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Topological Aspects of Low Dimensional Systems :  Session LXIX, 7-31 July 1998 - Alain Comtet

Topological Aspects of Low Dimensional Systems

Session LXIX, 7-31 July 1998

By: Alain Comtet (Editor), T. Jolicoeur (Editor), S. Ouvry (Editor), F. David (Editor)

Hardcover Published: March 2000
ISBN: 9783540669098
Number Of Pages: 911

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The book contains the worked out lecture notes of the courses delivered at the summer school in Les Houches. They address graduate students & are of interest for researchers as well. The book can be used as an introduction into three closely related fields: anyons & fractional statistics, fractional quantum Hall effect & knot theory. The theoretical papers are occasionally completed by reports on experimental techniques, & besides physics some application to biological systems can be found

Lecturersp. xi
Participantsp. xiii
Préfacep. xvii
Prefacep. xxi
Contentsp. xxiii
Electrons in a Flatlandp. 1
Introductionp. 3
Samples and measurementsp. 6
2D electrons at the GaAs/AlGaAs interfacep. 6
Magnetotransport measurement techniquesp. 10
Ground states of the 2D System in a strong magnetic fieldp. 10
Shubnikov-de Haas oscillations and the IQHEp. 10
FQHE and Wigner crystalp. 12
Composite fermionsp. 16
Ferromagnetic state at ¿ = 1 and Skyrmionsp. 19
Correlated bilayer electron statesp. 21
Overviewp. 21
Electron System in a wide, single, quantum wellp. 26
Evolution of the QHE states in a wide quantum wellp. 29
Evolution of insulating phasesp. 34
Many-body, bilayer QHE at ¿ = 1p. 41
Spontaneous interlayer Charge transferp. 44
Summaryp. 48
The Quantum Hall Effect: Novel Excitations and Broken Symmetriesp. 53
The quantum Hall effectp. 55
Introductionp. 55
Why 2D is importantp. 57
Constructing the 2DEGp. 57
Why is disorder and localization important?p. 58
Classical dynamicsp. 61
Semi-classical approximationp. 64
Quantum dynamics in strong B Fieldsp. 65
IQHE edge statesp. 72
Semiclassical percolation picturep. 76
Fractional QHEp. 80
The ¿ = 1 many-body statep. 85
Neutral collective excitationsp. 94
Charged excitationsp. 104
FQHE edge statesp. 113
Quantum hall ferromagnetsp. 116
Coulomb exchangep. 118
Spin wave excitationsp. 119
Effective actionp. 124
Topological excitationsp. 129
Skyrmion dynamicsp. 141
Skyrme latticesp. 147
Double-layer quantum Hall ferromagnetsp. 152
Pseudospin analogyp. 154
Experimental backgroundp. 156
Interlayer phase coherencep. 160
Interlayer tunneling and tilted field effectsp. 162
Lowest Landau level projectionp. 165
Berry's phase and adiabatic transportp. 168
Aspects of Chern-Simons Theoryp. 177
Introductionp. 179
Basics of planar field theoryp. 182
Chern-Simons coupled to matter fields - "anyons"p. 182
Maxwell-Chern-Simons: Topologically massive gauge theoryp. 186
Fermions in 2 + 1-dimensionsp. 189
Discrete symmetries: <$>{cal P}, {cal C}<$> and <$>{cal T}<$>p. 190
Poincaré algebra in 2 + 1-dimensionsp. 192
Nonabelian Chern-Simons theoriesp. 193
Canonical quantization of Chern-Simons theoriesp. 195
Canonical structure of Chern-Simons theoriesp. 195
Chern-Simons quantum mechanicsp. 198
Canonical quantization of abelian Chern-Simons theoriesp. 203
Quantization on the torus and magnetic translationsp. 205
Canonical quantization of nonabelian Chern-Simons theoriesp. 208
Chern-Simons theories with boundaryp. 212
Chern-Simons vorticesp. 214
Abelian-Higgs model and Abrikosov-Nielsen-Olesen vorticesp. 214
Relativistic Chern-Simons vorticesp. 219
NonabelianrelativisticChern-Simonsvorticesp. 224
Nonrelativistic Chern-Simons vortices: Jackiw-Pi modelp. 225
NonabeliannonrelativisticChern-Simonsvorticesp. 228
Vortices in the Zhang-Hansson-Kivelson model for FQHEp. 231
Vortex dynamicsp. 234
Induced Chern-Simons termsp. 237
Perturbatively induced Chern-Simons terms: Fermion loopp. 238
Induced currents and Chern-Simons termsp. 242
Induced Chern-Simons terms without fermionsp. 243
A finite temperature puzzlep. 246
Quantum mechanical finite temperature modelp. 248
Exact finite temperature 2 + 1 effective actionsp. 253
Finite temperature perturbation theory and Chern-Simons termsp. 256
Anyonsp. 265
Introductionp. 269
The concept of particle statisticsp. 270
Statistical mechanics and the many-body problemp. 273
Experimental physics in two dimensionsp. 275
The algebraic approach: Heisenberg quantizationp. 277
More general quantizationsp. 279
The configuration spacep. 280
The Euclidean relative space for two particlesp. 281
Dimensions d = 1, 2, 3p. 283
Homotopyp. 283
The braid groupp. 285
Schrödinger quantization in one dimensionp. 286
Heisenberg quantization in one dimensionp. 290
The coordinate representationp. 291
Schrödinger quantization in dimension d ≥ 2p. 295
Scalar wave functionsp. 296
Homotopyp. 298
Interchange phasesp. 299
The statistics vector potentialp. 301
The N-particle casep. 303
Chern-Simons theoryp. 304
The Feynman path integral for anyonsp. 306
Eigenstates for Position and momentump. 307
The path integralp. 308
Conjugation classes in SNp. 312
The non-interacting casep. 314
Duality of Feynman and Schrödinger quantizationp. 315
The harmonic oscillatorp. 317
The two-dimensional harmonic oscillatorp. 317
Two anyons in a harmonic oscillator potentialp. 320
More than two anyonsp. 323
The three-anyon problemp. 332
The anyon gasp. 338
The cluster and virial expansionsp. 339
First and second order perturbative resultsp. 340
Regularization by periodic boundary conditionsp. 344
Regularization by a harmonic oscillator potentialp. 348
Bosons and fermionsp. 350
Two anyonsp. 352
Three anyonsp. 354
The Monte Carlo methodp. 356
The path integral representation of the coefficients <$>G_{cal P}<$>p. 358
Exact and approximate polynomialsp. 362
The fourth virial coefficient of anyonsp. 364
Two polynomial theoremsp. 368
Charged particles in a constant magnetic fieldp. 373
One particle in a magnetic fieldp. 374
Two anyons in a magnetic fieldp. 377
The anyon gas in a magnetic fieldp. 380
Interchange phases and geometric phasesp. 383
Introduction to geometric phasesp. 383
One particle in a magnetic fieldp. 385
Two particles in a magnetic fieldp. 387
Interchange of two anyons in potential wellsp. 390
Laughlin's theory of the fractional quantum Hall effectp. 392
Generalized Statistics in One Dimensionp. 415
Introductionp. 417
Permutation group approachp. 418
Realization of the reduced Hilbert spacep. 418
Path integral and generalized statisticsp. 422
Cluster decomposition and factorizabilityp. 424
One-dimensional systems: Calogero modelp. 427
The Calogero-Sutherland-Moser modelp. 428
Large-N properties of the CSM model and dualityp. 431
One-dimensional systems: Matrix modelp. 433
Hermitian matrix modelp. 433
The unitary matrix modelp. 437
Quantization and spectrump. 438
Reduction to spin-particle systemsp. 443
Operator approachesp. 448
Exchange operator formalismp. 448
Systems with internal degrees of freedomp. 453
Asymptotic Bethe ansatz approachp. 455
The freezing trick and spin modelsp. 457
Exclusion statisticsp. 459
Motivation from the CSM modelp. 459
Semiclassics - Heuristicsp. 460
Exclusion statistical mechanicsp. 462
Exclusion statistics path integralp. 465
Is this the only "exclusion" statistics?p. 467
Epiloguep. 469
Lectures on Non-perturbative Field Theory and Quantum Impurity Problemsp. 473
Some notions of conformal field theoryp. 483
The free boson via path integralsp. 483
Normal ordering and OPEp. 485
The stress energy tensorp. 488
Conformal in(co)variancep. 490
Some remarks on Ward identities in QFTp. 493
The Virasoro algebra: Intuitive introductionp. 494
Cylindersp. 497
The free boson via Hamiltoniansp. 500
Modular invariancep. 502
Conformal invariance analysis of quantum impurity fixed pointsp. 503
Boundary conformal field theoryp. 503
Partition functions and boundary statesp. 506
Boundary entropyp. 509
The boundary sine-Gordon model: General resultsp. 512
The model and the flowp. 512
Perturbation near the UV fixed pointp. 513
Perturbation near the IR fixed pointp. 515
An alternative to the instanton expansion: The conformal invariance analysisp. 518
Search for integrability: Classical analysisp. 520
Quantum integrabilityp. 524
Conformal perturbation theoryp. 524
S-matricesp. 526
Back to the boundary sine-Gordon modelp. 531
The thermodynamic Bethe-ansatz: The gas of particles with "Yang-Baxter statistics"p. 532
Zamolodchikov Fateev algebrap. 532
The TBAp. 534
A Standard computation: The central Chargep. 536
Thermodynamics of the flow between N and D fixed pointsp. 538
Using the TBA to compute static transport propertiesp. 541
Tunneling in the FQHEp. 541
Conductance without impurityp. 542
Conductance with impurityp. 543
Quantum Partition Noise and the Detection of Fractionally Charged Laughlin Quasiparticlesp. 551
Introductionp. 553
Partition noise in quantum conductorsp. 554
Quantum partition noisep. 554
Partition noise and quantum statisticsp. 555
Quantum conductors reach the partition noise limitp. 557
Experimental evidences of quantum partition noise in quantum conductorsp. 558
Partition noise in the quantum Hall regime and determination of the fractional Chargep. 562
Edge states in the integer quantum Hall effect regimep. 562
Tunneling between IQHE edge channels and partition noisep. 563
Edge channels in the fractional regimep. 564
Noise predictions in the fractional regimep. 567
Measurement of the fractional Charge using noisep. 569
Beyond the Poissonian noise of fractional chargesp. 570
Mott Insulators, Spin Liquids and Quantum Disordered Superconductivityp. 575
Introductionp. 577
Models and metalsp. 579
Noninteracting electronsp. 579
Interaction effectsp. 582
Mott insulators and quantum magnetismp. 583
Spin models and quantum magnetismp. 584
Spin liquidsp. 586
Bosonization primerp. 588
2 Leg Hubbard ladderp. 592
Bonding and antibonding bandsp. 592
Interactionsp. 596
Bosonizationp. 598
d-Mott phasep. 601
Symmetry and dopingp. 603
d-Wave superconductivityp. 604
BGS theory re-visitedp. 604
d-wave symmetryp. 609
Continuum description of gapless quasiparticlesp. 610
Effective field theoryp. 612
Quasiparticles and phase flucutationsp. 612
Nodonsp. 618
Vorticesp. 623
ic/2e versus hc/e vorticesp. 623
Dualityp. 626
Nodal liquid phasep. 628
Half-fillingp. 628
Doping the nodal liquidp. 632
Closing remarksp. 634
Lattice dualityp. 635
Two dimensionsp. 636
Three dimensionsp. 637
Statistics of Knots and Entangled Random Walksp. 643
Introductionp. 645
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540669098
ISBN-10: 3540669094
Series: Les Houches - Ecole D'Ete de Physique Theorique
Audience: General
Format: Hardcover
Language: English , French
Number Of Pages: 911
Published: March 2000
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 22.71 x 16.2  x 4.42
Weight (kg): 1.58

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