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Tables of Contents for Topological Aspects of Low Dimensional Systems

Chapter/Section Title

Page #

Page Count

Lecturers

xi

Participants

xiii

Preface

xvii

Preface

xxi

Contents

xxiii

Electrons in a Flatland

1

52

M. Shayegan

Introduction

3

3

Samples and measurements

6

4

2D eletrons at the GaAs/AIGaAs interface

6

4

Magnetotransport measurement techniques

10

1

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

10

6

Shubnikov-de Haas oscillations and the IQHE

10

2

FQHE and Wigner crystal

12

4

Composite fermions

16

3

Ferromagnetic state at v = 1 and Skyrmions

19

2

Correlated bilayer electron states

21

32

Overview

21

5

Electron system in a wide single, quantum well

26

3

Evolution of the QHE states in a wide quantum well

29

5

Evolution of insulating phases

34

7

Many-body, bilayer QHE at v = 1

41

3

Spontaneous interlayer charge transfer

44

4

Summary

48

5

The Quantum Hall Effect: Novel Excitations and Broken Symmetries

53

124

S.M. Girvin

The quantum Hall effect

55

122

Introduction

55

2

Why 2D is important

57

1

Constructing the 2DEG

57

1

Why is disorder and localization important?

58

3

Classical dynamics

61

3

Semi-classical approximation

64

1

Quantum dynamics in strong B Fields

65

7

IQHE edge states

72

4

Semiclassical percolation picture

76

4

Fractional QHE

80

5

The v = 1 many-body state

85

9

Neutral collective excitations

94

10

Charged excitations

104

9

FQHE edge states

113

3

Quantum hall ferromagnets

116

2

Coulomb exchange

118

1

Spin wave excitations

119

5

Effective action

124

5

Topological excitations

129

12

Skyrmion dynamics

141

6

Skyrme lattices

147

5

Double-layer quantum Hall ferromagnets

152

2

Pseudospin analogy

154

2

Experimental background

156

4

Interlayer phase coherence

160

2

Interlayer tunneling and tilted field effects

162

3

Appendix A Lowest Landau level projection

165

3

Appendix B Berry's phase and adiabatic transport

168

9

Aspects of Chern-Simons Theory

177

88

G.V. Dunne

Introduction

179

3

Basics of planar field theory

182

13

Chern-Simons coupled to matter fields - ``anyons''

182

4

Maxwell-Chern-Simons: Topologically massive gauge theory

186

3

Fermions in 2 + 1-dimensions

189

1

Discrete symmetries: P, C an T

190

2

Poincare algebra in 2 + 1-dimensions

192

1

Nonabelian Chern-Simons theories

193

2

Canonical quantization of Chern-Simons theories

195

19

Canonical structure of Chern-Simons theories

195

3

Chern-Simons quantum mechanics

198

5

Canonical quantization of abelian Chern-Simons theories

203

2

Quantization on the torus and magnetic translations

205

3

Canonical quantization of nonabelian Chern-Simons theories

208

4

Chern-Simons theories with boundary

212

2

Chern-Simons vortices

214

23

Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices

214

5

Relativistic Chern-Simons vortices

219

5

Nonabelian relativistic Chern-Simons vortices

224

1

Nonrelativistic Chern-Simons vortices: Jackiw-Pi model

225

3

Nonabelian nonrelativistic Chern-Simon vortices

228

3

Vortices in the Zhang-Hansson-Kivelson model for FQHE

231

3

Vortex dynamics

234

3

Induced Chern-Simons terms

237

28

Perturbatively induced Chern-Simons terms: Fermion loop

238

4

Induced currents and Chern-Simons terms

242

1

Induced Chern-Simons terms without fermions

243

3

A finite temperature puzzle

246

2

Quantum mechanical finite temperature model

248

5

Exact finite temperature 2 + 1 effective actions

253

3

Finite temperature perturbation theory and Chern-Simons terms

256

9

Anyons

265

150

J. Myrheim

Introduction

269

11

The concept of particle statistics

270

3

Statistical mechanics and the many-body problem

273

2

Experimental physics in two dimensions

275

2

The algebraic approach: Heisenberg quantization

277

2

More general quantizations

279

1

The configuration space

280

6

The Euclidean relative space for two particles

281

2

Dimensions d = 1, 2, 3

283

1

Homotopy

283

2

The braid group

285

1

Schrodinger quantization in one dimension

286

4

Heisenberg quantization in one dimension

290

5

The coordinate representation

291

4

Schrodinger quantization in dimension d ≥ 2

295

11

Scalar wave functions

296

2

Homotopy

298

1

Interchange phases

299

2

The statistics vector potential

301

2

The N-particle case

303

1

Chern-Simons theory

304

2

The Feynman path integral for anyons

306

11

Eigenstates for position nd momentum

307

1

The path integral

308

4

Conjugation classes in SN

312

2

The non-interacting case

314

1

Duality of Feynman and Schrodinger quantization

315

2

The harmonic oscillator

317

21

The two-dimensional harmonic oscillator

317

3

Two anyons in a harmonic oscillator potential

320

3

More than two anyons

323

9

The three-anyon problem

332

6

The anyon gas

338

35

The cluster and virial expansions

339

1

First and second order perturbative results

340

4

Regularization by periodic boundary conditions

344

4

Regularization by a harmonic oscillator potential

348

2

Bosons and fermions

350

2

Two anyons

352

2

Three anyons

354

2

The Monte Carlo method

356

2

The path integral representation f the coefficients GP

358

4

Exact and approximate polynomials

362

2

The fourth virial coefficient of anyons

364

4

Two polynomial theorems

368

5

Charged particles in a constant magnetic field

373

10

One particle in a magnetic field

374

3

Two anyons in a magnetic field

377

3

The anyon gas in a magnetic filed

380

3

Interchange phases and geometric phases

383

32

Introduction to geometric phases

383

2

One particle in magnetic field

385

2

Two particles in a magnetic field

387

3

Interchange of two anyons in potential wells

390

2

Laughlin's theory of the fractional quantum Hall effect

392

23

Generalized Statistics in One Dimension

415

58

A.P. Polychronakos

Introduction

417

1

Permutation group approach

418

9

Realization of the reduced Hilbert space

418

4

Path integral and generalized statistics

422

2

Cluster decomposition and factorizability

424

3

One-dimensional systems: Calogero model

427

6

The Calogero-Sutherland-Moser model

428

3

Large-N properties of the CSM model and duality

431

2

One-dimensional systems: Matrix model

433

15

Hermitian matrix model

433

4

The unitary matrix model

437

1

Quantization and spectrum

438

5

Reduction to spin-particle systems

443

5

Operator approaches

448

11

Exchange operator formalism

448

5

Systems with internal degrees of freedom

453

2

Asymptotic Bethe ansatz approach

455

2

The freezing trick and spin models

457

2

Exclusion statistics

459

10

Motivation from the CSM model

459

1

Semiclassics - Heuristics

460

2

Exclusion statistical mechanics

462

3

Exclusion statistics path integral

465

2

Is this the only ``exclusion'' statistics?

467

2

Epilogue

469

4

Lectures on Non-perturbative Field Theory and Quantum Impurity Problems

473

78

H. Saleur

Some notions of conformal field theory

483

20

The free boson via path integrals

483

2

Normal ordering and OPE

485

3

The stress energy tensor

488

2

Conformal in(co)variance

490

3

Some remarks on Ward identities in QFT

493

1

The Virasoro algebra: Intuitive introduction

494

3

Cylinders

497

3

The free boson via Hamiltonians

500

2

Modular invariance

502

1

Conformal invariance analysis of quantum impurity fixed points

503

9

Boundary conformal field theory

503

3

Partition functions and boundary states

506

3

Boundary entropy

509

3

The boundary sine-Gordon model: General results

512

8

The model and the flow

512

1

Perturbation near the UV fixed points

513

2

Perturbation near the IR fixed point

515

3

An alternative to the instanton expansion: The conformal invariance analysis

518

2

Search for integrability: Classical analysis

520

4

Quantum integrability

524

8

Conformal perturbation theory

524

2

S-matrices

526

5

Back to the boundary sine-Gordon model

531

1

The thermodynamic Bethe-ansatz: The gas of particles with ``Yang-Baxter statistics''

532

9

Zamolodchikov Fateev algebra

532

2

The TBA

534

2

A standard computation: The central charge

536

2

Thermodynamics of the flow between N and D fixed points

538

3

Using the TBA to compute static transport properties

541

10

Tunneling in the FQHE

541

1

Conductance without impurity

542

1

Conductance with impurity

543

8

Quantum Partition Noise and the Detection of Fractionally Charged Laughlin Quasiparticles

551

24

D.C. Glattli

Introduction

553

1

Partition noise in quantum conductors

554

8

Quantum partition noise

554

1

Partition noise and quantum statistics

555

2

Quantum conductors reach the partition noise limit

557

1

Experimental evidence of quantum partition noise in quantum conductors

558

4

Partition noise in the quantum Hall regime and determination of the fractional charge

562

13

Edge states in the integer quantum Hall effect regime

562

1

Tunneling between IQHE edge channels and partition noise

563

1

Edge channels in the fractional regime

564

3

Noise predictions in the fractional regime

567

2

Measurement of the fractional charge using noise

569

1

Beyond the Poissonian noise of fractional charges

570

5

Mott Insulators, Spin Liquids and Quantum Disordered Superconductivity

575

68

Matthew P.A. Fisher

Introduction

577

2

Models of metals

579

4

Noninteracting electrons

579

3

Interaction effects

582

1

Mott insulators and quantum magnetism

583

5

Spin models and quantum magnetism

584

2

Spin liquids

586

2

Bosonization primer

588

4

2 Leg Hubbard ladder

592

12

Bonding and antibonding bands

592

4

Interactions

596

2

Bosonization

598

3

d-Mott phase

601

2

Symmetry and doping

603

1

d-Wave superconductivity

604

8

BCS theory re-visited

604

5

d-wave symmetry

609

1

Continuum description of gapless quasiparticles

610

2

Effective field theory

612

11

Quasiparticles and phase flucutations

612

6

Nodons

618

5

Vortices

623

5

hc/2e versus hc/e vortices

623

3

Duality

626

2

Nodal liquid phase

628

7

Half-filling

628

4

Doping the nodal liquid

632

2

Closing remarks

634

1

Appendix A Lattice duality

635

8

Two dimensions

636

1

Three dimensions

637

6

Statistics of Knots and Entangled Random Walks

643

92

S. Nechaev

Introduction

645

2

Knot diagrams as disordered spin systems

647

28

Brief review of statistical problem in topology

647

4

Abelian problems in statistics of entangled random walks and incompleteness of Gauss invariant

651

5

Nonabelian algebraic knot invariants

656

7

Lattice knot diagrams as disordered Potts model

663

6

Notion about annealed and quenched realizations of topological disorder

669

6

Random walks on locally non-commutative groups

675

26

Brownian bridges on simplest non-commutative groups and knot statistics

676

13

Random walks on locally free groups

689

3

Analytic results for random walks on locally fee groups

692

5

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

14

Construction of nonabelian connections for γ2 and PSL(2,ZZ) from conformal methods

702

5

Random walk on double punctured plane and conformal field theory

707

2

Statistics of random walks with topological constraints in the two-dimensional lattices of obstacles

709

6

Physical applications. Polymer language in statistics of entangled chain-like objects

715

12

Polymer chain in 3D-array of obstacles

716

3

Collapsed phase of unknotted polymer

719

8

Some ``tight'' problems of the probability theory and statistical physics

727

8

Remarks and comments to Section 2

728

1

Remarks and comments to Sections 3 and 4

728

1

Remarks and comments to Section 5

729

6

Twisting a Single DNA Molecule: Experiments and Models

735

32

T. Strick

J.-F. Allemand

D. Bensimon

V. Croquette

C. Bouchiat

M. Mezard

R. Lavery

Introduction

737

2

Single molecule micromanipulation

739

1

Forces at the molecular scale

739

1

Brownian motion: A sensitive tool for measuring forces

740

1

Stretching B-DNA is well described by the worm-like chain model

740

4

The Freely jointed chain elasticity model

740

3

The overstretching transition

743

1

The torsional buckling instability

744

10

The buckling instability at T = 0

744

2

The buckling instability in the rod-like chain (RLC) model

746

1

Elastic rod model of supercoiled DNA

746

5

Theoretical analysis of the extension versus supercoiling experiments

751

3

Critical torques are associated to phase changes

754

1

Unwinding DNA leads to denaturation

754

6

Twisting rigidity measured through the critical torque of denaturation

755

3

Phase coexistence in the large torsional stress regime

758

2

Overtwisting DNA leads to P-DNA

760

2

Phase coexistence of B-DNA and P-DNA in the large torsional stress regime

760

2

Chemical evidence of exposed bases

762

1

Conclusions

762

5

Introduction to Topological Quantum Numbers

767

76

D.J. Thouless

Preface

769

1

Winding numbers and topological classification

769

6

Precision and topological invariants

769

1

Winding numbers and line defects

770

2

Homotopy groups and defect classification

772

3

Superfluids and superconductors

775

11

Quantized vortices and flux lines

775

6

Detection of quantized circulation and flux

781

3

Precision of circulation and flux quantization measurements

784

2

The Magnus force

786

8

Magnus force and two-fluid model

786

2

Vortex moving in a neutral superfluid

788

4

Transverse force in superconductors

792

2

Quantum Hall effort

794

13

Introduction

794

1

Proportionality of current density and electric field

795

1

Bloch's theorem and the laughlin argument

796

3

Chern numbers

799

4

Fractional quantum Hall effect

803

3

Skyrmions

806

1

Topological phase transitions

807

12

The vortex induced transition in superfluid helium films

807

6

Two-dimensional magnetic systems

813

1

Topological order in solids

814

3

Superconducting films and layered materials

817

2

The A phase of superfluid 3He

819

7

Vortices in the A phase

819

4

Other defects and textures

823

3

Liquid crystals

826

17

Order in liquid crystals

826

2

Defects and textures

828

15

Geometrical Description of Vortices in Ginzburg-Landau Billiards

843

36

E. Akkermans

K. Mallick

Introduction

845

1

Differentiable manifolds

846

14

Manifolds

846

1

Differential forms and their integration

847

6

Topological invariants of a manifold

853

2

Riemannian manifolds and absolute differential calculus

855

3

The Laplacian

858

2

Bibliography

860

1

Fiber bundles and their topology

860

10

Introduction

860

1

Local symmetries. Connexion and curvature

861

1

Chern classes

862

3

Manifolds with a boundary: Chern-Simons classes

865

4

The Weitzenbock formula

869

1

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

870

2

The Ginzburg-Landau equations

870

1

The Bogomol'nyi identities

871

1

The superconducting billiard

872

7

The zero current line

873

1

A selection mechanism and topological phase transitions

874

1

A geometrical expression of the Gibbs potential for finite systems

874

5

The Integer Quantum Hall Effect and Anderson Localisation

879

16

J.T. Chalker

Introduction

881

1

Scaling theory and localisation transitions

882

3

The plateau transitions as quantum critical points

885

2

Single particle models

887

3

Numerical studies

890

2

Discussion and outlook

892

3

Seminar 5. Random Magnetic Impurities and Quantum Hall Effect

895

16

J. Desbois

Average density of states (D.O.S.) [1]

897

4

Hall conductivity [2]

901

3

Magnetization and persistent currents [3]

904

7

Seminars by participants

911