topological aspects of low-dimensional systems

Plasma physicsBlack holesFluids dynamicsMolecular fluids*Atomic and molecular physics and the interstellar matter*Structural analysis of collision amplitudes Frontiers in laser spectrosc

LES HOUCHESSession LXIX1998

ASPECTS TOPOLOGIQUES DE LA PHYSIQUE EN BASSE DIMENSION

TOPOLOGICAL ASPECTS OF LOW DIMENSIONAL SYSTEMS

G DUNNE

B DUPLANTIERM.P.A FISHER

S GIRVIN

J MYRHEIM

S NECHAEVA.P POLYCHRONAKOS

H SALEUR

M SHAYEGAN

D THOULESS

A AKKERMANSJ.T CHALKER

V CROQUETTE

J DESBOISD.C GLATTLI

a NATO Advanced Study Institute

LES HOUCHES

SESSION LXIX 7-31 July 1998

Aspects topologiques de la physique

7 avenue du Hoggar, PA de Courtabœuf,

B.P 112, 91944 Les Ulis cedex A, France

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Published in cooperation with the NATO Scientific Affair Division

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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

1951 -1997

Quantum mechanics Quantum field theoryQuantum mechanics Statistical mechanics Nuclearphysics

Quantum mechanics Solid state physics Statisticalmechanics Elementary particle physics

Quantum mechanics Collision theory Nucleon-nucleoninteraction Quantum electrodynamics

Quantum mechanics Non-equilibrium phenomena.Nuclear reactions Interaction of a nucleus with atomicand molecular fields

Quantum perturbation theory Low temperature physics.Quantum theory of solids; dislocations and plasticproperties Magnesium; ferromagnetism

Scattering theory Recent developments in field theory.Nuclear interaction; strong interactions High energyelectrons Experiments in high energy nuclear physicsThe 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 electronicsHigh energy physics

High energy astrophysics*

Many body physics*

Plasma physicsBlack holesFluids dynamicsMolecular 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 matterMembranes and intercellular communicationPhysical cosmology

Laser plasma interactionPhysics of defectsChaotic 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 shortdistances*

Signal processing*

Chance and matterAstrophysical fluid dynamicsLiquids at interfaces

Fields, strings and critical phenomenaOceanographic and geophysical tomographyLiquids, freezing and glass transitionChaos and quantum physics*

Fundamental systems in quantum optics*

Supernovae*

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

Particles in the nineties*

Strongly interacting fermions and high T c tivity

superconduc-Gravitation and quantizationsProgress in picture processing*

Computational fluid dynamicsCosmology and large scale structureMesoscopic quantum physicsFluctuating geometries in statistical mechanics andquantum field theory

Quantum fluctuations*

Quantum symmetries*

From cell to brain*

Trends in nuclear physics, 100 years laterModélisation du climat de la terre et de sa variabilitéParticules et interactions : le modèle standard mis àl'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 XII-XXV: Gordon and Breach - Sessions XXVI-LXVIII: North-Holland

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

S.M GIRVIN, Indiana University, Department of Physics, Bloomington,

A.P POLYCHRONAKOS, Institutionen för Teoretisk Fysik, Box 803,

751 08 Uppsala, Sweden, and Physics Department, University of Ionnina,

SEMINAR SPEAKERS

E AKKERMANS, Technion, Israel Institute of Technology, Department

of Physics, 32000 Haifa, Israel

J CHALKER, Theoretical Physics, Oxford University, 1 Keble road,Oxford, 0X1 3NP, U.K

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

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 deCiencias, c-v, Universidad Autonoma de Madrid, 28049 Madrid, Spain

D BAZZALI, Laboratoire de Physique Théorique et Modélisation, Université deCergy-Pontoise, 2 avenue Adolph Chauvin, 95302 Cergy-Pontoise,France

J BETOURAS, University of Oxford, Department of Physics, TheoreticalPhysics, 1 Keble Road, Oxford, OX1 3NP, U.K

M BOCQUET, Service de Physique Théorique, L'Orme des Merisiers, CEASaclay, 91191 G if-sur-Yvette Cedex, France

V BRUNEL, SPhT, L'Orme des Merisiers, CEA Saclay, 91191 G if-sur-YvetteCedex, France

J BÜRKI, Institut Romand de Recherche Numérique en Physique desMatériaux, École Polytechnique Fédérale de Lausanne, 1015 Lausanne,Suisse

D CARPENTIER, Laboratoire de Physique Théorique de l'École NormaleSupérieure, 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, CB3OHE, U.K

T FUKUI, Institut für Theoretische Physik, Universität zu Köln, Zülpicherstr.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

T HALL, University of Connecticut U-46, Physics Department, 2152 HillsideRoad, Storrs, CT 06269, U.S.A.

J.H HAN, APCTP, 207-43 Cheongryangri-Dong Dongdaemun-Gu, Seoul

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, UniversitätKarlsruhe, 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 OrsayCedex, France

D LILLIEHÖÖK, Department of Physics, Stockholm University, Box 6730, 113

M MILOVANOVIC, Technion, Physics Department, 32000 Haifa, Israel

G MISGUICH, LPTL, Université Pierre et Marie Curie, 4 place Jussieu, 75252Paris 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'ENSLyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

K.-V PHAM, Laboratoire de Physique des Solides, bâtiment 510, CentreUniversitaire 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, NJ08855-0849, U.S.A., Loomis Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, U.S.A

N SANDLER, Univ of Illinois at Urbana-Champaign, Dept of Physics, 1110West Green St., Urbana, IL 61801, U.S.A.

F SIANO, Univ of Southern California, Dept of Physics, Los Angeles, CA90089-0484, 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

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, SantaBarbara, 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 deSaclay, 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, andMedical Radiological Research Centre, Obninsk, Kaluga Region 249020,Russia

E TUTUC, Department of Elec Engineering, Princeton University, Princeton,

NJ 08544, U.S.A

L'utilisation en physique théorique de concepts empruntes a la topologie aconduit depuis plusieurs décennies à des développements intéressants dansdes directions variées.

En théorie quantique des champs, les travaux précurseurs de Skyrmesuivis de ceux sur les solutions classiques des équations de Yang-Mills-Higgs ont fait largement appel à ces notions et ont ainsi permis d'ex-plorer certains secteurs non perturbatifs des théories de jauge

Les concepts empruntés à la topologie ont trouvé d'autres champs plication, en particulier en physique de la matière condensée Citons parexemple les travaux sur la classification des défauts dans les milieux ordon-nés (Kleman, Toulouse, École des Houches XXXV, 1980)

d'ap-Plus récemment, un domaine ó topologie et physique de la matièrecondensée ont connu une synergie remarquable est l'effet Hall quantique.Les rapides et impressionnants progrès expérimentaux dans la fabricationd'hétérojonctions (par épitaxie moléculaire) ó un gaz bidimensionneld'électrons peut être piégé ont été accompagnés de progrès théoriques dans

la compréhension des systèmes tridimensionnels Ces développements jeurs, spécifiques des années 80-90, ont été couronnés en automne 1998 par leprix Nobel de Physique, attribué à deux expérimentateurs, Stornier et Tsui,pour la découverte expérimentale indirecte de porteurs de charges fraction-naires dans les systèmes Hall quantique, et à un théoricien, Laughlin, pourleur prédiction théorique

ma-Les notions de charge et de statistique fractionnaires ont précisément uneinterprétation 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 Lesthèmes principaux de l'École ont porté sur la physique de l'effet Hall, et lesconcepts théoriques spécifiques à la physique bidimensionnelle, tels les statis-tiques intermédiaires (modèle des anyons) ou les théories de Chern-Simons.Des incursions ont été effectuées dans les systèmes unidimensionnels, telsles liquides de Luttinger et les modèles de Calogero-Sutherland

contraintes topologiques peuvent en effet être décrites par des concepts pruntés à la théorie des noeuds et à la physique statistique C'est dans cecontexte qu'ont été abordés à l'École l'étude du mouvement Brownien etses relations avec la théorie des nœuds.

em-Le déroulement de l'Ecole a été le suivant : Steve Girvin a ouvert l'Ecolepar un cours théorique sur l'effet Hall quantique et certains développementsrécents comme les Skyrmions En parallèle, Mansour Shayegan couvraitles aspects expérimentaux de l'effet Hall Les théories de Chern-Simonsont été abordées par Gerald Dunne Le modèle des anyons et le problème

de la quantification d'un système de particules identiques en dimension 2ont été discutés en détail par Jan Myrheim Les aspects purement uni-dimensionnels des statistiques intermédiaires ont été couverts par AlexiosPolychronakos Hubert Saleur a donné un cours introductif aux théoriesconformes et à leurs applications au problème de la transmission tunnel àtravers 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 degrenaille du courant tunnel à travers l'échantillon Hall Il s'agit là de laconfirmation directe de l'existence de charges fractionnaires transportant lecourant Hall, entrevues dans les expériences de Störmer et Tsui du débutdes années 80 Certains développements expérimentaux de ce domaine par-ticulièrement chaud ont été couverts dans un séminaire donné par ChristianGlattli Serguei Nechaev et Bertrand Duplantier ont clôt l'École par deuxrevues sur le mouvement Brownien, le groupe des tresses et leurs relationsavec la théorie des nœuds Plusieurs de ces concepts interviennent dansl'étude des propriétés d'élasticité et de torsion des molécules d'ADN, sujetqui a fait l'objet d'un séminaire de Vincent Croquette

Il nous a paru opportun de replacer les considérations topologiquesévoquées dans les différents cours et séminaires dans un contexte plus géné-ral : c'était là l'objectif du cours de David Thouless La notion de nombretopologique a été illustrée par de nombreux exemples allant de la physique

de l'effet Hall à celle des superfluides Les vortex qui sont naturellement aucœur de ce dernier sujet sont réapparus dans le séminaire d'Éric Ackermansconsacré à la supraconductivité dans les systèmes mésoscopiques

Les questions de l'effet du désordre sur un gaz d'électrons nel en présence d'un champ magnétique jouent certainement un rôle central,encore mal compris, dans la compréhension de l'effet Hall Faute de pouvoir

bidimension-y consacrer un cours entier, ces problèmes ont été traités dans un séminairesur l'état de l'art par John Chalker, et dans un séminaire par Jean Desbois

sur un modèle dans lequel la source du désordre se trouve dans le champmagnétique.

Matthew Fisher a été malheureusement empêché à la dernière minute

de venir aux Houches donner son cours sur de nouvelles phases dans dessystèmes 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 cevolume 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 sent présenter des séminaires sur leur travail Deux sessions ont ainsi étéconsacrées 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 puparticiper à cette session Nous espérons que la publication rapide de cevolume leur permettra de profiter du programme de cette École

puis-Cette LXIXe session de l'École d'Été des Houches a été rendue possiblegrâce :

- au soutien de l'Université Joseph Fourier de Grenoble et aux tiens financiers du Ministère de l'Éducation Nationale, de la Recherche

sou-et de la Technologie (MENRT), du Centre National de la RechercheScientifique (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 laNational Science Foundation (NSF) des U.S.A ;

- aux orientations données par le Conseil d'Administration de l'École

The use of concepts borrowed from topology has led to major advances intheoretical 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 sification of defects in ordered media by homotopy theory is a well-knownexample (see e.g Kleman and Toulouse, Les Houches XXXV, 1980).More recently, topology and condensed matter physics have again met

Clas-in the realm of the fractional quantum Hall effect Experimental progress

in molecular beam epitaxy techniques leading to high-mobility samples lowed the discovery of this remarkable and novel phenomenon These de-velopments lead also to the attribution of the 1998 Nobel Prize in physics

al-to Laughlin, Störmer and Tsui

The notions of fractional charge as well as fractional statistics can beinterpreted by a topological interaction of infinite range So it is natural tofind in the Les Houches series a school devoted to quantum Hall physics,intermediate statistics and Chern-Simons theory This session also includedsome one-dimensional physics topics like the Calogero-Sutherland modeland some Luttinger-liquid physics

Polymer physics is also related to topology In this field topologicalconstraints may be described by concepts from knot theory and statisticalphysics Hence this session also included Brownian motion theory related

to knot theory

The school started with a theoretical survey by Steve M Girvin onthe quantum Hall effect, including recent developments on skyrmions Anexperimental review was given at the same time by Mansour Shayegan.Chern-Simons theories were discussed by Gerald Dunne The physics ofanyons and quantization in two dimensions was presented by Jan Myrheim.One-dimensional statistics was reviewed by Alexios Polychronakos HubertSaleur discussed conformai field theory and recent applications to impu-rity problems The evidence for fractional charge in shot noise measure-ments was presented by D Christian Glattli Serguei Nechaev and BertrandDuplantier presented Brownian motion, braid group theory and the linkwith knot theory A seminar by Vincent Croquette was devoted to recentapplications to DNA physics

A general overview of the role of topology in physics was given by DavidThouless The very notion of topological quantum numbers was illustrated

by various examples from quantum Hall physics to superfluids Vorticeswere also a common theme in a seminar given by Eric Akkermans

The all-important role of disorder in the quantum Hall effect was cussed in a review seminar by John Chalker and a more specialized talk byJean Desbois, who concentrated on a model with a random magnetic field.Matthew P.A Fisher was unfortunately unable attend the session asoriginally scheduled However, he kindly produced the lecture notes thatare included in this volume We are very grateful to him for this Thelectures by Bertrand Duplantier led to no written version at all, contrary

dis-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 allthe many highly qualified people who applied We hope that the quickpublication 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'EducationNationale, de la Recherche et de la Technologie (MENRT), the CentreNational de la Recherche Scientifique (CNRS) and the Commissariat

à l'Énergie Atomique (CEA);

- the Division for Scientific Affairs of NATO whose ASI program cluded this session;

in 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 thepreparation of this session as well as during the session Thanks are alsodue to "Le Chef as well as to all the people in Les Houches who made thiswonderful session possible

A Comtet

T Jolicceur

S Ouvry

F David

2.1 2D electrons at the GaAs/AlGaAs interface 6 2.2 Magnetotransport measurement techniques 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

5 Ferromagnetic state at ν = 1 and Skyrmions 19

6 Correlated bilayer electron states 21 6.1 Overview 21 6.2 Electron System in a wide, single, quantum well 26 6.3 Evolution of the QHE states in a wide quantum well 29 6.4 Evolution of insulating phases 34 6.5 Many-body, bilayer QHE at ν = 1 41 6.6 Spontaneous interlayer Charge transfer 44 6.7 Summary 48

Course 2 The Quantum Hall Effect: Novel Excitations

and Broken Symmetries

1.1 Introduction 55 1.2 Why 2D is important 57 1.3 Constructing the 2DEG 57 1.4 Why is disorder and localization important? 58 1.5 Classical dynamics 61 1.6 Semi-classical approximation 64 1.7 Quantum dynamics in strong B Fields 65 1.8 IQHE edge states 72 1.9 Semiclassical percolation picture 76 1.10 Fractional QHE 80 1.11 The ν = 1 many-body state 85 1.12 Neutral collective excitations 94 1.13 Charged excitations 104 1.14 FQHE edge states 113 1.15 Quantum hall ferromagnets 116 1.16 Coulomb exchange 118 1.17 Spin wave excitations 119 1.18 Effective action 124 1.19 Topological excitations 129 1.20 Skyrmion dynamics 141 1.21 Skyrme lattices 147 1.22 Double-layer quantum Hall ferromagnets 152 1.23 Pseudospin analogy 154 1.24 Experimental background 156 1.25 Interlayer phase coherence 160 1.26 Interlayer tunneling and tilted field effects 162 Appendix A Lowest Landau level projection 165

Appendix B Berry’s phase and adiabatic transport 168

Course 3 Aspects of Chern-Simons Theory

2.1 Chern-Simons coupled to matter fields - “anyons” 182 2.2 Maxwell-Chern-Simons: Topologically massive gauge theory 186 2.3 Fermions in 2 + 1-dimensions 189 2.4 Discrete symmetries: P, C and T 190

2.5 Poincar´ e algebra in 2 + 1-dimensions 192 2.6 Nonabelian Chern-Simons theories 193

3 Canonical quantization of Chern-Simons theories 195 3.1 Canonical structure of Chern-Simons theories 195 3.2 Chern-Simons quantum mechanics 198 3.3 Canonical quantization of abelian Chern-Simons theories 203 3.4 Quantization on the torus and magnetic translations 205 3.5 Canonical quantization of nonabelian Chern-Simons theories 208 3.6 Chern-Simons theories with boundary 212

4.1 Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices 214 4.2 Relativistic Chern-Simons vortices 219 4.3 Nonabelian relativistic Chern-Simons vortices 224 4.4 Nonrelativistic Chern-Simons vortices: Jackiw-Pi model 225 4.5 Nonabelian nonrelativistic Chern-Simons vortices 228 4.6 Vortices in the Zhang-Hansson-Kivelson model for FQHE 231 4.7 Vortex dynamics 234

5.1 Perturbatively induced Chern-Simons terms: Fermion loop 238 5.2 Induced currents and Chern-Simons terms 242 5.3 Induced Chern-Simons terms without fermions 243 5.4 A finite temperature puzzle 246 5.5 Quantum mechanical finite temperature model 248 5.6 Exact finite temperature 2 + 1 effective actions 253 5.7 Finite temperature perturbation theory and Chern-Simons terms 256

2.1 The Euclidean relative space for two particles 281 2.2 Dimensions d = 1, 2, 3 283 2.3 Homotopy 283 2.4 The braid group 285

3 Schr¨ odinger quantization in one dimension 286

4 Heisenberg quantization in one dimension 290 4.1 The coordinate representation 291

5 Schr¨ odinger quantization in dimension d 2 295 5.1 Scalar wave functions 296 5.2 Homotopy 298 5.3 Interchange phases 299 5.4 The statistics vector potential 301 5.5 The N-particle case 303 5.6 Chern-Simons theory 304

6 The Feynman path integral for anyons 306 6.1 Eigenstates for Position and momentum 307 6.2 The path integral 308 6.3 Conjugation classes in SN 312 6.4 The non-interacting case 314 6.5 Duality of Feynman and Schr¨ odinger quantization 315

7.1 The two-dimensional harmonic oscillator 317 7.2 Two anyons in a harmonic oscillator potential 320 7.3 More than two anyons 323 7.4 The three-anyon problem 332

8.1 The cluster and virial expansions 339 8.2 First and second order perturbative results 340 8.3 Regularization by periodic boundary conditions 344 8.4 Regularization by a harmonic oscillator potential 348 8.5 Bosons and fermions 350 8.6 Two anyons 352 8.7 Three anyons 354 8.8 The Monte Carlo method 356 8.9 The path integral representation of the coefficients GP 358 8.10 Exact and approximate polynomials 362 8.11 The fourth virial coefficient of anyons 364 8.12 Two polynomial theorems 368

9 Charged particles in a constant magnetic field 373 9.1 One particle in a magnetic field 374 9.2 Two anyons in a magnetic field 377 9.3 The anyon gas in a magnetic field 380

10 Interchange phases and geometric phases 383 10.1 Introduction to geometric phases 383 10.2 One particle in a magnetic field 385 10.3 Two particles in a magnetic field 387 10.4 Interchange of two anyons in potential wells 390 10.5 Laughlin’s theory of the fractional quantum Hall effect 392

Course 5 Generalized Statistics in One Dimension

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 433 4.1 Hermitian matrix model 433 4.2 The unitary matrix model 437 4.3 Quantization and spectrum 438 4.4 Reduction to spin-particle systems 443

5.1 Exchange operator formalism 448 5.2 Systems with internal degrees of freedom 453 5.3 Asymptotic Bethe ansatz approach 455 5.4 The freezing trick and spin models 457

6.1 Motivation from the CSM model 459 6.2 Semiclassics – Heuristics 460 6.3 Exclusion statistical mechanics 462 6.4 Exclusion statistics path integral 465 6.5 Is this the only “exclusion” statistics? 467

Course 6 Lectures on Non-perturbative Field Theory

and Quantum Impurity Problems

1 Some notions of conformal field theory 483 1.1 The free boson via path integrals 483 1.2 Normal ordering and OPE 485 1.3 The stress energy tensor 488 1.4 Conformal in(co)variance 490 1.5 Some remarks on Ward identities in QFT 493 1.6 The Virasoro algebra: Intuitive introduction 494 1.7 Cylinders 497 1.8 The free boson via Hamiltonians 500 1.9 Modular invariance 502

2 Conformal invariance analysis of quantum impurity fixed

2.1 Boundary conformal field theory 503 2.2 Partition functions and boundary states 506 2.3 Boundary entropy 509

3 The boundary sine-Gordon model: General results 512 3.1 The model and the flow 512 3.2 Perturbation near the UV fixed point 513 3.3 Perturbation near the IR fixed point 515 3.4 An alternative to the instanton expansion: The conformal

6 The thermodynamic Bethe-ansatz: The gas of particles with

6.1 Zamolodchikov Fateev algebra 532 6.2 The TBA 534 6.3 A Standard computation: The central Charge 536 6.4 Thermodynamics of the flow between N and D fixed points 538

7 Using the TBA to compute static transport properties 541 7.1 Tunneling in the FQHE 541 7.2 Conductance without impurity 542 7.3 Conductance with impurity 543

Seminar 1 Quantum Partition Noise and the Detection

of Fractionally Charged Laughlin Quasiparticles

2 Partition noise in quantum conductors 554 2.1 Quantum partition noise 554 2.2 Partition noise and quantum statistics 555 2.3 Quantum conductors reach the partition noise limit 557 2.4 Experimental evidences of quantum partition noise in quantum

conductors 558

3 Partition noise in the quantum Hall regime and determination

3.1 Edge states in the integer quantum Hall effect regime 562 3.2 Tunneling between IQHE edge channels and partition noise 563 3.3 Edge channels in the fractional regime 564 3.4 Noise predictions in the fractional regime 567 3.5 Measurement of the fractional Charge using noise 569 3.6 Beyond the Poissonian noise of fractional charges 570

Course 7 Mott Insulators, Spin Liquids and Quantum

3 Mott insulators and quantum magnetism 583 3.1 Spin models and quantum magnetism 584 3.2 Spin liquids 586

5.1 Bonding and antibonding bands 592 5.2 Interactions 596 5.3 Bosonization 598 5.4 d-Mott phase 601 5.5 Symmetry and doping 603

6 d-Wave superconductivity 604 6.1 BGS theory re-visited 604 6.2 d-wave symmetry 609 6.3 Continuum description of gapless quasiparticles 610

7.1 Quasiparticles and phase flucutations 612 7.2 Nodons 618

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

Course 8 Statistics of Knots and Entangled Random

incompleteness of Gauss invariant 651 2.3 Nonabelian algebraic knot invariants 656 2.4 Lattice knot diagrams as disordered Potts model 663 2.5 Notion about annealed and quenched realizations of topological

disorder 669

3 Random walks on locally non-commutative groups 675 3.1 Brownian bridges on simplest non-commutative groups and knot

statistics 676 3.2 Random walks on locally free groups 689 3.3 Analytic results for random walks on locally free groups 692 3.4 Brownian bridges on Lobachevskii plane and products

of non-commutative random matrices 697

4 Conformal methods in statistics of random walks with topological

4.1 Construction of nonabelian connections for Γ 2 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

Seminar 2 Twisting a Single DNA Molecule: Experiments

and Models

by T Strick, J.-F Allemand, D Bensimon, V Croquette,

C Bouchiat, M M ´ezard and R Lavery 735

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

experiments 751 4.5 Critical torques are associated to phase changes 754

5 Unwinding DNA leads to denaturation 754 5.1 Twisting rigidity measured through the critical torque

of denaturation 755 5.2 Phase coexistence in the large torsional stress regime 758

6 Overtwisting DNA leads to P-DNA 760 6.1 Phase coexistence of B-DNA and P-DNA in the large torsional

stress regime 760 6.2 Chemical evidence of exposed bases 762

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

3.1 Magnus force and two-fluid model 786 3.2 Vortex moving in a neutral superfluid 788 3.3 Transverse force in superconductors 792

4.1 Introduction 794 4.2 Proportionality of current density and electric field 795 4.3 Bloch’s theorem and the Laughlin argument 796 4.4 Chern numbers 799 4.5 Fractional quantum Hall effect 803 4.6 Skyrmions 806

5.1 The vortex induced transition in superfluid helium films 807 5.2 Two-dimensional magnetic Systems 813 5.3 Topological order in solids 814 5.4 Superconducting films and layered materials 817

6 The A phase of superfluid3He 819 6.1 Vortices in the A phase 819 6.2 Other defects and textures 823

3 Fiber bundles and their topology 860 3.1 Introduction 860 3.2 Local symmetries Connexion and curvature 861 3.3 Chern classes 862 3.4 Manifolds with a boundary: Chern-Simons classes 865 3.5 The Weitzenb¨ ock formula 869

4 The dual point of Ginzburg-Landau equations for an infinite

4.1 The Ginzburg-Landau equations 870 4.2 The Bogomol’nyi identities 871

5.1 The zero current line 873 5.2 A selection mechanism and topological phase transitions 874 5.3 A geometrical expression of the Gibbs potential for finite Systems 874

Seminar 4 The Integer Quantum Hall Effect

and Anderson Localisation

2 Scaling theory and localisation transitions 882

3 The plateau transitions as quantum critical points 885

Seminar 5 Random Magnetic Impurities

and Quantum Hall Effect

ELECTRONS IN A FLATLAND

M SHAYEGAN

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

1 Introduction 3

2.1 2D electrons at the GaAs/AlGaAs interface 62.2 Magnetotransport measurement techniques 10

3 Ground states of the 2D system in a strong magnetic field 10

3.1 Shubnikov-de Haas oscillations and the IQHE 103.2 FQHE and Wigner crystal 12

5 Ferromagnetic state at
= 1 and Skyrmions 19

6 Correlated bilayer electron states 21

6.1 Overview 216.2 Electron system in a wide, single, quantum well 266.3 Evolution of the QHE states in a wide quantum well 296.4 Evolution of insulating phases 346.5 Many-body, bilayer QHE at ν = 1 416.6 Spontaneous interlayer charge transfer 446.7 Summary 48

of current are shown, and the vertical markings denote the Landau-level

filling factor (ν) Look how the behavior of ρ xx with temperature (T ),

shown schematically in the inset, changes as a function of the magneticfield At certain fields, markedA, ρ xx 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

(ρ xy) becomes quantized near these fields The QHE is best described as anincompressible quantum liquid which often possesses a high degree of short-

range electron correlation Next, look at the T -dependence of ρ xx at thefields markedB (near 13 and 14 T for this sample) Here ρ xxexponentially

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 positionalorder Now look at what happens at the magnetic field marked C At this

field, ρ xxshows a nearly temperature-independent behavior, reminiscent of

a metal It turns out that at this particular field there are two flux quantaper each electron The electron magically combines with the two flux quantaand forms the celebrated “composite Fermion”, a quasiparticle which nowmoves 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 ety of ground states ranging from insulating to metallic to “superconducting-like” And, as it turns out, these ground states are stabilized primarily bystrong electron-electron correlations The data of Figure 1 reveals the ex-treme richness of this system, one which has rendered the field of 2D carriersystems in a high magnetic field among the most active and exciting in solidstate physics It has already led to two physics Nobel prizes, one in 1985

vari-c

EDP Sciences, Springer-Verlag 1999

Fig 1 Low-temperature magnetotransport coefficients of a high-quality

(low-disorder) 2D electron system in a modulation-doped GaAs/AlGaAs

heterostruc-ture with a 2D density of 6.6 × 1010cm−2 The longitudinal (ρ xx ) and Hall (ρ xy)resistivities at a temperature of 40 mK are shown in the main figure The Landau-

level filling factors (ν) are indicated by vertical markings The right upper inset

shows the typical measurement geometry while the left inset schematically

illus-trates the widely different temperature dependences of ρ xxat different magneticfields (filling factors)

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], butsurprises 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, haveunveiled 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

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 provideelegant 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, thesimilarity of the IQHE and FQHE, the transitions between QHE states andthe transitions between QHE and insulating states at low fillings

The purpose of these notes is to provide a glimpse of some of the excitingrecent experimental results in this field I will focus on the following fiveareas; I will be very brief when covering these topics except in the partdealing with the bilayer systems, where I will go a bit more in depth:

1 a quick summary of some of the sample parameters and experimentalaspects;

2 some basic and general remarks on the ground states of a 2DES in astrong magnetic field;

3 a simple magnetic focusing experiment near ν = 12 which provides aclear demonstration of the presence of a composite Fermion Fermi sur-face 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

free-dom 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 theimportant and exciting aspects of the QHE and related phenomena Theyprovide only a limited and selective sample of recent experimental devel-opments Readers interested in more details are referred to the originalpapers 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 gest reading the comprehensive and illuminating notes by Steve Girvin inthis volume and those by Allan MacDonald in proceedings of the 1994 LesHouches Summer School [9]

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 undopedGaAs 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 tominimize 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 structureshown in (c))

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 interfacebetween 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,