UJF NATO ASI LES HOUCHES Session LXIX 1998
ASPECTS TOPOLOGIQUES DE LA PHYSIQUE EN BASSE DIMENSION
TOPOLOGICAL ASPECTS ...
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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
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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.
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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.
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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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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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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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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
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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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
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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+(
5HHQWUDQW (OHFWURQ6ROLG 3KDVH
:&
)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.
ρ[\>KH @
5[[>NΩ@
%>7@ Fig. 10. Magnetotransport data, taken at T 30 mK, for a 750 ˚ A-wide well with n = 1.03 × 1011 cm−2 (main figure) and n = 1.55 × 1011 cm−2 (inset), showing well-developed even-denominator FQH states at ν = 12 and 32 . These unique FQH states are stabilized by both interlayer and intralayer correlations. (After Suen et al. [49].)
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
ρ[\>KH @
−
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ν
ρ[[>NΩV@
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%>7@ Fig. 11. Data for the same sample of Figure 10 but with n = 1.26 × 1011 cm−2 . Here we observe an insulating phase (IP) reentrant around the ν = 12 FQH state. The inset shows the temperature dependence of resistivity: at ν = 12 , ρxx vanishes as T → 0 indicative of a FQH state while at slightly higher and lower ν, ρxx shows an insulating behavior as it diverges with decreasing T . (After Manoharan et al. [51].)
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
−
× FP
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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
*PH9
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6
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M. Shayegan: Electrons in a Flatland
*
29
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Fig. 15. Summary of sample and measurement geometries. Top figure: The sample has front and back electrodes (gates) which can be biased by applying voltages (VFG and VBG ) with respect to the electron system to change its density and charge distribution symmetry. (a) through (c) illustrate how in various measurements we can tune the bilayer system by varying three parameters, the density, the charge distribution symmetry or “balance” (characterized by density imbalance nt ), and the tilt angle between the direction of the magnetic field and sample normal. (After Manoharan [28].)
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
D
5[[>Ω@
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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 .
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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
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.
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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 −→ ¯
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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.
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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].
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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
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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
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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)
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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,
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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
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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).
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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.
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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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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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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.
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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).
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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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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
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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
94
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
S.M. Girvin: The Quantum Hall Effect
95
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
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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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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.
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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.
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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
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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.
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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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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].
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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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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
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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
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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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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.
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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)
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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
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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)
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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.
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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.
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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
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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
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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)
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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)
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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.
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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)
S.M. Girvin: The Quantum Hall Effect
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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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
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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
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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
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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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143
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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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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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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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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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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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.
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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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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.
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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.
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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.
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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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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
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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.
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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.
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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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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.
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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.
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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)
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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
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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)
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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)
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^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
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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)
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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)
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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 ∆
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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.
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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
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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 . . . . . . . . . . . . .
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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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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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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
G.V. Dunne: Aspects of Chern-Simons Theory
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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Topological Aspects of Low Dimensional Systems
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)
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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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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)
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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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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)
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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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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 .
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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)
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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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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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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
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_ 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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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
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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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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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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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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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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
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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.
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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.
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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)
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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).
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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
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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)
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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)
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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
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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,
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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)
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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)
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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).
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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
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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)
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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
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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
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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)
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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)
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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.
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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
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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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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
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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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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
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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)
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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
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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.
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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).
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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)
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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
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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.
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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.
G.V. Dunne: Aspects of Chern-Simons Theory
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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Topological Aspects of Low Dimensional Systems
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.
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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
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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].
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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
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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.
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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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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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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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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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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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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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'$
'$
? 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)
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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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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)
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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].
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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.
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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
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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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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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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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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
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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
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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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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
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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
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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
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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
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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
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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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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.
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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)
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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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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
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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)
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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)
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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)
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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.
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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)
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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)
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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)
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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 .
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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)
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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
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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].
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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.
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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 .
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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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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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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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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
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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
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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)
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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.
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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
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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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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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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)
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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.
J. Myrheim: Anyons
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
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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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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
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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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Topological Aspects of Low Dimensional Systems
• 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
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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
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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 .).
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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
427
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.
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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 =
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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)
446
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
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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).
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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.
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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
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(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 s
