Part.Ⅰ Quantum Field Theory 1
1．Quantum Mechanics 3
1.1 Hilbert Spaces 3
1.2 Canonical Formalism 4
1.3 Creation and Annihilation Operators 6
1.4 Uncertainty Principle 10
1.5 CoherentStates andvonNeumannLattice 11
1.6 Squeezed Coherent State 14
1.7 Particle Number and Phase 16
1.8 Macroscopic Coherence 18
2．Quantum Field Theory 20
2.1 One-Body Hamiltonian 20
2.2 Many-Body Hamiltonian 21
2.3 Boson Field Operators 23
2.4 Quantum Field Theory 25
2.5 Fermion Field Operators 27
2.6 Electrons Interacting with Electromagnetic Field 29
3. Canonical Quantization 31
3.1 Relativistic Particles and Wayes 31
3.2 Schr？dinger Field 32
3.3 Real Klein-Gordon Field 37
3.4 Complex Klein-Gordon Field 42
3.5 N？ther Currents 44
4．Spontaneous Symmetry Breaking 47
4.1 Ferromagnets 47
4.2 Real Klein-Gordon Field 51
4.3 Complex Klein-Gordon Field 55
4.4 Sigma Model 56
4.5 Schr？dinger Field 58
4.6 Superfluidity 62
4.7 Goldstone Theorem 64
5．Electromagnetic Field 67
5.1 Maxwell Equations 67
5.2 Canonical Quantization 70
5.3 Interaction with Matter Field 77
5.4 Anderson-Higgs Mechanism 79
5.5 Massive Vector Field 81
5.6 Superconductivity 83
5.7 Aharonov-Bohm Effect 89
6．Dirac Field 94
6.1 Dirac Equation 94
6.2 Plane Wave Solutions 97
6.3 Canonical Quantization 99
6.4 Interaction with Electromagnetic Field 103
6.5 Weyl Field(Massless Dirac Field) 105
6.6 Dirac Electrons in Magnetic Field 108
7．Topological Solitons 113
7.1 Topological Sectors 113
7.2 Classical Fields 115
7.3 Solitary Waves,Kinks and Solitons 117
7.4 Sine-Gordon Solitons 119
7.5 Vortex Solitons 123
7.6 Homotopy Classes 127
7.7 O(3)Skyrmions 132
7.8 CPN-1Skyrmions 138
8．Anyons 144
8.1 Spin and Statistics 144
8.2 Fractional Statistics 148
8.3 Quantum Mechanics 150
8.4 Chern-Simons Gauge Theory 153
8.5 Anyon Field Operators 157
Part．Ⅱ Monolayer Quantum Hall Systems 161
9．Overview of Monolayer QH Systems 163
10．Landau Quantization 177
10.1 Planar Electrons 177
10.2 Cvclotron Motion 181
10.3 Symmetric Gauge 187
10.4 Landau Gauge 190
10.5 von Neumann Lattice 193
10.6 Electrons in Nth Landau Level 195
10.7 Hall Current 199
11．Quantum Hall Effects 204
11.1 Incompressibility 204
11.2 Integer Quantum Hall Effects 205
11.3 Fractional Quantum Hall Effects 207
11.4 Quasiparticles 210
11.5 Hall Plateaux 213
12．Quasiparticles and Activation Energy 215
12.1 Impurity Potentials 215
12.2 Gap Energies 216
12.3 Dispersion Relation 219
12.4 Thermal Activation 222
13．Field Theory of Composite Particles 227
13.1 Composite Particles 227
13.2 Statistical Transmutation 230
13.3 Effective Magnetic Field 233
13.4 Dressed Composite Particles 235
13.5 Composite Particles in Lowest Landau Level 239
13.6 Composite Fermions in Lowest Landau Level 242
14．Composite Bosons and Semiclassical Analysis 246
14.1 GroundState andLaughlinWaveFunction 246
14.2 Perturbative Excitations 248
14.3 Vortex Excitations 250
14.4 Field Theory of Vortex Solitons 255
14.5 Haldane-Halperin Hierarchy 258
15．Quantum Hall Ferromagnets 261
15.1 Spin Coherence 261
15.2 Spin Degree of Freedom 263
15.3 Composite Bosons and Spin-Charge Separation 265
15.4 Spin Field,Sigma Field and CP1 Field 268
15.5 Effective Hamiltonian 271
16．Spin Textures 274
16.1 Spin Excitations 274
16.2 Factorizable Skyrmions 277
16.3 Skyrmion Excitation Energy 284
16.4 Experimental Evidence 287
17．Hierarchy of Fractional QH States 291
17.1 Jain Hierarchy 291
17.2 Landau Levels of Composite Fermions 294
17.3 Beyond Principal Sequences 296
17.4 Spin Polarization 298
17.5 Gap Energies 301
18．Edge Effects 307
18.1 Edge Currents and Bulk Currents 307
18.2 Shot Noises of Fractional Charges 308
18.3 Chiral Edge Excitations 311
18.4 Chiral Tomonaga-Luttinger Liquid 312
18.5 Electrodynamics on Edge 316
18.6 EdgeTunneling and Sine-Gordon Solitons 320
19．Stripes and Bubbles in Higher Landau Levels 325
19.1 Higher Landau Levels 325
19.2 Haldane's Pseudopotentials 327
19.3 Effective Coulomb Interactions 328
19.4 Density Matrix Renormalization Group Method 331
19.5 Stripes,Bubbles and Wigner Crystal 332
20．Quantum Hall Effects in Graphene 335
20.1 Unconventional QH Effects 335
20.2 Graphene and Dirac Electrons 338
20.3 Dirac Hamiltonian and Supersymmetry 343
20.4 Effective Coulomb Interactions 346
20.5 Excitonic Condensation 352
20.6 Valley Polarization 355
20.7 Multilayer Graphene Systems 360
20.8 Berry Phase and Index Theorem 365
Part.Ⅲ Bilayer Quantum Hall Systems 367
21．Overview of Bilayer QH Systems 369
22．SU(2)Pseudospin Structure 383
22.1 Bilayer Planar Electrons 383
22.2 Pseudospins 385
22.3 Tunneling Interaction 387
22.4 Imbalanced Configuration 388
22.5 Capacitance Energy 392
22.6 Compound States 393
22.7 Charge-Transferable States 394
23．Bilayer-Locked States 400
23.1 Composite-Boson Field 400
23.2 Wave Functions 404
23.3 Ground State 405
23.4 Vortex Excitations 406
24．Interlayer Coherence 409
24.1 Pseudospin Ferromagnet 409
24.2 Effective Hamiltonian 413
24.3 Pseudospin Waves 415
24.4 Anomalous Bilayer QH Currents 417
24.5 Pseudospin Texture 424
25．SU(4)Quantum Hall Ferromagnets 428
25.1 SU(4)Isospin Structure 428
25.2 SU(4)Isospin Fields 431
25.3 SU(4)Isospin Waves 433
25.4 SU(4)Isospin Textures 436
25.5 ExcitationEnergyofSU(4)Skyrmions 440
25.6 Activation Energy Anomaly 446
26．Bilayer Quantum Hall Systems atv=2 454
26.1 Spin Phase,Ppin Phase and Canted Phase 454
26.2 Ground-State Energy 457
26.3 Ground-State Structure 460
26.4 Phase Diagrams 463
26.5 Experimental Data 466
26.6 SU(4)Breaking and Grassmannian Fields 471
26.7 Grassmannian G42Solitons 475
26.8 Genuine Bilayer versus Two-MonolayerSystems 479
26.9 Experimental Indication of Biskyrmions 482
27．Bilayer Quantum-Hall Junction 485
27.1 Josephson-Like Phenomena 485
27.2 Parallel Magnetic Field 487
27.3 Effective Hamiltonian 493
27.4 Commensurate-Incommensurate Phase Transition 495
27.5 Soliton Lattice 499
27.6 Anomalous Diagonal Resistivity 508
27.7 Plasmon Excitations 514
27.8 Josephson-Like Effects 516
Part.Ⅳ Microscopic Theory 521
28．Overview of Microscopic Theory 523
29．Noncommutative Geometry 534
29.1 Noncommutative Coordinate 534
29.2 Weyl Operator and Symbo1 535
29.3 Magnetic Translation 541
29.4 Density Operators 543
29.5 SU(N)-Extended W∞Algebra 549
29.6 Classical Fields 552
29.7 Topological Charge Density 557
29.8 Kac-Moody Algebra on Edges 559
30．Landau Level Proj ection 562
30.1 Projected Coulomb Interactions 562
30.2 Monolayer QH System 565
30.3 Electron-Hole Pair Excitations 569
30.4 ElectronExcitationandHoleExcitation 570
30.5 Bilayer System withoutSpin(v=1) 573
30.6 Bilayer System with Spin(v=1) 575
30.7 Bilayer System with Spins(v=2) 578
31．Noncommutative Solitons 584
31.1 Topological Charge and Electric Charge 584
31.2 Microscopic Skyrmion States 585
31.3 Noncommutative CP1 Skyrmion 589
31.4 Hole and Skyrmion 592
31.5 Skyrmion Wave Functions 593
31.6 Hardcore Interaction 595
31.7 Coulomb Interaction 600
31.8 SU(4)Skyrmions 605
32．Exchange Interactions and Effective Theory 607
32.1 Exchange Hamiltonian 607
32.2 Decomposition Formula 612
32.3 Spontaneous Symmetry Breaking 615
32.4 Classical Equations of Motion 616
32.5 Four-Layer Condenser Model 618
32.6 Derivative Expansion and Effective Theory 622
32.7 Noncommutative CpN-1 Model 627
32.8 Equations of Motion and Hall Currents 631
32.9 HallCurrentsinPseudospinQHFerromagnet 634
Appendices 643
A Energy Scales 643
B Hausdorff Formulas 645
C Group SU(2)and Pauli Matrices 647
D Groups SU(N)and SU(2N) 647
E Cauchy-Riemann Equations 650
F Green Function 651
G Bogoliubov Transformation 652
H Energy-Momentum Tensor 654
I Exchange Interaction 655
J Mermin-Wagner Theorem 659
K Lorentz Transformation 663
L One-Dimensional Soliton Solutions 666
M Field-Theoretical Vortex Operators 669
N Bosonization in One-Dimensional Space 672
O Coulomb Energy Formulas 677
P Woo Algebra 679
References 681
Index 699