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
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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
