ISBN: 3-540-66909-4
TITLE: Aspects topologiques de la physique en basse dimension. Topological aspects of low dimensional systems
AUTHOR: Comtet, A.; Jolicoeur, T.; Ouvry, S.; David, F. (Eds.)
TOC:

Lecturers xi
Participants xiii
Prface 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 nu = 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 nu = 1 41
6.6 Spontaneous interlayer Charge transfer 44
6.7 Summary 48
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 nu = 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
2.5 Poincar 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 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 Schrdinger quantization in one dimension 286
4 Heisenberg quantization in one dimension 290
4.1 The coordinate representation 291
5 Schrdinger 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 S_N 312
6.4 The non-interacting case 314
6.5 Duality of Feynman and Schrdinger quantization 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
8.9 The path integral representation of the coefficients G_P 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 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
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
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
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
4 Conformal methods in statistics of random walks with topological constraints 701
4.1 Construction of nonabelian connections for gamma_2 and PSL(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 twodimensional 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. Mzard 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
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
6 The A phase of superfluid ^3He 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 Weitzenbck formula 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
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
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