Topological Insulators (eBook)

Professor Shun-Qing Shen, an expert in the field of condensed matter physics, is distinguished for his research works on topological quantum materials, spintronics of semiconductors, quantum magnetism and orbital physics in transition metal oxides, and novel quantum states of condensed matter. He proposed topological Anderson insulator, theory of weak localization and antilocalization for Dirac fermions, spin transverse force, resonant spin Hall effect and the theory of phase separation in colossal magnetoresistive (CMR) materials. He proved the existence of antiferromagnetic long-range order and off-diagonal long-range order in itinerant electron systems.  Professor Shun-Qing Shen has been a professor of physics at The University of Hong Kong since July 2007. Professor Shen received his BS, MS, and PhD in theoretical physics from Fudan University in Shanghai. He was a postdoctorial fellow (1992 – 1995) in China Center of Advanced Science and Technology (CCAST), Beijing, Alexander von Humboldt fellow (1995 – 1997) in Max Planck Institute for Physics of Complex Systems, Dresden, Germany, and JSPS research fellow (1997) in Tokyo Institute of Technology, Japan. In December 1997 he joined Department of Physics, The University of Hong Kong. He was awarded Croucher Senior Research Fellowship (The Croucher Award) in 2010.

Preface to the Second Edition 6
Preface to the First Edition 7
Contents 9
1 Introduction 14
1.1 From the Hall Effect to the Quantum Spin Hall Effect 14
1.2 Topological Insulators as a Generalization of the Quantum Spin Hall Systems 19
1.3 Beyond Band Insulators: Disorder and Interaction 21
1.4 Topological Phases in Superconductors and Superfluids 22
1.5 Topological Dirac and Weyl Semimetals 24
1.6 Dirac Equation and Topological Insulators 25
1.7 Topological Insulators and Landau Theory of Phase Transition 25
1.8 Summary 26
1.9 Further Reading 27
References 27
2 Starting from the Dirac Equation 30
2.1 Dirac Equation 30
2.2 Solutions of Bound States 32
2.2.1 Jackiw-Rebbi Solution in One Dimension 32
2.2.2 Two Dimensions 35
2.2.3 Three and Higher Dimensions 36
2.3 Why not the Dirac Equation? 36
2.4 Quadratic Correction to the Dirac Equation 37
2.5 Bound State Solutions of the Modified Dirac Equation 38
2.5.1 One Dimension: End States 38
2.5.2 Two Dimensions: Helical Edge States 40
2.5.3 Three Dimensions: Surface States 42
2.5.4 Generalization to Higher-Dimensional Topological Insulators 44
2.6 Summary 44
2.7 Further Reading 45
References 45
3 Minimal Lattice Model for Topological Insulators 46
3.1 Tight Binding Approximation 46
3.2 Mapping from a Continuous Model to a Lattice Model 48
3.3 One-Dimensional Lattice Model 50
3.4 Two-Dimensional Lattice Model 53
3.4.1 Integer Quantum Hall Effect 53
3.4.2 Quantum Spin Hall Effect 55
3.5 Three-Dimensional Lattice Model 55
3.6 Parity at the Time Reversal Invariant Momenta 57
3.6.1 One-Dimensional Lattice Model 58
3.6.2 Two-Dimensional Lattice Model 59
3.6.3 Three-Dimensional Lattice Model 61
3.7 Summary 63
References 63
4 Topological Invariants 64
4.1 Bloch's Theorem and Band Theory 64
4.2 Berry Phase 65
4.3 Quantum Hall Conductance and the Chern Number 68
4.4 Electric Polarization in a Cyclic Adiabatic Evolution 72
4.5 Thouless Charge Pump 74
4.6 Fu--Kane Spin Pump 77
4.7 Integer Quantum Hall Effect: The Laughlin Argument 79
4.8 Time Reversal Symmetry and the Z2 Index 81
4.9 Generalization to Two and Three Dimensions 86
4.10 Phase Diagram of the Modified Dirac Equation 88
4.11 Further Reading 91
References 92
5 Topological Phases in One Dimension 93
5.1 Su--Schrieffer--Heeger Model for Polyacetylene 93
5.2 Topological Ferromagnet 98
5.3 p-Wave Pairing Superconductor 98
5.4 Ising Model in a Transverse Field 100
5.5 One-Dimensional Maxwell's Equations in Media 101
5.6 Summary 102
References 102
6 Quantum Anomalous Hall Effect and Quantum Spin Hall Effect 103
6.1 Quantum Anomalous Hall Effect 103
6.1.1 Two-Dimensional Dirac Model and the Chern Number 103
6.1.2 Haldane Model 104
6.1.3 Experimental Realization 107
6.2 From the Haldane Model to the Kane-Mele Model 110
6.3 Transport of Edge States 113
6.3.1 Landauer-Büttiker Formalism 114
6.3.2 Transport of Edge States 116
6.4 Stability of Edge States 119
6.5 Realization of the Quantum Spin Hall Effect in HgTe/CdTe Quantum Wells 121
6.5.1 Band Structure of HgTe/CdTe Quantum Wells 121
6.5.2 Exact Solution of Edge States 124
6.5.3 Experimental Measurement 127
6.6 Quantized Conductance in InAs/GaAs Bilayer Quantum Well 129
6.7 Quantum Hall Effect and Quantum Spin Hall Effect: A Case Study 130
6.7.1 Quantum Hall Effect (?=2) 130
6.7.2 Quantum Spin Hall Effect 131
6.8 Coherent Oscillation Due to the Edge States 132
6.9 Further Reading 134
References 134
7 Three-Dimensional Topological Insulators 136
7.1 Family Members of Three-Dimensional Topological Insulators 136
7.1.1 Weak Topological Insulators: PbxSn1-xTe 136
7.1.2 Strong Topological Insulators: Bi1-xSbx 137
7.1.3 Topological Insulators with a Single Dirac Cone: Bi2Se3 and Bi2Te3 138
7.1.4 Strained HgTe 138
7.2 Electronic Model for Bi2Se3 140
7.3 Effective Model for Surface States 142
7.4 Physical Properties of Topological Insulators 145
7.4.1 Absence of Backscattering 145
7.4.2 Weak Antilocalization 146
7.4.3 Shubnikov-de Haas Oscillation 147
7.5 Surface Quantum Hall Effect 148
7.6 Surface States in a Strong Magnetic Field 151
7.7 Topological Insulator Thin Film 153
7.7.1 Effective Model for Thin Film 153
7.7.2 Structural Inversion Asymmetry 157
7.7.3 Experimental Data of ARPES 159
7.8 HgTe Thin Film 159
7.9 Further Reading 161
References 162
8 Impurities and Defects in Topological Insulators 164
8.1 One Dimension 164
8.2 Integral Equation for Bound State Energies 166
8.2.1 ?-Potential 167
8.3 Bound States in Two Dimensions 168
8.4 Topological Defects 172
8.4.1 Magnetic Flux and Zero Energy Mode 172
8.4.2 Wormhole Effect 174
8.4.3 Witten Effect 175
8.5 Disorder Effect on Transport 179
8.6 Further Reading 181
References 181
9 Topological Superconductors and Superfluids 183
9.1 Complex (p+ip)-Wave Superconductor for Spinless ƒ 184
9.2 Spin Triplet Pairing Superfluidity: 3He-A and -B Phases 188
9.2.1 3He: Normal Liquid Phase 189
9.2.2 3He-B Phase 189
9.2.3 3He-A Phase: Equal Spin Pairing 192
9.3 Spin-Triplet Superconductor: Sr2RuO4 194
9.4 Superconductivity in Doped Topological Insulators 195
9.5 Further Reading 196
References 196
10 Majorana Fermions in Topological Insulators 198
10.1 What Is a Majorana Fermion? 198
10.2 Majorana Fermions in p-Wave Superconductors 199
10.2.1 Zero Energy Mode Around a Quantum Vortex 199
10.2.2 Majorana Fermions in Kitaev's Toy Model 202
10.2.3 Quasi-One-Dimensional Superconductors 204
10.3 Majorana Fermions in Topological Insulators 207
10.4 Sau--Lutchyn--Tewari--Das Sarma Model for Topological Superconductors 208
10.5 4?-Josephson Effect 211
10.6 Non-Abelion Statistics and Topological Quantum Computing 213
10.7 Further Reading 215
References 215
11 Topological Dirac and Weyl Semimetals 216
11.1 Weyl Equations and Weyl Fermions 216
11.1.1 Weyl Equations 216
11.1.2 A Single Node and Magnetic Monopole 217
11.2 Emergent Dirac and Weyl Semimetals 218
11.2.1 Dirac Semimetal 219
11.2.2 Topological Dirac Semimetal 220
11.2.3 Topological Weyl Semimetal 221
11.3 Graphene: A Topological Dirac Semimetal 221
11.4 Two-Node Model 223
11.4.1 Model 224
11.4.2 The Chern Number and Fermi Arc 225
11.4.3 Quantum Anomalous Hall Effect 227
11.5 Tight-Binding Model and Topological Phase Transition 229
11.6 Chiral Anomaly 231
11.7 Exotic Magnetotransport 232
11.7.1 Three-Dimensional Weak Antilocalization 232
11.7.2 Negative Magnetoresistance 233
11.7.3 Linear Magnetoconductance Near the Weyl Nodes 236
11.7.4 High Mobility and Large Magnetoresistance 237
11.8 Further Reading 238
References 238
12 Topological Anderson Insulator 239
12.1 Band Structure and Edge States 239
12.2 Quantized Anomalous Hall Effect 241
12.3 Topological Anderson Insulator 243
12.4 Effective Medium Theory for Topological Anderson Insulator 245
12.5 Band Gap or Mobility Gap 246
12.6 Summary 248
12.7 Further Reading 248
References 249
13 Summary: Symmetries and Topological Classification 250
13.1 Ten Symmetry Classes for Non-interacting Fermion Systems 250
13.2 Physical Systems and the Symmetry Classes 252
13.2.1 Standard (Wigner--Dyson) Classes 252
13.2.2 Chiral Classes 253
13.2.3 Bogoliubov-de Gennes (BdG) Classes 253
13.3 Characterization in the Bulk States 254
13.4 Five Types in Each Dimension 255
13.5 Conclusion 256
13.6 Further Reading 257
References 257
Appendix A Derivation of Two Formulae 258
Appendix B Time Reversal Symmetry 264
Appendix C The Dirac Matrices and the Dirac Gamma Matrices 268
Index 269