CONTENTS



Lecturers                                                                                                  xi

Participants                                                                                              xiii

  ´
Preface                                                                                                  xvii

Preface                                                                                                   xxi

Contents                                                                                                 xxiii




Course 1. Electrons in a Flatland
by M. Shayegan                                                                                              1
1 Introduction                                                                                              3
2 Samples and measurements                                                                                  6
  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
4 Composite fermions                                                                                      16
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
xxiv

Course 2. The Quantum Hall Effect: Novel Excitations
and Broken Symmetries
by S.M.Girvin                                                                                                         53
1 The    quantum Hall effect                                                                                            55
  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
by G.V. Dunne                                                                                                         177
1 Introduction                                                                                                        179
2 Basics of planar field theory                                                                                        182
  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
                                                                                                                  xxv

   2.5   Poincar´ algebra in 2 + 1-dimensions . . . . . . . . . . . . . . . . . . 192
                e
   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 Chern-Simons vortices                                                                                           214
  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 Induced Chern-Simons terms                                                                                      237
  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

Course 4. Anyons
by J. Myrheim                                                                                                     265

1 Introduction                                                                                                    269
  1.1 The concept of particle statistics . . . . . . . . . .          ..      .   .   .   .   .   .   .   .   .   270
  1.2 Statistical mechanics and the many-body problem                  .      .   .   .   .   .   .   .   .   .   273
  1.3 Experimental physics in two dimensions . . . . .                ..      .   .   .   .   .   .   .   .   .   275
  1.4 The algebraic approach: Heisenberg quantization                 ..      .   .   .   .   .   .   .   .   .   277
  1.5 More general quantizations . . . . . . . . . . . . .            ..      .   .   .   .   .   .   .   .   .   279

2 The    configuration space                                                                                       280
  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¨dinger quantization in one dimension
      o                                                                                                           286
xxvi

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


                                                      2
5 Schr¨dinger quantization in dimension d
      o                                                                                                                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
         Conjugation classes in SN . . . . . . . . . . . . . .             .   .   .   .   .   .   .   .   .   .   .   312
  6.3
  6.4    The non-interacting case . . . . . . . . . . . . . .              .   .   .   .   .   .   .   .   .   .   .   314
  6.5    Duality of Feynman and Schr¨dinger quantization
                                        o                                      .   .   .   .   .   .   .   .   .   .   315


7 The    harmonic oscillator                                                                                           317
  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 The    anyon gas                                                                                                     338
  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
                                                                                       .   .   .   .   .   .   .   .   358
  8.9    The path integral representation of the coefficients GP
  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
                                                                                                                      xxvii

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
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                                                                                   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 Operator approaches                                                                                                     448
  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 Exclusion statistics                                                                                                    459
  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

7 Epilogue                                                                                                                469
xxviii

Course 6. Lectures on Non-perturbative Field Theory
and Quantum Impurity Problems
by H. Saleur                                                                                                        473

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
  points                                                                         503
  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
         invariance analysis . . . . . . . . . . . . . . . . . . . . . . .                  . . . . . . 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”                                                                                          532
  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
                                                                                                                                   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                                                                                     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
  of the fractional Charge                                                                                                         562
  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
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                                                                                                              588

5 2 Leg Hubbard ladder                                                                                                             592
  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
xxx

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 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                                             647
  2.1 Brief review of statistical problems in topology . . . . . . . . .         . . . 647
  2.2 Abelian problems in statistics of entangled random walks and
      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
                                                                                       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, Z) 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          the probability theory and statistical
  physics                                                                            727
  6.1 Remarks and comments          to Section 2 . . . . . . . . . . . . . . . . . . 728
  6.2 Remarks and comments          to Sections 3 and 4 . . . . . . . . . . . . . . 728
  6.3 Remarks and comments          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. Mezard 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    torsional buckling instability                                                  744
  4.1    The buckling instability at T = 0 . . . . . . . . . . . . . . .       . . . . . 744
  4.2    The buckling instability in the rod-like chain (RLC) model            . . . . . 746
  4.3    Elastic rod model of supercoiled DNA . . . . . . . . . . . .          . . . . . 746
  4.4    Theoretical analysis of the extension versus supercoiling
         experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . 751
   4.5   Critical torques are associated to phase changes . . . . . . .        . . . . . 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     Magnus force                                                                                     786
  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 Quantum Hall effect                                                                                       794
  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 Topological phase transitions                                                                            807
  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
                                                                                                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                                                                           846
  2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   846
  2.2 Differential forms and their integration . . . . . . . . . .       .   .   .   .   .   .   .   847
  2.3 Topological invariants of a manifold . . . . . . . . . . . .      .   .   .   .   .   .   .   853
  2.4 Riemannian manifolds and absolute differential calculus            .   .   .   .   .   .   .   855
  2.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . .   .   .   .   .   .   .   .   858
  2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .    .   .   .   .   .   .   .   860

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¨ck formula . . . . . . . . . . . . . .
                   o                                             ....   .   .   .   .   .   .   .   869

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    superconducting billiard                                                      872
  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
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