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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