m m −n = 2 j · dR + A · dR = 2 j · dR + B · dS . (2.9) 2e e ns e ns The quantity on the right hand side of this equation is known as the ﬂuxoid, and it is the quantized quantity. Deep in the interior of the superconductor the current density is zero, apart from terms exponentially small in the ratio of the depth in the sample to the penetration depth, and so the ﬂux through a surface whose edge lies well inside the superconductor is h/2e times the winding number of the phase. In the case of low density ns of superconducting electrons, where the penetration depth is very large, the ﬁrst term on the right hand side of this equation dominates, and it reduces

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to the quantization of circulation for a neutral superﬂuid given by equation (2.2). This result, without the factor of 2 in the electron pair charge, was obtained by London [22]. This expression, like the expression for the circulation of a neutral superﬂuid, still involves the winding number of the phase of a condensate wave function. However, it results from an integral round a loop of the canonical momentum density, so that the integrand is gauge-dependent, although the integral is not. Where the integral of the current density is negligible, which it is inside the superconductor at depths greater than a small multiple of the penetration depth, except for Type II superconductors in ﬁelds greater than Bc1 , the quantized quantity is magnetic ﬂux, and this is something that can be measured far more readily than the circulation of a neutral ﬂuid. In Type I superconductors magnetic ﬂux is completely expelled for weak ﬁelds, and when it begins to penetrate the positive interface energy between the ﬂux-free superconductor and the ﬂux-carrying normal metal causes the magnetic ﬁeld to be aggregated in domains carrying many quanta of ﬂux. For a Type II superconductor the magnetic ﬁeld begins to penetrate the superconductor at ﬁelds above Bc1 , and the negative interface energy favors singly quantized ﬂux lines each carrying ﬂux h/2e. It is this mixed state of the Type II superconductor, with uniformly spaced ﬂux lines, that closely resmbles a rotating superﬂuid with a uniform array of singly quantized vortex lines. 2.2 Detection of quantized circulation and ﬂux There are three techniques that have been used to show quantized circulation and the properties of quantized vortices in superﬂuid 4 He directly. The ﬁrst was developed by Vinen [2]. In this experiment there is a straight wire under tension along the axis of a cylinder ﬁlled with liquid helium. The helium is set into rotation by initially rotating the whole system above the λ-point, cooling the helium through the transition to the superﬂuid state, and then bringing the apparatus to rest, leaving the superﬂuid circulating around the wire. The circulation is measured by using the Magnus force that the circulating superﬂuid exerts on the wire. A derivation of this is given in the book by Putterman [23]. The component of the force transverse to the direction of motion has the form, very similar to the form known from classical hydrodynamics, (2.10) FM = ρs κs (vL − vs ) × dlL + ρn κn (vL − vn ) × dlL , where ρs , ρn are the superﬂuid density and normal ﬂuid density respectively, vs , vn are the velocities with which the superﬂuid and normal ﬂuid components are ﬂowing past the wire, vL is the velocity of the wire, dlL

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Topological Aspects of Low Dimensional Systems

an element of its length, and κs , κn are the circulations of the two components around the wire. The superﬂuid circulation should have its quantized value κ0 = h/m, while normal ﬂuid viscosity should reduce the normal ﬂuid circulation to zero, so the expected form of the Magnus force would be (2.11) FM = ρs κ0 (vL − vs ) × dlL . In addition there is a dissipative force in the direction of motion if ρn is nonzero. There is, however, dispute about the correct form of the Magnus force when the normal ﬂuid density is non-negligible; a recent summary of the situation has been given by Sonin [24]. The Magnus force breaks the degeneracy of the two fundamental vibrational modes of the wire, giving a splitting of the circularly polarized modes proportional to ρs κ0 , so that a direct measure of this quantity is made. Vinen’s experiments showed a splitting that agreed with the expected value with a precision of about 3%. In one later version of this experiment, Whitmore and Zimmermann [25] worked at relatively high temperatures, where ρn is signiﬁcant, and conﬁrmed that the transverse force is proportional to ρs . Zieve et al. [26] recently repeated Vinen’s experiment, with somewhat higher precision, and used the 4 He measurements as a calibration for a similar experiment on the B phase of superﬂuid 3 He [27], for which they conﬁrmed that the quantum of circulation is indeed h/2m3 . Rayﬁeld and Reif [3, 28] used the trapping of ions on vortex rings to detect single vortex rings. The total momentum associated with a ring is h ρs ¯ gradS d3 r = ρs κ0 dA , (2.12) m4 where the double integral is over an area bounded by the vortex ring. For a circular ring of radius R and a vortex core of radius a, this gives ρs κ0 π(R − a)2 in the direction normal to the plane of the ring. The expression for the speed of the ring is analogous to the expression for the magnetic ﬁeld acting on a circular loop due to a current ﬂowing round the loop, and is κ0 8R 1 − v= ln · (2.13) 4πR a 4 The expression for the energy is analogous to the expression for the magnetic energy of a current-carrying loop, and is 1 8R 7 − E = κ20 ρs R ln + const. (2.14) 2 a 4 The equations for energy, momentum and velocity are of the form that make the area of the ring and the position of the plane of the ring conjugate variables. This was exploited by Volovik [29] in his discussion of the quantum

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tunneling of vortex rings. This Lagrangian formulation of vortex dynamics goes back more than a hundred years to Kirchhoﬀ. Rayﬁeld and Reif worked at relatively low temperatures, around 0.5 K, and found that at such temperatures ions could ﬂow without losing energy to the phonon system, apparently because they were trapped on vortex rings. The energies of the ions could be changed by known amounts by passing them through voltage drops, and the speed could be measured by a resonance technique that involved the coherent voltage pulses applied as they moved. The results show that the energy and speed are roughly reciprocals of one another, as larger energy means a larger vortex ring and so a lower speed. In terms of current–voltage relations this means that the product of current and voltage is roughly constant. The quantity obtained most directly is 8R 7 8R 1 κ 3 ρs − − ln ln , (2.15) vE = 0 8π a 4 a 4 but more careful ﬁtting gives the best value of the vortex core radius a (apparently somewhat less than the interatomic spacing at those temperatures), and so gives κ30 ρs . The precision with which κ0 is determined in these experiments is comparable with the precision of the Vinen experiments. Although it is much easier to detect magnetic ﬂux than circulating superﬂuid, direct measurements of quantized ﬂux in superconductors were not particularly easy. Two experimental measurements were published in 1961, by Deaver and Fairbank [30] and by Doll and N¨ abauer [31]. These had an accuracy of about 20%. A short time later, measurements of the ﬂuxoid, as given in equation (2.9), were made by Parks and Little [32]. Modern measurements are somewhat more precise, and a measurement of the ﬂux quantum for copper oxide superconductors by Gough et al. [33] showed that changes in ﬂux were quantized to a value of h/2e with a precision of about 4%. One reason for the poor precision is that a direct measurement of ﬂux usually depends on a detailed knowledge of the geometry of the sample, and the position of the magnetometer. Abrikosov’s [34] prediction that the magnetic ﬂux should penetrate a Type II superconductor as a regular lattice of ﬂux lines was ﬁrst veriﬁed by Cribier et al. [35] using neutron diﬀraction. Essmann and Tr¨ auble [36] developed a technique of decorating the regions of strong magnetic ﬁeld with magnetic particles to show the ﬂux lattice, which usually has dislocations and other defects, directly. These measurements were also used to compare the ﬂux density with the number of lattice points per unit area, to conﬁrm the magnitude of the ﬂux quantum in this context. Recent work on the copper oxide superconductors using neutron scattering can be found in the work of Cubitt et al. [37], and using the decoration method can be found in the work of Bishop et al. [38]. In these materials the very large anisotropy

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between directions normal to and parallel to the copper oxide planes is important, and vortices that are well localized to lattice points within a plane may wander between planes. The ﬂux lattice is extremely important for understanding the electrical resistance of Type II superconductors in the mixed state. In a nearly perfect material the ﬂux lattice would ﬂow with the electric current, just as vortices in superﬂuid 4 He are carried along by the superﬂuid ﬂow. Irregularities would then cause the vortices to lower their energy by drifting transeversely to the current direction, and such transverse motion can be shown to generate a voltage in the direction of the current, so that the motion is dissipative. Increased disorder pins the vortex lattice, so that the current can ﬂow through it without dissipation. Thus the ﬂux ﬂow resistivity is decreased by increasing disorder, and good materials for superconducting magnets are highly disordered. Use of the decoration method for detecting vortex lines in rotating superﬂuids, and took many years of work by by Packard et al. [39, 40]. The method involved trapping ions on the equilibrium (or steady state) vortices of a rotating cylinder of superﬂuid, and then ejecting the ions to get a photographic image of the positions of the vortices. The structure of neutral superﬂuid rotation is much more diﬃcult to stabilize and to display than the structure of magnetized superconductors. 2.3

Precision of circulation and ﬂux quantization measurements

Theoretical arguments suggest that the only limit to the precision with which ﬂux is quantized in a ring of superconductor below the critical ﬁeld (Bc1 in the case of a Type II superconductor) is set by the magnitude of the term j · dR on the right side of equation (2.9). Since the current density is governed by the London equation (2.7), it becomes exponentially small in the interior of a sample large compared with the penetration depth. The Josephson eﬀects [41–43] depend essentially on the quantization of ﬂux. The SQUID magnetometer gives a response which is periodic in the ﬂuxoid, which is almost equal to the ﬂux except for a small contribution from the current density in the neighborhood of the weak link itself. I am not aware of any very precise absolute calibration of a SQUID, or of precise comparison between SQUIDs made of diﬀerent materials. A less direct application of ﬂux quantization is provided by the use of the ac Josephson eﬀect to measure voltages by the relation V =

h ν 2e

(2.16)

between voltage and frequency. The connection of this with ﬂux quantization is that the emf (electromotive force) round a circuit can be written

D.J. Thouless: Introduction to Topological Quantum Numbers

as

785

E · dr = −(d/dt)

A · dr

(2.17)

in a gauge in which the scalar potential vanishes, which is the rate of change of ﬂux, as is immediately apparent from Faraday’s law. The equation gives a connection between the number of ﬂux quanta per unit time as given by Faraday’s law and the ac frequency generated. This has been used for high precision measurements. The Josephson voltage standard is the best voltage standard that there is, whose adoption led to revisions of the accepted values of fundamental constants [44,45]. Voltage balance between Josephson junctions made from diﬀerent materials have shown a relative precision of a few parts in 1017 [46–48]. In a neutral system the circulation of the superﬂuid velocity is a topological quantum number, and is therefore exact in principle. However, there is a sense in which this is tautologous, at nonzero temperatures or in thin ﬁlms, since the superﬂuid velocity is defined as the gradient of a phase angle. Physically measurable variables are the total ﬂuid density, the average mass ﬂow, and the normal ﬂuid velocity, which is set by the physical boundaries. Superﬂuid density is determined by combining these variables together, for example by equating the mass ﬂow to ρn vn + ρs vs . In the Vinen experiment it is not the circulation of the superﬂuid velocity itself which is measured, but, if equation (2.10) is accepted, it is the circulation of mass ﬂow (momentum density). Since this question is the focus of much of our current work, it is discussed separately in Section 3. Since the vortex ring experiments also depend on the energy–momentum relations of vortex rings, I think that they are dependent on the same sort of relation. Even at low temperatures in a bulk system it is not clear with what precision this type of experiment could be used to determine it even if the experimental diﬃculties could all be overcome. The results seem to depend not only on the quantized circulation, but also on rather speciﬁc details of the two-ﬂuid dynamics. It is not a fundamental problem that the superﬂuid density is needed in equation (2.11), since this can be measured independently, and, for superﬂuid 4 He, it rapidly approaches the mass density at low temperatures. A more serious problem is that the frequency shift is proportional to the ratio of the superﬂuid density to the eﬀective mass per unit length of the wire, and this eﬀective mass includes the hydrodynamic mass of the surrounding ﬂuid. This hydrodynamic mass is not simply the mass of ﬂuid displaced by the wire, as it is for an ideal incompressible classical ﬂuid, but there are uncertain corrections due to boundary layer eﬀects, and, as has been pointed out by Duan [49] and Demircan et al. [50], there is a correction due to the compressibility of the ﬂuid that is logarithmically divergent for low frequencies and large systems. Since helium has a much

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lower density than the wire this is not a major correction, but it is one factor that makes it diﬃcult for the experiment to be very precise. Another possible source of imprecision is due to the eﬀects of the boundary on the ﬂow in the interior. Under ideal geometrical conditions, with a very small wire at the axis of a cylinder of radius R, displacement of the wire by a small distance a from the axis of the cylinder produces an eﬀect that can be represented by an image of the vortex at a distance R2 /a, so there is a backﬂow of magnitude κ0 a/2πR2 , which leads to a correction of the Magnus force on the wire whose relative magnitude is hT /m4 R2 , where T is the period of oscillation of the wire. For the conditions of Vinen’s experiment this gives a correction of a few parts in 105 . Such eﬀects are due to the fact that the controling equation for S is the Laplace equation in a neutral system, whereas for a superconductor long range eﬀects are exponentially reduced for distances larger than the penetration depth. 3

The Magnus force

3.1 Magnus force and two-ﬂuid model This section is largely based on work that my collaborators and I have done in recent years. There is a brief review of this work contained in a paper written by ﬁve of us in the autumn of 1997 [51]. Before discussing the details of the theory I want to give a brief review of the theory of the Magnus force in classical hydrodynamics, and then discuss the modiﬁcations which may be needed as a result of the two-ﬂuid picture of superﬂuidity. It is an old paradox of classical hydrodynamics that potential ﬂow around an object gives no drag in the direction of the ﬂuid ﬂow and no lift perpendicular to it. Drag is provided by the eﬀects of viscosity and by the creation of turbulence, and is very complicated, but lift is produced by the interaction of circulation of the ﬂuid round the object with its motion, and has, to lowest order, a very simple form. A partial explanation is given in many textbooks of elementary physics, and is applied to problems like the lift on the wing of an airplane or the curved trajectory of a spinning ball. The usual explanation is given in terms of the diﬀerent Bernoulli pressures on the two sides of the object. Actually a rather more detailed explanation is needed, and a detailed explanation shows that the result is very general, and quite independent of details of the ﬂuid such as whether it is compressible or incompressible. Consider a cylinder, perhaps a solid cylinder, or perhaps the hollow core of a vortex, with circulation κ around it, held in a ﬁxed position with ﬂuid ﬂowing past it with asymptotic velocity v0 in the x direction. At a large distance R from the cylinder the components of velocity will be (v0 − κ sin θ/2πR, κ cos θ/2πR, 0). This gives a Bernoulli pressure which is

D.J. Thouless: Introduction to Topological Quantum Numbers

approximately

−ρv02 /2 + ρv0 κ sin θ/2πR ,

787

(3.1)

which gives a net force per unit length −ρv0 κ/2 in the y direction, acting on the ﬂuid inside a cylinder of radius R. The total force acting on the cylindrical volume of ﬂuid is this Bernoulli force plus the force which must be applied to keep the cylinder or vortex stationary, and this total force must equal the rate of change of momentum of the ﬂuid which is instantaneously in the cylinder of radius R. The downward momentum density ρκ cos θ/2πR on the left side of the cylinder is replaced by upward momentum density on the right side as the cylinder of ﬂuid moves from left to right with speed v0 , so the rate of change of momentum per unit length is 1 dP = L dt

0

2π

ρκv0 cos2 θ dθ = ρκv0 /2 . 2πR

(3.2)

Comparison of this with the Bernoulli force shows that an additional force ρκv0 must be applied in the y direction, or, alternatively, that the moving ﬂuid exerts force −ρκv0 on the vortex. The Galilean invariant form of this is ˆ × (vV − v0 ) . (3.3) FM = ρκk The argument depends only on the asymptotic properties of the ﬂow, and on momentum conservation. Despite its generality, this argument cannot be directly taken over to the case of a superﬂuid. A superﬂuid is described, both hydrodynamically and thermodynamically, by the two-ﬂuid picture of Landau and Tisza. The phase of the condensate wave function determines the superﬂuid velocity vs through equation (2.1). At nonzero temperature there will be excitations from the condensate, phonons with a linear energy–momentum relation, and, in the case of 4 He, rotons with a nonzero wavenumber of the order of the reciprocal of the interatomic spacing, and an energy around 8 K. These excitations interact with one another to form a local equilibrium, and all the entropy of the system is concentrated in this normal ﬂuid component. The spectrum is determined by the local value of vs , but the average velocity is not, but is determined by the boundary conditions. In particular, there can be an equilibrium state in which the normal ﬂuid velocity vn is zero because the boundaries are static, even when the superﬂuid velocity is nonzero. This was actually the situation in the experiment of Vinen [2] described in Section 2.2, where, in equilibrium (before the wire was made to vibrate) the normal ﬂuid was at rest, but the superﬂuid was circulating around the wire down the axis of the cylinder. The velocities vs and vn are essentially deduced from the boundary conditions rather than being directly measured. Since vs is deﬁned as the gradient of a phase it does not make much sense to ask if its integral round

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a closed loop deviates slightly from an integral multiple of h/m4 . Quantities that can be directly measured are the total ﬂuid density ρ = ρs + ρn ,

(3.4)

and the total momentum density (mass ﬂow) p = ρs vs + ρn vn ,

(3.5)

and quantities such as the free energy density F = F0 + ρs vs2 /2 + ρn vn2 /2 ,

(3.6)

and the entropy ﬂow. From measurements of such quantities the variation of ρs and ρn with temperature and pressure can be deduced, and then the equation of state can be used to analyse other measurements. The essential feature of two-ﬂuid hydrodynamics that should be used to generalize equation (3.3) to the superﬂuid case is that the two ﬂuids coexist without interfering with one another, as is shown by equation (3.6). Therefore if the superﬂuid and normal ﬂuid circulations round a vortex are κs and κn , while the asymptotic superﬂuid and normal ﬂuid velocities are vs and vn , the two components will contribute independently to the transverse force, and give the result quoted from Putterman’s book [23] in equation (2.10). Furthermore, the theory suggests that the superﬂuid circulation κs should be quantized, and the normal ﬂuid circulation κn should not be stable, but should eventually be dissipated by the normal ﬂuid viscosity. 3.2 Vortex moving in a neutral superﬂuid The Magnus force itself provides some interesting connections between quantized variables. In classical mechanics such a nondissipative force linear in the velocity can be represented by a term in the Lagrangian which is linear both in the velocity and in the displacement. There is a lot of ambiguity in the deﬁnition of such a Lagrangian, since any total derivative of the form r˙ · ∇f (r, t) + ∂f (r, t)/∂t can be added to it, but there is no ambiguity in the action round a closed path. This ambiguity is very similar to that introduced by a choice of gauge in electrodynamics. In quantum mechanics such a term in the Lagrangian translates into a Berry phase [52], a phase that depends on the path of the system but not on the speed with which the path is traversed – again, this phase depends on a choice of gauge, but the phase associated with a closed path is gauge independent. It was observed by Haldane and Wu [53] that the Berry phase associated with a vortex in a two-dimensional superﬂuid is an integer multiple of 2π when the vortex is taken on a closed path that surrounds an integer number of atoms.

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This can be seen from equation (2.11), which gives a force derivable from a Lagrangian of the form L=

1 ρs κ0 (vs − vL ) × rL , 2

(3.7)

where ρs is now the superﬂuid density per unit area. Integration of this round a closed loop in the superﬂuid (where I take for simplicity vs = 0) gives an action equal to ρs κ0 times the area of the loop, or, using κ0 = h/m4 , h times the number of atoms enclosed, rescaled by the factor ρs /ρ. Since the Berry phase is the action divided by h ¯ this gives a Berry phase equal to 2π times the number of atoms enclosed. There is a similar relation between the Lorentz force on an electron in a magnetic ﬁeld and the magnetic ﬂux quantum h/e. The Berry phase associated with taking an electron on a closed path is equal to 2π times the number of ﬂux quanta which the path surrounds. These connections between diﬀerent quantum numbers, circulation and number of atoms in one case, electric charge and magnetic ﬂux in the other, have led my collaborators, Ao and Niu, and me to look more closely into the question [54]. The two-dimensional result of Haldane and Wu has an obvious generalization to three dimensions, where the vortex is deﬁned not just by a point in two-dimensional space, but as a curve in three-dimensional space. When such a curve is moved around and then returned to its original position it sweeps out a two-dimensional surface, and the Berry phase should be equal to 2π times the number of atoms surrounded by this surface. This simple statement hides a number of diﬃculties that we have tried to address. We think we understand what is meant by the path of an electron, but what is meant by the path of a vortex, an object whose microscopic deﬁnition is obscure? The number of atoms inside a geometrical surface is not ﬁxed, but is a quantity subject to zero-point as well as thermal ﬂuctuations, so what number should be used in this context? The density of the superﬂuid is reduced at the vortex core by the Bernoulli pressure, so does this reduction in density reduce the Berry phase? In a more recent paper, Thouless et al. [55] have tried to sharpen some of the questions by considering the eﬀect of pinning the vortex core to a certain curve x0 (z, t), y0 (z, t) by centering a short-ranged potential (repulsive to the atoms) on this curve – the reduced density at the core should cause the core to be attracted to the curve. This has enabled us to study the dynamics of the vortex cleanly by studying the eﬀects of moving the pinning potential. To determine the coeﬃcient of vV , we consider an inﬁnite system with superﬂuid and normal ﬂuid asymptotically at rest (vn = 0 = vs ) in the presence of a single vortex which is constrained to move by moving the pinning potential. For simplicity we describe the two-dimensional problem of a vortex in a superﬂuid ﬁlm, but the three-dimensional generalization is straightforward. Also we restrict this discussion to the ground

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Topological Aspects of Low Dimensional Systems

state of the vortex, but the generalization to a thermal equilibrium state is straightforward. The reaction force on the pinning potential is calculated to lowest order in the vortex velocity vV . This can be studied as a timedependent perturbation problem, but this can be transformed into a steady state problem, with the perturbation due to motion of the vortex written as ivV · grad0 . The force in the y direction on a vortex moving with speed vV in the x direction can then be written as

∂V ∂

P

Fy = ivV Ψr0 (3.8)

Ψr + comp. conj.,

∂x0 E0 − H ∂y0 0 where P projects oﬀ the ground state of the vortex. Since ∂V /∂x0 is the commutator of H with the partial derivative ∂/∂x0 , the denominator cancels with the H in the denominator, and so the expression is equal to the Berry phase form ∂Ψ ∂Ψ ∂Ψ ∂Ψ Fy = −ivV | | + ivV · (3.9) ∂x0 ∂y0 ∂y0 ∂x0 Since the Hamiltonian consists of kinetic energy, a translation invariant interaction between the particles of the system, and the interaction with the pinning center, which depends on the diﬀerence between the pinning center coordinates and the particle coordinates, the derivatives ∂/∂x 0 , ∂/∂y0 , can be replaced by the total particle momentum operators − ∂/∂xj , − ∂/∂yj . This gives the force as a commutator of components, Px , Py of the total momentum,

Fy = −ivV Ψr0 [Px , Py ] Ψr0 · (3.10) At ﬁrst sight one might think that the two diﬀerent components of momentum commute, but this depends on boundary conditions, since the momentum operators are diﬀerential operators. Actually this expression is the integral of a curl, and can be evaluated by Stokes’ theorem to get

gradj Ψr0 · dr = js · dr, (3.11) Fy = vV Ψr0 −i

j

where the integral is taken over a loop at a large distance from the vortex core. This gives the force in terms of the circulation of momentum density (mass current density) at large distances from the vortex. There is actually a striking resemblance between the expression (3.10) for the coeﬃcient of the Magnus force and the expression for the Hall conductance in terms of a Chern number which is discussed in Section 4.4. Our result, that the transverse force is equal to vV times the line integral of the mass current, is independent of the nature or size of the pinning

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potential. The general form of this is Ft = ρs Ks × vV + ρn Kn × vV ,

(3.12)

where Kn represents the normal ﬂuid circulation, in agreement with the vV -dependent part of equation (2.10). Since our result is controversial, Tang and I have checked the method that we use by using it also to calculate the dissipative part of the force, the component in the direction of motion [56]. Although we are not able to evaluate this in the general case, we ﬁnd that for symmetrical potentials acting on noninteracting particles the method gives the usual expression for the longitudinal force on a moving potential in terms of the transport cross section. By using this technique we verify part of equation (2.10) without using any speciﬁc model of a vortex, independently of whether its core is a solid cylinder or the consequence of a mathematical singularity in the pure ﬂuid. We ﬁnd that the coeﬃcient of κ0 vV is indeed ρs , the asymptotic value of the superﬂuid density at large distances from the vortex core. Only the coherent part of the wave function contributes to the Magnus force, so at nonzero temperatures the Berry phase is reduced by a factor ρs /ρ. The other half of equation (2.11), which gives the coeﬃcient of κ0 vs , has been veriﬁed in recent work by Wexler [57]. Wexler considers superﬂuid contained in a ring, scuh as the one shown in Figure 2 with n quanta of circulation trapped in the ring, so that the superﬂuid velocity is vs = hn/m4 L, where L is the perimeter of the ring. If an additional vortex is greated on the outer edge of the ring, dragged slowly across the ring by a pinning force, and annihilated on the inner edge, the number of trapped quanta of cirulation is increased to n + 1, changing the superﬂuid velocity by h/m4 L. This leads to an increase in the free energy of the superﬂuid circulating aroung the ring by hvs LA = ρs κ0 vs A , ∆F = ρs (3.13) m4 L where A is the area of cross section of the ring, since the superﬂuid density ρs is deﬁned in terms of the dependence of the free energy on superﬂuid velocity. The work done against the transverse force in moving a vortex across the ring is equal to the force per unit length times the area of cross-section A. Comparison of this with equation (3.13) shows that the transverse force per unit length acting on a nearly stationary vortex when superﬂuid ﬂows past it with velocity vs has magnitude ρs κ0 vs , in agreement with equation (2.10). This argument has considerable analogies with the argument given by Laughlin for the integer quantum Hall eﬀect [58], which is discussed in Section 4.3. In that argument the ﬂux through a ring is changed, and electrons are moved from one edge of the ring to the other. Comparison is

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Topological Aspects of Low Dimensional Systems

made between the work done by the transverse force on the ﬂux produced by the electric current and the energy change of the electrons moving from one edge to the other. The surprising feature of these results is that the normal ﬂuid component does not seem to enter into the expression for the transverse force on a vortex. If the coeﬃcients of κs vV and κs vs are respectively ±ρs , Galilean invariance shows that the coeﬃcient of κs vn for the transverse force must be zero. Volovik [59] argues that we are wrong, and the coeﬃcient of κs vV should be ρ rather than ρs , which then gives the coeﬃcient of κs vn as −ρn , again from Galilean invariance. Our result is in contradiction with widely accepted results, going back to the work of Iordanskii [60], which show a transverse force, proportional to the normal ﬂuid velocity, due to the scattering of phonons or rotons from a vortex. Sonin [24] has given a recent survey of this argument. The arguments developed by us suggest that the integral of momentum density round a closed loop is actually the topological quantum number that can be measured, rather than the integral of superﬂuid velocity. The superﬂuid density, the factor by which these two quantities diﬀer, is, however, a quantity that can itself vary, not only with temperature, but also with velocity, so the quantization of this quantity is not very precise. 3.3 Transverse force in superconductors The problem of how to generalize to a superconductor our arguments about the transverse force on a moving vortex in a neutral superﬂuid raised some diﬃculties. Translation invariance plays an essential part in the result of Thouless, Ao and Niu [55], and there are various features of superconductivity that make translation invariance diﬃcult to apply. The regular lattice of ions even in a perfect metal has only discrete translation invariance, not the continuous translation invariance that is needed for this argument. The impurities that exist in any real metal, and are essential for getting a ﬁnite conductivity in the normal state, destroy even the discrete translation invariance. Finally, magnetic eﬀects, which are essential for understanding properties of superconductivity such as the Meissner eﬀect, are usually put in by means of a vector potential, and a choice of gauge for this hides the fundamental translation invariance, even if it does not destroy it. We generalized the argument for an unrealistic model in which impurities and disorder are ignored, and in which the background of a regular array of positive ions is replaced by a uniform positive background, and some short-ranged pairing interaction between the electrons is put in to give superconductivity at low temperatures [61]. The magnetic ﬁeld is not put in explicitly, but, in addition to the Coulomb interaction between the electrons, and between the electrons and the background, the current–

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current interaction is included in the form Hmag

(rn − rn )i (rn − rn )j e2 µ0 i δ ij + =− pn ) pjn . (3.14) 3 16πm2 |r − r | |r − r | n n n n n=n

This gives a complete account of electromagnetic eﬀects, apart from the spin-orbit interaction and the relativistic mass correction, up to second order in v 2 /c2 . It is interesting to note that this approach was taken by Dirac et al. [62] in responding to criticism by Eddington of the relativistic theory of the hydrogen atom – of course they also included spin-orbit eﬀects and the second order mass correction, since it was the Dirac equation which was in question. We can ignore the mass correction and spin-orbit interaction because they are not important for superconductivity theory. We have shown that such an approach to superconductivity theory, without explicit introduction of magnetic ﬁelds produced by the electrons, does lead to the Meissner eﬀect and to magnetic screening with the usual penetration depth. Just as we do for the neutral superﬂuid, we introduce a pinning potential to control the position of the ﬂux line. This is the only part of the Hamiltonian that breaks translation symmetry. The argument for the force on the pinning potential goes through as before, and we get the same result that the transverse force on a moving vortex is equal to the vortex velocity times the asymptotic value of the circulation of momentum density. The magnitude of this transverse force for a vortex moving in an idealized superconductor is not surprising. It is equal and opoposite to the transverse Lorentz force which is obtained when a supercurrent ﬂows past a stationary vortex. The value of the Lorentz force due to a supercurrent can be derived by an adaptation of Wexler’s argument for the force due to ﬂow past a vortex in a neutral superﬂuid [57], but there has been no serious doubt of this result. In combination these results tell us that in the absence of other forces, such as pinning and frictional forces, the vortices will ﬂow with the average velocity of the supercurrent. The form of the result is surprising, since at large distances the circulation of canonical momentum density does not correspond to any current density, and there is no Bernoulli pressure imbalance or net momentum ﬂux which could be used to explain this force, in the way that the classical Magnus force is explained. The explanation was given thirty years earlier in a paper by Nozi`eres and Vinen [63]. At distances small compared with the penetration depth the moving magnetic ﬂux line does behave like a vortex, and the forces are mostly hydrodynamic. The magnetic ﬂux that moves with the line does, however, generate an electric ﬁeld, a dipolar ﬁeld, which exerts a net force on the rigid positive substrate. Therefore this transverse force, which was hydrodynamic close to the vortex core, is transmitted to large distances as an elastic stress in the positive background.

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In fact the arguments and results of Nozi`eres and Vinen are very close to ours. Like us they assumed that the positive background was uniform. We have not had to assume, as they had to, that the superconductor is very strongly Type II. 4

Quantum Hall eﬀect

4.1 Introduction The quantum Hall eﬀect was discovered in 1980 by von Klitzing et al. [64]. Systems which display the quantum Hall eﬀect are generally twodimensional electron systems such as are found at the interface between silicon and silicon dioxide in an MOS (metal-oxide-semiconductor) device, or at the interface of a heterojunction such as the GaAs-Alx Ga1−x As system. A strong magnetic ﬁeld (5 T or more) and low temperatures (helium temperatures or less) are needed to observe the eﬀect. One of the striking things about this initial report is that quantization of the Hall conductance I/VH was observed to be an integer multiple of e2 /h with very high precision, better than one part in 105 . Later measurements have shown an absolute precision of 1 part in 107 [65]. A comparison in which the Hall voltage generated in a silicon MOS device against the Hall voltage generated by the same current in a GaAs-Alx Ga1−x As device has shown consistency between diﬀerent devices of a few parts in 1010 [66]. The quantum Hall eﬀect is suﬃciently reproducible that it provides the best available secondary standard of electrical resistance, and its value is included in the adjustments of fundamental constants [67]. This precision suggests that a topological explanation of the quantum Hall eﬀect is appropriate. The earliest theoretical interpretations of the observed quantization showed that the plateaus in the Hall conductivity came from ﬁlled Landau levels, and that the Fermi level was pinned between Landau levels by localized states produced by the disorder of the substrate; these localized states make no contribution to the low-temperature conductivity. The quantized value is unaltered by disorder and interactions to all orders in perturbation theory [68–70]. These arguments, although sound, do not connect the quantum Hall eﬀect with other phenomena that have a very high precision, but Laughlin [58] gave an argument which is much more general, and which revealed a topological basis for the integer quantization. Later work [71–75] has interpreted the topological aspects of the eﬀect diﬀerently, but Laughlin’s argument remains one of the most powerful ways of understanding the quantum Hall eﬀect. As soon as the reasons for integer quantization of the Hall conductance were understood clearly, experiments by Tsui et al. [76] showed that the Hall conductance could be a fractional multiple of e2 /h. The initial work was not very precise, but later the fractional values were shown very clearly as

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plateaus in the Hall conductance, whose precision was less well determined than that of the integer eﬀect only because lower temperatures and higher magnetic ﬁelds were needed. 4.2 Proportionality of current density and electric ﬁeld The proportionality of electric ﬁeld and electric current density perpendicular to the ﬁeld for a two-dimensional electron system can be explained in a simple and straightforward manner, and should have led to a prediction of the quantum Hall eﬀect before it was discovered experimentally. For a system of noninteracting electrons in the absence of a substrate potential the Hamiltonian can be written as 2 2 1 ∂ ∂ 1 + eAx + + eAy + V (x, y) −i¯ h −i¯ h H = 2m ∂x 2m ∂y = −

¯ 2 ∂2 h 1 (¯ hky + eBx)2 + eEx , + 2m ∂x2 2m

(4.1)

where the Landau gauge Ax = 0, Ay = Bx has been used, the electric ﬁeld is E in the x-direction, and the y-dependence of the wave function is taken to hky + be exp(iky y). The y-component of the current density operator is e(¯ eBx), and so this Hamiltonian leads to a current density corresponding to an electron drift velocity −E/B in the y-direction, just as is found in the classical theory of charged particle motion. For n full Landau levels the electron density is nBe/h, n times the density of single electron ﬂux quanta, so the current density is jy = (nBe/h)(eE/B) = (ne2 /h)E .

(4.2)

This gives the right result, but is not yet an adequate explanation of the observed quantization of the Hall conductivity σxy , because it does not give plateaus as the magnetic ﬁeld is varied for ﬁxed electron density, since there is a partially ﬁlled Landau level for general values of the magnetic ﬁeld B. Under some conditions, such as in silicon MOS devices, it is the chemical potential rather than the electron density that is controlled, but in other systems, such as optically excited GaAs systems, the recombination time is very long, and the electron number is kept constant. A reﬁnement of this argument [68–70] shows that weak disorder makes no change in this result, to all orders in perturbation theory. The disorder produces localized states, lying between the mobile states associated with each Landau level, and these localized states serve to pin the Fermi energy between Landau levels without changing the Hall current associated with the mobile states. The Kubo formula [77, 78], which relates the conductivity of a material to its current-current correlation function, can be used to display this result

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in a form in which its insensitivity to perturbations is made manifest. For the longitudinal conductivity in the dc limit the Kubo formula involves a delta-function on the Fermi surface, and is rather sensitive to the order of limits, but the result for the Hall conductivity is less sensitive. In a manyelectron state |Ψ0 of energy E in which no current ﬂows, the electron ˆ is, to lowest current in the y-direction induced by the perturbation eE X order in perturbation theory, EeΨ0 |Jˆy

P ˆ E−H

ˆ 0 + EeΨ0 |X ˆ X|Ψ

P ˆ E−H

Jˆy |Ψ0 ,

(4.3)

ˆ Since the where P is the operator that projects oﬀ the eigenstate |Ψ0 of H. ˆ X] ˆ = −i¯hJˆx /e, current operator Jˆx is given by the commutation relation [H, this gives the current in the form P P ˆ ˆ ˆ ˆ Jy = −i¯ J − Jx J |Ψ0 hEΨ0 | Jy ˆ 2 x ˆ 2 y (E − H) (E − H) 1 hE ¯ 1 ˆ 1 ˆ J J , (4.4) = dzTr ˆ yz −H ˆ xz −H ˆ 2π z−H where the integral over the complex variable z goes round only the lowest ˆ the ground state and neighborof the eigenvalues of the Hamiltonian H, ing states that involve localized excitations from the ground state. The integrand is closely related to a Green function at energy z, and such a Green function is exponentially localized both when the imaginary part of z is nonzero, and when z lies in a gap in the spectrum or in a region of the spectrum where eigenstates are localized by disorder [79] (in a mobility gap). This argument, or a simple modiﬁcation of it, shows that the result is unchanged by local perturbations that are not strong enough to push extended states through the Fermi surface, and that the current density is a local function of the applied ﬁeld when the Fermi energy lies in a mobility gap. 4.3 Bloch’s theorem and the Laughlin argument In the early 1930s Bloch proved a result that he claimed showed that all existing theories of superconductivity were wrong – this was just before the Meissner eﬀect was discovered. Bloch’s theorem states that the free energy F of the equilibrium state for a loop or other nonsimply connected piece of conductor is a periodic function of the ﬂux Φ enclosed by the loop, with period h/e, so that the current ∂F/∂Φ is periodic and has zero average. This result was widely known, but was never published by Bloch, and one of the best discussions was given by Bohm in 1949 [80]. The oscillations about zero of the equilibrium current are generally expected to be small,

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but the example of the SQUID magnetometer shows that such oscillatory equilibrium currents can be relatively large under certain circumstances. Laughlin’s argument for the quantization of Hall conductance is compact and powerful, but perhaps too compact to be readily understood. A more transparent version of Laughlin’s argument was given by Halperin [81]. I have rephrased Laughlin’s argument as a generalization of Bohm’s version of the Bloch theorem [82]. I consider an annulus of two-dimensional conductor in a uniform magnetic ﬁeld, with a strength such that the Fermi energy lies in a mobility gap. The Bloch theorem tells us that in equilibrium the only current ﬂowing is the mesoscopic current which averages to zero when the ﬂux threading the annulus is varied. In Laughlin’s argument the electrochemical potentials at the two edges of the annulus are allowed to have slightly diﬀerent values, µi on the inner edge and µo on the outer edge. Because there are no mobile electron states with energies close to the range between µi and µo , this nonequilibrium state can be maintained. I suppose that the electrochemical potentials are deﬁned by keeping reservoirs of electrons with Fermi energies µi and µo in contact with the two edges. If the ﬂux Φ threading the annulus is now changed by one quantum unit δΦ = h/e the annulus returns to its former state, apart from trivial gauge changes of the electron states by the factor exp −i(e/¯ h) δA(r) · dr · (4.5) Such a gauge change is allowed even for electron states that extend round the annulus, since the wavefunction remains single valued. Since the annulus has returned to its original state, apart from this gauge change, the only signiﬁcant thing that could have happened is that an integer number n of electrons might have passed across the system from the inner reservoir to the outer. The change in free energy of the system is therefore n(µo − µi ). The work done is the time integral of the current times the voltage around the annulus, and this voltage is dΦ/dt, by Faraday’s law, so that the equality of these two gives dΦ dt = J dΦ = n(µo − µi ) . (4.6) J dt ¯ The left hand side is J(h/e), where J¯ is the current round the ring, averaged over the fractional part of the ﬂux, and the right side is neV , where V = (µo − µi )/e is the voltage between the two edges, so the conclusion of Laughlin’s argument is that 2

e J¯ = n V , h and the conductance is quantized as an integer multiple of e2 /h.

(4.7)

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Topological Aspects of Low Dimensional Systems

At ﬁrst sight it is surprising that electrons can be moved across the system in the absence of any mobile states close to the Fermi energy, but a more detailed explanation is given in Halperin’s paper [81]. Each occupied Landau level has edge states that are close to the Fermi energy, and there is a quasicontinuum of mobile states between the two edges, with the states in the interior well below the Fermi energy. The eﬀect of changing the vector potential is to make an adiabatic change that maps this continuum of states into one another in such a way that n states on the inner edge are emptied and n states on the outer edge are ﬁlled. For example, if we take the case of an ideal annulus and use the symmetric gauge centered on the center of the annulus, the states of the lowest Landau level can be written in the form −eΦ/h s 2 eB (x + iy) exp −|x + iy| , (4.8) ψ(x, y) = |x + iy| 4¯h where s is an integer. This wave function is concentrated around a circle of radius rs , where Bπrs2 = −Φ + sh/e ; (4.9) this is a good wave function provided the value of rs lies between the inner and outer radii of the annulus. Decrease of Φ by h/e pushes each wave function out to the former position of the next one, and so moves one electron from the inner edge to the outer edge. A similar argument holds for higher occupied Landau levels. This argument is very general, and all it seems to require is that the Fermi energy lies in a mobility gap. It does not, for example, require that the voltage diﬀerence between the two edges should be vanishingly small. All that is required is that the voltage diﬀerence should be small enough that there is no appreciable tunneling of electrons between the two edges or to higher unoccupied levels. The only unsatisfactory feature of the argument is that it gives only the current J¯ averaged over the ﬂux in the annulus, whereas the actual current could include a mesoscopic contribution. Such a mesoscopic contribution could only come from the edge states, since the bulk Green functions are exponentially localized at the Fermi energy. In the edges of typical devices used for studying the quantum Hall eﬀect there are many levels contributing to edge currents and to diamagnetic susceptibility, and there is no reason to expect strong interference eﬀects that could give rise to a signiﬁcant mesoscopic correction to equation (4.7). Because one does not expect corrections to be propagated over long distances, one also expects that the details of the geometry should not aﬀect the current-voltage relation. The same relation that Laughlin showed for the annulus should also hold for a more typical arrangement where current is fed into a Hall bar at one end and removed from it at the other. An important aspect of this argument is that it suggests sources for departures from exact quantization, as well as the order of magnitudes

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to be expected for them. Firstly, there are processes that lead to currents between the edges as well as round the annulus, and these can alter the estimates of free energy changes used in equation (4.6). These longitudinal currents can be produced by thermally activated hopping of localized electrons, thermal activation of electrons in unﬁlled Landau levels or of holes in ﬁlled Landau levels, or by quantum tunneling. The last of these will fall exponentially as the width of the annulus increases, and the others fall exponentially as the temperature is lowered, so these eﬀects can be made exponentially small by increasing the width of the system and reducing the temperature. The other obvious source of deviations from exact quantization is that the actual current may deviate from its value J¯ averaged over a ﬂux period. There is no reason to expect such deviations to be large, as they are in a Josephson junction, and the simplest picture of them is that they come about as the closely spaced edge levels move across the Fermi energy under the inﬂuence of changing values of the ﬂux Φ. The energy spacing between such edge levels should be inversely proportional to the circumferences of the edges, and might give rise to corrections inversely proportional to the circumference at zero temperature, but I would expect thermal broadening to reduce these corrections to something exponential once the temperature exceeds the mean energy spacing between edge states. This argument contains the essential features as the more explicitly topological argument which is presented in the next subsection. It uses the gauge covariance of quantum mechanics in regions that are not simply connected, together with the quantization of electron charge in the reservoirs. It is clearly invariant under small perturbations, so long as they do not bring the energies of mobile electrons in the interior close to the electrochemical potential range. It also has some features similar to the argument for the force on a vortex in a supercurrent which was presented in Section 3.2 [57]. A charge carrier in the theory of the quantum Hall eﬀect plays much the same role as a vortex in a superﬂuid.

4.4 Chern numbers The ﬁrst expression for the Hall conductance which gave explicitly a topological invariant was obtained for the case of electrons moving simultaneously in a uniform magnetic ﬁeld and a periodic potential [71]. This is a problem for which very interesting results had previously been obtained by Azbel [83] and Hofstadter [84]. A weak periodic potential splits each Landau level into q subbands, where there are q ﬂux quanta for every p unit cells of the periodic potential. Each of these subbands carries an integer Hall conductance, and these integers can be diﬀerent from unity. For example, when p/q = 3/5 the Hall conductances of the 5 subbands alternate between −1 and 2.

800

Topological Aspects of Low Dimensional Systems (a)

V B J

A

C

J

D

(b)

Voltage Loop

ΦV Solenoid

Solenoid

Current Loop

ΦJ Fig. 3. The Hall bar, with current and voltage leads, shown in (a) can be replaced by the arrangement shown in (b), where the voltage is supplied by changing ﬂux ΦV through one loop, and the current is monitored by observing changes of the ﬂux ΦJ through the other loop.

The quantum number characterizing the Hall conductance of a subband turns out to be the topological invariant known as the Chern number. Later work showed that this method could be extended to a much more general situation. In order to observe the Hall eﬀect in the usual way one takes a bar of the two-dimensional electron system in a strong magnetic ﬁeld, passes a ﬁxed current through it from a pair of current leads, and measures the voltage across the sample by connecting two voltage leads on opposite edges of the sample to a voltmeter. This is the set-up shown in idealized form in Figure 3a. In the work of Avron and Seiler [73], and of Niu and Thouless [85], the leads connected to current source and voltmeter are replaced by leads of the same material as the Hall bar, connected in pairs as shown in Figure 3b. Through the voltage loop there passes a solenoid which has a variable ﬂux ΦV , while there is another solenoid with ﬂux ΦJ passing through the current loop. If the ﬂux ΦV is changed at a uniform rate, the solenoid will maintain a constant electromotive force dΦV /dt around the voltage loop, and the other solenoid can be used as a pick-up to monitor the current that is generated around the current loop. One may notice

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that while experimentalists like to measure the voltage resulting from the passage of a ﬁxed current, theorists prefer to ask about the current that will be generated by a given voltage. The current operators in the current and voltage leads are ∂H/∂ΦJ and ∂H/∂ΦV . The Hall conductance can be calculated from the Kubo formula [77, 78] for the conductivity, which gives, by means of standard perturbation theory, the current density which is the linear response to an applied electric ﬁeld or electrochemical potential gradient. The result can be written in terms of a current-current correlation function, as it was in equation (4.4). For the Hall conductance we want the Jx , Jy correlation function for the many-electron ground state wave function |Ψ0 (ΦJ , ΦV ), and so the Hall conductance can be written as ∂H P ∂H SH (ΦJ , ΦV ) = i¯ hΨ0 (ΦJ , ΦV )| ∂ΦV (E − H)2 ∂ΦJ P ∂H ∂H − (4.10) |Ψ0 (ΦJ , ΦV ), ∂ΦJ (E − H)2 ∂ΦV where H is the hamiltonian for the system, depending on the parameters ΦJ and ΦV , and E is the corresponding energy of the ground state. The operator P projects oﬀ the ground state. Perturbation theory for the wave function gives ∂H P ∂Ψ0 = |Ψ0 , (4.11) | ∂ΦV E − H ∂ΦV and the corresponding equation for the perturbation due to the ﬂux through the current loop, so the the equation for the Hall conductance is ∂Ψ0 ∂Ψ0 ∂Ψ0 ∂Ψ0 SH (ΦJ , ΦV ) = i¯ h | − | · (4.12) ∂ΦV ∂ΦJ ∂ΦJ ∂ΦV This must be periodic in each of the ﬂuxes with period h/e, and the ﬂuxes have the eﬀect of changing the phase of quasiperiodic boundary conditions round each of the current and voltage loops. The quantities that appear in this equation can be written in terms of the Green function for the many-body system, integrated over the two spatial coordinates of the system, and integrated around a contour in the complex energy plane which surrounds a part of the real axis that includes no mobile excited states. Since the Fermi energy lies in a mobility gap, we expect the Green function to fall oﬀ exponentially with distance at a rate that depends on the localization length at the Fermi energy, except at the edge of the system where there will be extended electron edge states [81]. In this system the edge states cannot contribute to the total current around the current loop, since there is only one edge, and any current that ﬂows along the edge goes in opposite directions on the two edges of the current leads.

802

Topological Aspects of Low Dimensional Systems

The system with its two sets of leads looping around has the topology of a torus pierced by a single hole. I do not think that this argument has been properly worked out in detail, but it suggests that the expression (4.12) is independent of the quasiperiodic boundary conditions determined by ΦJ , ΦV , up to corrections exponentially dependent on the ratio of the width (or length) of the Hall bar to the localization length at the Fermi energy. If this is the case we can write the Hall conductance as h/e h/e ∂Ψ0 ∂Ψ0 ie2 ∂Ψ0 ∂Ψ0 SH = dΦJ dΦV | − | · (4.13) 2πh 0 ∂ΦV ∂ΦJ ∂ΦJ ∂ΦV 0 The wave function |Ψ0 (ΦJ , ΦV ) gives a mapping of the torus deﬁned by ΦJ , ΦV (physical quantities are periodic in these variables) onto the complex projective space of normalized wave functions with arbitrary phase. The integral is 2π times the integer invariant that deﬁnes the ﬁrst Chern class of this mapping [86]. For a physicist, a more familiar way of getting the desired result is to argue that the integrand is the curl of the vector whose components in ΦJ , ΦV space are 1 ∂Ψ0 ∂Ψ0 ∂Ψ0 ∂Ψ0 |Ψ0 − Ψ0 | , |Ψ0 − Ψ0 | , (4.14) 2 ∂ΦJ ∂ΦJ ∂ΦV ∂ΦV which is −i times the gradient of the phase η of |Ψ0 in this ﬂux space. This therefore gives e2 ne2 , (4.15) SH = gradΦ η · dΦ = 2πh h where the integral is taken round the boundary of the two-dimensional integration in equation (4.13). The phase, which is very like a Berry phase [52], must be deﬁned in some unambiguous way, such as by parallel transport, or by ﬁxing the phase of the wave function to be zero at some point in the space of electron coordinates – but then ambiguities arise at those values of ΦJ , ΦV which give a zero at this chosen point. However the phase is deﬁned, it must return to the same value around the path in equation (4.15) up to a multiple of 2π. It is this winding number of the Berry phase that gives the integer n on the right side of the equation. Various implementations of the phase have been discussed by Thouless [87], Kohmoto [88], and by Arovas et al. [89]. Although the result can be reduced to the winding number of a Berry phase round the perimeter of a unit cell in two-dimensional ﬂux space (ΦJ , ΦV ), yet the argument seems to be intrinsically two-dimensional, involving simultaneously what is happening in the current leads and in the voltage leads. The same could be said of the Laughlin argument [58], where

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there needs to be a simultaneous consideration of the change in vector potential around the annulus and the transfer of electrons across the annulus. Again there is some analogy between the two-form in equation (4.14) that gives the Hall conductance and the expectation value of a commutator in equation (3.10) that gives the transverse force on a vortex moving in a superﬂuid in Section 3.2. 4.5 Fractional quantum Hall eﬀect The arguments that have been presented in the last two subsections show that the Hall conductance is an integer multiple of e2 /h if the Fermi energy lies in a mobility gap, and if the ground state wave function is unique. The discovery of fractional values of the Hall conductance number by Tsui et al. [76] was therefore surprising. Subsequent work has shown that many diﬀerent simple odd-denominator fractions occur, and that the fractional quantization is fairly precise. Simple modiﬁcations of the theory for noninteracting or weakly interacting electrons did not seem to give this eﬀect, and Laughlin [90] argued that it must be due to the existence of a new sort of correlated many-electron ground state, and proposed the sort of ground state that should reduce the repulsive Coulomb energy of the electrons and display fractional quantization with odd-denominator fractions. In the central gauge, with the vector potential equal to A = (−By/2, Bx/2), the degenerate many-body ground state wave function for N noninteracting electrons all in the lowest Landau level has the form N 1 |zi |2 , (4.16) f (z1 , z2 , . . . zN ) exp − 2 4l0 i=1 where zi represents xi + iyi , f is any multinomial antisymmetric in the variables, and l0 is the magnetic length ¯h/eB. If f has the form f=

N (zj − zi ),

(4.17)

i<j

the particles are concentrated in a disk whose area is approximately 2πN l02 , and this represents a fully occupied Landau level with a density, inside the disk, of one electron per ﬂux quantum. Laughlin [90] suggested that the wave function obtained by setting the multinomial equal to f (z1 , z2 , . . . zN ) =

N (zj − zi )q ,

(4.18)

i<j

with q an odd integer, would be particularly eﬀective at keeping the electrons apart and so reducing the Coulomb energy. It gives a uniform one-electron

804

Topological Aspects of Low Dimensional Systems

density within the disk, and has no pairs of electrons whose relative angular momentum is less than q¯ h. Haldane and Rezayi [91] have shown that the generalization of this wave function to spherical geometry gives the true and unambiguous ground state for a model interaction similar to the Coulomb interaction. It is also in very close agreement with ﬁnite system calculations for the Coulomb interaction. For such a wave function, given by equations (4.16) and (4.18), the highest power of zj in the multinomial is (N − 1)q, so the area of the disc is increased by a factor of q, and the density of electrons is now 1/q per ﬂux quantum. This ground state is separated from excited states by an energy gap for the creation of quasiparticle excitations. The lowest energy quasiparticles are presumably localized by the substrate disorder, so the ﬁlling does not have to be exactly 1/q for the Hall conductance to be determined by the properties of this ground state. This fractionally occupied Landau level will give a Hall conductance equal to e2 /qh whether or not there are additional localized quasiparticles present. Laughlin showed that there should be a hole-like quasiparticle with fractional charge e/q formed at the point z0 by adding an extra ﬂux line at that point; this can be done by multiplying the ground state wave function by a constant times i (zi − z0 ). Similarly a quasiparticle with charge −e/q can be formed by removing a ﬂux line, which can be represented approximately by the operator i (∂/∂zi − qz0 /l02 ) acting on the multinomial f . Jain [92–94] has generalized this idea in the composite fermion model. Start with the wave function for p full Landau levels in some magnetic ﬁeld, then attach ±2m ﬂux quanta to each electron, multiplying the wave function by (zi − zj )±2m . Project the result back onto the lowest Landau level. This gives a ﬁlling factor, the number of electrons per ﬂux quantum, equal to p/(2mp ± 1). For m = 1 this gives the two sequences tending to 1/2 which are prominent in the experimental data, 1/3, 2/5, 3/7, . . . and its particle-hole conjugate. For m = 2 the sequences of odd-denominator fractions tend to 1/4. Further manipulation with particle-hole conjugation gives other fractions like the sequences tending to 3/4. The arguments that explain the integer quantum Hall eﬀect are rather general, and have to be reconciled with the widespread occurrence of the fractional eﬀect. After Laughlin’s [90] explanation of the fractional quantum Hall eﬀect was generally accepted, it was suggested that the ground state should have a discrete broken symmetry [74,95,96]. If there are q equivalent ground states, then, in Laughlin’s argument for the integer eﬀect [58], q ﬂux quanta have to be introduced before the wave function returns to its original form. If only one ﬂux quantum is introduced a diﬀerent ground state is obtained. This seemed to be contradicted by the apparent nondegeneracy of Laughlin’s trial wave function for electrons conﬁned to a disk [90], and by

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the proved nondegeneracy of the wave function for electrons on a sphere [91]. However, one cannot do a measurement of the quantum Hall eﬀect on the sphere without introducing probes of some sort through the surface, even if magnetic monopoles are available to provide a uniform magnetic ﬁeld through the surface, and once probes are introduced one must worry about boundary eﬀects. We have analysed this problem for the sphere [97] and the annulus [98]. For an annulus threaded by ﬂux Φ the wave function takes the form N i=1

  n1 −Φe/h  q 2 2 zi |zi | (zi − zj ) exp − |zi | /4l0 , (ij)

(4.19)

i

which is a superposition of determinants made up from one-electron wave functions of the form given in equation (4.8). The powers of |zi | in the prefactor of the exponential range from n1 − Φe/h to n1 − Φe/h + q(N − 1), and so the electrons are concentrated in the range 2(n1 − Φe/h)l02 < |z|2 < 2(n1 − Φe/h + qN )l02 .

(4.20)

The eﬀect of increasing Φ adiabatically by h/e is to pull the whole electron distribution inwards, producing a state which is diﬀerent from, and inaccessible from, the new quasiequilibrium state, which is obtained from equation (4.19) by simultaneously increasing Φe/h and n1 by unity. It is only when Φ has been increased by q that the new quasiequilibrium state can be reached by transferring one electron from the inner edge to the reservoir and one electron from the outer edge to the reservoir. In this picture, whether applied to the annulus connected to two diﬀerent reservoirs, or to the torus which was used to related Hall conductance to Chern numbers, the fractionally charged quasiparticles serve as point topological defects whose migration across the system can enable transitions to occur between these diﬀerent ground states [98, 99]. An excitation for with fractional charge e/q (quasihole) located at z0 can be generated the annulus by multiplying the wave function (4.19) by A(z0 ) i (zi − z0 ), where A is a normalization factor. When z0 is very large, A is 1/(−z0 )N , and this factor does not change the wave function. For z0 = 0 it increases n1 by unity, pushing the electron distribution outwards, so that there is an extra charge e/q on the inner edge and −e/q on the outer edge. This state can be reached by moving z0 continuously from the outside of the annulus to the inside. While z0 is in the interior of the annulus there is a quasihole of charge e/q crossing the annulus with some compensating charge density on each of the edges.

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4.6 Skyrmions In most of the discussion of the fractional quantum Hall eﬀect up to this point it has been assumed that there is a single Landau level involved, with a single spin orientation. In certain systems this assumption is inappropriate, and two or more coupled Landau levels have to be considered. In most systems the Zeeman energy is relatively small, and in some systems it can be made particularly small, so the assumption that the spins of the electrons are completely polarized is inappropriate. A similar situation arises when the Zeeman energy is large, but a bilayer consisting of two adjacent layers of electrons is formed. In this case a pseudospin can be used to describe in which layer an electron lies. Halperin [100] introduced a modiﬁcation of Laughlin’s wave function (4.18) which allows for equal occupation of the two spin or peseudospin states. This has the form f{m,m,n} (z1 , z2 , . . . ; z1 , . . .) =     N N N  (zj − zi )m   (zj − zi )m   (zi − zi )n  , i<j

i <j

(4.21)

i,i

where the unprimed variables are the coordinates of spin up electrons and the the primed variables are the coordinates of spin down particles. Power counting shows that this gives the occupation number ν, the ratio of the total number of electrons to the number of ﬂux quanta in the area they occupy, as ν = 2/(m + n). The exponents of (zj − zi ) and (zj − zi ) must be the same, or else the spin up and spin down electrons will be occupying diﬀerent areas, so this cannot be used to describe partial spin polarization. States of the form {m, m, m} in equation (4.21) are particularly interesting. For m = 1 this gives a singly occupied Landau level rotated by π/2 in pseudospin space, and for m = 3, 5, . . . it gives the Laughlin wave functions (4.18) rotated by π/2 in pseudospin space. All the pseudospins are aligned, so this is a pseudospin, or, for the case of vanishing Zeeman splitting, real spin ferromagnet. In regular Heisenberg ferromagnets, in two dimensions or in three, the lowest-lying part of the excitation spectrum consists of spin waves, in which a single spin is reversed. In two dimensions a skyrmion excitation is possible. This is a texture in the r, θ plane such as Sx = sin f (r) cos θ, Sy = sin f (r) sin θ, Sz = cos f (r),

(4.22)

with f (r) a function that is zero at inﬁnity and goes smoothly to π at the origin. This texture gives a mapping of the plane (plus the point at inﬁnity) onto the sphere of spin directions that wraps round the sphere and has a unit topological quantum number. The winding number is given by equation (1.3), with the integral over the unit sphere replaced by an

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integral over the inﬁnite plane. In the case given in equation (4.22) the winding number is Nw = 1. For the Heisenberg model the skyrmions have rather a high energy, but it has been argued that for the quantum Hall systems that are being considered in this section the skyrmion should give the lowest excitations for ν = 1 [101–104]. As is shown in the work of Fertig et al. [105], the topological charge, the winding number, gives the electric charge of the excitation. Several experimental groups have seen eﬀects that seem to be due to such skyrmions at ν = 1. Barrett et al. [106] and Aifer et al. [108] found a strong peak in the spin polarizability at ν = 1 Schmeller et al. [107] found multiple spin ﬂips (as many as 7) accompanying elementary excitations for ν = 1, but not for ν = 1/3 or 1/5. Bayot et al. [109] found an enhancement of the speciﬁc heat, which are attributed to the skyrmion excitations at ν = 1. Maude et al. [110] made electrical transport measurements, and found an enhanced spin gap even for g = 0, as would be expected if the excitations were skyrmionic.

5

Topological phase transitions

5.1 The vortex induced transition in superﬂuid helium ﬁlms In the conventional phase transitions there is usually some order parameter which has a nonzero value in the low temperature phase and is zero in the high temperature phase. The susceptibility associated with this order parameter diverges at a critical point, and this gives singularities in other quantities such as the speciﬁc heat. The two-dimensional Ising model was the ﬁrst model solved in these terms, but renormalization group allowed good accounts of three-dimensional models such as the Ising, planar spin and Heisenberg models to be given. Also the theory of conformal invariance gave results for critical exponents in other discrete two-dimensional models such as the three-state Potts model. In such conventional continuous (second order) phase transitions, ﬂuctuations away from the equilibrium value of the order parameter decay exponentially with a negative exponent whose value is the ratio of the distance to a temperature-dependent correlation length. At the critical point where the phase transition occurs the correlation length is inﬁnite, and the correlations fall oﬀ as a power of the distance, rather than as an exponential function. It was known from the work of Landau and Peierls [111] in the 1930s that two-dimensional systems with continuous order parameters could not have conventional long range order, and this was established rigorously by Mermin and Wagner [112] and by Hohenberg [113]. The possibility of some other type of order was not ruled out – it was known, for example, that a solid could have algebraic divergence of its X-ray peaks at the Bragg angles

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[114], or it might have orientational long-range order without positional long-range order. Onsager [1] and Feynman [20] suggested that the phase transition in bulk 4 He from the superﬂuid to the normal state might not be characterized by the disappearance of the magnitude |Ψ0 | of the condensate wave function, but rather by a loss of phase coherence produced by a tangle of vortex lines. This does not ﬁt well with modern renormalization group ideas, whereas the observed critical exponents of this transition do ﬁt the renormalization group results for a two-component vector order parameter rather well. The results for a three-dimensional system with a twocomponent vector lie smoothly between the results for the Ising model and the Heisenberg model. Also it is technically diﬃcult to handle a phase transition driven by vortex lines, because it involves a model like a directed polymer with a long-range interaction. In two dimensions the situation is much simpler, since vortices are just point defects with a long-ranged interaction between them. We refer to a phase transition driven by such defects as a topological phase transition. Helium ﬁlms provide the simplest example of a topological phase transition. The order parameter, the condensate wave function, is a single complex function of the two-dimensional position, so the system is equivalent to a planar spin model in two dimensions. The singularities of this system, the vortices, which play a vital role in this theory, are simply point singularities where the order parameter goes to zero, and around which the phase changes by ±2π. This type of system was studied in the work of Berezinskii [115,116] and of Kosterlitz and Thouless [117, 118]. The development of the theory is simpliﬁed if we consider a lattice gas model of the superﬂuid, in which there is a magnitude |Ψi | and a phase θi of the order parameter associated with each site i of the lattice. At low temperatures there will only be small ﬂuctuations of the magnitude |Ψi |, and the phase ﬂuctuations will be controlled by an energy term of the form H = −K cos(θi − θj ), kB T

(5.1)

(ij)

where (ij) denotes a pair of nearest neighbor sites. In the Gaussian approximation, the cosine in this formula is replaced by its quadratic approximation, to get the exponent of the Boltzmann factor as −

H0 1 H =− − K (θi − θj )2 . kB T kB T 2

(5.2)

(ij)

The2 angles θj are then written as their Fourier transforms, proportional to d k c(k) exp(ik · Rj ), so that equation (5.2) becomes a diagonal quadratic form in the Fourier components c(k). The correlation function for cos θi can

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then be expressed in terms of an integral of an exponential whose exponent is quadratic and linear in these Fourier components, and the evaluation of this integral gives the correlation function as 1 1 − cos(q.R a2 ) ij d2 q , (5.3) cos θi cos θj = exp − 2 2 4π 4 − 2 cos(qx a) − 2 cos(qy a) where I have taken the lattice to be a square lattice of side a, and Rij is the vector between the sites i and j. For large values of Rij the integrand is of order 1/q 2 a2 for q between 1/Rij and 1/a, so the integral depends logarithmically on the ratio of these two quantities, and the result η(T ) a cos θi cos θj ≈ , (5.4) Rij with

1 , (5.5) 2πK is obtained. This power law behavior of the correlation function at all low temperatures was discussed by Wegner [119] and by Jancovici [120], and means that the whole low-temperature region can be regarded as a critical line. For the superﬂuid the energy associated with variation of the phase can be written as h2 ¯ 1 (5.6) H= d2 rρs 2 (gradθ)2 , 2 m η(T ) =

and comparison of equations (5.1, 5.5) and (5.6) shows that the correlation exponent is given by m2 kB T η(T ) = , (5.7) 2π¯ h2 ρs where ρs is the superﬂuid density per unit area. This algebraic fall-oﬀ of the correlation function implies that the usual order parameter, related to the inﬁnite distance limit of the correlation function, must be zero. In accordance with the discussion of superﬂuid ﬂow in Section 2.1, the unbounded ﬂuctuation of the relative phase at large distances implied by equation (5.4) will not be enough to destroy superﬂuid ﬂow in an annulus, which is represented by a texture of the phase in which the phase changes by a multiple of 2π round the annulus. The phase can only lose this twist by the passage of vortices across the system, and it turns out that, in the thermodynamic limit, there is an inﬁnite barrier to the creation of the necessary vortices. The energy of an isolated vortex in a system of linear dimension L is L 1 h2 1 ¯ πρs ¯h2 L 2πr dr 2 2 = ln , (5.8) Ev = ρs 2 2 m r m a0 a0

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Topological Aspects of Low Dimensional Systems

where a0 is the vortex core radius. There is also an entropy associated with the possible positions of a vortex. If the number of possible positions in a system of area L2 is L2 /πa21 , then the entropy per vortex is Sv = kB ln(L2 /a21 ) .

(5.9)

At low temperatures the free energy per vortex is positive, and depends logarithmically on the size of the system, so the concentration of free vortices is zero in the thermodynamic limit at low temperatures. Above the h2 /2m2 the logarithmic term in the free temperature given by kB T = πρs ¯ energy of a vortex changes sign, so free vortices appear, and supercurrents are unstable. This argument leads to the conclusion obtained by Nelson and Kosterlitz [121] that a necessary condition for the stability of superﬂuid ﬂow is 2m2 kB T , (5.10) ρs ≥ π¯ h2 or, from equation (5.7), η(T ) ≤ 1/4 . (5.11) In the more detailed theory this inequality becomes an equality at the transition temperature, and the critical line ends at η = 1/4. We described this type of phase transition as a topological phase transition because the most prominent feature of the ordered low-temperature phase is the absence of topological defects and the stability of states with nonzero topological quantum number in the thermodynamic limit, rather than the existence of a conventional order parameter. In the detailed theory [118], the eﬀect of creation of bound pairs of positive and negative vortices on the superﬂuid density was taken into account. The energy of interaction of two vortices with quantum numbers n1 , n2 , giving phases θ1 (r), θ2 (r), is R12 ρs ¯ 2πρs ¯h2 h2 dr d2 r 2 gradθ1 (r) · gradθ1 (r) = − n n 1 2 m m2 r a0 =

−

2πρs ¯h2 R12 n1 n2 ln · (5.12) m2 a0

The vortices therefore behave like a Coulomb gas (with a two-dimensional Coulomb interaction) of classical charges.The bound vortex pairs are like classical molecules. The Hamiltonian for this classical Coulomb gas can be written in lattice gas form as µ 2 ri − rj L H = −πK + ni nj ln ni + πK ni nj ln · (5.13) kB T τ kB T i τ i,j i=j

Here K is ρs ¯ h2 /m2 kB T , τ is the lattice spacing, of order a0 , µ is a chemical potential to represent the short-range contributions to the vortex energy,

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and the ﬁnal term represents the long range contributions to the energy in a system of linear dimension L when ni is nonzero. The eﬀective dielectric constant K is assumed to be a function of distance, because the smaller vortex pairs are polarized by the ﬁelds acting between more distant pairs. A crude treatment of the polarizability gives it as ∞ 3 R dR exp[−2πK ln(R12 /τ )] πK − 1 2 · (5.14) = τ2 = r0∞ R12 πK − 2 RdR exp[−2πK ln(R /τ )] 12 r 0

This diverges at K = 2/π, where η(T ) = 1/4, which is the bound given in the expression (5.11); with divergent polarizability there is no barrier to the dissociation of vortex pairs. Renormalization arguments originally developed by Anderson and Yuval [122] were applied by Kosterlitz [123] to this problem of interacting vortices or interacting charges in two dimensions. This method was developed further by Jos´e et al. [124]. The basic idea of this method is to renormalize the free vortex (or free charge) fugacity y(L) = exp(−µ/kB T ) at each length scale L in terms of the dimensionless superﬂuid density (or reciprocal of the dielectric constant) K(L). The dielectric constant on the length scale L is determined by those dipoles whose size is less than L, so the change of the dielectric constant when the length scale is varied is proportional to the square of the concentration of free charges found on length scales L. This leads to the ﬁrst of the Kosterlitz equations L

dK −1 (L) = 4π 3 y 2 (L) + O(y 4 ) . dL

(5.15)

The self-energy of a charge (or vortex) on a length scale L depends on the polarizability of the medium on that length scale and on the concentration of free vortices, so that leads to the second Kosterlitz equation L

dy(L) = [2 − πK(L)]y(L) + O(y 3 ) . dL

(5.16)

More detailed derivations of these equations can be found in the review by Nelson [125]. If the concentration y of defects (charges or vortices) is suﬃciently small that higher order terms in y can be ignored, the two equations can be combined to get dK(L) 4π 3 yK 2 =− , (5.17) dy(L) 2 − πK which can immediately be integrated to get the relations between K(L) and y(L) πK(L) 2 ln + = 2π 2 y 2 (L) + 1 + α , (5.18) 2 πK(L)

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Topological Aspects of Low Dimensional Systems

Fig. 4. Flow diagram for the renormalization of K and y from equations (5.15) and (5.16).

where α is a constant of integration. These solutions, which represent the renormalization ﬂow of K and y, are shown in Figure 4. Those curves for which α is positive ﬂow into solutions for large L with y tending to zero, whereas if α is negative the ﬂow may initially be to smaller concentrations y, but eventually goes to larger concentrations, so that there are free defects on a large length scale. The explicit equation (5.18) for y as a function of K can be substituted back into equation (5.15) to get a set of expressions for K as a function of L. To use this scaling relation to obtain long-range properties it is necessary to start with known properties at length scales of order a0 , the vortex core radius. The superﬂuid density K(a0 ) and the vortex fugacity y(a0 ), unmodiﬁed by the speciﬁc two-dimensional properties we have been discussing, will both be functions of temperature, as will the core radius itself. From where this trajectory K(a0 (T ), T ), y(a0 (T ), T ) cuts the ﬂow lines (5.18), the long-range superﬂuid density K(∞, T ) for T < Tc and the transition temperature Tc can be read oﬀ. The most striking result is the one already mentioned [121], that, at the temperature where free vortices ﬁrst appear and the superﬂuidity is destroyed, K(∞, Tc ) is always 2/π, and the superﬂuid density always has the value ρs (Tc ) =

2m2 kB Tc · π¯ h2

(5.19)

This relation was veriﬁed by Rudnick [126] studying the propagation of third sound, the waves which propagate on superﬂuid ﬁlms, and by Bishop and Reppy [127, 128] using an Andronikashvilii oscillator constructed of a roll of mylar ﬁlm with a thin coating of liquid helium. Both these experiments

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use a nonzero measuring frequency; this also implies a ﬁnite length scale of the order of the wavelength of third sound at that frequency. The theory used to interpret such measurements at nonzero frequency measurements was developed by Amegaokar et al. [129]. At such nonzero frequencies the transition is rounded and pushed up to higher temperatures, and a good ﬁt to the experimental data was obtained with this theory, and an extrapolation can be made to the discontinuity of superﬂuid density and the transition temperature at zero frequency or inﬁnite length scale. Later work by McQueeney et al. [130] on solutions of 3 He in 4 He carried this relation down to rather low temperatures.

5.2 Two-dimensional magnetic systems Theoretical studies of critical phenomena usually relate the universality class to an equivalent magnetic model. The liquid-vapor critical point and the magnetic system with a single preferred axis of magnetization both are in the same universality class as the Ising model. The superﬂuid transition, with an order parameter that is a complex scalar quantity, resembles a magnetic system with a preferred plane of magnetization but no preferred direction in the plane. This is the planar spin model. The three-dimensional planar spin model is similar in many ways to other three-dimensional magnetic models, intermediate in its critical exponents between the Ising model with a single preferred axis and the Heisenberg model with no preferred direction of magnetization. In two dimensions, however, the planar spin model has very special properties. There is no net magnetization in zero ﬁeld, and the spin-spin correlation function tends to zero at large distances except at zero temperature. However, below the critical temperature for vortex pair unbinding the spin-spin correlation function decays algebraically with distance, rather than exponentially, and the spin-wave stiﬀness is nonzero. For spins on a lattice one can deﬁne vortices uniquely, as points on the dual lattice around which the adjacent spins rotate through an angle of ±2π, if one deﬁnes the angle between two neighboring spins to be in the range from −π to π. Isolated vortices have an energy that depends logarithmically on the size of the system, and vortex-antivortex pairs have an energy that depends logarithmically on their separation. The phase transition is driven by the dissociation of vortex-antivortex pairs, and is precisely analogous to the superﬂuid transition discussed in the previous subsection. The spin-wave stiﬀness is the quantity that is analogous to the superﬂuid density.

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Topological Aspects of Low Dimensional Systems

When the rotational symmetry in the plane is broken various thing can happen. If there is a two-fold symmetry – two equivalent minima in the energy at an angle π apart – the magnetic system has the critical behavior of the Ising model. With a three-fold symmetry axis, so that the minima in energy are at an angle 2π/3 apart, the critical behavior is that of the three-state Potts model. If the symmetry axis perpendicular to the plane has ﬁve-fold or higher symmetry, as it will if the anisotropy is due to crystal ﬁelds in a triangular lattice, then the anisotropy can be shown to lock the magnetization into one of the preferred directions at suﬃciently low temperatures, but at intermediate temperatures the anisotropy becomes irrelevant, so that the system behaves like an isotropic planar spin system, with algebraically decaying long-range order, whose exponent changes continuously with temperature until it reaches the critical value at which free vortices can form spontaneously [124]. The case of anisotropy with four equivalent axes in the plane lies on the boundary between these two types of behavior, and algebraically decaying order occurs only at one temperature, but the exponent at this temperature has a nonuniversal value. The topological nature of the planar spin transition is shown also by considering what happens in the Heisenberg model in two dimensions, with the spin free to point in any direction in space [118, 131]. In this case there is no longer a metastable vortex, a singular point around which the spin rotates by an angle 2π, because the spin is free to be directed anywhere on the sphere. This allows the vortex to be replaced by a texture in which the spins near the center of the texture are tilted out of the xy plane towards the z direction, so that the direction of the spins is a smooth function of position. The energy of this texture, instead of being proportional to ln(L/a0 ), where a0 is of the order of the lattice spacing, is proportional to ln(L/ξ0 ), where ξ0 is the length scale of this central region in which the spins are tilted out of the plane. Since ξ0 is continuously variable this energy is continuously variable down to zero, and so there is no energy barrier to the creation or annihilation of vortex pairs. There is no phase transition for the two-dimensional Heisenberg model at nonzero temperature. 5.3 Topological order in solids A solid can be characterized in a number of diﬀerent ways. It has positional long-range order which is characterized by the existence of a reciprocal lattice and shown up by the sharp Bragg peaks in X-ray scattering. It has orientational long range order, in that the directions from one atom to its nearest neighbors in one part of the crystal are correlated with those in some other part of the crystal. It is rigid, in the sense that it resists a shearing stress and has inﬁnite viscosity – it may yield to a shearing stress, but the rate of yield in an ideal solid rises much slower than linearly with the applied stress.

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Associated with the symmetry breaking from continuous to discrete translational symmetry there is an order parameter which is the local displacement of the actual lattice from the ideal lattice, which is the quantity whose Fourier transform determines the sharpness of the Bragg peaks. In a two-dimensional system the peaks become algebraic rather than δfunction peaks [114], because thermal ﬂuctations of this order parameter grow logarithmically with distance. The topological defects associated with this order parameter are dislocations. A typical edge dislocation in a square lattice is shown in Figure 5. In three dimensions this can be formed by the Volterra process, in which a half-plane of atoms bounded by the dislocation core is removed, and then the atoms are rejoined across the cut. If we deﬁne the vector u to be the displacement of the local lattice relative to the closest point of the ideal lattice, then the gradient of u round a closed loop gives a lattice vector, which is the Burgers vector of the dislocation. In the case of an edge dislocation this vector is in the plane perpendicular to the core. One can ﬁnd the Burgers vector by mapping a closed path round the dislocation core onto an ideal lattice; the Burgers vector is the amount by which the map of the path fails to close. The dislocation is known as a screw dislocation if the Burgers vector is parallel to the dilocation core, but this is not relevant to the two-dimensional case, where the core is a point defect, and the Burgers vector must lie in the plane. The topological description of dislocations has been given in a paper by Kl´eman [132].

Fig. 5. Edge dislocation formed by the Volterra process. A Burgers circuit is shown, and the Burgers vector is the amount by which it fails to close.

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Topological Aspects of Low Dimensional Systems

The rigidity of a solid is unrelated to the sharpness of the Bragg peaks or the thermal ﬂuctuations of the order parameter, but depends on the absence of dislocations in the crystal lattice [133]. Dislocations can move under the inﬂuence of shearing stress to relieve the stress, and if this process is thermally activated under conditions of low shearing stress it will lead to a viscous ﬂow with a rate of strain proportional to stress. The dislocation produces a strain ﬁeld, both longitudinal and transverse, which falls oﬀ linearly with the reciprocal of the distance from the core. As a result, the energy of a single dislocation depends logarithmically on the size of the system in two dimensions, and the interaction between two dislocations also depends logarithmically on the distance between them. However, this interaction also depends on the Burgers vectors of the dislocations; the logarithmic interaction is proportional to the scalar product of the two Burgers vectors, but there is an additional term independent of distance that is proportional to the projection of the Burgers vectors in the direction of the displacement of one dislocation relative to the other. The theory of dislocation mediated melting in two dimensions was worked out by Young [134] and by Halperin and Nelson [135]. The theory is superﬁcially similar to the theory of superﬂuidity in two dimensions, but there are important diﬀerences introduced by the vector character of the Burgers vector that characterizes the dislocations. For example, in place of equation (5.10) there is the more complicated relation 4πkB T µ(µ + λ) > , 2µ + λ a20

(5.20)

where µ, λ are the elastic moduli characterizing transverse and isotropic strain and a0 is the lattice constant. This gives a lower bound on the rigidity modulus µ in the solid phase, just as equation (5.10) gives a lower bound for the superﬂuid density in the superﬂuid phase. The same combination of elastic moduli is involved in the generalization of the Kosterlitz scaling relations (5.15) and (5.16) [134,136]. Instead of a single exponent η characterizing the algebraic decay of correlations there is a set of exponents, one for each Bragg peak. Above the dislocation unbinding temperature there may still be orientational order [136], but it now has an algebraic decay with distance, instead of having a nonzero long-range limit as it has below the transition temperature. Orientational order can also be deﬁned in terms of a topological property. At each point in the dislocated crystal, except very close to the dislocation cores, the short range order of the atoms deﬁnies a local set of crystal axes. A parallel translation of these local axes can be made along a long closed path, and this deﬁnes a continuous mapping of the path on the space of axis orientations. If this mapping cannot be shrunk to a point there must be a singularity enclosed by the path. Such singularities are

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disclinations, similar to the disclinations found in liquid crystals and in the A phase of superﬂuid 3 He. They can be constructed from the ideal lattice by removing or adding a slice of atoms, for example a π/3 sector in a triangular lattice, or a π/2 sector in a square lattice, and rejoining the edges smoothly. At the center of a disclination in a triangular lattice a lattice site has ﬁve or seven nearest neighbors instead of the six which is usual. In a normal solid the elastic energy of a disclination is very high, proportional to the area of the system, but once the dislocation unbinding transition has occurred this energy depends logarithmically on the size of the system, so that the statistical mechanics of disclinations in this region is very similar to the statistical mechanics of vortices in a superﬂuid. Nelson and Halperin [135,136] called this phase hexatic. Dislocations can be regarded as tightly bound pairs of disclinations of opposite sign, and the transition to an unoriented liquid goes by the process of unbinding of disclination pairs. In the hexatic phase the orientational order falls oﬀ algebraically with an exponent η6 , and the transition occurs when η6 has the critical value 1/4, in agreement with equation (5.11). The most commonly studied form of two-dimensional solid is a physisorbed layer, such as a noble gas adsorbed on a graphite or other solid surface. In this case the substrate provides orientational ordering at all temperatures. The adsorbate may be in registry with the substrate, but it may also form its own crystal lattice incommensurate with that of the substrate. In this case it can undergo a dislocation unbinding transition to a ﬂuid state, and this has been studied by Young [134]. Most systems studied by experiment or in computer simulations actually undergo a ﬁrst order phase transition to a ﬂuid state before they reach the bound given by equation (5.20). A possible exception is the two-dimensional Coulomb solid, which was ﬁrst found in experiments by Grimes and Adams [137]. At least both experimental [138] and theoretical [139] studies have shown a rigidity which comes very close to the theoretical bound given by equation (5.20). A review both of the theory and of its possible applications has been given by Strandburg [140]. 5.4 Superconducting ﬁlms and layered materials The vortices in a charged superﬂuid generate a magnetic ﬁeld, so that, as we have seen in Section 2.1, it is not the circulation which is quantized but the ﬂuxoid, a combination of the circulation round a loop and the magnetic ﬂux enclosed by it. On length scales small compared with the London penetration depth the vortices in a superconducting thin ﬁlm are very similar to those in a neutral superﬂuid thin ﬁlm, but on length scales larger than the penetration depth they are quantized ﬂux lines. One result of this is that the logarithmic interaction between a vortex and an antivortex is cut oﬀ at a distance of the order of the penetration depth, to be replaced by

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Topological Aspects of Low Dimensional Systems

an interaction that falls oﬀ as the inverse cube of the distance, characteristic of the interaction between localized magnetic dipoles. Since the logarithmic interaction cuts oﬀ at the penetration depth we supposed that vortex pair unbinding would occur at arbitrarily low temperatures, and that there should be no phase transition in superconducting thin ﬁlms [118]. In fact, however, the London penetration depth in very thin ﬁlms is inversely proportional to the ﬁlm thickness, and can be rather large, comparable with the size of the sample. Therefore, as was argued by Beasley et al. [141] and by Doniach and Huberman [142], there can be a vortex unbinding transition in superconducting thin ﬁlms. It will not be a sharp transition, but will have a rounding determined by the penetration depth or by the sample size, whichever is smaller. The resistivity and surface impedance of such superconducting ﬁlms have properties which are characteristic of the vortex-unbinding transition, such as a current-voltage relation with an exponent that passes through the value 1/3 at the transition, and this behavior has been observed in a number of experiments [143–145]. There is a review of the theory and its applications to superconductivity by Minnhagen [146]. For a superconducting ﬁlm in a magnetic ﬁeld a number of issues arise, which were ﬁrst discussed by Huberman and Doniach [147] and by Fisher [148]. In an ideal superconductor the ﬂux lines will form a lattice at low temperatures, but this lattice should either melt when it becomes unstable to the dissociation of dislocation pairs, or should have a ﬁrst order phase transition at a lower temperature. Once the lattice is melted it may be in a hexatic phase, which would then become isotropic at higher temperatures, or it may go directly into an isotropic ﬂuid phase if there is a ﬁrst order transition. The vortex-driven transition from the superconducting state has also been extensively studied in two-dimensional arrays of Josephson junctions. One way of fabricating such a device is to deposit a regular array of superconducting islands on a normal metal substrate, such as copper. At low temperatures, in the absence of an external magnetic ﬁeld, not only will all the islands be superconducting, but their phases will be locked together by Josephson tunneling through the normal metal substrate. If an external magnetic ﬁeld is applied normal to the plane of the array there will be diamagnetic currents in the superconducting array which will cause vortices to form in the junction array. Energetically it is far more favorable for a vortex to form in a normal region between neighboring superconducting islands, a point in the lattice dual to the lattice of islands, than in the superconducting islands themselves. Only the small Josephson currents between the islands are aﬀected by the vortex, whereas a vortex in one of the superconducting islands would lead to suppression of the superconductivity in that island. Even in the absence of an external magnetic ﬁeld vortex-antivortex pairs

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can form on neighboring or close sites of the dual lattice. The phase of the superconducting order parameter provides an order parameter similar to that of the planar spin model, so the energy of a vortex pair depends logarithmically on the separation of the vortex from the antivortex. In principle there is a London penetration distance for the network, beyond which the magnetic ﬁeld is screened, and the interaction between vortices is power law rather than logarithmic, but in practice the Josephson currents are too small for this to be important. The thermally driven dissociation of the vortex pairs therefore leads to a zero-ﬁeld phase transition with the same properties as the other transitions discussed in this chapter, such as the superﬂuid transition in 4 He ﬁlms and the melting of the hexatic phase in two dimensions. At temperatures below this vortex-unbinding transition, current can be passed from one side of the network to the other with no resistance just as it can in a bulk superconductor. Above the unbinding transition there is a voltage generated when a current is passed. This voltage is produced by the free vortices moving to one edge of the array, across the current, and the antivortices moving to the other edge. These vortices that move to one edge of or the other are replaced by thermally excited vortex-antivortex pairs. Although the network cannot pass a supercurrent free of dissipation above this transition temperature, the islands of which it is composed still have the usual properties of small superconducting regions. Thus the superconducting order parameter still exists locally, although its phase is not locked across the sample. This is the same picture which is used to describe the vortex-driven transition in superconducting ﬁlms and in superﬂuid ﬁlms. This transition in superconducting networks was observed by Abraham et al. [149], who reported that when they studied the voltagecurrent relation they found V ∝ I α , where the exponent α is greater than 3 in the superconducting phase, and drops sharply to unity at the transition temperature. This is the same behavior that was observed in superconducting ﬁlms.

6

The A phase of superﬂuid 3 He

6.1 Vortices in the A phase The atoms of 3 He are fermions, with nuclear spin 1/2, and the superﬂuid condensate is formed from a P -state, which implies that the nuclear spins must also be coupled together in a triplet state. This P -state pairing results in a very rich behavior of the order parameter, with far more possibilities than exist in superﬂuid 4 He or in conventional superconductors. The triplet pairing of the nuclear spins also allows nuclear magnetic resonance to be used as a tool to explore details of this behavior.

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Topological Aspects of Low Dimensional Systems

The A phase of the superﬂuid is stable in a region that extends along the solidiﬁcation pressure (33 atm.) from 2.7 mK to 2.1 mK, and down to a point at 20 atm. and 2.5 mK on the phase boundary between normal ﬂuid and superﬂuid. In this A phase the orbital state of the pair is described by a state with ML = 1 along some axis which is denoted by l. The spin state of the pair has MS = 0 along some other axis which is denoted by d. The hyperﬁne magnetic coupling between the nuclear spins tends to align the vectors l and d parallel or antiparallel to one another, but this hyperﬁne coupling is relatively weak, and under many circumstances they can take up directions which are independent of one another. This superﬂuid phase is therefore anisotropic in space, with two distinguished axes, or one axis if the directions of l and d coincide. Early treatments of the topological properties of the A phase of 3 He, together with discussions of liquid crystals, were given in the papers of Toulouse and Kl´eman [150] and of Volovik and Mineev [151]. There is a detailed treatment in the book by Volovik [152]. To understand how this type of order parameter aﬀects the topological properties of the phase we will ﬁrst of all neglect the spin part of the order parameter. The axis of quantization of the orbital angular momentum of the pair can be chosen in any direction in space. Since this is an ML = 1 state, rotation about this axis changes the phase, and we can represent phase and direction together by taking an element of the group SO(3); in concrete terms by choosing three Euler angles to set direction and phase. The group SO(3) is not simply connected, since a path which gives a rotation by 2π cannot be shrunk to zero. It is more convenient therefore to work with the simply connected group SU (2) which gives a twofold covering of SO(3). There are two convenient ways of representing the topology of SU (2). One is in terms of the surface of a 3-sphere – a unit vector in the four-dimensional space of real coeﬃcients of the unit matrix and the three Pauli matrices. Another way is in terms of a point in the interior of a sphere of radius 2π. The distance of the point from the center of the sphere gives the angle of rotation about an axis whose direction is given by the vector from the center to the point; in this representation all points on the surface of the sphere are equivalent to a 2π rotation. Each possible orientation of the l axis and phase is represented by two points in SU (2), in accordance with the double covering of SO(3). One is separated from the other by a rotation of 2π, so they are represented by points lying on the same diameter of the 2-sphere separated by a distance 2π. Closed paths in this space can either be trivial loops in the space, or paths leading from one point to its equivalent point. The homotopy group π1 is therefore equivalent to the group Z2 , the group given by multiplication of ±1, or addition of the integers 0 and 1 modulo 2.

D.J. Thouless: Introduction to Topological Quantum Numbers

(2)

e

821

(2)

e

(1)

e

l

(a)

(b)

e(2) e

(1)

(c)

Fig. 6. This shows the continuous unwinding of a π twist in the order parameter for the A phase of 3 He by rotation about the tangential direction. In (a) and (c) the directions of twist are reversed, and in the intermediate case (b) it is obvious that there is no superﬂuid circulation.

It is clear that such a quantum number, with values 0 and 1, cannot correspond to a classical variable like circulation. For the A phase the superﬂuid velocity is given by h/2m3 ) vs = (¯

3 j=1

(1)

(2)

ej grad ej

,

(6.1)

where e(1) , e(2) are the two coordinate axes perpendicular to l. When l has constant direction this corresponds to equation (2.1), with m4 is replaced by 2m3 . Anderson and Toulouse [153] argued that circulation can always be changed in a continuous way. An example of how this can be done is shon in Figure 6. Figure 6a shows the variation of the orientation of the axis-system when there is unit circulation round a channel. The direction of l is shown as out of the plane. If the orientation of the axis-system is slowly rotated, for example about the tangential direction, the circulation is reduced and can be reversed. Figure 6b shows the situation when a π/2 rotation has been made, so that l is along the inward radius and e(1) is in

822

Topological Aspects of Low Dimensional Systems

Fig. 7. Possible orientation of the order parameter for a nonsingular doubly quantized vortex in the A phase of 3 He. The directions of e(1) and e(2) are shown with solid and hollow arrowheads, and of l with a broken line.

the axial direction. In this case equation (6.1) shows that the ﬂow velocity is zero. With the further rotation by π/2 shown in Figure 6c the circulation has been reversed. This process shows why the topological quantum number −1 is not distinct from 1 for the A phase. Two important consequences follow from this. The ﬁrst is that circulation is not quantized around channels unless some constraint is placed on the direction of l by the boundaries, as it is, for example in a thin ﬁlm where l is constrained to be perpendicular to the ﬁlm. The second is that when vortices do occur, for example in a rotating system, they can form an array of nonsingular doubly quantized vortices, where the circulation is 2h/2m3 around a path far from the vortex, where the path in the angle-axis space is a diameter of the 2-sphere or an equator of the 3-sphere. Closer in to the axis the path followed by the axis-system can shrink, so there is no singularity at the axis, where this path in order parameter space has shrunk to a point. An isolated doubly quantized vortex could have the three axes represented by the columns of the matrix  sin φ sin f cos2 φ + sin2 φ cos f sin φ cos φ(1 − cos f )  sin φ cos φ(1 − cos f ) cos2 φ cos f + sin2 φ − cos φ sin f  , − sin φ sin f cos φ sin f cos f 

(6.2)

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where φ is the azimuthal angle measured from the vortex center, and f (r) is a function of the distance r from the vortex center that goes smoothly from 0 at r = 0 to π for r > a0 . In such a conﬁguration the axis system rotates by an angle 2π when one goes between two points on opposite sides of the vortex, but the rotation is about the z-axis for a path that avoids the vortex center, while it is about an axis in the xy plane for a path that goes straight through the center. Such a nonsingluar double vortex is illustrated in Figure 7. These textures are characterized by topological quantum numbers, but the quantum numbers are stabilized not by their topological structure, but by the external symmetry-breaking perturbation such as the imposed rotation. Recent work by Parts et al. [154] has shown regions of singly and doubly quantized vortices coexisting in the same rotating containers of A-phase superﬂuid. The nuclear spin of the pair adds an extra complication to the classiﬁcation of line defects in the A phase. If the direction of quantization of the spin is rotated by an angle π about a perpendicular axis, the spin wave function changes sign. This sign change can compensate for the sign change which occurs for a ±π rotation about the l direction. There are therefore two more equivalent points for each pair of equivalent points in SU (2). Since the combination of two ±π rotations of the orbital system about l and two π rotations of the real vector d gives a zero or 2π rotation of the orbital angle-axis system and zero rotation of the nuclear system, the homotopy group is isomorphic to the group Z4 , the group of multiplications of fourth roots of unity. The topological quantum numbers of linear defects can take on the four values, which are assigned the numbers 0, ±1/2, 1. This assignment of the quantum numbers ensures that the familiar single vortex, in which the phase of the order parameter changes by 2π round the vortex, has quantum number unity. The half-integer quantum numbers can correspond to vortices with circulation ±h/4m3, combined with disclinations in the direction of d by an angle π. Such a half-integer vortex is shown in Figure 8. As it is shown there, l is out of the plane, and the phase rotates by π on a path round the vortex. At the same time the direction d of spin quantization also rotates by π, so that the apparent discontinuity across the half-plane is removed by two factors of −1. The hyperﬁne energy of such a defect will be large, because d cannot match the direction of l at large distances. A survey of the properties of vortices in superﬂuid 3 He can be found in a paper by Krusius [155]. 6.2 Other defects and textures In accordance with the discussion in Section 1.3, topologically stable interface defects exist only when the states on the two sides of the defect have a diﬀerence in discrete symmetry. At the A − B phase boundary the two

824

Topological Aspects of Low Dimensional Systems

Fig. 8. Half-integer vortex for the A phase, with a π rotation of the orbital angular momentum or phase variable combined with a π disclination of the spin axis.

phases have the same free energy, but diﬀerent and incompatible discrete symmetries, such that one symmetry cannot pass smoothly into the other. The interface between them is therefore topologically stable. One example of such a phase coexistence is provided by rotating B phase He. It may be energetically favorable for the vortex cores to take advantage of the breaking of time reversal symmetry produced by the rotation to undergo a transition to the A phase. There have been both theoretical [156, 157] and experimental [158] studies of this. 3

In general a vortex sheet, separating two regions of the same superﬂuid with diﬀerent uniform velocities, is both topologically and energetically unstable, and will break up into a row of vortex lines, separated by smoothly joined regions between them. As was mentioned in Section 1.3, there is an exception to this in the A phase of 3 He, which has been the subject of theoretical and experimental work recently [10, 11]. The vortex sheet that this work describes is stabilized by the locking of the orbital vector l to the spin orientation d parallel and antiparallel on the two sides of the vortex sheet, so that l · d has the values ±1 on the two sides of the vortex sheet, and both of these orientations minimize the dipolar hyperﬁne energy. This is topologically stable, because the two diﬀerent orientations of the orbital angular momentum relative to the spin give two disjoint stable regions for the order parameter. A texture that joins two regions of opposite orbital angular mo¯ k1 /2m3 , v2 = h ¯ k2 /2m3 , mentum, each with diﬀerent ﬂow velocities v1 = h across a planar vortex sheet, could have the unit vectors e(1) , e(2) given by

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the column vectors     (1 − p) cos(k1 x) + p cos(k2 x) (1 − p) sin(k1 x) + p sin(k2 x)  −(1 − p) sin(k1 x) + p sin(k2 x)  ,  (1 − p) cos(k1 x) − p cos(k2 x)  . −2 p(1 − p) sin[ 12 (k2 + k1 )x] 2 p(1 − p) cos[ 12 (k2 + k1 )x] (6.3) Here p is a parameter that interpolates smoothly from zero to unity on the two sides of the vortex sheet, and the spin axis d is the same on the two sides of the sheet. The orbital anglular momentum vector l is the vector product of these two vectors, and reverses its direction as p goes from zero to unity. In the rotating systems that have been studied experimentally there seems to be a single vortex sheet that spirals through the system in such a way as to make the circulation density almost uniform, just as a vortex array gives an almost uniform circulation density in the more usual situation. Point defects are characterized by the way the order parameter behaves on a closed surface, with the topology of a sphere, surrounding the defect. The possible topologies are characterized by the homotopy group π2 , which classiﬁes the diﬀerent mappings of the order parameter on the surface of a sphere S 2 . The mappings of the surface of the sphere onto the interior of the sphere representing rotations SO(3) of three-dimensional axes, for the combination of orbital order parameter and phase in the A phase, are all trivial. The only nontrivial topology arises when the sphere surrounding the point defect is mapped onto the sphere that represents the orientation of the spin axis d in the A phase. Such a mapping can be classiﬁed in terms of the winding number deﬁned in equation (1.3), which describes how many times one sphere is wrapped around the other. Boundary conditions can stabilize defects and textures which are unstable in a uniform medium. One obvious example of this is that 4 He in a rotating container has a vortex array as its equilibrium state. Similarly in the A phase of 3 He in a rotating container the equilibrium state either has an array of single vortices, or a texture made up of doubly quantized vortices like the one shown in Figure 7. The singly quantized vortices have a higher core energy than the texture, but the doubling of the circulation increases the energy contribution from regions a little further from the center of the texture. A particularly interesting case of a texture imposed on the A phase by boundary conditions was discussed by Mermin [159]. In the A phase the orbital order parameter l tends to be lined up normal to the walls of the container. However, if one considers a simple container with the topology of a sphere there is no continuous way of arranging the axes e(1) , e(2) so that they are always parallel to the surface of the sphere. Singularities of some sort have to be introduced on the surface. If there is a single singularity on the surface of the sphere, the rest of the surface, apart from

826

Topological Aspects of Low Dimensional Systems

the immediate neighborhood of the singular point, can be opened up into a disk, and constant directions of e(1) , e(2) corresponds to a rotation of these axes by 4π about the singularity. Alternatively one can have two singularities, opening up the sphere into something which is topologically equivalent to a ﬁnite cylinder, and parallel axes on this cylinder gives a phase change of 2π around each of the singularities, like the compass bearings we use on the earth’s surface, with their singularities at each pole. If there are two singly quantized vortices on the surface these must extend into the interior of the container. The two surface singularities must be connected by a line singularity. These two singularities on the surface can approach one another and merge, to form a single doubly quantized vortex on the surface, with no singular lines in the interior, only a texture which is known as a boojum [159]. Close to the surface the axes e(1) and e(2) rotate through an angle 4π, as they do in the outer parts of the texture shown in Figure 7. Further away from the surface the orientation of the vectors will change round a loop in the kind of way that they do on one of the inner loops round the center of this ﬁgure.

7

Liquid crystals

7.1 Order in liquid crystals Liquid crystals, or mesophases, have some of the properties of solids and some of liquids. Typically they are anisotropic in space, but lack the rigidity of crystals, and can ﬂow more or less like liquids. The molecules of a liquid crystal are generally quite complicated, but a physicist usually thinks of them as inﬂexible rod-like or disk-like objects whose two ends are indistinguishable – this is not because the two ends of the molecule are in fact indistinguishable, but because, although the molecules are aligned with their axes parallel to a certain direction, they are randomly pointing in two opposite directions. A survey of the properties of liquid crystals can be found in the book of de Gennes and Prost [160]. The simplest liquid crystal phase is known as the nematic phase. In nematics there is a preferred direction for the orientation of the molecular axis, but the order is otherwise like that of a liquid, with no ordering in space. The order parameter of the nematic phase is known as a director, which is like a vector, but with the two opposite directions equivalent. The space in which a director lives is a projective 2-sphere, a sphere with an equivalence relation between any two diametrically opposite points. In the equilibrium state of a bulk nematic the director is aligned everywhere in the same direction, but boundary conditions may impose other conditions on the director, such as that it should make a ﬁxed angle with the boundary. Such boundary conditions mean that there will in general be a space-dependence

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827

of the director, and there will usually be some sort of surface or interior defects asociated with changes in the director. For the nematic phase the free energy density depends on the rate of change of the director n, according to the formula F=

1 1 1 K1 (∇ · n)2 + K2 (n · ∇ × n)2 + K3 [n × (∇ × n)]2 , 2 2 2

(7.1)

where the three Frank constants K1 , K2 , K3 control the free energies of splay, twist and bend respectively. Most nematics are uniaxial, but there have been recent experimental studies of biaxial nematics, and these are particularly interesting from the topological point of view. We can think of the molecules of these as like rectangular disks, which, under suitable conditions, get stacked in such a way that both the normal to the disk and the orientation of the disk within the plane are determined. Their orientation is therefore deﬁned by a triplet of axes, but with equivalence between those orientations that diﬀer by a π rotation about one of the three axes. The order parameter therefore lives in the space SO(3)/D2 , where D2 is the dihedral group of π rotations about the coordinate axes. In the equilibrium state of a cholesteric liquid crystal the molecules are aligned along a direction that varies periodically. There are planes in which the director is constant, lying in the plane, but the director rotates uniformly around the axis perpendicular to these planes, maintaining a π/2 angle with the axis. Thus the director can be written as ˆ 0) , ˆ 0 ) + ˆj sin(2πr · k/l n(r) = ˆi cos(2πr · k/l

(7.2)

ˆ form a triplet of mutually perpendicular unit vectors. There is where ˆi, ˆj, k therefore a ﬁnite period l0 /2 in space between planes where the molecules are similarly oriented; the factor 1/2 arises because we are dealing with a director, not a vector. The cholesteric has a broken U (1) symmetry associated with this changing angle of orientation, as well as the director order parameter. From a practical point of view the cholesterics are particularly useful, as the periods of rotation can be comparable with optical wavelengths and can give dramatic optical interference eﬀects which can be controlled by external electric ﬁelds or varying temperatures. Blue phases of cholesteric liquid crystals appear to have a threedimensional ordering of the director, generally in a cubic lattice. Smectic phases of liquid crystals have some sort of spatial order as well as the orientational order of nematics. In the two phases I discuss here, smectic A and smectic C, the molecules lie on planes with a regular spacing between them, but there is no regular ordering of the molecules within the planes. The molecules therefore have one-dimensional ordering but no ordering in the other two dimensions. In smectic A the director along which

828

Topological Aspects of Low Dimensional Systems

the molecules are preferentially aligned is normal to the planes on which the molecules lie, but in the smectic C phase the molecules are tilted relative to the normal to the planes in which the molecules lie. There are also columnar phases of liquid crystals, in which there is spatial order in two dimensions, with the disk-like molecules stacked in columns. There is no long-range order within the columns, but the columns are arranged in a regular two-dimensional array. 7.2 Defects and textures There are many similarities between the topological theory of defects in liquid crystals and the theory of defects in superﬂuid 3 He, which was discussed in Section 6. These theories were developed in parallel about twenty years ago in a series of papers by Toulouse et al. [132, 150, 161] and by Volovik and Mineev [151,162]. A review of the topological study of defects in liquid crystals has been written by Kl´eman [163], and there is a more recent review by Kurik and Lavrentovich [164]. Defects in liquid crystals can be seen with a polarizing microscope, and so they have been studied for a long time. In fact the nematics get their name from the worm-like structures which can be seen threading them – these are simply disclinations. The director, the order parameter of uniaxial nematic liquid crystals, lies on the surface of a projective sphere, a 2-sphere on which opposite points are regarded as identical. The only closed paths that cannot be shrunk to a point in this space are those paths that join two opposite points of the sphere. The homotopy group π1 of this space is therefore the group Z2 with two elements. There is one single type of line defect in a uniaxial nematic, which is a disclination around which the director rotates by an angle π. A disclination of this sort was shown for the spin of the A phase of superﬂuid 3 He in Figure 8. The energy of a disclination in a nematic liquid crystal is much smaller than the energy of a disclination in a solid, because there is no regular arrangement of molecules to be disturbed by a strain ﬁeld. Instead, equation (7.1) shows that the energy density depends on ﬁelds that fall oﬀ at least as fast as the reciprocal of the distance from the disclination core, so the energy of a disclination in a nematic liquid crystal, like the energy of a dislocation in a solid, is proportional to the length of the system in the direction of the disclination times the logarithm of the width of the system. The point defects for a uniaxial nematic are deﬁned by the mappings of a sphere surrounding the defect onto the projective sphere that represents the director. The situation is rather similar to that discussed for the A phase of superﬂuid 3 He discussed in Section 6.2, and the only diﬀerence that the sign ambiguity of the director makes is that the quantum numbers are ambiguous in sign. The quantum number is given again by equation (1.3), apart from this sign ambiguity, and the simplest form of defect is

D.J. Thouless: Introduction to Topological Quantum Numbers

829

again the hedgehog, with quantum number unity, with, for example, the director normal to the surface of the sphere at all points. The energy of a hedgehog is proportional to the radius of the system, as can be seen from equation (7.1), since the divergence of n is 2/r for this simple conﬁguration of the hedgehog. The order parameter for biaxial nematics deﬁnes a set of axes in space, as does the order parameter for the A phase of superﬂuid 3 He, but with the vital diﬀerence that rotation by π about any of the axes produces an equivalent orientation of the order parameter. This was discussed in by Po´enaru and Toulouse [165]. This is particularly interesting, as it turns out that the residual symmetry of the order parameter is described by a noncommutative group, and the noncommutativity has some important consequences. In the previous subsection I referred to the space of this order parameter as SO(3)/D2 , but actually the topology of paths in SO(3) is better described in terms of the covering group SU (2), since SO(3) does not distinguish rotation by 2π about an axis from no rotation, yet they are topologically distinct. There are equivalences of the order parameter given by the subgroup which is generated by the π rotations about the axes of the molecular ordering. This is the quaternion group Q, which is isomorphic to the eight elements e, −e, ±iσx, ±iσy , ±iσz , where the σs are the Pauli matrices and e is the 2 × 2 identity matrix. Because this is a nonabelian group, with noncommuting elements, the defects behave in a diﬀerent way from those we have discussed earlier. The space of the order parameter is SU (2)/Q, and the topologically distinct closed paths in this space are those that go from e to each of the ﬁve classes of this group. Rotations by ±π about the same axis are in the same class, because they are related by unitary transformations such as −iσz = (iσx )(iσz )(−iσx ) .

(7.3)

Therefore a path from the identity element e to iσz is not distinct from a path from e to −iσz , but it is distinct from paths from e to iσx or from e to iσy . The linear defects for a biaxial nematic are therefore of four sorts, corresponding to the four classes of the quaternion group apart from the identity. Three are disclinations of strength 1/2, corresponding to a π rotation about each of the three symmetry axes of the molecules, and one is a disclination of strength 1 corresponding to a 2π rotation about any axis. One possible way the molecular orientation can change on a loop round the defect is shown in Figure 9. These may look quite diﬀerent if the molecules are diﬀerently oriented relative to the defect, but the diﬀerent appearances can be continuously transformed into one another. For example, if each of the molecules shown in the ﬁgure is rotated by π about an axis through its

830

Topological Aspects of Low Dimensional Systems

Fig. 9. Possible form for a disclination in which the orientation of molecules in a biaxial nematic rotates by π about the normal to the plane of the molecule.

centre in a ﬁxed direction in the plane of the ﬁgure, the sense of rotation about the molecular axis is reversed. The law of combination of such defects is given by the class multiplication law of the group. Two disclinations from the class {±σz } can combine together either to annihilate one another, giving the group element e, or can combine to give a disclination of strength 1, corresponding to the group element −e. The same pair of defects can therefore be combined together in two completely diﬀerent ways. One disclination from the class {±σx } and one from the class {±σy } will combine to give one from the class {±σz }. Another result of this special behavior of disclinations in biaxial nematics was pointed out by Toulouse [166]. Two diﬀerent π disclinations cannot cross one another without generating a linking 2π disclination, as a careful examination of the changes generated by going round loops will show, so the disclinations give biaxial nematics a topological stiﬀness. The topology of cholesterics is similar to that of biaxial nematics. The director at a given point is determined, in equation (7.2), by the triplet of ˆ and this is invariant under sign reversal of any two of the vectors ˆi, ˆj, k,

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831

Fig. 10. Disclinations in a cholesteric liquid crystal. The dashes show curves in which the director is in the plane, while the dots show where the director is normal to the plane; in between the director is at an intermediate angle. A λ+ is shown in (a), a τ + in (b), a λ− in (c) and a τ − in (d). One particular plane normal to ˆ axis has the director oriented as shown in (e) for the χ disclination, and the k the director rotates steadily as one moves out of the plane, so that the surfaces of constant direction form a screw dislocation. In (f) a 2π disclination dissociated into two λ disclinations is shown.

three vectors, so the space in which the order parameter lives is SO(3)/D2 , for which the covering space describing the topology is SU (2)/Q. There are again four topologically distinct types of line defects corresponding to the four classes of the quaternion group Q distinct from the identity. Disclinations in a cholesteric liquid crystal were analysed by Kl´eman and Friedel [167] in terms of Volterra processes, and their classiﬁcation is widely

832

Topological Aspects of Low Dimensional Systems

Fig. 11. A π disclination for a smectic A is shown in (a). When a disclination of opposite sign is added to this with a displacement of one lattice vector from the original disclination, the dislocation shown in (b) is obtained.

used. The disclinations τ + , λ+ are made by removing all the material ˆ plane (so that the director in beyond a certain plane normal to the k that plane is everywhere in the same direction, which we identify as the ˆ direction). If the remaining material is folded about an axis parallel x to the director so that the exposed surfaces are rejoined smoothly, a λ+ disclination is obtained, as shown in Figure 10a. In this case the director is continuous even at the core of the disclination. If the remaining material is folded about an axis perpendicular to the director and the exposed surfaces are rejoined smoothly the τ + disclination shown in Figure 10b is obtained. In this case the director has singular behavior at the disclination core, and, as a result, the τ disclination core has a higher energy than the λ disclination core. The λ− , τ − disclinations can be constructed by the Volterra process in which a semi-inﬁnite cut is made in a plane of constant director. If the cut terminates on an axis parallel to the director the λ− disclination is obtained, but if it terminates on a line perpendicular to the director a τ − disclination is obtained. The two faces exposed by the cut are then opened up by a π rotation until they form a plane, the remaining space is ﬁlled smoothly with undeformed material, then the system is allowed to relax until the angle is 2π/3. The resulting disclinations are shown in Figures 10c and d. Again, the director is continuous at the core for the λ disclination and discontinuous at the core for the τ disclination. This is a manifestation of the fact that the λ± disclinations both belong to the class {±iσx } of the homotopy group π1 (a rotation of the pattern about the direction of the director), while the τ ± disclinations belong to the class {±iσy }

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(rotation about an axis in the plane of constant director perpendicular to the director). Diﬀerent patterns may be obtained if the disclination bends, or if there are other disclinations in its neighborhood, but the assignment to the class of this group is invariant. The χ disclination could be formed by making a cut normal to the planes of constant phase and introducing a twist π of the director in any circle that goes round the core of the disclination. One of these planes, originally of constant director, is shown in Figure 10e. This defect could also be regarded as a screw dislocation with Burgers vector l0 /2, since surfaces of constant direction of the director have the form that one gets by introducing a screw dislocation into a set of parallel planes. It belongs to the class {±iσz }, ˆ the normal to the planes of constant director. since the rotation is about k, Like the τ disclination this has singular behavior of the director at the core. In principle it could dissociate into a λ − τ pair of disclinations, and the direction of the π change in angle can be reversed by bends in the line or the presence of other defects in the neighborhood. Figure 10f shows a defect belonging to the class {−e}, a 2π disclination, in this case dissociated into a λ+ − λ− pair, so that there is no singular behavior of the director at the core. This pair of disclinations half a period apart forms the same pattern that one would get with an edge dislocation, an extra period of the pattern inserted to the right of the position of the λ− . Neither biaxial nematics nor cholesterics have topologically stable point defects, since the mappings of a sphere onto SO(3) are all trivial. In the smectic A phase the direction of the molecules and the normal to the planes on which the molecules lie are the same, so the director has similar properties to the director of a uniaxial nematic. In the smectic C phase there is a pair of directions deﬁned by the normal to the plane and the direction of the molecules. This is unchanged by a π rotation about the normal to the two, so the space of the directional order parameter is SU (2)/Z2 . Because the molecules are arranged on approximately planar surfaces there is a complicated interplay between directional order and positional order of these surfaces. Both disclinations and dislocations can exist. In a regular solid the energy per unit length of a dislocation is proportional to the logarithm of the cross sectional area of the material, while the energy per unit length of a disclination is proportional to the cross-sectional area. In a smectic liquid crystal the positions of molecules can adjust to remove strain energy, but the spacings between surfaces must remain constant. As a consequence, the normal to one surface must also be normal to the neighboring surfaces. The splay ∇ · n of a smectic A is given by the derivatives of the director in the plane of the surface. For a dislocation the splay falls oﬀ like 1/r2 , while for a disclination it falls oﬀ like 1/r, and so the energy per unit length of a dislocation is independent of the sample size, and the energy per unit length of a disclination depends logarithmically on

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the cross-sectional area of the sample. Figure 11a shows a π disclination in a smectic A liquid crystal. If a second disclination of opposite sign is formed by cutting away everything to the right of the middle of the surface of atoms immediately to the right of the disclination core, and then rejoining the two halves of this surface, the edge dislocation shown in Figure 11b is formed. Thus an edge dislocation can be regarded as the sum of two opposite disclinations displaced by a lattice spacing from one another. There is therefore an interplay between the algebras of dislocations and of disclinations. The condition that the surfaces on which the molecules lie should have common normals, and, indeed, common centers of curvature, imposes severe restraints on the nature of point defects for A phase of a smectic. The only form of a true point defect is a hedgehog surrounded by spherical surfaces. More complicated structures may have terminating lines of defects. The C phase of a smectic cannot have isolated point defects, since there are no nontrivial mappings of a sphere onto the order parameter. References [1] Onsager L., Nuovo Cimento 6 (1949) 249-250. [2] Vinen W.F., The detection of single quanta of circulation in liquid helium II, Proc. Roy. Soc. (London) A 260 (1961) 218-236. [3] Rayﬁeld G.W. and Reif F., Evidence for the creation and motion of quantized vortex rings in superﬂuid helium, Phys. Rev. Lett. 11 (1963) 305-308. [4] Dirac PAM, Quantized singularities in the electromagnetic ﬁeld, Proc. Roy. Soc. (London) 133 (1931) 60-72. [5] Dirac PAM, The theory of magnetic poles, Phys. Rev. 74 (1948) 817-830. [6] Piccard A. and Kessler E., Determination of the ratio between the electrostatic charges of the proton and of the electron, Arch. Sci. Phys. Nat. 7 (1925) 340-342. [7] Petley B.W., The fundamental physical constants and the frontier of measurement (A. Hilger, Bristol, 1985) pp. 282-287. [8] Donnelly R.J., in The collected works of Lars Onsager: with commentary (World Scientiﬁc, Singapore, 1996) pp. 693-696. [9] London F., Superﬂuids, Vol. II, (1954) pp. 151-155. [10] Thuneberg E.V., Introduction to the vortex sheet of superﬂuid 3 He, Physica B 210 (1995) 287-299. ¨ Ruutu V.M.H., Koivuniemi J.H., Krusius M., Thuneberg E.V. and [11] Parts U., Volovik G.E., Measurements on the vortex sheet in rotating superﬂuid 3 He-A, Physica B 210 (1995) 311-333. [12] Pitaevskii L.P., Vortex lines in an imperfect Bose gas, Zhur. Eksp. Teor. Fiz. 40 (1961) 454-477 [Translation in Soviet Physics JETP 13 (1961) 451]. [13] Gross E.P., Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961) 454-477. [14] Hall H.E. and Vinen W.F., The rotation of liquid helium II. I: Experiments on the propagation of second sound in uniformly rotating helium II, Proc. Roy. Soc. (London) A 238 (1956) 204.

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[125] Nelson D.R., Defect-mediated phase transitions, in Phase transitions and critical phenomena Vol. 7, pp. 1-99, edited by C. Domb and J.L. Lebowitz (Academic Press Ltd., London and New York, 1983). [126] Rudnick I., Critical surface density of the superﬂuid component in 4 He ﬁlms, Phys. Rev. Lett. 40 (1978) 1454-1455. [127] Bishop D.J. and Reppy J.D., Study of the superﬂuid transition in two-dimensional 4 He ﬁlms, Phys. Rev. Lett. 40 (1978) 1727-1730. [128] Bishop D.J. and Reppy J.D., Study of the superﬂuid transition in two-dimensional 4 He ﬁlms, Phys. Rev. B 22 (1979) 5171-5185. [129] Ambegaokar V., Halperin B.I., Nelson D.R. and Siggia E.D., Dynamics of superﬂuid ﬁlms, Phys. Rev. B 21 (1980) 1806-1826. [130] McQueeney D., Agnolet G.T. and Reppy J.D., Surface superﬂuidity in dilute 4 He–3 He mixtures, Phys. Rev. Lett. 52 (1984) 1325-1328. [131] Polyakov A.M., Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills ﬁelds, Phys. Lett. B 59 (1975) 79-81. [132] Kl´eman M., Relationship between Burgers circuit, Volterra process and homotopy groups, J. Phys. Lett. (Paris) 38 (1977) L199-202. [133] Nabarro F.R.N., Theory of crystal dislocations (Clarendon Press, Oxford, 1967). [134] Young A.P., Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19 (1979) 1855-1866. [135] Halperin B.I. and Nelson D.R., Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978) 121-124 and 1519. [136] Nelson D.R. and Halperin B.I., Dislocation-mediated melting in two dimensions, Phys. Rev. B 19 (1979) 2457-2484. [137] Grimes C.C. and Adams G., Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons, Phys. Rev. Lett. 42 (1979) 795-798. [138] Gallet F., Deville G., Valdes A. and Williams F.I.B., Fluctuations and shear modulus of a classical two-dimensional electron solid: experiment, Phys. Rev. Lett. 49 (1982) 212-215. [139] Morf R.H., Temperature dependence of the shear modulus and melting of the twodimensional electron solid, Phys. Rev. Lett. 43 (1979) 931-935. [140] Strandburg K.J., Two-dimensional melting, Rev. Mod. Phys. 60 (1988) 161-207. [141] Beasley M.R., Mooij J.E. and Orlando T.P., Possibility of vortex–antivortex pair dissociation in two-dimensional superconductors, Phys. Rev. Lett. 42 (1979) 11651168. [142] Doniach S. and Huberman B.A., Topological excitations in two-dimensional superconductors, Phys. Rev. Lett. 42 (1979) 1169-1172. [143] Hebard A.F. and Fiory A.T., Evidence for the Kosterlitz-Thouless transition in thin superconducting aluminum ﬁlms, Phys. Rev. Lett. 44 (1981) 291-294. [144] Fiory A.T., Hebard A.F. and Glaberson W.I., Superconducting phase transitions in indium/indium-oxide thin-ﬁlm composites, Phys. Rev. B 28 (1983) 5075-5087. [145] Hebard A.F. and Fiory A.T., Critical-exponent measurements of a two-dimensional superconductor, Phys. Rev. Lett. 50 (1983) 1603-1606. [146] Minnhagen P., The two-dimensional Coulomb gas, vortex unbinding, and superﬂuid–superconducting ﬁlms, Rev. Mod. Phys. 59 (1987) 1001-1066. [147] Huberman B.A. and Doniach S., Melting of two-dimensional vortex lattices, Phys. Rev. Lett. 43 (1979) 950-952. [148] Fisher D.S., Flux-lattice melting in thin-ﬁlm superconductors, Phys. Rev. B 22 (1980) 1190-1199.

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[149] Abraham D.A., Lobb C.J., Tinkham M. and Klapwijk T.M., Resistive transition in two-dimensional arrays of superconducting weak links, Phys. Rev. B 26 (1982) 5268-5271. [150] Toulouse G. and Kl´eman M., Principles of a classiﬁcation of defects in disordered media, J. Phys. Lett. France 37 (1976) L149-151. [151] Volovik G.E. and Mineev V.P., Investigation of singularities in superﬂuid He3 and liquid crystals by homotopic topology methods, Zhur. Eksp. Teor. Fiz. 72 (1977) 2256-2274 [translation in Soviet Phys. JETP 45 (1977) 1186-1196]. [152] Volovik G.E., Exotic Properties of Superﬂuid 3 He (World Scientiﬁc, Singapore, 1992). [153] Anderson P.W. and Toulouse G., Phase slippage without vortex cores: vortex textures in superﬂuid 3 He, Phys. Rev. Lett. 38 (1977) 508-511. ¨ Avilov V.V., Koivuniemi J.H., Krusius M., Ruohio J.J. and Ruutu [154] Parts U., V.M.H., Vortex arrays of coexisting singly and doubly quantized vortex lines in 3 He-A, Czechoslovak J. Phys. 46 (1996) 13-14. [155] Krusius M., The vortices of superﬂuid 3 He, J. Low Temp. Phys. 91 (1993) 233-273. [156] Thuneberg E.V., Identiﬁcation of vortices in superﬂuid 3 He-B, Phys. Rev. Lett. 56 (1986) 359-362. [157] Salomaa M.M. and Volovik G.E., Topological transition of v-vortex core matter in 3 He-B, Europhys. Lett. 2 (1986) 781-787. [158] Kondo Y., Korhonen J.S., Krusius M., Dmitriev V.V., Mukharsky Y.M., Sonin E.B. and Volovik G.E., Direct observation of the nonaxisymmetric vortex in superﬂuid 3 He-B, Phys. Rev. Lett. 67 (1991) 81-84. [159] Mermin N.D., Surface singularities and superﬂow in 3 He–A, in Quantum Fluids and Solids, edited by S.B Trickey, E.D. Adams and J.W. Dufty (Plenum Press, New York, 1977), pp. 3-22. [160] de Gennes P.G. and Prost J., The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993). [161] Kl´eman M. and Michel L., Spontaneous breaking of Euclidean invariance and classiﬁcation of topologically stable defects and conﬁgurations of crystals and liquid crystals, Phys. Rev. Lett. 40 (1978) 1387-1390. [162] Volovik G.E. and Mineev V.P., Line and point singularities in superﬂuid He3 , Pis’ma Zh. Eksp. Teor. Fiz. 24 (1976) 605-608 [translation in JETP Lett. 24 (1976) 561-563]. [163] Kl´eman M., Points, Lines and Walls (John Wiley & Sons, Chichester, 1983). [164] Kurik M.V. and Lavrentovich O.D., Defects in liquid crystals: homotopy theory and experimental studies, Usp. Fiz. Nauk 154 (1988) 381-431 [translation in Sov. Phys. Usp. 31 (1988) 196-224]. [165] Po´enaru V. and Toulouse G., The crossing of defects in ordered media and the topology of 3–manifolds, J. Phys. (Paris) 38 (1977) 887-895. [166] Toulouse G., On biaxial nematics, J. Phys. Lett. (Paris) 38 (1977) L67-68. [167] Kl´eman M. and Friedel J., J. Phys. France Colloq 30 (1969) 43.

SEMINAR 3

GEOMETRICAL DESCRIPTION OF VORTICES IN GINZBURG-LANDAU BILLIARDS

E. AKKERMANS Laboratoire de Physique des Solides and LPTMS, 91405 Orsay Cedex, France and Physics Dept. Technion, Israel Institute of Technology, Haifa 32000, Israel

Contents 1

Introduction

845

2 Differentiable manifolds 2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Differential forms and their integration . . . . . . . . . . 2.3 Topological invariants of a manifold . . . . . . . . . . . 2.4 Riemannian manifolds and absolute differential calculus 2.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

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846 846 847 853 855 858 860

3 Fiber bundles and their topology 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Local symmetries. Connexion and curvature . . . . 3.3 Chern classes . . . . . . . . . . . . . . . . . . . . . 3.4 Manifolds with a boundary: Chern-Simons classes 3.5 The Weitzenb¨ ock formula . . . . . . . . . . . . . .

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860 860 861 862 865 869

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4 The dual point of Ginzburg-Landau equations for an infinite system 870 4.1 The Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . 870 4.2 The Bogomol’nyi identities . . . . . . . . . . . . . . . . . . . . . . 871 5 The 5.1 5.2 5.3

superconducting billiard 872 The zero current line . . . . . . . . . . . . . . . . . . . . . . . . . . 873 A selection mechanism and topological phase transitions . . . . . . 874 A geometrical expression of the Gibbs potential for finite systems . 874

GEOMETRICAL DESCRIPTION OF VORTICES IN GINZBURG-LANDAU BILLIARDS E. Akkermans1,2 and K. Mallick2 ,3

1

Introduction

In these notes we discuss the topological nature of some problems in condensed matter physics. This topic has been widely studied in various contexts. In statistical mechanics, the possible stable defects in an ordered system have been classified according to the nature of the order parameter (e.g. scalar, vector, matrix) and the space dimensionality of the system using homotopy groups [1]. Then, the discovery of the quantum Hall effects and the role played by stable integers or rational numbers for systems with few or no conserved quantum symmetries have motivated several topological models of quantum condensed matter systems [2, 4]. A combination of these two ideas of defects classification and microscopic quantum models has been used in the description of superfluid 3 He [5]. Here, instead of trying an exhaustive review of problems where topological ideas may play a role, we present the basic constituents needed in a geometrical description and calculate the related topological numbers. The ideas and methods developed in mathematics and mathematical physics to solve problems in geometry and topology are pretty sophisticated and sometimes expressed in a way unfamiliar to the physicist. We adopt the language of differential geometry to present this subject, since it is adapted to develop some intuition towards more elaborate concepts like fiber bundles, connexions and topological invariants. In the last two sections, we shall discuss the problem of superconducting billiards within the Ginzburg-Landau approximation. This problem is interesting for several reasons. First, it is a non trivial example on which topological methods naturally apply to give an elegant solution to the calculation of the ground state energy. It is also a situation for which one can 1 Lab.

de Physique des Solides, LPTMS, 91405 Orsay Cedex, France. Dept. Technion, Israel Institute of Technology, Haifa 32000, Israel. 3 Service de Physique Th´ ´ eorique, Centre d’Etudes Nucl´eaires de Saclay, 91191 Gif-surYvette Cedex, France. 2 Physics

c EDP Sciences, Springer-Verlag 1999

846

Topological Aspects of Low Dimensional Systems

address the question of transition between different values of a topological number controlled by the boundary of the system. This question is similar to the transition between quantum Hall plateaus. Such problems are not only of academic interest. Our motivation has been triggered by a set of new experimental results obtained on small size aluminium disks [6, 7] in a regime where their radius R is comparable with both the coherence length ξ and the London penetration depth λ. The magnetization, as a function of the applied magnetic field, presents a series of jumps with an overall shape reminiscent of type-II superconductors, although a macroscopic sample of aluminium is a genuine type-I superconductor. These notes end with a theoretical analysis of these experimental results.

2

Differentiable manifolds

2.1 Manifolds A differentiable manifold M of dimension n is a space which looks locally like an open set of Rn . On the vicinity of each point of M one can define a local coordinate system where each point is represented by a set of n real numbers (x1 , ..., xn ). There exists geometrical spaces which are not differentiable manifolds like arithmetical ensembles (Q, Z/5Z), fractals, objects with branching points (Feynman diagrams). On the other hand vector spaces, spheres, projective spaces, matrix groups GL(n, R), SO(2), SO(3) (direct isometries), SU (2), SU (n) (complex isometries) are differentiable manifolds. In order to do differential calculus on a manifold, different coordinate systems are required to be compatible with each other: the local transformations between one system of coordinates to another have to be smooth and invertible. A quantity attached to a manifold is said to be geometrical (or intrinsic) if it is independent of the choice of a coordinate system. Example: The sphere S 2 in Euclidian space R3 can be endowed with local cartesian coordinates, or with spherical coordinates (latitude and longitude). None of these two systems is global, for instance the spherical coordinates are singular at the poles, although from a geometrical point of view all points on the sphere are equivalent. Therefore the special role played by the poles is an artifact of the spherical coordinate system. On the other hand the tangent plane passing through a point of the sphere is an intrinsic object, although its equation looks very different in cartesian and spherical coordinates.

E. Akkermans and K. Mallick: Geometrical Description of Vortices 847

2.2 Differential forms and their integration 2.2.1 Tangent space Consider a manifold M and p one of its points. Let γ be a curve in M passing through p. In a local coordinate system, the curve γ is given by γ : [−1, 1] → M t

7→ (x1 (t), ..., xn (t)).

(1)

For t = 0, γ passes through the point p represented by the coordinates (x1 (0), ..., xn (0)). A tangent vector to the manifold M at the point p is by definition a tangent vector to a curve passing through p. For instance, dγ with components: the curve γ defines a tangent vector at p ~v = dt vi =

i

dx dt

t=0

t=0

·

(2)

If one uses a different set of local coordinates (X 1 , ..., X n ) in the vicinity of p, one finds that the components of ~v are given by dX i i · (3) V = dt t=0 The transformation law from one set of components to the other is ∂X i vj . (4) Vi = ∂xj p We use Einstein’s convention of summation upon repeated indexes. The transformation law of the components of ~v is inverse of that of the partial ∂ derivatives ∂x j . Hence in “classical” (XIX century) mathematical literature, a tangent vector was said to be contravariant and the transformation formula was used as a definition: a tangent vector is an object whose components transform according to (4) under a change of the local coordinate system. The set of all the tangent vectors at p to all curves γ included in M and passing through p is a vector space, called the the tangent space of M at p and denoted by Tp M . It has the same dimension as the manifold itself. With the help of (4), one can define a tangent vector, as an invariant object that does not depend on the coordinate system: ∂ ∂ i i =V · (5) ~v = v ∂xi p ∂X i p

The vectors ei =

∂ ∂xi

i = 1, ..., n p

(6)

848

Topological Aspects of Low Dimensional Systems

are a basis of the tangent space Tp M . This basis is local and depends on the point p. Usually one does not write explicitly the dependence on p. However it is extremely important to keep in mind that two tangent spaces Tp M and Tp0 M at two different points p and p0 are different vector spaces. It is not possible a priori to compare (i.e. add, subtract) two tangent vectors ~v and v~0 at p and p0 . A simple way to understand that is to realise that the transformation law (4) is point dependent. Therefore ~v and v~0 could by chance have the same components in a special coordinate system but different components in another system. So it would be a mistake to say that they are “equal”. The total tangent space T M of M isSthe collection of all tangent vectors at any point of M . Hence T M = p Tp M . There is a natural function π from T M to M called projection: to each ~v ∈ T M is associated p = π(~v ) which is the point of M at which ~v is tangent to M . It can be shown that T M is a manifold of dimension 2n: if ~v belongs to T M one needs n components to specify at which point p the vector ~v is tangent and n more components to specify ~v in Tp M . By definition a vector-field is a smooth field of tangent vectors to the manifold M , i.e. to each point p is associated a vector F~ (p) tangent to M at p. In a local coordinate system a vector-field is expressed as ∂ F~ (x1 , ..., xn ) = F i (x1 , ..., xn ) i · ∂x

(7)

2.2.2 Forms A 0-form is a smooth scalar valued function on the manifold M . A 1form is a linear function on vectors. Consider first the case of R3 . A vector ~v = (x, y, z) in R3 is written in a basis ~v = x~i + y~j + z~k. Let ω be a 1-form on R3 . By linearity, ω(~v ) is given by ω(~v ) = xω(~i) + yω(~j) + zω(~k).

(8)

Thus the question of calculating ω(~v ) boils down to the question of evaluating it on the vectors of the basis ~i, ~j, ~k. In R3 , we define the 1-form dx by dx(~i) = 1, dx(~j) = 0 and dx(~k) = 0 and equivalently for the 1-forms dy and dz. We have shown that ω = ω(~i)dx + ω(~j)dy + ω(~k)dz. The triplet (dx,dy,dz) is a basis for 1- forms called the dual-basis of (~i, ~j, ~k). More generally, one can consider a field of 1-forms on R3 i.e. a 1-form ω with coefficients that vary from point to point. It can be written as ω = A(x, y, z)dx + B(x, y, z)dy + C(x, y, z)dz.

E. Akkermans and K. Mallick: Geometrical Description of Vortices 849 The action of ω on a vector field F~ = f (x, y, z)~i + g(x, y, z)~j + h(x, y, z)~k is ω(F~ )(x, y, z) = Af + Bg + Ch.

(9)

A k-form is a smooth multilinear and antisymmetric function on k-tuples of tangent vectors to M , all of them tangent at the same point. This can be implemented by considering two vectors in the plane, ~v1 = x1~i + y1~j and ~v2 = x2~i + y2~j. Let φ be a 2-form, then, φ(~v1 , ~v2 ) = =

x1 y1 φ(~i,~i) + x1 y2 φ(~i, ~j) + x2 y1 φ(~j,~i) + x2 y2 φ(~j, ~j) (10) (x1 y2 − x2 y1 )φ(~i, ~j)

because the antisymmetry condition implies φ(~i,~i) = φ(~j, ~j) = 0 and φ(~i, ~j) = −φ(~j,~i). The requirement that differential forms be antisymmetric comes from the fact that we need to keep track of orientation. Examples: 1. In R3 the 2-form dx∧ dy is defined by dx∧ dy(~i, ~j) = 1 , dx∧ dy(~j, ~k) = 0 and dx∧ dy(~k,~i) = 0 and equivalently for the 2-forms dy∧ dz and dz∧ dx. 2. More generally, in Rn , the dual basis of the canonical basis (e1 , ..., en ) is denoted by (dx1 , ..., dxn ). It satisfies dxi (ej ) = δji , where δji is the Kronecker delta. One defines the k-form dxi1 ∧ dxi2 ∧ ... ∧ dxik , such that if (~v1 , ~v2 , ..., ~vk ) is a k-tuple of vectors of Rn , the quantity dxi1 ∧ dxi2 ∧ ... ∧ dxik (~v1 , ~v2 , ..., ~vk ) is the k × k determinant of the components of the vectors ~vi along the directions defined by (ei1 , ei2 ... eik ). The geometric interpretation of this number is known: it is the volume of the projection of the parallelepiped generated by (~v1 , ~v2 , ..., ~vk ) on the linear space spanned by (ei1 , ei2 , ..., eik ). From example 2, one can prove that the set of k-forms (dxi1 ∧ dxi2 ∧ ... ∧ dxik ) with i1 < i2 < ... < ik is a basis of the vector space n! . of the k-forms in Rn . Hence the space of k-forms has dimension k!(n−k)! n 1 2 n In particular all n-forms on R are proportional to (dx ∧ dx ∧ ... ∧ dx ), which is nothing but the determinant. On an general manifold M , one constructs k-forms locally for each tangent space Tp M . For instance, a 1-form ω on M is a smooth linear function on Tp M for every point p on M . Similarly, a k-form w on a manifold M is a smooth collection of k-forms on each tangent space Tp M . For 1 n each p one defines a local basis ((dx )p , ...,(dx )p ) dual to the basis (6) ∂ , ..., en = ∂x∂ n p of Tp M . Then any k-form w on M can be e1 = ∂x 1 p written in a local system of coordinates as follows: X (11) w= Ai1 ,...,ik dxi1 ∧ dxi2 ∧ ... ∧ dxik where the Ai1 ,...,ik (p) are real functions. Once again, the p dependence is not explicitly stated.

850

Topological Aspects of Low Dimensional Systems

2.2.3 Wedge-product A 2-form ψ can be constructed by forming the wedge-product ψ = ω1 ∧ ω2 of two smooth 1-forms ω1 and ω2 via: ψ(~u, ~v ) = ω1 (~u)ω2 (~v ) − ω1 (~v )ω2 (~u)

(12)

for any vectors ~u and ~v . More generally, if φ is a k-form and ψ a l-form one can construct a (k + l)-form φ ∧ ψ which acts on (k + l)-tuples of tangent vectors at a point, by antisymmetrizing correctly the product φψ. We shall not write the explicit formula here. It is the same as the one used to construct a fermionic (antisymmetric) wave-function of (k + l) variables starting with two fermionic wave functions of respectively k and l variables. One has in particular (13) φ ∧ ψ = (−1)kl ψ ∧ φ. 2.2.4 The exterior derivative The exterior derivative d generalizes the usual operations of vector calculus. The exterior derivative d transforms k-forms into (k + 1)-forms. Let for instance A be a 0-form (a function), its exterior derivative dA is the 1-form dA =

∂A ∂A 1 dx + ... + n dxn · 1 ∂x ∂x

(14)

If ~v is a tangent vector then dA(~v ) =

∂A i v ∂xi

is the rate of variation of the function A in the direction of ~v . This quantity is usually denoted by (∇A).~v or ∇~v A. We have seen that a k-form w can be written in a local basis X w= Ai1 ,...,ik dxi1 ∧ dxi2 ∧ ... ∧ dxik where the Ai1 ,...,ik are real functions. The exterior derivative operates on w by acting on each of the coefficients Ai1 ,...,ik via (14). For example, for a 1-form φ = A dx + B dy + Cdz, one obtains dφ = dA ∧ dx + dB ∧ dy + dC ∧ dz = (Cy −Bz )dy ∧ dz + (Az −Cx )dz ∧ dx + (Bx −Ay )dx ∧ dy. (15) An important property of the exterior derivative is that it gives 0 when applied twice, (16) d2 φ = 0

E. Akkermans and K. Mallick: Geometrical Description of Vortices 851

for any k-form φ. This follows from the Schwartz identity for partial derivatives: ∂2A ∂2A = · ∂xi ∂xj ∂xj ∂xi There is a Leibniz rule for the exterior derivative of a k-form φ and a l-form ψ (17) d(φ ∧ ψ) = (dφ) ∧ ψ + (−1)k φ ∧ dψ. Examples: In the Euclidean space R3 , a vector field F~ is defined at each point (x, y, z) by F~ = A~i + B~j + C~k, where A, B, C are smooth functions of the coordinates. And let f be a function. Its gradient is the vector field ∇f = fx~i + fy~j + fz~k. Similarly, the application of the exterior derivative d on the 0-form (function) f gives the 1-form df = fx dx + fy dy + fz dz. Hence, to the 1-form df is associated the vector field ∇f . The action of the rotational on the vector field F~ gives another vector field ∇ × F~ = (Cy − Bz )~i + (Az − Cx )~j + (Bx − Ay )~k, and the exterior derivative d operates on a 1-form φ = A dx + B dy + C dz to give the corresponding 2-form dφ = (Cy − Bz )dy ∧ dz + (Az − Cx )dz ∧ dx + (Bx − Ay )dx ∧ dy. Finally, one can check that the divergence of a vector field corresponds to d acting on a 2-form to generate a 3-form. In summary, gradient, curl and divergence result from the application of d to 0-forms, 1-forms and 2-forms respectively. The relations ∇ × ∇A = ~0 and ∇.∇ × F = 0 are simply a consequence of d2 = 0. 2.2.5 Closed and exact forms If φ is a differential form defined on a manifold M with the property d φ = 0, then φ is said to be closed. If it has the property that φ = d ψ for some form ψ on each point in M , then φ is exact. It follows from these definitions and from d2 = 0, that every exact form is closed. But the reciprocal is not true with the important exception that on a simply connected domain M , i.e. a domain in which every closed curve can be continuously deformed to a point through deformations that remain in M , every closed 1-form is exact. We shall discuss later in more detail the 1-form ω=

x −y dx + 2 dy. x2 + y 2 x + y2

It is closed (dω = 0) but not exact on M = R2 \(0, 0).

(18)

852

Topological Aspects of Low Dimensional Systems

2.2.6 Integration of forms A k-dimensional submanifold (or k-chain) of a manifold M of dimension n ≥ k is a subset which can be parametrized with only k coordinates. In this terminology, a 0-chain is a point, a 1-chain is a curve and a 2-chain a surface. k-forms are the right objects to be integrated over k-chains. Indeed k-forms were invented by Elie Cartan for this purpose! (Their similarity with determinants is not by chance: when changing variables in an oriented integral one must multiply the integrand by a determinant and k-forms were built to produce such a determinant by a change of the local coordinate system.) We shall simply consider the integral of 1-forms over 1-chains in the Euclidean space R3 . Let C be an oriented smooth curve (i.e. a 1-chain) parameterized by ~r(t) = (x(t), y(t), z(t)) for t ∈ I = [−1, 1] and let ω = A dx + B dy + C dz be a smooth 1-form on C. Then, Z 1 Z ω= (A(x, y, z)x0 (t) + B(x, y, z)y 0 (t) + C(x, y, z)z 0 (t)) dt. (19) C

−1

Let F~ = A~i + B~j + C~k be the vector field which corresponds to the 1-form ω. We obtain the more familiar expression Z Z F~ .~dr (20) ω= C

C

which represents the work of F~ along the curve C. Exercise: The winding number 1. Show that on the unit circle S 1 parameterized by R r(θ) = (cos θ, sin θ) in the plane, the 1-form ω given by (18) is such that S 1 ω = 2π. 2. Show that if γ is aR path connecting two points P and Q of the plane R2 \(0, 0) the integral γ ω mesures the difference of the polar angles of Q and P , the center of the polar coordinates being (0,0). 3. Deduce from 2. that ifR γ is a closed path that encircles n times the point 1 ω = n. In particular if γ is a closed path that (0,0), W (γ, (0, 0)) ≡ 2π γ does not encircle (0,0), this integral is equal to zero. 4. The mapping W (γ, (0, 0)) defined from the space of closed curves in R2 \(0, 0) to the set of rational integers Z is called the winding number. It allows to classify different type of curves in R2 \(0, 0). It is a simple example of a topological invariant (see Sect. 2.3). Two curves with the same winding number are said to be homologous. 2.2.7 Theorem of Stokes Let M be a compact oriented smooth manifold of dimension n with boundary ∂M (possibly empty) and let ∂M be given the induced orientation [8,9].

E. Akkermans and K. Mallick: Geometrical Description of Vortices 853 For a (n − 1)-form φ, we have Z

Z dφ =

M

φ.

(21)

∂M

R The integral C ω of a k-form is said to be path-independent if the value of this integral depends R only on the boundary ∂C of the oriented k-chain C. This implies that C ω = 0 for every closed k-chain. This property can be used to state the important result: a form ω defined on a manifold M is R exact iff C ω is path-independent on M . For instance, the 1-form (18) is not exact. All these properties are generalizations of well-known results in vector calculus. For instance, a vector field F~ is conservative if F~ = ∇f for some function f (the potential). Let ω be the 1-form associated to F~ . A conservative F~ corresponds to ω = df so that ω is exact. This implies that ω is closed and since dω corresponds to ∇ × F~ , this implies ∇ × F~ = 0 as well. Electrostatics results from the fact that in R3 a closed 1-form is exact, hence the electric field is conservative. Another consequence is the widely used result (e.g. in thermodynamics) that a 1-form φ = pdx + q dy cannot ∂q = ∂p be exact unless ∂x ∂y . 2.3 Topological invariants of a manifold 2.3.1

Motivations

Two manifolds are said to be homeomorphic if there is a continuous mapping from each other, with continuous inverse mapping. A topological invariant is an intrinsic characteristic of a manifold conserved by homeomorphism. These invariants reveal important features and help to classify different types of manifolds. Topological invariants can be numbers, scalars, polynomials, differential forms or more general algebraic sets such as groups, or algebras. Their importance for condensed matter physics was recognized in a seminal paper of Toulouse and Kleman [1] for the study of defects (vortices, nodal lines, textures, anomalies) and their stability as a function of external parameters. The analysis of [1] depends on general characteristics of the system under study (e.g. dimensionality of space, nature and symmetries of the order parameter) and not on the precise form of the equations governing the system. Loosely speaking, different types of defects correspond to different (non homeomorphic) geometrical structures. Therefore topological invariants can help to distinguish between them. Of course this general scheme does not tell how to compute the relevant invariant in a given problem. A nice example is the Aharonov-Bohm effect in an infinite plane where the relevant invariant is the winding number (defined in the exercise of Sect. 2.2.6). A thorough study was done in [10]. In the last chapter of these notes we discuss two dimensional superconductors.

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Topological Aspects of Low Dimensional Systems

There the winding number measures the circulation of the phase of the order parameter around the vortices. In the following paragraphs we merely give a taste of the vast subject of invariants in a manifold.

2.3.2

The Euler-Poincar´e characteristic

The Euler-Poincar´e characteristic χ(M ) of a manifold M is the oldest and the most celebrated topological invariant. We shall explain how to calculate it for a surface. Any given surface S can be tiled by triangles (of different sizes and shape). Such a partition is called a triangulation. For any triangulation of S, denote by V , E and F the number of vertices, edges and faces respectively. Then χ(S) is defined by χ(S) = V − E + F.

(22)

This number does not depend on the chosen triangulation and is a characteristic of the surface S [11]. Exercise: Show for the sphere S 2 that χ(S 2 ) = 2. Show for the torus T 2 that χ(T 2 ) = 0. For a three dimensional manifold one needs to use tetrahedra to perform a “triangulation” and the formula for χ becomes χ = V − E + F − T , where T is the number of tetrahedra. It is possible to generalize this notion to higher dimensional manifolds [9].

2.3.3 De Rham’s cohomology We have seen (Sect. 2.2.5) that any exact form is closed but that the reverse is not true. Forms which are closed but not exact reveal important topological features and help to classify different types of manifolds. The kth cohomology group of De Rham of the manifold M , H k (M ), is the set of closed k-forms modulo the exact k-forms. It is a finite dimensional vector space, its dimension bk is called the kth Betti number. This integer is a topological invariant of the manifold M . A beautiful result [9,12] is that the alternating sequence of Betti numbers is the Euler-Poincar´e characteristics of the manifold, namely χ(M ) =

n X

(−1)r br .

(23)

r=0

This shows that topological properties, as χ, can sometimes be calculated using analytical tools such as differential forms.

E. Akkermans and K. Mallick: Geometrical Description of Vortices 855

2.4 Riemannian manifolds and absolute differential calculus 2.4.1 Riemannian manifolds A Riemannian manifold is a differentiable manifold M with a local scalar ~ are two tangent product defined on each tangent space Tp M . If ~v and w vectors at the same point p then ~ = gij (p)v i wj . h~v |wi ~ p = ~v .w

(24)

In particular the norm of ~v is given by the square-root of gij (p)v i v j . The quantities gij (p) are the local components of the metric and are called the metric tensor. They allow to compute the length L(γ) of a curve γ parameterized as in (1). We recall that dγ dt is the tangent vector to γ at the point p = γ(t). !1/2

1/2 Z L(γ) dxi dxj dt gij (p) = ds. dt dt −1 0 (25) The infinitesimal arc length is denoted by ds. In classical books the metric is written as: (26) ds2 = gij dxi dxj . Z

~ ~ dγ dγ dt h | ip L(γ) = dt dt −1 1

Z

1

=

A geodesic curve between two points p and p0 of M is a path of minimal length. Using variational calculus, one can find the Euler-Lagrange equations for a geodesic curve (x1 (s), ..., xn (s)) parameterized by its arc length s (this is a good exercise!). One obtains a system of differential equations: i j d2 xi i dx dx =0 + Γ jk ds2 ds ds

(27)

where the Christoffel symbols Γijk are given by Γijk

1 = g il 2

∂gjl ∂gkl ∂gjk + − ∂xk ∂xj ∂xl

with g il glj = δji ,

(28)

i.e. the matrix g il is the inverse of the metric tensor. 2.4.2 Covariant derivative. Connexion and curvature forms We have emphasized that on a general manifold there is no way to compare tangent vectors at different points. However, given a smooth vector field, a rather natural question to ask is “What is the infinitesimal variation of a ~ tangential vector field Y~ at point p if one moves in the direction of a vector V at the point p to the manifold?” A possible way out is to immerse the

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Topological Aspects of Low Dimensional Systems

manifold in a larger space where coordinates are defined. But this is not intrinsic: it depends on the surrounding space. One needs an additional structure, called a connexion, that allows to compare tangent vectors at different points and to differentiate vector fields (or more generally tensor fields). This method was invented by Levi-Civita and called absolute differential calculus. It is possible to give a general definition of a connexion without refering to the metric, as these concepts are independent. However we shall restrict ourselves to Riemannian manifolds. A Riemannian manifold can canonically be endowed with a connexion ~ on provided by the Christoffel symbols Γijk (28). Consider a vector field Y ~ along the direction the manifold. One defines its covariant derivative ∇V~ Y ~ of the vector V tangential at the point p to the manifold by i ~ = ∂Y V j + Γi V j Y k ei (29) ∇V~ Y jk ∂xj ∂ th vector of the basis (6) of the tangent space Tp M at where ei = ∂x i is the i ~ is a vector, tangent to the manifold M at the the point p. The quantity ∇V~ Y ~ along the direction of point p. It represents the total rate of variation of Y ~ ~ along V . The first term in (29) is nothing but the convective derivative of Y ~ the direction of V . This term is familiar in hydrodynamics (e.g. in Euler and Navier-Stokes equations). The second term represents the derivative ~ . Such of the vectors ei of the local basis of Tp M along the direction of V a term is familiar from mechanics when one uses non-cartesian coordinates (e.g. polar or spherical), it reflects that the coordinate system is local and point dependent. One can rewrite (29) as follows:

~ ∇V~ Y

~ )ei + Y k ∇ ~ ek = ∇V~ Y i ei = dY i (V V

with ∇V~ ek

= V j ∇ej ek

and ∇ej ek

= Γijk ei .

(30)

It is analogous to the well-known expression in mechanics of the derivative of a vector in the rotating frame d ∂~v ~ d~v = (v i ei ) = + Ω × ~v . dt dt ∂t ~ *Parallel Transport: The formula (29) for the covariant derivative ∇V~ Y allows to compute the variation of a vector field along a given direction. ~0 at a point p0 and a curve γ going from Inversely, from a tangent vector Y p0 to p1 , one can construct from (29) a vector field Y~ along the curve γ ~0 . Such an operation is called the parallel transport of ~ (P0 ) = Y such that Y

E. Akkermans and K. Mallick: Geometrical Description of Vortices 857 ~ is determined by ~0 along γ. The vector field Y Y ~ with V ~ = dγ · 0 = ∇V~ Y dt It is therefore obtained by solving the system of differential equations: 0=

dY i dxj k dxj k ∂Y i dxj + Γijk Y = + Γijk Y j ∂x dt dt dt dt

(31)

~ =V ~ = ~0 . If one considers the special case Y with initial condition Y~ (0) = Y dγ ds where s is the arc-length of the curve γ, equation (31) becomes identical to the formula defining a geodesic (27). The equation of a geodesic is thus ~ = 0 with V~ = dγ . In other words a geodesic is a curve which is ∇V~ V ds parallel transported along itself, as this is well-known for straight lines in Euclidean spaces. Remark: We did not give any explanation for the appearance of the Christoffel symbols in the formula (30) for the covariant derivative of the vectors of the basis. This is in fact the heart of Levi-Civita’s construction. We encourage the reader to develop an intuition on this formula through his/her own readings [11, 13, 14]. A nice (and historical) approach is pro~ at a point p of the vided by surface theory in R3 . A tangent vector V surface is transported to an infinitesimally close point p0 by first bringing ~ has no reason to be tangent to the it parallel to itself from p to p0 . As V ~ on the tangent plane at p0 and obtains a new surface at p0 , one projects V ~ through this operation provides ~ 0 tangent at p0 . The variation of V vector V the formula for the covariant derivative (this is explained in [11]). Another approach is via “Tensor Calculus”: the first term in the r.h.s. of (29) is not a tensor, so one has to add to it something to have a tensorial derivative. The Christoffel symbols are not tensors themselves but adding them to the first term turns the sum into a tensor. This approach is well explained in [15], or in books about General Relativity [16, 17].

We now give a more compact expression for the covariant derivative. ~ is linear in V ~ , i.e. ∇Y ~ can thus be viewed as One first notices that ∇V~ Y ~ produces another tangent vector at p. a linear operator which acting on V ~ Equivalently, ∇Y is tangent-vector valued 1-form and can be written as ~ = dY i + Γi dxj Y k ei . ∇Y jk From this expression we see that ∇ itself acts as a linear operator on the ~ . Recalling that a matrix L acts on Y~ as LY ~ = Li Y k ei , we write: vector Y k ∇=d+ω

(32)

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Topological Aspects of Low Dimensional Systems

where ω is a matrix whose coefficients are 1-forms. Precisely ω has matrix elements (33) ωki = Γijk dxj . The matrix-valued 1-form ω is called the connexion 1-form and ∇ the covariant derivative. The curvature Ω of a Riemannian Manifold is defined by the matrix valued 2-form Ω = dω + ω ∧ ω

(34)

~ Y ~ on the manifold, Ω(X, ~ Y ~ ) is a matrix. Notice that i.e. for two vectors X, in general the connexion 1-form can be written as ω = Aj dxj where the Aj ’s are matrices. Then, ω ∧ ω = when the matrices do not commute.

(35) P i<j

dxi ∧ dxj is non zero

Exercise: Find in a book the formula of the Riemann curvature tensor (see e.g. Spivak [13], Vol II, Chap. 4, p. 189) in terms of the Christoffel symbols. Using (33), verify that Ω defined in (34) is indeed the Riemann tensor. 2.5 The Laplacian We saw that the spaces of k-forms and (n − k)-forms over a manifold of dimension n have the same dimension. It is possible to define a duality transformation ∗, called the Hodge star, between these two spaces. In the Euclidean space R3 the Hodge duality is given by: ∗1 = ∗dx = dy ∧ dz, ∗dy = ∗(dx ∧ dy) = dz, ∗(dy ∧ dz) = ∗(dx ∧ dy ∧ dz) =

dx ∧ dy ∧ dz dz ∧ dx, ∗dz = dx ∧ dy, dx, ∗(dz ∧ dx) = dy 1.

(36)

More generally the duality ∗ operation on a manifold depends on the local metric, and is given by: ∗(dxi1 ∧ dxi2 ∧ ... ∧ dxik ) = |g|1/2 dxik+1 ∧ ... ∧ dxin

(37)

where (i1 , i2 , ..., ik , ik+1 , ..., in ) is a permutation of (1, 2, ..., n) with positive signature and |g| is the determinant of the metric (gij ) (i.e. |g|1/2 is the volume element). If φ is a k-form, one can check (exercise!) that ∗ ∗ φ = (−1)k(n−k) φ. The Hodge duality allows to define a scalar product in the space of k-forms by Z φk ∧ ∗ψk (38) (φk , ψk ) = M

E. Akkermans and K. Mallick: Geometrical Description of Vortices 859

where the integral is over the manifold. Exercise: Show that the scalar product of two 1-forms φ = A1 dx + B1 dy + C1 dz and ψ = A2 dx + B2 dy + C2 dz in R3 is Z (A1 A2 + B1 B2 + C1 C2 )dxdydz. (φ, ψ) = R3

The exterior derivative d maps k-forms onto (k + 1)-forms. One defines the adjoint exterior derivative δ which maps k-forms onto (k − 1)-forms via (φk , dψk−1 ) = (δφk , ψk−1 ). Exercise: 1. Show by integrating by part [18] that in a n-dimensional vector space δ = (−1)nk+n+1 ∗ d∗. 2. Deduce that δ 2 = 0. A co-exact k-form ω satisfies the global condition ω = δφ where φ is a (k + 1)-form. A co-closed k-form ω satisfies δω = 0.

The Laplacian is an operation which takes k-forms onto k-forms and generalizes the usual Laplacian on functions. It is defined by 4 = (d + δ)2 = dδ + δd.

(39)

that if f is a function, one has 4f = Exercise: In R3 show − ∂x 2 f + ∂y 2 f + ∂z 2 f . A k-form ωk such that 4ωk = 0 is called harmonic. For a smooth enough k-form, a necessary and sufficient condition for harmonicity is to be closed and co-closed: 4ωk = 0 ⇐⇒ dωk = 0 and δωk = 0. Hodge’s theorem: On a compact manifold without boundary any k-form ωk can always be decomposed into the sum of an exact form dαk−1 , a coexact form δβk+1 and a harmonic form γk [19, 20]: ωk = dαk−1 + δβk+1 + γk .

(40)

This very important result is well-known in 3d vector analysis of electromagnetism of continuous media as the Helmholtz decomposition according to which for a closed and compact manifold or for a controlled growth ~ can be decomposed as of the fields at infinity, any smooth vector field B ~ ~ ~ ~ B = ∇φ + ∇ × M + H, where H has both vanishing curl and divergence.

860

Topological Aspects of Low Dimensional Systems

Remark: if ωk is closed then dδβk+1 = 0, which implies, as d and δ are adjoint, that δβk+1 = 0. Hence ωk = dαk−1 + γk . Therefore ωk and γk belong to the same cohomology class. Denoting by Harmk (M ) the space of harmonic k-forms on M , the Hodge theorem establishes an isomorphism between the two spaces H k (M ) and Harmk (M ) and according to (23), the Euler-Poincar´e characteristic is given by χ(M ) =

n X

(−1)k dim Harmk (M ).

(41)

k=0

This relation shows that properties of the Laplacian probe the topology of the manifold M [20]. 2.6 Bibliography We do not intend to be exhaustive, we just list books we used and give some (subjective) comments. A classical and outstanding introduction to Geometry in general is [21]. A “modern-classic” more inclined toward topology and analysis is [22]. A good way to start with differential geometry is to study curves and surfaces. A very pleasant book on these topics, full of results, figures and historical comments is [23]. Some books begin with curves and surfaces and then introduce the general concept of n dimensional manifolds. This can be a very helpful point of view since it is not too formal [11, 15, 24]. More recent works are [25] with emphasis on applications, [19] more inclined towards algebra and [20] devoted to analysis on manifolds and well adapted to theoretical physicists. An exhaustive treatment is to be found in [8, 9, 13] and the following volumes. We owe a lot to the review paper [26] written for high energy physics and Yang-Mills theories. For a pedagogic introduction to differential forms, see [27]. 3

Fiber bundles and their topology

3.1 Introduction The concepts of connexion and curvature that we have defined for Riemannian manifolds can be extended to a more general structure called a fiber bundle. A fiber bundle is a manifold X that locally looks like the product of two simpler manifolds. For instance the torus T 2 = S 1 ×S 1 is globally the product of two 1-dimensional circles S 1 . A cylinder is also a global product obius strip is locally the product of an arc of S 1 by [−1, 1] S 1 × [−1, 1]. A M¨ but not globally. More precisely, a fiber bundle is a triplet (X, M, π) where X is a manifold (the total space), M (the base space) a submanifold of X and π (the projection) a smooth function from X to M : π:X

→ M

E. Akkermans and K. Mallick: Geometrical Description of Vortices 861

x

7→ π(x) = p.

(42)

The inverse image π −1 (p) of any point p ∈ M is called the fiber above p and is isomorphic to a given manifold F . Intuitively, a fiber bundle is a collection of identical (isomorphic) manifolds F which depend on a parameter p belonging to the base manifold M . When the fiber F is a vector-space Rn or Cn the bundle is called a vector-bundle (see e.g. [28]). A section s of a fiber bundle is a smooth function which associates to each point p ∈ M a element of the fiber above p. Therefore one has π ◦ s = IdM where IdM is the Identity function in M . Examples: 1. The product M1 × M2 of two manifolds M1 and M2 is a fiber bundle called the trivial bundle. One can take either M1 or M2 to be the base and the other one to be the fiber. 2. The total tangent space of M is a vector bundle also called the tangent bundle. The base space is the manifold M itself and the fiber above the point p is Tp M which is homeomorphic to Rn . The projection is exactly the one defined in Section 2.2.1. And what is a section of the tangent bundle? A vector-field! 3. More generally the space of k-forms is also a vector bundle over the manifold M. The fiber bundles described here are assumed to be locally trivial: for any point p of the base M there is a neighborhood Up such that the restriction of the bundle over Up , i.e. π −1 (Up ), is homeomorphic to Up × F . 3.2 Local symmetries. Connexion and curvature In physical applications one considers vector bundles where the base space is the physical space and the fiber is a representation of a continuous symmetry group, i.e. a Lie group G with Lie algebra G. At each point, the order parameter is an element of the fiber over that point and a local symmetry group acts upon it. The order parameter is precisely a section of a vector bundle. It is natural to ask about the variation of the order parameter when one moves from one point of the base to another. However, although all fibers are homeomorphic to the same vector space, there is no intrinsic way to identify two fibers over two different points p and p0 of the base. In other words, given a section s, the two vectors s(p) and s(p0 ) can not be compared a priori. This is exactly the same problem that we encountered for vectors fields, which are sections of the tangent bundle. The extra-structure needed to compute the variation of a section s along a given direction tangent to the base is called a fiber bundle connexion and is again denoted by ∇. The expression of ∇ is obtained by reinterpreting the formula (32): ∇ = d + ω.

(43)

862

Topological Aspects of Low Dimensional Systems

Here d is the usual operation of differentiation of functions and ω is a matrixvalued 1-form (33) on the base space M that can be written ω = Ai dxi where (x1 , ..., xn ) is a local system of coordinates on the base M . We impose that ω represents at each point an infinitesimal transformation of the group G i.e. the matrices Ai belong to the Lie algebra G. The curvature Ω of the connexion is also given by the same formula as above (34): Ω = dω + ω ∧ ω. Important examples: 1. For the tangent bundle, we take the symmetry Lie group to be the group of invertible matrices GL(n, R). The associated Lie algebra is just the group of all n × n matrices. Connexions on a Riemannian manifold are a special case of fiber bundle connexions. 2. We study now a 2-dimensional physical system M = R2 where at each point an order parameter (or wavefunction) is defined and is a complex number ψ = |ψ|eiχ . Hence the fiber is F = C. The Lie group for Maxwell electromagnetism is U (1) and the Lie algebra is iR. Then, the connexion 1-form (or gauge potential) can be written ω = −iA where A = Ax dx + Ay dy, Ax and Ay being real functions (1 × 1 matrices). Since the Lie group is abelian, the curvature reduces to Ω = dω = −iB where B = (∂x Ay − ∂y Ax ) dx∧ dy is nothing but the magnetic field. The covariant derivative given by (43) is then ∇ = d − iA. This is indeed the differential operator that appears when one studies the Schr¨ odinger equation in a magnetic field, or the Ginzburg-Landau model for a superconductor. 3. In the Gross-Pitaevskii description for rotating superfluid 4 He the con~ × ~r, Ω ~ being here the angular velocity. nexion is the Coriolis term Ω 4. The Yang-Mills connexion is obtained by the same construction as before with a non-abelian Lie group (typically SU (2)). The reader is referred to the original paper of Yang and Mills [29]. The theory of fiber bundles is the right setting for gauge symmetries in high energy physics [30, 31]. 3.3 Chern classes The cylinder and the Moebius strip have both S 1 as base space and [−1, 1] as fiber, but they are not homeomorphic. One would like to classify different types of fiber bundles with given base and fiber. Characteristic classes are cohomology classes of the base space that are topological invariants of the fiber bundle [32]. We shall study the Chern classes for fiber bundles with Lie group G = GL(n, C), the group of invertible complex n × n matrices. The ChernWeil theorem gives an explicit construction of these classes starting from

E. Akkermans and K. Mallick: Geometrical Description of Vortices 863

a connexion 1-form ω and the associated curvature Ω. This theorem also proves that the invariants obtained do not depend on the chosen connexion (see [33] for a most enlightning exposition). We recall that the curvature Ω is a 2-form whose coefficients are n × n matrices with complex coefficients. A polynomial P (Ω) is said invariant if for any matrix g in GL(n, C), P (Ω) = P (g −1 Ωg). For example, det(1 +

i 2π Ω)

iΩ

and tr(e 2π ) are invariant polynomials.

Theorem (Chern-Weil): If P is an invariant polynomial and Ω curvature 2-form, then (i) P (Ω) is a closed differential form (dP = 0). (ii) If ω 0 is another connexion on the same fiber bundle and Ω0 the associated curvature, then there exists a form Q such that P (Ω0 ) − P (Ω) = dQ.

Property (i) shows that P (Ω) defines a cohomology class of the base space M and (ii) proves that this class does not depend on the chosen connexion Ω. P (Ω) is called a characteristic class. In particular, integrals of P (Ω) over cycles (i.e. submanifolds without boundaries) will provide topological invariants [33, 34]. This construction can be generalized to other Lie groups and the polynomial P has a different expression according to the associated the Lie algebra. Here, we shall consider only the Chern classes defined from i Ω = 1 + c1 (Ω) + c2 (Ω) + ... (44) P (Ω) = det 1 + 2π where ci (Ω) is a scalar-valued 2i-form called the ith Chern class. From the expansion of the determinant we obtain the expressions c0

= 1

c1

=

c2

=

i tr(Ω) 2π 1 (trΩ ∧ Ω − trΩ ∧ trΩ) 8π 2

(45)

and ci = 0 when 2i is greater than the dimension of M . Since dP = 0, each Chern form is closed as well: dci (Ω) = 0 . The ith Chern class belongs to the cohomology group H 2i M. A remarkable fact is that the Chern forms define integer cohomology classes: the integral of ci (Ω) over any 2i-cycle C i.e. any oriented submanifold C of M, of dimension 2i and without boundary is an

864

Topological Aspects of Low Dimensional Systems

integer, depending upon C but independent from the connexion 1-form ω over the fiber bundle: Z ci (Ω) = n ∈ Z. (46) C

This integer n is a topological invariant of M called a Chern number. Remark: if P (Ω) is homogeneous of degree r (see e.g. (45)) an explicit formula can be given for the form Q that appears in the Chern-Weil theorem: Z Q=r

1

dtP (ω 0 − ω, Ωt , ..., Ωt )

(47)

0

where Ωt is the curvature associated to the interpolating connexion ωt = tω 0 + (1 − t)ω. Examples: 1. Consider the U (1) abelian bundle described by the curvature Ω = −iB where B is the magnetic field. On a two dimensional base space, we have from (44) B P (Ω) = 1 + 2π and the only non-zero Chern class is c1 (B) = surface without boundary Z 1 B=n∈Z 2π M

B 2π .

Then, if M is a closed

(48)

corresponds to a well-known flux quantization condition [36]. 2. More generally, on a manifold M , all the k-fold exterior products Ω, Ω ∧ Ω, Ω ∧ Ω ∧ Ω etc. (k ≤ 12 dimM ) lead to topological invariants. If dimM = 4, only the two first are relevant I Z → ~ − dS = n ∈ Z Ω ∝ B. M Z Z ~ Bd ~ 4 x = m ∈ Z. Ω ∧ Ω ∝ E. (49) M

To end this section, we discuss the issue of what makes certain physical quantities topological numbers and some others not. To that purpose, we consider two examples taken from geometry and electromagnetism. 1 2 Example 1: Consider the 2-sphere R S of radius2 R. Its curvature is K = R2 . Then the area of the sphere S 2 dS = 4πR is a metric invariant. But

E. Akkermans and K. Mallick: Geometrical Description of Vortices 865 R Ω = S 2 KdS = 4π is a topological invariant and is conserved by smooth deformations of the sphere that change the metric. R

S2

Example 2: For electromagnetism, the electric charge inside a closed surface S is given by Z Z 1 1 → ~ ~ − ∇.EdV. E. dS = 4π S 4π ~ = ρ (charge density). The conservaThis statement is equivalent to ∇.E H → d ~ − E. dS = 0. It is equivalent to the tion of the electric charge in time is dt S Maxwell-Faraday equation. These two relations do not refer to topological invariance although it is sometimes asserted that a topological invariant is obtained by integrating a total divergence. The electric charge is a counterexample of this statement. 3.4 Manifolds with a boundary: Chern-Simons classes 3.4.1 The Gauss-Bonnet theorem We recall some facts concerning a surface S embedded in R3 [23]. To investigate geometrical properties, Euler had the idea to study curves drawn on S. More precisely, he considered normal sections at a point p, i.e. plane curves obtained by intersecting the surface S with its normal plane at a point p. Euler proved that such all normal sections have a curvature κ at p limited by two extremal values κ1 and κ2 called principal curvatures. Moreover, if the principal curvatures are not equal, there is one curve C1 having curvature κ1 and another C2 , perpendicular to the first one, having curvature κ2 . If a normal section makes an angle φ with C1 , its curvature is given by κ = κ1 cos2 φ + κ2 sin2 φ. Exercise: Prove this theorem of Euler (hint: write the local equation of the surface as z = z(x, y) and Taylor-expand to the second order [13, 23]). The product of the two principal curvatures κ1 and κ2 at a point p is denoted by K = κ1 κ2 . The scalar K is the Gaussian Curvature of the surface S at the point p. The Gaussian curvature is conserved under deformations that preserve the metric (e.g. bending a surface). It is a metric invariant, that only depends on the intrinsic geometry of the surface and not on the surrounding space, although the above definition involves this embedding space. Gauss proved this fact by discovering an explicit formula for K in terms of the local metric gij and its derivatives. This is the Theorema Egregium (1827), a most “remarkable theorem” [11, 15, 23]. Suggestion: Do not try to prove the Theorema Egregium. Exercise: Check that the Gaussian curvature of a plane is 0. Prove that

866

Topological Aspects of Low Dimensional Systems

the Gaussian curvature of the sphere S 2 of radius R is K = 1/R2 . Since S 2 and the plane have different Gaussian curvatures it is not possible to make an isometric mapping, even local, between them. There is no faithful plane map of the Earth that would respect the distances.

The Gauss-Bonnet theorem [11] for a closed surface S establishes a relation between the curvature, which is a metric invariant and the EulerPoincar´e characteristic of S: Z Z 1 KdS = χ(S). (50) 2π S RR 1 KdS where dS is the surface element, is sometimes The quantity 2π S called curvatura integra. The Gauss-Bonnet theorem establishes that the curvatura integra is a topological invariant of closed surfaces. It has numerous and profound implications. For example [15], as the reader can check, it implies that no metric of negative curvature can be defined on sphere, or that no metric with strictly positive or strictly negative curvature can be defined on a torus (for a torus, χ = 0). One should notice the similarity between the theorem of Gauss-Bonnet (50) and the Chern numbers (46) defined as topological invariants obtained by integrating a Chern class. The curvature K of a surface can be defined as a characteristic class (called the Euler class). So in fact these two expressions are not only similar: they have the same origin. Chern developed the theory of characteristic classes while studying higher dimensional generalizations of the Gauss-Bonnet theorem [35]. 3.4.2 Surfaces with boundary We now consider that the surface S has a boundary and is oriented. An ~ can be defined orientation means that an outward unitary normal vector N in a coherent manner throughout the surface (for instance this is not possible for a Moebius strip). Let γ(s) be a space curve parameterized by its arclength s, with ds2 = dx2 +dy 2 +dz 2. Suppose that γ is drawn on the surface S. To each point p ∈ S on the curve, one associates a local, orthonormal ~ ) where ~t = dγ is a unitary tangent vector to the and direct frame (~t, ~u, N ds curve γ, ~u is perpendicular to ~t and belongs to the tangent plane of S at p ~ is the normal vector at p. One has in particular and N ~ × ~t. ~u = N The curvature vector ~k of the curve γ is defined by ~ ~k = dt · ds

(51)

E. Akkermans and K. Mallick: Geometrical Description of Vortices 867 ~ . Its As ~t is unitary, the curvature vector ~k has components on ~u and N ~ projection on N is called the normal curvature of γ, whereas the projection on ~u defines the geodesic curvature kg : d~t ~. .N kg = ~k.~u = ~t × ds

(52)

The geodesic curvature kg vanishes for a geodesic curve of the surface S [15, 23]. We can now give the generalization of the Gauss-Bonnet theorem if the the boundary ∂S is not empty. As the curvatura integra is not an integer in general, a boundary contribution must be added [11]: Z Z I 1 1 KdS + kg dl = χ(S) (53) 2π 2π ∂S S where K and kg are respectively the curvature of the manifold S and the geodesic curvature of its boundary. Example: A spherical cap in the north hemisphere, containing the north pole and limited by circle of latitude θ0 , is topologically equivalent to a disk i.e. to a triangle. Hence its Euler-Poincar´e characteristic is equal to 1. The geodesic curvature of the circle of latitude θ0 is kg = R1 cotanθ0 . The Gauss-Bonnet theorem (53) gives indeed 1 Area of the cap 2πRkg sin θ0 = 1. + 2 2πR 2π

(54)

For θ0 = π2 , i.e. at the equator (half-sphere case), the boundary is a geodesic and kg = 0 so that only the first term in (53) contributes. 3.4.3 Secondary characteristic classes We showed that the Gauss-Bonnet theorem need to be generalized to incorporate the contribution of a boundary. Is it possible to modify the characteristic classes theory accordingly? The answer is again contained in the Chern-Weil theorem. Since P (Ω0 ) − P (Ω) = dQ, the integral of dQ vanishes if the manifold has no boundary. But if there is a boundary ∂M , then by Stokes’ theorem Z Z dQ = Q (55) M

∂M

needs not be zero and corresponds precisely the contribution of a “geodesic curvature”. Hence (46) rewrites [34]: Z Z ci (Ω) − Q(ω, ω0 ) = n ∈ Z (56) M

∂M

868

Topological Aspects of Low Dimensional Systems

where ω is the connexion 1-form associated to the curvature Ω and ω0 is a well chosen connexion that compensates boundary effects. The forms Q associated to each Chern forms ci are called Chern-Simons classes or secondary characteristic classes. We now present some explicit calculations. The characteristic class associated to Ω = dω +ω ∧ω for the Yang-Mills case is tr(Ω ∧ Ω). Consider another connexion ω 0 and the interpolation between ω and ω 0 defined by ωt Ωt

= tω 0 + (1 − t)ω = dωt + ωt ∧ ωt

(57)

for t ∈ [0, 1]. Define α = ω − ω 0 so that by (47), Q is given by Q(ω, ω 0 ) = 2

Z

1

tr(α ∧ Ωt )dt.

(58)

0

Since Ωt = Ω − tdα + t2 α ∧ α − tα ∧ ω − tω ∧ α, then 1 1 1 1 Q(ω, ω 0 ) = 2tr α ∧ Ω− α ∧ dα+ α ∧ α ∧ α− α ∧ α ∧ ω − α ∧ ω ∧ α . 2 3 2 2 The cyclicity of the trace together with the property (13) of the wedgeproduct provides 2 0 Q(ω, ω ) = tr 2α ∧ Ω − α ∧ dα − 2α ∧ ω ∧ α + α ∧ α ∧ α . 3 Taking ω 0 = 0 we recover a famous expression for the Chern-Simons connexion [44] namely 2 (59) Q(ω) = tr ω ∧ dω + ω ∧ ω ∧ ω . 3 When applied to the abelian connexion U(1) on a domain M of the 2d plane, we obtain Q(A, A0 ) = A − A0 and the curvature is related to the magnetic field Ω = −iB. The vector potential A0 to be equal to ∇χ, where χ is the phase of the order parameter ψ = |ψ|eiχ . Then, according to (56) we have I Z B+ (∇χ − A) = n ∈ Z (60) M

∂M

which corresponds to the well-known fluxoid expression. By analogy with the Gauss-Bonnet theorem for a surface with boundary (53), we can say that B is a curvature and that (∇χ − A), which we shall interpret in Section 5, plays the role of a geodesic curvature.

E. Akkermans and K. Mallick: Geometrical Description of Vortices 869

3.5 The Weitzenb¨ock formula The definition of the Laplacian on forms (39) is not always adapted to describe problems we aim to solve. For a quantum particle moving in a magnetic field described by a U (1)-connexion or for the related GinzburgLandau equation, the Hamiltonian H (or the free energy) is given in terms of the covariant derivative ∇ = d − iA and of its adjoint ∇∗ with respect to the scalar product on 1-forms defined in (38), e.g. H = 12 ∇∗ ∇ in units where both the mass of the particle and ~ are set to one. Thus the question arises to relate the covariant derivative to the Laplacian. We first denote by D the operator D = d + δ so that the Laplacian in the dual basis i = dxi , we have ∆ = dδP+ δd = D2 . On a flat space P and 2 2 Df = i i ∂i f so that D f = − i ∂i f coincides with the usual Laplacian on functions. But more generally for a Riemannian manifold a curvature term, P the local basis vectors i are also dynamical variables so we write Df = i i ∇i f , where ∇i is the covariant derivative along the i-direction, then, X X ∇2i f + j i (∇j ∇i − ∇i ∇j )f. D2 f = − i

j
The second term in the rhs corresponds precisely to the curvature K so that we have the general expression D2 f = ∇∗ ∇f + Kf

(61)

known as the Weitzenb¨ ock formula [20, 37]. Coming back to the example of a quantum particle in a magnetic field, we can identify the curvature K with the magnetic field. For the case of a constant magnetic field, the K-term in (61) is just a constant so that the Hamiltonian coincides with the Laplacian. Therefore, on a compact manifold without boundary, the Schr¨ odinger equation is geometrical and admits topological invariants, namely the total quantized magnetic flux. This is not true anymore in the presence of boundaries. For a non-uniform magnetic field, the curvature K becomes a local function of the coordinates and the problem defined by the Hamiltonian H = 12 ∇∗ ∇ has no geometrical features. To recover a geometrical formulation, Aharonov and Casher [38] ~ playing the role of K in (61), must proved that a Zeeman term ( 12 ~σ .B), be added. Similar non-geometrical features appear for the Pauli or Dirac equations. These equations admit zero-modes solutions of the type Df = 0 whose number n is an Atiyah-Singer Index. But this number does not correspond to a Chern number of the type (48) since the magnetic field is not a dynamical variable of the problem but only a parameter and its magnetic flux is not quantized. To obtain an example of a U (1)-connexion with topological invariants, we consider now the case of a two-dimensional superconductor described by the Ginzburg-Landau equations, where both the order parameter and the magnetic field are dynamical variables.

870

4

Topological Aspects of Low Dimensional Systems

The dual point of Ginzburg-Landau equations for an infinite system

The existence and stability of vortices in superfluid or superconducting systems have been mainly studied for the case of infinite systems or in a limit where boundary effects do not play an essential role. Among the large variety of methods available to study vortices in superfluids or superconductors, we choose to work in the framework of the Ginzburg-Landau expression for the free energy. We consider the case of a finite 2d superconducting bounded domain (a billiard) and study the existence and stability of vortices. The superconducting state is characterized by a complex order parameter. For infinite systems, nonlinear functionals given by Ginzburg-Landau, GrossPitaevskii or Higgs expressions admit vortex like solutions. These solutions are characterized by topological numbers e.g. the number n of vortices. How can these results be extended to finite size systems? Is there a mechanism by which boundary conditions may allow to select a state with a given number of vortices? 4.1 The Ginzburg-Landau equations The Ginzburg-Landau equations describe a superconducting billiard if both the order parameter and the vector potential have a slow spatial variation. The expression of the Ginzburg-Landau energy density a is 2e ~ 2 B 2 ~ ψ + −i A a = a0 + a2 |ψ|2 + a4 |ψ|4 + a1 ∇ ~c 8π

(62)

where ψ = |ψ|eiχ is the complex-valued order parameter, B is the magnetic field and the ai ’s are real parameters. Defining [39] ξ 2 = |aa12 | , λ2 = √ ~c 2 a 1 4 4π 2 2e a1 |a2 | , the dimensionless free energy F is Z F=

1

Ω2

~ − iA)ψ| ~ 2 |B|2 + κ2 |1 − |ψ|2 |2 + |(∇

(63)

q φ0 2| where ψ is measured in units of ψ0 = |a 2a4 , B in units of 4πλ2 , and the √ √ lengths in units of λ 2. The numerical factor 2 is for further convenience. The ratio κ = λξ is the only free parameter in (63) and it determines, in the limit of an infinite system, whether the sample is a type-I or type-II superconductor [39]. The integral is over the two-dimensional domain Ω of the superconducting sample. The Ginzburg-Landau equations for the ~ =∇ ~ ×A ~ are obtained from order parameter ψ and for the magnetic field B a variation of F . They are nonlinear second order differential equations. Their solutions are not known except for some particular cases.

E. Akkermans and K. Mallick: Geometrical Description of Vortices 871

4.2 The Bogomol’nyi identities ~ can be reduced For the special value κ = √12 , the equations for ψ and A to first order differential equations. This special point was first used by Sarma [41] in his discussion of type-I vs. type-II superconductors and then identified by Bogomol’nyi [40] in the more general context of stability and integrability of classical solutions of some quantum field theories. This special point is also called a duality point. We first review some properties of the Ginzburg-Landau free energy at the duality point. We use the following identity true for two dimensional systems ~ × ~ + B|ψ|2 ~ − iA)ψ| ~ 2 = |Dψ|2 + ∇ |(∇

(64)

~ ~ is the current density and the operator D is − |ψ|2 A where ~ = Im(ψ ∗ ∇ψ) defined as D = ∂x + i∂y − i(Ax + iAy ). This relation is a relative of the Weitzenb¨ ock formula (61). At the duality point κ = √12 the expression (63) for F can be rewritten using (64) as I Z 1 ~ ~dl |B − 1 + |ψ|2 |2 + |Dψ|2 + (~ + A). (65) F= 2 Ω ∂Ω where the last integral over the boundary ∂Ω of the system results from Stokes theorem. For an infinite system, we impose [40] the usual conditions for a superconductor, namely |ψ| → 1 and ~ → 0 at infinity. The boundary term in (65) then becomes I I ~ ~ ~ ~ (~ + A).dl = + A .~dl. (66) |ψ|2 ∂Ω ∂Ω This last integral is the fluxoid. It is quantized and is equal to I ~ ~dl = 2πn. ∇χ. ∂Ω

The integer n is the winding number of the order parameter ψ and as such is a topological characteristic of the system. Using (66), we see that n is also the total magnetic flux through the system: Z B = n. (67) Ω

As we interpreted B as a first Chern class, this relation is similar to (46). The extremal values of F , are obtained when the two Bogomol’nyi [40] equations are satisfied Dψ B

= =

0 1 − |ψ|2 .

(68)

872

Topological Aspects of Low Dimensional Systems

In this case, the free energy F is given by 1 F = n. 2π

(69)

Therefore the free energy itself is a topological invariant. The two Bogomoln’yi equations can be decoupled and |ψ| is a solution of the second order nonlinear equation ∇2 ln|ψ|2 = 2(|ψ|2 − 1)

(70)

which is related to the Liouville equation. It should be noticed that the set of equations (68, 70) has been obtained without any assumption on the nature of the magnetic field and appears in various other situations, e.g. Higgs [43], Yang-Mills [42] and ChernSimons [44] field theories. It was proven that these equations admit families of vortex solutions [43]. For infinite systems, it can be shown that each vortex carries one flux quantum and that the winding number n is equal to the number of vortices in the system. However for an infinite system there is no mechanism to select the value of n. It will be precisely the role of the boundary of a finite system to introduce such a selection mechanism and to determine n, according to the applied magnetic field. 5

The superconducting billiard

From now on, we shall study finite size systems in an external magnetic field. The question then arises to know if they can sustain stable vortex solutions and what their behaviour is, as a function of the applied field. An interesting approach has been developed by Bethuel and coworkers [47]. They considered the case of a billiard Ω without applied magnetic field but with the boundary condition for the order parameter ψ|Ω = g(θ) with R 2π ∂g 1 g(θ) = eiφ(θ) and a prescribed winding number 2iπ 0 ∂θ dθ = n . In the London limit, i.e. κ → ∞, |ψ| is 1 almost everywhere but because of the degree n on the boundary, |ψ| must vanish n times in the bulk therefore leading to vortices. An extension of this approach to the case where a magnetic field is applied on the system has been proposed in [48] where it is shown by a variational argument that vortex solutions have a lower energy when the magnetic field is increased. By the same method, it is also possible to discuss the type of vortices and their distribution as a function of the geometry of the billiard. Numerical simulations [45] of the Ginzburg-Landau equations for a long parallelepiped in a uniform magnetic field show that the physical picture derived for κ = √12 , namely the existence of stationary vortex solutions whose number depends on the magnetic field, remains valid for quite a large range of values of κ, and the corresponding change of free energy is small [46]. We shall therefore study the case κ = √12 , i.e. the

E. Akkermans and K. Mallick: Geometrical Description of Vortices 873

duality point and extend the previous approach to a system with finite size where boundary effects are important. 5.1 The zero current line In a finite system, there are in general non-zero edge currents and the order parameter is not equal to 1 on the boundary. Hence, the identification of the boundary integral in (65) with the fluxoid (66) is not possible anymore, and the free energy can not be minimized just by imposing Bogomol’nyi equations (68). However, the currents on the boundary of the system screen the external magnetic field and therefore produce a magnetic moment (a circulation) opposite to the direction of the field, whereas vortices in the bulk of the system produce a magnetic moment along the direction of the applied field. Hence currents in the bulk circulate in a direction opposite to those at the boundary. If one assumes cylindrical symmetry, ~ has only an azimuthal component, with opposite signs in the bulk and on the edge of the system (the radial component is zero since ~ is divergence free). Thus, there exists a circle Γ on which ~ vanishes. This allows us to separate the domain Ω into two concentric subdomains Ω = Ω1 ∪ Ω2 such that the boundary ∂Ω1 is the curve Γ. On ∂Ω1 , the current density ~ is zero, therefore I I ~ ~ ~ ~.dl = .dl = 0. (71) 2 |ψ| ∂Ω1 ∂Ω1 Thus one deduces as above that Bogomol’nyi and Liouville equations are valid in the finite domain Ω1 as in the case of the infinite plane. The magnetic flux Φ(Ω1 ) is calculated using the fluxoid and (71) so that I ~.~dl = n. Φ(Ω1 ) = n − 2 ∂Ω1 |ψ| H ~ ~dl = 2πn, as well as the As before n is the winding number, i.e. ∂Ω1 ∇χ. number of vortices [49] in Ω1 . The free energy in Ω1 is F (Ω1 ) = 2πn.

(72)

The contribution of Ω2 to the free energy is given by (2) and can be expressed using the phase and the modulus of the order parameter ψ Z 2 2 2 ~ − A| ~ 2 + B + (1 − |ψ| ) · (73) (∇|ψ|)2 + |ψ|2 |∇χ F (Ω2 ) = 2 2 Ω2 The boundary conditions for both the magnetic field B(R) and the vector potential A(R) adapted to a flat disk geometry are provided by the condition 2 and φ = φe where φe is the flux of the applied field Be namely, φe = Bφe R 0 hc φ0 = 2e . It implies that at the boundary B(R) is larger than the applied field Be due to the distorsion of the flux lines near the edge of the system.

874

Topological Aspects of Low Dimensional Systems

5.2 A selection mechanism and topological phase transitions The analysis presented in [49] leads for the total Gibbs potential of the billiard to the expression 1 2λ2 1 G(n, φe ) = F (n, φe )− 2 φe 2 2π 2π R √ !3 √ λ λ 2 1 2λ2 . 2 (n−φe )2 − = n+ (n − φe )4 − 2 φe 2 (74) R 2 R R This relation consists in a set of quartic functions indexed by the integer n. The minimum of the Gibbs potential is the envelop curve defined by the equation ∂G ∂n |φe = 0, i.e. the system chooses its winding number n in order to minimize G. This provides a relation between the number n of vortices in the system and the applied magnetic field φe . In the limit of a large enough R λ , the quartic term is negligible and the Gibbs potential reduces to a set of parabolas. The winding number n is then given by 1 R · (75) n = φe − √ + 2 2λ 2 ∂G is given by The magnetization of the system M = − ∂φ e

√ 4λ2 2 2λ (φe − n) − 2 φe . −M = R R

(76)

For φe smaller that 2√R2λ , we have n = 0 and (−M ) increases linearly with the external flux. This corresponds to the London regime before the first vortex enters the system. The field H1 at which the first vortex enters the φ0 0 + 2πR system corresponds to G(n = 0) = G(n = 1), i.e. to H1 = 2π√φ2Rλ 2. The subsequent vortices enter one by one for each crossing G(n + 1) = G(n); this happens periodically in the applied field, with a period equal to ∆H = √ φ0 2 2λ πR2 . This gives rise to a discontinuity of the magnetization ∆M = R . 5.3 A geometrical expression of the Gibbs potential for finite systems For infinite system, with the boundary conditions |ψ| → 1 and ~ → 0 at infinity, we showed that the free energy at the dual point κ = √12 is a topological invariant proportional (69) to the fluxoid n, which represents also the number of vortices in the system: Z 1 F = B = n. (77) 2π

E. Akkermans and K. Mallick: Geometrical Description of Vortices 875

This relation is a property of the dual point at which the Ginzburg-Landau functional has a geometrical interpretation and admits the quantized magnetic flux as a topological invariant. For a finite system the fluxoid quantification can be expressed as Z I ~ 1 ~ ~ B.dS + .d~l = n. (78) 2 2π Ω |ψ| ∂Ω This relation is analogous to the Gauss-Bonnet theorem for a surface with a boundary (53) or more generally to a topological invariant obtained by summing a Chern class in the bulk and a Chern-Simons class on the boundary (56). Recalling (60), we see that the magnetic field B plays the role ~  of a curvature K, and the quantity |ψ| 2 = ∇χ − A is similar to a geodesic curvature kg . For a system with cylindrical symmetry, one can show [49] ~  that |ψ| 2 = n − φe . Therefore equation (74) can be rewritten as Z Z Z I ~ 1 F= B+ η (79) ≡ K + η(kg ). 2π |ψ|2 Ω ∂Ω Comparing (79) with (77) we conclude that the boundary correction is a functional of the geodesic curvature. In the preceeding section, we obtained an explicit formula for the function η as an even fourth order polynomial in the geodesic curvature. This geometric interpretation makes us believe that an expression such as (79) is fairly general. It could be well suited, as an Ansatz, to describe finite systems which are known to have a topological description in the infinite limit; for example, a suitable generalization of (79) to SU (2) symmetry could describe superfluid 3 Helium in a bounded domain. We have presented a geometrical formulation for the Ginzburg-Landau problem which we now briefly summarize. i) For a certain theory, like Ginzburg-Landau or other functionals of this type, it may appear stable singular solutions (e.g. vortices) whose nature is determined by the related homotopy groups of the Toulouse-Kleman [1] approach. To these solutions are associated topological numbers in the Bogomol’nyi limit. ii) The existence of topological numbers signals the occurence of a geometrical description of the problem. What is it good for? First, we notice that topological quantities describe global features of the problem i.e. behaviour in the large by opposition to the local behaviour obtained from solutions of differential equations. Then, by identifying physical quantities in terms of global topological invariants, we do not need to solve the local equations to obtain the behaviour of the system. As a result, we may say that an important goal of a geometrical description is to obtain physical quantities in terms of global topological expressions. When it is possible, it is very much rewarding.

876

Topological Aspects of Low Dimensional Systems

It is our pleasure to thank A. Joets for a critical reading of the manuscript.

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E. Akkermans and K. Mallick: Geometrical Description of Vortices 877

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SEMINAR 4

THE INTEGER QUANTUM HALL EFFECT AND ANDERSON LOCALISATION

J.T. CHALKER Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, UK

Contents 1 Introduction

881

2 Scaling theory and localisation transitions

882

3 The plateau transitions as quantum critical points

885

4 Single particle models

887

5 Numerical studies

890

6 Discussion and outlook

892

THE INTEGER QUANTUM HALL EFFECT AND ANDERSON LOCALISATION

J.T. Chalker

1

Introduction

The existence of the Integer Quantum Hall Eﬀect (IQHE) depends crucially on Anderson localisation, and, conversely, many aspects of the delocalisation transition have been studied in most detail in quantum Hall systems. The following article is intended to provide a introduction to the IQHE from this viewpoint, as a supplement to the broader accounts of quantum Hall experiment and theory by Shayegan [1] and by Girvin [2], which appear elsewhere in this volume. To give an overview of what is to come, we start by outlining some of the experimental facts and some of the theoretical ideas used in their interpretation. Consider a two-dimensional electron gas (2DEG) in a magnetic ﬁeld, with some (but not too much) scattering from disorder. Suppose the ﬁlling factor ν (the ratio of electron density to ﬂux density) is increased from one integer value to the next, at low temperature. The Hall conductivity, σxy , as a function of ν has the form of a staircase, consisting of broad plateaus at integer multiples of e2 /h (with e the electron charge and h Planck’s constant), separated by narrow risers. The dissipative conductivity, σxx , is very small at electron densities for which the Hall conductivity is quantised, but has Shubnikov – de Haas peaks which coincide with the risers in σxy . Both the transitions between Hall plateaus and the Shubnikov – de Haas peaks become sharper at lower temperatures, as indicated schematically in Figure 1. It was appreciated [3] rather quickly after von Klitzing’s [4] discovery of the IQHE that some understanding of this observed dependence of the conductivity tensor on ν can be got by starting from a single-particle description of the electron states which involves an Anderson delocalisation transition. More speciﬁcally, the picture is as follows. In a clean system, the spectrum for a charged particle moving in two dimensions in a perpendicular magnetic ﬁeld consists of a series of macroscopically degenerate Landau levels. A realistic model of the experimental sample should include scattering from impurities and inhomogeneities, which lifts the degeneracy and c EDP Sciences, Springer-Verlag 1999

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broadens the Landau levels. Provided the magnetic ﬁeld is strong enough, however, the disorder-broadening is less than the cyclotron energy, and the levels retain their identity. It turns out that the states within each disorderbroadened Landau level fall into two categories, for reasons that we shall outline heuristically in Section 4. States in the tails of Landau levels are Anderson localised, meaning that each is trapped within a particular, microscopic region of the system. By contrast, states near the centre of each Landau level have wavefunctions that extend throughout that sample. The distinction is crucial for the conductivity, since, within linear response, a constant external electric ﬁeld cannot induce transitions between diﬀerent localised states. Thus, only those electrons occupying extended states participate in current ﬂow. Given these features of the eigenstates, one can deduce the essentials in the dependence of conductivity on electron density. To do so, imagine adding electrons at zero temperature to a system with the spectrum sketched in Figure 1b. Initially, all occupied states are localised and transport is impossible. Addition of more electrons brings the Fermi energy into a region of extended states, so that σxy increases with ν. In this situation, σxx is non-zero, because scattering of electrons close to the Fermi energy gives rise to dissipation. A further increase in ν brings the Fermi energy into the localised states in the upper tail of the Landau level. Now, a variation in electron density does not alter the number of extended, current-carrying states that are occupied. Correspondingly, the Hall conductivity exhibits a plateau. Also, since the occupied, current-carrying states are buried at some depth beneath the Fermi energy, dissipation is necessarily activated, and vanishes in the low temperature limit. In turn, this implies that the Hall angle is 90◦ , and thus that σxx = 0. The sequence is repeated on moving the Fermi energy through higher Landau levels, and generates the behaviour represented in Figure 1a. Several questions arise from this account of the IQHE. In particular, one can ask what evidence there is for the suggested delocalisation transition, how the transition ﬁts into the general theory of critical phenomena, and how interactions might change the single-particle picture. We summarise some of the current answers in the remaining sections of this article. More detail can be found in the following sources: for the quantum Hall eﬀect in general, the book edited by Prange and Girvin [5]; for the IQHE and localisation, the article by Huckestein [6]; for the IQHE as a quantum phase transition, the article by Sondhi et al. [7]. 2

Scaling theory and localisation transitions

Scaling theory provides a framework for discussing delocalisation transitions, both in systems of real, interacting electrons, and in single-particle

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σ

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n ,ξ

(a)

(b)

ν

E

Fig. 1. (a) Schematic behaviour σxy and σxx as a function of filling factor (full lines); the plateau transition becomes sharper at lower temperature (dashed lines). (b) Density of states, n, (full line) and localisation length, ξ, (dashed line) as a function of energy, E, in a disorder-broadened Landau level.

models. It was ﬁrst developed for transitions in zero or weak magnetic ﬁeld, in a celebrated paper by Abrahams et al. [8] shortly before the discovery of the IQHE. Although a substantial extension [9] of these original ideas is necessary in order to encompass the IQHE, we discuss ﬁrst the zero ﬁeld problem in order to provide some perspective. A general review of disordered electronic systems is given in the article by Lee and Ramakrishnan [10]; a more recent account is by Kramer and MacKinnon [11]. The basis of scaling theory is to identify a quantity that plays the role of a coupling constant for the problem under consideration, and to discuss the change in this coupling constant with a change in the length scale at which it is measured. Both of these steps are most transparently described for a non-interacting system, though most of the ideas developed in that context should also apply to interacting systems. To be concrete, suppose that a change in length scale is accomplished in d-dimensions by combining 2d systems, individually of size Ld , to form a single system of size (2L)d . Each single-particle state of the large system can be considered as a superposition of states from the smaller systems. The states entering one such superposition will be drawn mainly from a window in energy around the level in question. Let the width of this window be , and let the mean level spacing (in, say, the smaller systems) be ∆. From a microscopic point of view, the dimensionless ratio /∆ is a good candidate for a coupling constant: if /∆ 1, each state of the large system should be spread over all 2d subsystems, while if each state if the large system is localised in a particular subsystem, one expects /∆ 1. Crucially, this ratio also has a macroscopic signiﬁcance. As Thouless argued [12], should be related

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by the uncertainty principle to the time required for a particle to travel a distance L: with diﬀusion constant D, ∼ ¯ hD/L2 . Expressing ∆ in terms of the density of states, n, and recalling the Einstein expression for the conductivity, σ = e2 nD, one has (¯ hD/L2 ) h ∼ ∼ 2 σLd−2 ≡ g(L) d −1 ∆ (L n) e

(1)

where g(L) is the conductance of a hypercube of size L2 , measured in units of the conductance quantum, e2 /h. Given g(L) as a coupling constant, the next step is to consider its scaledependence, which (treating L now as a continuous variable) is characterised by the function d ln(g) · (2) β(g) ≡ d ln(L) The approach rests on the assumption that β(g) is a function only of g, and independent of L or the microscopic details of the system. A formal justiﬁcation for this assumption comes from the derivation of a ﬁeld-theoretic description for the localisation transition from microscopic models of disordered conductors (see [10]); this route also provides a way to calculate β(g), at least close to two dimensions. The essential features of the function β(g) can be determined [8], however, from its asymptotic behaviour at large and small g, and the requirement that, since it is deﬁned in a system of ﬁnite size, it should interpolate smoothly between these limits. For g 1, one expects the size-dependence of the conductance to be given correctly by Ohm’s law, so that g ∝ Ld−2 and β(g) = d − 2, while for strong disorder, one expects localisation on a length scale ξ, g ∼ exp(−L/ξ) 1 for L ξ, and β(g) ∼ ln(g) + const. These ideas (together with perturbation theory in g −1 at large g) lead (in the absence of spin-orbit scattering) to the behaviour sketched in Figure 2 [8]. In this way, the two dimensional problem is identiﬁed as marginal: in more than two dimensions, β(g) has a zero at g = gc . Flow is towards metallic behaviour for g > gc , and to insulating behaviour for g < gc . In or below two dimensions, ﬂow as illustrated is always towards the insulator. These ideas have been conﬁrmed by extensive experiments in zero or weak magnetic ﬁeld (see [10]). The existence of the IQHE, however, shows that a strong magnetic ﬁeld must be capable of changing the behaviour. The mechanism was identiﬁed, at the level of a ﬁeld-theoretic description, by Pruisken et al. [13], who showed that σxy appears as a second coupling constant. A scaling ﬂow diagram in the σxx − σxy plane, incorporating the IQHE, was suggested by Khmel’nitskii [9], and is illustrated in Figure 3. In this diagram each ﬂow line indicates how the two components of the conductivity tensor change as a quantum Hall system is probed at increasing length scales, for ﬁxed ν, and diﬀerent ﬂow lines correspond to diﬀerent values of

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β (g)

d=3 gc ln(g) d=2

d=1

Fig. 2. The scaling function β(g) as a function of ln(g) for dimensions d = 1, 2, and 3, in the absence of a magnetic field.

ν. Experimentally, the length scale of measurements can be increased, and scaling ﬂow lines traced out, by reducing the temperature [14], while by varying ν at ﬁxed high temperature, the system is swept along a trajectory in the conductance plane that intersects a range of ﬂow lines. The ﬂow as a whole is periodic in σxy , and its asymptotic behaviour is controlled by ﬁxed points, which are of two kinds. Almost all scaling ﬂow lines end on stable ﬁxed points, located at σxx = 0 for σxy = N , with N integer. These ﬁxed points represent quantum Hall plateaus and (for N = 0) the zero magnetic ﬁeld localised phase. The experimental result that (with some idealisation) the Hall conductance is quantised and dissipation vanishing in the low temperature limit, for almost almost all ﬁlling factors, has its correspondence in the fact that nearly all ﬂow is towards these points. The fact that ﬂow on the σxy = 0 axis is towards a ﬁxed point with σxx = 0 is the representation in Figure 3 of the implications of Figure 2 for two-dimensional systems without a magnetic ﬁeld. A discrete set of exceptional ﬂow lines end at unstable ﬁxed points, which have σxx non-zero and σxy = N + 1/2. These ﬁxed points represent the plateau transitions. Flow in their vicinity has one unstable direction, leading to the adjacent stable ﬁxed points on either side. It is principally the nature of this ﬂow from the unstable ﬁxed point that is probed in studies of the quantum Hall plateau transition as a critical point.

3

The plateau transitions as quantum critical points

The quantum Hall plateau transition is one of the best examples we have at present of a quantum critical point in a disordered system. Viewing it

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σxx

σxy Fig. 3. The scaling flow diagram for the IQHE.

in this way brings certain additional expectations to those that follow from the scaling ﬂow of Figure 4. In this section we summarise the ideas involved, which are reviewed in the article by Sondhi et al. [7]. Consider a system that is close to a plateau transition, with, for example, magnetic ﬁeld strength as the control parameter that tunes the system through the transition: let ∆B be the deviation of this control parameter from its critical value. We expect the correlation length, ξ, to be ﬁnite away from the critical point, and to diverge as the critical point is approached, with a critical exponent denoted by ν (but not to be confused with the ﬁlling factor) (3) ξ ∼ |∆B|−ν . This correlation length corresponds in a single-particle description to the localisation length at the Fermi energy, and is the scale at which ﬂow leaves the vicinity of the unstable ﬁxed point and reaches that of the stable ﬁxed point. The interacting quantum system is also expected to have a characteristic correlation time, τ , and this too will diverge as the critical point is approached. The dynamical scaling exponent z relates the diverging spatial and temporal scales via (4) τ ∼ ξz . Correspondingly, the characteristic energy scale shrinks as the critical point is approached, with the dependence h/τ ∼ ξ −z ∼ |∆B|νz . ¯

(5)

At the critical point itself, this energy scale vanishes. If some other energy scale remains non-zero, set for example by temperature, T , or by the

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frequency, ω, at which the system is probed, then the transition is rounded: because of this, the plateau transition is a zero temperature phase transition. Close to the critical point, scaling theory constrains the dependence of physical quantities on ∆B and external scales. The situation is particularly simple for the components of the conductivity tensor, which in a two-dimensional system have a magnitude ﬁxed by e2 /h [15]: they should be given by scaling functions, the arguments of which can be chosen to be ratios of the various energy scales, so that |∆B|νz ω , , ... . (6) σij = Fij T T From this, one expects the width, ∆B ∗ , of the transition to scale, for example, with temperature at ω = 0 as ∆B ∗ ∼ T −1/(νz) . A number of experiments have probed this and other aspects of scaling behaviour, determining the width in magnetic ﬁeld of the Shubnikov - de Haas peak in the dissipative resistivity, or the width of the riser between two plateaus in the Hall resistivity. Wei et al. ﬁnd power-law scaling of ∆B ∗ with T over about one and a half decades, and obtain the value νz ≈ 2.4 [16]; Engel et al. demonstrate that ∆B ∗ is independent of ω for ω < T and ﬁnd dependence consistent with ∆B ∗ ∼ ω −1/(νz) and the same value of νz for ω > T [17]. Determination of ν and z separately requires diﬀerent approaches. One, employed by Koch et al. [18], is to work at small T and ω using mesoscopic samples, so that broadening of the plateau transition is a consequence of ﬁnite sample size, rather than of an external energy scale. In this way ν ≈ 2.3 is obtained [18], implying z ≈ 1. An alternative is to work at small T and ω in a macroscopic sample, and to use ﬁnite electric ﬁeld strength, E, to broaden the transition. Since eEξ sets an energy scale, one expects the transition width to satisfy Eξ ∼ (∆B ∗ )νz , and hence ∆B ∗ ∼ E 1/(ν[z+1]) ; in combination with temperature scaling, this allows ν and z to be determined separately. By this route Wei et al. obtain ν ≈ 2.3 and z ≈ 1 [19]. 4

Single particle models

To arrive at a satisfactory scaling theory of the plateau transition as a quantum phase transition, beginning from a microscopic description, would necessarily involve a treatment of the many electron system with interactions and disorder. While some progress (which we summarise in Sect. 6) has been made in this direction, the single-particle localisation problem provides a useful and very much simpler starting point. Even in this case, the obstacles to analytic progress are formidable. The size of the relevant coupling constant is the value of σxx (or, strictly, its inverse) at the unstable ﬁxed points of the scaling ﬂow diagram of Figure 4; since this is O(1)

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(and, in fact, the ﬁxed point is invisible in perturbation theory), a nonperturbative approach is presumably required. So far most known quantitative results have been obtained from numerical simulations, which we outline in Section 5. In the present section we introduce models that have been studied numerically, and describe a semiclassical picture of the transition. The Hamiltonian for a particle moving in two dimensions with a uniform magnetic ﬁeld and random scalar potential should provide a rather accurate description of the experimental system, apart from the neglect of electronelectron interactions. It is characterised by two energy scales and two length scales, with in each case one scale set by the magnetic ﬁeld and one by the disorder. The energy scales are the cyclotron energy and the amplitude of ﬂuctuations in the random potential (if necessary, averaged over a cyclotron orbit). The IQHE occurs only when the ﬁrst of these is the larger; a natural but limited simpliﬁcation is to take it to be much larger, in which case inter-Landau level scattering is suppressed and the potential ﬂuctuations establish the only energy scale of importance. The length scales are the magnetic length and the correlation length of the disorder, and varying their ratio provides some scope for theoretical simpliﬁcation, as we shall explain. Experimentally, both limits for the ratio can realised: disorder on atomic length scales is presumably dominant in MOSFETs, while in heterostructures the length scale of the potential experienced by electrons is set by their separation from remote ionised donors, and this may be larger than the magnetic length. A semiclassical limit for the localisation problem is reached if the potential due to disorder is smooth on the scale of the magnetic length. This limit has the advantage that it can be used to make the existence of a delocalisation transition intuitively plausible [20], and to construct a simpliﬁed model for the transition, known as the network model [21]. If the potential is smooth, then the local density of states at any given point in the system will consist of a ladder of Landau levels, displaced in energy by the local value of the scalar potential. As a function of position in the system, the displaced Landau levels form a series of energy surfaces, which are copies of the potential energy, V (x, y), itself, having energies V (x, y) + (N + 1/2)¯ hωc . Suppose one Landau level, and for simplicity the lowest, is partially occupied, so that the ﬁlling factor is 0 < ν < 1. This value of the ﬁlling factor arises, for a smooth potential, from a spatial average over some regions in which the local ﬁlling factor is νlocal = 1, (those places at which, with chemical potential µ, the potential satisﬁes V (x, y) + h ¯ ωc /2 < µ) and others in which the local ﬁlling factor is νlocal = 0 (because at these places V (x, y) + h ¯ ωc /2 > µ). As illustrated in Figure 5, for small average ﬁlling factors, there will be a percolating region with νlocal = 0, dotted with isolated, ﬁnite “lakes”, in which νlocal = 1. By contrast, for average ﬁlling factors close to 1, a region with νlocal = 1 will percolate, and this “sea” will

J.T. Chalker: The Integer Quantum Hall Eﬀect

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 0000 1111 0000 1111 0000 1111 0000 1111

1111 0000 0000 1111 0000 1111 0000 1111

111111111111 00000000 0000 00000000 11111111 0000 000000001111 11111111 0000 1111 00000000 11111111 0000 000000001111 11111111 0000 1111 00000000 11111111 0000 000000001111 11111111 0000 1111 00000000 11111111 0000 000000001111 11111111 00000000 11111111 00000000 11111111 00000000 11111111

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111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111

Fig. 4. Snapshots of a quantum Hall system with smooth random potential at three successive values of the average filling factor. Shaded regions have local filling factor νlocal = 1, and unshaded regions have νlocal = 0.

contain isolated “islands” in which νlocal = 0. A transition between these two situations must occur at an intermediate value of ν. (In particular, if the random potential distribution is symmetric under V (x, y) → −V (x, y), the critical point is at ν = 1/2.) To connect this geometrical picture with the nature of eigenstates in the system, recall that states at the chemical potential lie on the boundary between the regions in which νlocal = 0 and those in which νlocal = 1, so that one has a Fermi surface in real space. There are two components to the classical dynamics of electrons on the Fermi surface, and they have widelyseparated time scales in the smooth potential we are considering. The fast component involves cyclotron motion around a guiding centre: when quantised, it contributes (N + 1/2)¯ hωc to the total energy. The slow component involves drift of the guiding centre in the local electric ﬁeld that arises from the gradient of the potential V (x, y). Since this gradient is almost constant on the scale of the magnetic length, the guiding centre drift is analogous to the Hall current that ﬂows when a uniform electric ﬁeld is applied to an ideal system, and therefore carries the guiding centres along contours of constant potential. If one imagines quantising this classical guiding centre drift, say by a Bohr-Sommerfeld procedure, then eigenstates result which have their probability density concentrated in strips lying around contours of the potential, with width set by the magnetic length. States in the low-energy tail of a Landau level are associated with contours that encircle minima in the potential, while states in the high-energy tail belong to contours around maxima of the potential. At a critical point between these two energies, the characteristic size of contours diverges, and one has the possibility of extended states. An obvious factor which complicates the simple association of eigenstates with closed contour lines is the possibility of tunneling near saddlepoints in the potential, between disjoint pieces of a given energy contour. Equally, once tunneling is allowed for, there may be more than one path by which electrons can travel between two points, and interference eﬀects

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can become important. Away from the critical point, tunneling and interference are unimportant provided the potential is suﬃciently smooth, but as the critical point is approached their inﬂuence always dominates. It is for this reason that the delocalisation transition is not in the same universality class as classical percolation. The network model [21] provides a simple way to incorporate these quantum eﬀects. In this model, portions of a given equipotential are represented by links, which carry probability ﬂux in one direction, corresponding to that of guiding centre drift. The wavefunction is caricatured by a complex current amplitude, deﬁned on links. On traversing a link, a particle acquires an Aharonov-Bohm phase: if zi and zj are amplitudes at opposite ends of the link k, zj = eiφk zi . Tunneling at saddlepoints of the random potential is included in the model at nodes, where two incoming and two outgoing links meet. The amplitudes (say, z1 and z2 ) on the outgoing links are related to those on incoming links (z3 and z4 ) by a scattering matrix, which must be unitary for current conservation and can be made real by a suitable choice of gauge. Then cos β sin β z3 z1 = , (7) z2 z4 − sin β cos β where the single real parameter β characterises the node. The model as a whole is built by connecting together these two elements, links and nodes, to form a lattice. The simplest choice is a square lattice: thinking of this as a chess board, the black squares represent regions in which νlocal = 1, while for the white squares νlocal = 0. Guiding centre drift, and the direction of links of the model, is (for a one sense of the magnetic ﬁeld) clockwise around the white squares and anticlockwise around the black squares. Randomness is introduced into the model by taking the link phases, φk , to be independent random variables, for simplicity uniformly distributed between 0 and 2π. Variation of the node parameter from β = 0 to β = π/2 corresponds to sweeping the Fermi energy through a Landau level, and a delocalisation transition occurs at β = π/2. This transition has been studied numerically, using the approach described in Section 5. In addition, the model itself has been mapped onto other descriptions of the problem, notably a supersymmetric quantum spin chain in 1 + 1 dimensions [22]. 5

Numerical studies

The quantitative information available on the delocalisation transition in models of the IQHE without interactions comes from numerical simulations, reviewed by Huckestein [6]. The most important results of these calculations are as follows. First, there is universality, in the sense that most choices of model lead to the same results (and those choices that do not are plausibly argued to be plagued by a slow crossover, preventing one from

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reaching asymptotic behaviour in the available system sizes [23]). Second, it is conﬁrmed that the scaling ﬂow diagram of Figure 4 is correct (at least for a non-interacting system), in particular, in the sense that the localisation length is divergent only at one energy within a disorder-broadened Landau level. Third, a value ν = 2.3 ± 0.1 is obtained for the localisation length exponent. This value is in agreement with experimental results, as summarised in Section 3, a fact that raises several questions, which we touch on in Section 6. A necessary ﬁrst step in numerical calculations on the delocalisation transition is to discretise the problem. There are several ways of doing this. One is to project the Hamiltonian onto the subspace spanned by states from a single Landau level. A second is to study the network model, described in the previous section, and a third is to treat a tight-binding model, with a magnetic ﬁeld introduced by including Peierls phases in the hopping matrix elements. Given a suitable model, there are two approaches to simulations. The most direct is simply to diagonalise the Hamiltonian for a square sample, and use an appropriate criterion to distinguish localised and extended states. The method has the potential disadvantage that, in a system of ﬁnite size, all states having a localisation length larger than the sample size will appear extended. For that reason it is important to examine the fraction of apparently extended states as a function of system size, L: at the plateau transition this fraction tends to zero as L−1/ν . Bhatt et al. have used this method extensively [24], introducing boundary conditions which include phase shifts, so that Chern numbers can be deﬁned for each state. Extended states are identiﬁed as those with non-zero Chern number. An alternative approach is to study transmission properties of systems that are in the shape of long cylinders. This transmission problem (or a calculation of the Green function) has computational advantages, since there exist for it algorithms [25] which are much less demanding on computer memory than matrix diagonalisation. Since the geometry is quasi-one dimensional, all states are localised along the length of the sample for any ﬁnite radius. Because of this, transmission amplitudes, for example, decay exponentially with sample length. The computational procedure is to calculate the mean decay rate (or, technically, the smallest Lyapunov exponent). The inverse of this is the localisation length, ξ(E, M ) which in general depends on the circumference, M , of the cylinder as well as the energy, E, under consideration. At energies for which states in the two-dimensional system are localised, the localisation length in the quasi-one dimensional system tends to the bulk localisation length, ξ(E), as the circumference is taken to inﬁnity, while at the critical energy, the quasi-one dimensional localisation length remains proportional to the cylinder circumference for arbitrarily large values of M . Finite size scaling theory provides a framework for analysing data from these

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Topological Aspects of Low Dimensional Systems

calculations: one expects ξ(M, E) = M f (ξ(E)/M )

(8)

where f (x) is a function of the single scaling variable x = ξ(E)/M , rather than of ξ(E) and M separately. Calculations of this kind for the network model [21, 26] and for a Hamiltonian projected onto the lowest Landau level [27] both lead to the same scaling function, and to the value for ν quoted above. 6

Discussion and outlook

A number of important open questions remain. Recent experiments by Shahar and collaborators have produced several intriguing results which are not fully understood. Examining the plateau transition in the lowest Landau level, they ﬁnd a reﬂection symmetry in the current-voltage characteristics which they interpret in terms of charge-ﬂux duality [28]. Following properties to higher magnetic ﬁelds, and therefore into the insulating phase, they ﬁnd the Hall resistance to be nearly quantised, even far from the transition [29]. Finally, and disturbingly, in the samples they study, the transition apparently retains a ﬁnite width ∆B ∗ , even in the low-temperature limit [30], in serious contradiction to the scaling ideas of Section 3. On the theoretical side, one of the important problems is to understand better the eﬀect of interactions on the plateau transition. In view of the apparent agreement between the value of ν determined from experiment and that from simulations of models without electron-electron interactions, it is initially tempting to think that interactions might be irrelevant in the renormalisation group sense. In fact, this is not directly tenable, since in a non-interacting system with a ﬁnite density of states at the mobility edge, the dynamical exponent necessarily takes the value z = 2. A Hartree-Fock study of interacting electrons in a Landau level with disorder [31], and a numerical calculation of the scaling dimension of interaction strength at the non-interacting ﬁxed point [32], both suggest that it may be possible to attribute z = 1 to interaction eﬀects, whilst retaining the value of ν found in the non-interacting system. I am grateful to EPSRC for support, and to many colleagues for collaborations and discussions.

References [1] Shayegan M., this volume. [2] Girvin S.M., this volume.

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[3] Prange R.E., Phys. Rev. B 23 (1981) 4802; Aoki H. and Ando T., Sol. State Comm. 38 (1981) 1079. [4] von Klitzing K., Dorda G. and Pepper M., Phys. Rev. Lett. 45 (1980) 494. [5] Prange R.E. and Girvin S.M., The Quantum Hall Eﬀect (Springer, New York, 1990). [6] Huckestein B., Rev. Mod. Phys. 67 (1995) 357. [7] Sondhi S.L., Girvin S.M., Carini J.P. and Shahar D., Rev. Mod. Phys. 69 (1997) 315. [8] Abrahams E., Anderson P.W., Liciardello D.C. and Ramakrishnan T.V., Phys. Rev. Lett. 42 (1979) 673. [9] Khmel’nitskii D.E., Piz’ma Zh. Eksp. Teor. Fiz. 82 (1983) 454; [JETP Lett. 38 (1983) 552]. [10] Lee P.A. and Ramakrishnan T.V., Rev. Mod. Phys. 57 (1985) 287. [11] Kramer B. and MacKinnon A., Rep. Prog. Phys. 56 (1993) 1469. [12] Edwards J.T. and Thouless D.J., J. Phys. C 5 (1972) 807. Thouless D.J., Phys. Rev. Lett. 39 (1977) 1167. [13] Levine H., Libby S.B. and Pruisken A.M.M., Phys. Rev. Lett. 51 (1983) 1915; and Pruisken A.M.M. in reference [5] [14] Wei H.P., Tsui D.C. and Pruisken A.M.M., Phys. Rev. B. 33 (1986) 1488. [15] Fisher M.P.A., Grinstein G. and Girvin S.M., Phys. Rev. Lett. 64 (1990) 587. [16] Wei H.P., Tsui D.C., Paalanen M. and Pruisken A.M.M., Phys. Rev. Lett. 61 (1988) 1294. [17] Engel L.W., Shahar D., Kurdak C. and Tsui D.C., Phys. Rev. Lett. 71 (1993) 2638. [18] Koch S., Haug R., von Klitzing K. and Ploog K., Phys. Rev. Lett. 67 (1991) 883. [19] Wei H.P., Engel L.W. and Tsui D.C., Phys. Rev. B 50 (1994) 14609. [20] Tsukada M., J. Phys. Soc. Jpn. 41 (1976) 1466; Iordansky S.V., Sol. State Comm. 43 (1982) 1; Kazarinov R.F. and Luryi S., Phys. Rev. B 25 (1982) 7626; Prange R.E. and Joynt R., Phys. Rev. B 25 (1982) 2943; Trugman S.A., Phys. Rev. B 27 (1983) 7539; Shapiro B., Phys. Rev. B 33 (1986) 8447. [21] Chalker J.T. and Coddington P.D., J. Phys. C 21 (1988) 2665. [22] Read N., (unpublished); Zirnbauer M.R., Annalen der Physik 3 (1994) 513. [23] Chalker J.T. and Eastmond J.F.G. (unpublished); Huckestein B., Phys. Rev. Lett. 72 (1994) 1080. [24] Huo Y. and Bhatt R.N., Phys. Rev. Lett. 68 (1992) 1375; Huo Y., Hetzel R.E. and Bhatt R.N., Phys. Rev. Lett. 70 (1993) 481. [25] MacKinnon A. and Kramer B., Phys. Rev. Lett. 47 (1981) 1546; Pichard J.L. and Sarma G., J. Phys. C 14 (1981) L127; MacKinnon A. and Kramer B., Z. Phys. B 53 (1983) 1. [26] Lee D.-H., Wang Z. and Kivelson S., Phys. Rev. Lett. 70 (1993) 4130. [27] Huckestein B. and Kramer B., Phys. Rev. Lett. 64 (1990) 1437. [28] Shahar D., Tsui D.C., Shayegan M., Shimshoni E. and Sondhi S.L., Science 274 (1996) 589. [29] Hilke M., Shahar D., Song S.H., Tsui D.C., Xie Y.H. and Monroe D., Nature 395 (1998) 675. [30] Shahar D., Hilke M., Li C.C., Tsui D.C., Sondhi S.L., Cunningham J.E. and Razeghi M., Solid. State Comm. 107 (1998) 19. [31] Yang S.R.E., Macdonald A.H. and Huckestein B., Phys. Rev. Lett. 74 (1995) 3229. [32] Lee D.-H. and Wang Z., Philos. Mag. Lett. 73 (1996) 145.

SEMINAR 5

RANDOM MAGNETIC IMPURITIES AND QUANTUM HALL EFFECT

J. DESBOIS Laboratoire de Physique Th´eorique et Mod`eles Statistiques, 91406 Orsay Cedex, France

Contents 1 Average density of states (D.O.S.)

897

2 Hall conductivity

901

3 Magnetization and persistent currents

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RANDOM MAGNETIC IMPURITIES AND QUANTUM HALL EFFECT

J. Desbois

Abstract In this talk, we present works done in collaboration with Ouvry et al. [1–3]. Random magnetic fields have already been studied in several papers [4]. Here, we will consider a model where the disorder is contained in the definition of the magnetic field itself. By magnetic impurities, we will mean infinitely thin vortices carrying a flux φ (α ≡ eφ/2π), perpendicular to a plane. Those vortices are randomly dropped according to a Poisson’s law with average density ρ. In a first part, we will consider the average density of states (D.O.S.) of a charged particle coupled to the impurities. In particular, we will show that this system exhibits broadened Landau Levels for small α values. This fact has motivated us to study the Hall Conductivity (part II) and, finally, persistent currents and magnetization.

1

Average density of states (D.O.S.) [1]

We consider Hamiltonians for an electron minimally coupled to a vector ~ r ) with the additional coupling of the electron spin up or down potential A(~ σz = ±1 to the local magnetic field B(~r) (we set the electron mass me = ¯h = 1) 2 1 ~ r ) − eB(~r) σz . p~ − eA(~ (1) H= 2 2 It rewrites 1 (2) σz = +1 Hu = Π− Π+ 2 1 (3) σz = −1 Hd = Π+ Π− 2 where Π± = (px − eAx ) ± i(py − eAy ) ≡ vx ± ivy are the covariant momentum operators. In the homogeneous field case, the spin coupling is a trivial constant shift, but, in general, it has important effects. In the one vortex or magnetic impurity cases, it is a sum of δ(~r −~ri ) functions, which is c EDP Sciences, Springer-Verlag 1999

898

Topological Aspects of Low Dimensional Systems

needed to define in a non ambiguous way [5, 6] the short distance behavior of the wavefunctions at the location of the impurities ~ri . It can be attractive or repulsive and in the sequel we will only be concerned with the repulsive case (σz = −1, H = Hd ). ~ r ) = 2παδ(~r ), ~k ~ r ) = α~k × ~r/r2 , eB(~ For one vortex located in O (eA(~ is the unit vector perpendicular to the plane), it is easy to realize that the partition function reads: Zα (t) = Z0 (t)heiα2πn i{C}

(4)

where 2πn is the angle wound around O by the closed brownian curve C of length t. h· · ·i{C} stands for averaging over the set of all such curves and Z0 (t) is the free partition function. Zα (t) is unchanged when α → α + 1 and α → −α; so, we can restrict α to the interval [0, 1/2] when there is no additionnal magnetic field. The D.O.S. exhibits a depletion at the bottom of the spectrum: ρα (E) − ρ0 (E) =

α(α − 1) δ(E), 2

ρ0 (E) =

V 2π

(5)

(V is the (infinite) area of the system). ~ r) = Turning now to magnetic impurities located in ~ri , i = 1, 2, ..., N , (eA(~ PN PN ~ 2 ~ α i=1 k × (~r −~ri )/|~r −~ri | ), eB(~r) = 2πα i=1 δ(~r −~ri ), we get for a given configuration of the N vortices: PN i 2πnj α j=1 (6) Zα (t) = Z0 (t) e {C}

2πnj is the angle wound around vortex j by C. Averaging over disorder, we are left with: P 2iπαn −1) i{C} Zα (t) = Z0 (t)heρ n Sn (e

(7)

Sn is the arithmetic area of the n-winding sector (Sn ≥ 0; −∞ < n < +∞). Remarking that the random variables Sn scale like t, we rewrite Zα (t) as: Zα (t) = Z0 (t)he−ρt(S−iA) i{C} S=

2X Sn sin2 (παn), t n

A=

hSi = πα(1 − α)

1X Sn sin(2παn), t n

hAi = 0.

(8) (9) (10)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

899

In this formalism, the rescaled algebraic area enclosed by C should be written: 1X nSn . (11) A= t n From (8), it is easy to deduce that the average D.O.S. ρ(E) is a function of E/ρ and α.

i) When α → 0, ρ → ∞ with 2πρα (≡ ehBi) fixed, we get after a careful analysis [1]: (12) Zα (t) →α→0 e−tehBi/2 ZhBi (t) ZhBi (t) is the Landau partition function forPthe average magnetic field hBi ∞ (hωc i ≡ ehBi/2). (12) shows that ρ(E) = n=1 δ(E − 2nhωc i) i.e. we get the Landau spectrum shifted by hωc i.

ii) α = 1/2. (10) shows that A ≡ 0 and (8) leads to: Z

E/ρ

ρ(E) = ρ0 (E)

P (S)dS

(13)

0

where P (S) is the probability distribution of S. ρ(E) is a monotonically growing function of E with a depletion of states at the bottom of the spectrum.

iii) 0 < α < 1/2. Using an argument based on the specific heat c (≡ kt2 d2 ln Z/dt2 , k is the Boltzmann constant), we can show that the D.O.S. surely oscillates when α is smaller than some value α0 . The argument runs as follows. With the expression (8), we get: c − c0 ∼t→0 kt2 (hS 2 i{C} − hSi2{C} − hA2 i{C} )

(14)

(c0 is the free specific heat). Numerically, one obtains that (c − c0 ) < 0 for ≈ .28. On another hand, Zα (t) is the Laplace Transform of 0 < α < αnum 0 ρ(E). Integrating by parts, we show that: Z c − c0 ∼t→0 kt 2π 2

2 0

∞

Z 0

∞

dEdE 0

dρ(E)/V dρ(E 0 )/V (E − E 0 )2 . dE dE 0

(15)

Thus, we deduce that ρ(E)/V is a non-monotonic function of E for 0 < α < . αnum 0

900

Topological Aspects of Low Dimensional Systems

Fig. 1. Average density of states of the random magnetic impurity model.

To precise the shape of ρ(E), we remark that variables |A| and |A| are strongly correlated, especially for small α values. Assuming the linear relation: p p |A| = (eBeff /ρ)|A| with eBeff /ρ = hA2 i/hA2 i = 12hA2 i, (Beff →α→0 hBi; Beff = 0 when α = .5), and introducing the new variable S 0 = S − µ|A| such that S 0 and |A| are uncorrelated, we can write: 0

Zα (t) ≈ he−ρtS i{C} he−ρtµ|A| cos(eBeff tA)i{C} .Z0 (t).

(16)

Performing the inverse Laplace Transform, we get ρ(E) as shown in Figure 1. Now, let us show briefly how the critical α0 value can be recovered analytically. With the concentration expansion: Z=

∞ X e−ρV (ρV )N ZN N!

(17)

N =0

ZN = hT r e−tHN i/V

(18)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

901

(ZN is the average N impurities partition function per unit volume), we see that α0 is given by the equation: 2 Z1 Z1 Z2 −2 +1= −1 . (19) Z1 Z0 Z0 Computing Z2 diagrammatically to fourth order in α, we finally get that α0 is solution of: 2 1 1 7 1 α(α − 1) 4 = (α(α − 1))2 + (α(1 − α))3 + 1 − ζ(3) (α(α − 1)) . 2 6 3 8 2 (20) In the interval ]0, 1/2], one obtains: α0 ≈ .29. To end up with this part, let us mention what happens when we consider correlated impurities that are spatially distributed like fermions at T = 0 [1]. Diagrammatic computations lead to: Zα (t) =

1 (1 + πα(α − 1)ρt + 0(ρt)2 + ...(ρt)3 + . . .) 2πt

(21)

and for the specific heat: c − c0 ∼t→0 −k(tρπα(1 − α))2 < 0

(22)

when 0 < α ≤ 1/2. From this, we conclude that the D.O.S. has always oscillations. 2

Hall conductivity [2]

For a review on the Integer Quantum Hall Effect, see, for instance, reference [7]. In this part, we develop a Kubo inspired formalism and compute the linear response of the system to a small uniform electric field applied in the ~ = δ(t)E ~ o . The local current ~x direction, E ~j(~r) = e {~v |~r ih~r | + |~r ih~r |~v } 2

(23)

~ is proportional to the conductivity (~v is the velocity operator ~v = p~ − eA) ρ[ji (~r, t), rj ]} σij (~r, t) = iθ(t)e T r {ˆ

(24)

where θ(t) is the Heaviside function. T rρˆ · · · is the thermal Boltzmann or Fermi-Dirac average. ji (~r, t) is the current density operator in the Heisenberg representation ~j(~r, t) = eiHt~j(~r)e−iHt .

(25)

902

Topological Aspects of Low Dimensional Systems

Considering the combination σ − (~r, t) = σxx (~r, t) − iσyx (~r, t)

(26)

and ρˆ = e−βH /Zβ (Boltzmann statistics), the global conductivity averaged over volume reads: Z 1 (27) d~r σβ− (~r, t) σβ− (t) ≡ V Z e2 1 d~rd~r 0 Π− Git (~r, ~r 0 )x0 Gβ−it (~r 0 , ~r) − (it → it + β) = iθ(t) V Zβ (G is the thermal propagator). To deduce from (27) the conductivity of a gas of electrons at zero temperature and Fermi energy EF , one uses the integral representation of the step function θ(EF − H) Z ∞ 0 dt0 eiEF t Zβ→it0 +0 σβ→it0 +0 (t) (28) σEF (t) = lim η 0 ,0 →0+ −∞ 2iπ t0 − iη 0 where 0 and η 0 are regulators which have to be set to zero at the end. In general, it will be more convenient to calculate the derivative σ(t) ˙ of σ(t) with respect to time, rather than σ(t) itself. In the case of the thermal Boltzmann conductivity, one gets Z e2 1 e2 − δ(t) − θ(t) d~rd~r 0 e B(~r) Π− Git (~r, ~r 0 )x0 Gβ−it (~r 0 , ~r) σ˙ β (t) = V V Zβ −(it → it + β) . (29) To derive (29), the identity e B(~r) ± V (~r) [H, Π± ] = ∓ eB(~r)Π± + Π± , 2

(30)

has been used, which is valid in general for an Hamiltonian H = (~ p− 2 ~ eA) /2 + V (~r). The appearance of the local magnetic field B(~r) in (29) – in the magnetic impurity case, it is a sum of δ(~r −~ri ) functions – greatly simplifies the space integrals. We now discuss some examples:

i) homogeneous magnetic field (29) rewrites as σ˙ βL− (t) =

e2 δ(t) + 2iωcσβL− (t) V

(31)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

903

leading to: σβL− (ω) =

1 e2 · V − i(ω + 2ωc )

(32)

For a gas of electrons at T = 0: L (ω)|yx = −N (EF ) Re σE F

e2 2ωc · V 4ωc2 − ω 2

(33)

In (33), the limit ωc → 0 is properly defined only if one keeps ω 6= 0, in which case it vanishes, as it should. The Hall conductivity finally reads L (ω = 0)|yx = −N (EF ) Re σE F

e 1 · V B

(34)

This is the classical straight line, showing no plateaus in the Hall conductivity as a function of the number of electrons N (EF ), or of the inverse magnetic field 1/B.

ii) Hall conductivity for one vortex With the standard Aharonov-Bohm propagator 0 +∞ X 1 0 rr 1 − 2β (r 2 +r 02 ) e I|m−α| eim(θ−θ ) Gβ (~r, ~r ) = 2πβ β m=−∞ 0

(35)

(Iν (z)’s are the modified Bessel functions), one gets: e2 1 e2 sin(πα) α δ(t) + θ(t) 2 eiπα (t (t + iβ)1−α − t1−α (t − iβ)α ). V V β Zβ π (36) Its Fourier transform reads i e2 1 iπα sin(πα) Γ(1 + α)Ψ(1 + α, 3; β(ω + i)) e 1− σβ− (ω) = V (ω + i) Zβ π −Γ(2 − α)Ψ(2 − α, 3; −β(ω + i)) (37) σ˙ β− (t) =

where the Ψ(a, b, z)’s are the unregular confluent hypergeometric functions. In the ω → 0 limit, the Hall conductivity reads:

Re σβ (ω)|yx =

¯ e2 sin(2πα) h α(1 − α) 2 (¯ h βω) ln(¯ h βω) + · · · . (38) 1 + m2e V 2 ω2 2

904

Topological Aspects of Low Dimensional Systems

For a gas of electrons coupled to the vortex at zero temperature, one gets, in the limit ω EF , Re σEF (ω)|yx = N (EF )

e2 1 e2 2πα sin(2πα) 2 'α→0 N (EF ) 2 2 2 V ω V ω

(39)

consistent with the homogeneous magnetic field result (Eq. (33) with 2πα/V = eB = 2ωc ). It is possible to generalize those results when an external uniform B field is added to the vortex. When α is small: ie πα2 d − (ω = 0) = ) + N (E ) + ··· (40) N (E σE F F F V hBtot i V dEF (ehBtot i = eB + 2πα/V ).

iii) Perturbative hall conductivity for magnetic impurities Considering the nonunitary wavefunction redefinitions 1

ψ(~r) = e− 2 hωc ir

2

N Y

|~r − ~ri |α ψ˜0 (~r) ≡ U 0 d (~r)ψ˜0 (~r)

(41)

i=1

one obtains the Hamiltonian acting on the new wavefunctions ψ˜0 1 hLi hLi hLi H˜ 0 d = Π+ Π− − iα(Ω − hΩi)Π− (42) 2 P hLi z /2), Ω = N where Π− = −2i(∂z − hωc i¯ i=1 1/(z − zi ) and hΩi = πρz. It allows for perturbative computations. Skipping details [2], we get the final simple result: Re σEF (ω = 0)|yx = −N (EF + αhωc i)

1 e · V hBi

(43)

In Figure 2, σxy exhibits small oscillations above the classical straight line. It is worthwhile to notice that Hall plateaus shifted above the classical straight line have already been observed experimentally when the Quantum Hall device contains repulsive impurities [8]. 3

Magnetization and persistent currents [3]

Since the pioneering work of Bloch [9], several questions concerning persistent currents have been answered. The conducting ring case has been

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

905

Fig. 2. Hall conductivity in unit of e2 /h of the random magnetic impurity model h F) at first order in α for α = 0.01 as a function of the filling factor ν = N(E : V ehBi straight line = classical result, full line = perturbative result.

largely discussed in the literature [10], as well as the persistent current due to a point-like vortex [11]. Let us start with the definition of the total magnetization Z e 1 d~r (~r × h~j(~r)i) · ~k + hσz i (44) M= 2 2 where h i means average over Boltzmann or Fermi-Dirac distributions. In the Boltzmann case, one obtains the thermal magnetization Zβ ≡ T r e−βH n o e T r e−βH (~r × ~v ) · ~k + σz · (45) Mβ = 2Zβ It is easy to realize that Mβ =

1 ∂ ln Zβ (B 0 ) lim 0 β B →0 ∂B 0

(46)

where Zα (B 0 ) is the partition function when an uniform magnetic field, B 0 , perpendicular to the plane, is added to the system.

906

Topological Aspects of Low Dimensional Systems

Dropping the spin term, the orbital part of the magnetization reads e Mβorb = Mβ − σz . 2

(47)

Let us now turn to the persistent current and consider in the plane a semi infinite line D starting at ~r0 and making an angle θo with the horizontal x-axis. The orbital persistent current I orb (~r0 , θ0 ) through the line is Z I orb (~r0 , θ0 ) ≡

D

d|~r − ~r0 |

(~r − ~r0 ) × h~j(~r)i ~ · k. |~r − ~r0 |

(48)

Consider now systems rotationnally invariant around ~r0 . I orb (~r0 , θ0 ) no longer depends on θ0 and, without loss of generality, can be averaged over θ0 . So Z 1 (~r − ~r0 ) × h~j(~r)i ~ ·k (49) d~r I orb (~r0 ) = 2π |~r − ~r0 |2 and for the Boltzmann distribution e (~r − ~r0 ) × ~v ~ T r e−βH · k · Iβorb (~r0 ) = 2πZβ |~r − ~r0 |2

(50)

It is possible to show that [3]: Iβorb (~r0 ) =

e ∂ ln Zβ (α0 ) e Gβ (~r0 , ~r0 ) lim − σz 2πβ α0 →0 ∂α0 2 Zβ

(51)

where Gβ is the thermal propagator and Zβ (α0 ) the partition function when a fictitious vortex of strength α0 is added in ~r0 . The last term in equation (51) emphasizes the importance of the spin coupling in the Hamiltonian for a correct definition of persistent currents. For systems that are both invariant by translation and rotation, we can write, using (45, 50), Z 1 1 orb = (52) d~r0 I(~r0 ) = M orb . I V V Let us now discuss some specific examples:

i) Point vortex in O + uniform magnetic field case Using the corresponding partition function (b ≡ βωc ) e−b V be−b −(α−1)b sinh αb + α−e Zβ (B, α) = 2πβ sinh b 2 sinh b sinh b

(53)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

one obtains Mβorb

=

1 eπβ 1 sinh αb − coth b − α − e−(α−1)b b 2V b2 sinh b b sinh αb e−(2α−1)b (α − e(α−1)b ) + ··· + sinh b sinh b e 2

907

Iβorb (~0)

e = 2V

1 2e−2αb − b 1 − e−2b

(54)

(55)

with, of course, Mβorb 6= Iβorb V .

ii) Magnetic impurities After averaging over disorder, the system is invariant by translation and rotation, so it is sufficient to compute M orb to get the orbital persistent current. In the Brownian motion approach, we get:

−βρS e sin(βρA)A {C} orb Mβ = − −βρS he cos(βρA)i{C}

(56)

where A, S and A have been defined previously in equations (9-11). (56) allows for numerical computations. Mβorb is actually only a function of βρ and α, odd in α. Thus, for α ∈ [0, 1/2], necessarily Mβorb = (1 − 2α)F (βρ, α(1 − α)) = (1 − 2α)

∞ X

(βρ)n

n=1

X

amn (α(1 − α))m

m≥n

(57) which can in principle be obtained in perturbation theory. Setting 1 e 0 orb − coth hbi Mβ |mean = (1 − 2α) 2 hbi0

(58)

0

with hbi = βπρα(1 − α) and using the previous example i), one obtains the result (mean field + one vortex corrections) [3]: Mβorb

=

Mβorb |mean + eα(1 − α)(1 − 2α)

×

hbi − 1 − e−2hbi 2hbi (1 − hbi )e−2hbi 1 + + 0 0 0 1 − e−2hbi (1 − e−2hbi )2 2hbi

0

0

0

0

0

! + · · ·(59)

Topological Aspects of Low Dimensional Systems

Mβorb

908

0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05

βρ =1

0

0.2

0.4

α

0.6

0.8

1

Fig. 3. The orbital magnetization in unit e = 1 in the magnetic impurity problem for βρ = 1. Comparison between the analytical computation, solid curve, and numerical simulations.

Figure 3 (βρ = 1), shows a rather good agreement between (59) – full curve – and the numerical simulations – points – based on (56). However, the situation becomes less transparent for higher βρ values. Clearly, the perturbative analytical approach needs more and more corrections coming from hBi + two vortices, . . . References [1] Desbois J., Furtlehner C. and Ouvry S., Nucl. Phys. B[FS] 453 (1995) 759; J. Phys. I France 6 (1996) 641; J. Phys. A: Math. Gen. 30 (1997) 7291. [2] Desbois J., Ouvry S. and Texier C., Nucl. Phys. B[FS] 500 (1997) 486. [3] Desbois J., Ouvry S. and Texier C., Nucl. Phys. B[FS] 528 (1998) 727. [4] Pryor C. and Zee A., Phys. Rev. B 46 (1992) 3116; Lusakopwski A. and Turski A., Phys. Rev. B 48 (1993) 3835; G. Gavazzi, J. M. Wheatley and A. J. Schonfield, Phys. Rev. B 47 (1993) 15170; Kiers K. and Weiss J., Phys. Rev. D 49 (1994) 2081; Emparan R. and Valle Basagoiti M.A., Phys. Rev. B 49 (1994) 14460; Geim A.K., Bending S.J. and Grigorieva I.V., Phys. Rev. Lett. 69 (1992) 2252; Geim A.K., Bending S.J., Grigorieva I.V. and Balmire M.G., Phys. Rev. B 49 (1994) 5749; Brey L. and Fertig H.A., Phys. Rev. B 47 (1993) 15961; Khaetskii A.V., J. Phys. C 3 (1991) 5115. [5] Bergman O. and Lozano G., Ann. Phys. 229 (1994) 416; Emparan R. and Valle Basagoiti M.A., Mod. Phys. Lett. A 8 (1993) 3291; Valle Basagoiti M.A., Phys. Lett. B 306 (1993) 307. [6] Comtet A., Mashkevich S. and Ouvry S., Phys. Rev. D 52 (1995) 2594; Ouvry S., Phys. Rev. D 50 (1994) 5296. [7] Janssen M., et al., “Introduction to the theory of the Integer Quantum Hall Effect”, edited by J. Hadju (VCH, Weinheim, 1994) and references therein. [8] Haug R.J., Gerhardts R.R., Klitzing K.V. and Ploog K., Phys. Rev. Lett. 59 (1987) 1349.

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[9] Bloch F., Phys. Rev. 137 (1965) A787. [10] L´ evy L.P., Dolan G., Dunsmuir J. and Bouchiat H., Phys. Rev. Lett. 64 (1990) 2074; Cheung H.F., Gefen Y., Riedel E.K. and Shih W.H., Phys. Rev. B 37 (1988) 6050; Cheung H.F., Riedel E.K. and Gefen Y., Phys. Rev. Lett. 62 (1989) 587; Buttiker M., Phys. Scr. T54 (1994) 104; Buttiker M., Imry Y. and Landauer R., Phys. Lett. 96A (1983) 365; Avishai Y. and Kohmoto M., Ben Gurion University Report for a review and references see Narevich R., Technion Haifa Thesis 5757 (July 1997). [11] Akkermans E., Auerbach A., Avron J.E. and Shapiro B., Phys. Rev. Lett. 66 (1991) 76; Comtet A., Moroz A. and Ouvry S., Phys. Rev. Lett. 74 (1995) 828; Moroz A., Phys. Rev. A 53 (1996) 669; for a review and references see R. Narevich, Technion Haifa Thesis 5757 (july 1997); see also Sitenko Yu.A. and Babansky A.Yu., Bogolyubov Institute Report (1997).