Topological Insulators and Topological Superconductors (Hardcover)

<內容簡介>

This graduate-level textbook is the first pedagogical synthesis of the field of topological insulators and superconductors, one of the most exciting areas of research in condensed matter physics. Presenting the latest developments, while providing all the calculations necessary for a self-contained and complete description of the discipline, it is ideal for graduate students and researchers preparing to work in this area, and it will be an essential reference both within and outside the classroom.

The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana p-wave wires. Additionally, the book covers zero modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book also analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. Topological Insulators and Topological Superconductors will provide graduate students and researchers with the physical understanding and mathematical tools needed to embark on research in this rapidly evolving field.
 
 <章節目錄>

1 Introduction 1
2 Berry Phase 6
2.1 General Formalism 6
2.2 Gauge-Independent Computation of the Berry Phase 8
2.3 Degeneracies and Level Crossing 10
2.3.1 Two-Level System Using the Berry Curvature 10
2.3.2 Two-Level System Using the Hamiltonian Approach 11
2.4 Spin in aMagnetic Field 13
2.5 Can the Berry Phase Be Measured? 14
2.6 Problems 14
3 Hall Conductance and Chern Numbers 15
3.1 Current Operators 15
3.1.1 Current Operators from the Continuity Equation 16
3.1.2 Current Operators from Peierls Substitution 17
3.2 Linear Response to an Applied External Electric Field 18
3.2.1 The Fluctuation Dissipation Theorem 20
3.2.2 Finite-Temperature Green’s Function 22
3.3 Current-Current Correlation Function and Electrical Conductivity 23
3.4 Computing the Hall Conductance 24
3.4.1 Diagonalizing the Hamiltonian and the Flat-Band Basis 25
3.5 Alternative Formof the Hall Response 29
3.6 Chern Number as an Obstruction to Stokes’ Theorem over the Whole BZ 30
3.7 Problems 32
4 Time-Reversal Symmetry 33
4.1 Time Reversal for Spinless Particles 33
4.1.1 Time Reversal in Crystals for Spinless Particles 34
4.1.2 Vanishing of Hall Conductance for T-Invariant Spinless Fermions 35
4.2 Time Reversal for Spinful Particles 35
4.3 Kramers’ Theorem 36
4.4 Time-Reversal Symmetry in Crystals for Half-Integer Spin Particles 37
4.5 Vanishing of Hall Conductance for T-Invariant Half-Integer Spin
Particles 39
4.6 Problems 40
5 Magnetic Field on the Square Lattice 41
5.1 Hamiltonian and Lattice Translations 41
5.2 Diagonalization of the Hamiltonian of a 2-D Lattice in aMagnetic Field 44
5.2.1 Dependence on ky 46
5.2.2 Dirac Fermions in the Magnetic Field on the Lattice 47
5.3 Hall Conductance 49
5.3.1 Diophantine Equation and Streda Formula Method 49
5.4 Explicit Calculation of the Hall Conductance 51
5.5 Problems 59
6 Hall Conductance and Edge Modes: The Bulk-Edge Correspondence 60
6.1 Laughlin’s Gauge Argument 60
6.2 The TransferMatrixMethod 62
6.3 Edge Modes 65
6.4 Bulk Bands 65
6.5 Problems 69
7 Graphene 70
7.1 Hexagonal Lattices 70
7.2 Dirac Fermions 72
7.3 Symmetries of a Graphene Sheet 72
7.3.1 Time Reversal 73
7.3.2 Inversion Symmetry 73
7.3.3 Local Stability of Dirac Points with Inversion and Time Reversal 74
7.4 Global Stability of Dirac Points 76
7.4.1 C3 Symmetry and the Position of the Dirac Nodes 76
7.4.2 Breaking of C3 Symmetry 79
7.5 Edge Modes of the Graphene Layer 80
7.5.1 Chains with Even Number of Sites 82
7.5.2 Chains with Odd Number of Sites 85
7.5.3 Influence of Different Mass Terms on the Graphene Edge Modes 89
7.6 Problems 90
8 Simple Models for the Chern Insulator 91
8.1 Dirac Fermions and the Breaking of Time-Reversal Symmetry 91
8.1.1 When the Matrices r Correspond to Real Spin 91
8.1.2 When the Matrices r Correspond to Isospin 92
8.2 Explicit Berry Potential of a Two-Level System 92
8.2.1 Berry Phase of a Continuum Dirac Hamiltonian 92
8.2.2 The Berry Phase for a Generic Dirac Hamiltonian in Two Dimensions 93
8.2.3 Hall Conductivity of a Dirac Fermion in the Continuum 94
8.3 Skyrmion Number and the Lattice Chern Insulator 95
8.3.1 M > 0 Phase and M < .4 Phase 96
8.3.2 The .2 < M < 0 Phase 96
8.3.3 The .4 < M < .2 Phase 98
8.3.4 Back to the Trivial State for M < .4 98
8.4 Determinant Formula for the Hall Conductance of a Generic Dirac
Hamiltonian 99
8.5 Behavior of the Vector Potential on the Lattice 99
8.6 The Problem of Choosing a Consistent Gauge in the Chern Insulator 100
8.7 Chern Insulator in aMagnetic Field 102
8.8 Edge Modes and the Dirac Equation 103
8.9 Haldane’s Graphene Model 104
8.9.1 Symmetry Properties of the Haldane Hamiltonian 106
8.9.2 Phase Diagram of the Haldane Hamiltonian 106
8.10 Problems 107
9 Time-Reversal-Invariant Topological Insulators 109
9.1 The Kane andMeleModel: Continuum Version 109
9.1.1 Adding Spin 110
9.1.2 Spin ↑ and Spin ↓ 112
9.1.3 Rashba Term 112
9.2 The Kane and Mele Model: Lattice Version 113
9.3 First Topological Insulator: Mercury Telluride QuantumWells 117
9.3.1 Inverted QuantumWells 117
9.4 Experimental Detection of the Quantum Spin Hall State 120
9.5 Problems 121
10 Z2 Invariants 123
10.1 Z2 Invariant as Zeros of the Pfaffian 123
10.1.1 Pfaffian in the Even Subspace 124
10.1.2 The Odd Subspace 125
10.1.3 Example of an Odd Subspace: da = 0 Subspace 125
10.1.4 Zeros of the Pfaffian 126
10.1.5 Explicit Example for the Kane and Mele Model 127
10.2 Theory of Charge Polarization in One Dimension 128
10.3 Time-Reversal Polarization 130
10.3.1 Non-Abelian Berry Potentials at k, .k 133
10.3.2 Proof of the Unitarity of the Sewing Matrix B 134
10.3.3 A New Pfaffian Z2 Index 134
10.4 Z2 Index for 3-D Topological Insulators 138
10.5 Z2 Number as an Obstruction 141
10.6 Equivalence between Topological Insulator Descriptions 144
10.7 Problems 145
11 Crossings in Different Dimensions 147
11.1 Inversion-Asymmetric Systems 148
11.1.1 Two Dimensions 149
11.1.2 Three Dimensions 149
11.2 Inversion-Symmetric Systems 151
11.2.1 ga = gb 151
11.2.2 ga = .gb 152
11.3 Mercury Telluride Hamiltonian 154
11.4 Problems 156
12 Time-Reversal Topological Insulators with Inversion Symmetry 158
12.1 Both Inversion and Time-Reversal Invariance 159
12.2 Role of Spin-Orbit Coupling 162
12.3 Problems 163
13 Quantum Hall Effect and Chern Insulators in Higher Dimensions 164
13.1 Chern Insulator in Four Dimensions 164
13.2 Proof That the Second Chern Number Is Topological 166
13.3 Evaluation of the Second Chern Number: From a Green’s Function
Expression to the Non-Abelian Berry Curvature 167
13.4 Physical Consequences of the Transport Law of the 4-D Chern
Insulator 169
13.5 Simple Example of Time-Reversal-Invariant Topological Insulators with
Time-Reversal and Inversion Symmetry Based on Lattice Dirac Models 172
13.6 Problems 175
14 Dimensional Reduction of 4-D Chern Insulators to 3-D Time-Reversal
Insulators 177
14.1 Low-Energy Effective Action of (3 + 1)-D Insulators and the
Magnetoelectric Polarization 177
14.2 Magnetoelectric Polarization for a 3-D Insulator with Time-Reversal
Symmetry 181
14.3 Magnetoelectric Polarization for a 3-D Insulator with Inversion
Symmetry 182
14.4 3-D Hamiltonians with Time-Reversal Symmetry and/or Inversion
Symmetry as Dimensional Reductions of 4-D Time-Reversal-Invariant
Chern Insulators 184
14.5 Problems 185
15 Experimental Consequences of the Z2 Topological Invariant 186
15.1 Quantum Hall Effect on the Surface of a Topological Insulator 186
15.2 Physical Properties of Time-Reversal Z2-Nontrivial Insulators 187
15.3 Half-Quantized Hall Conductance at the Surface of Topological
Insulators with Ferromagnetic Hard Boundary 188
15.4 Experimental Setup for Indirect Measurement of the Half-Quantized
Hall Conductance on the Surface of a Topological Insulator 189
15.5 Topological Magnetoelectric Effect 189
15.6 Problems 191
16 Topological Superconductors in One and Two Dimensions by
Taylor L. Hughes 193
16.1 Introducing the Bogoliubov-de-Gennes (BdG) Formalism for s-Wave
Superconductors 193
16.2 p-Wave Superconductors in One Dimension 196
16.2.1 1-D p-WaveWire 196
16.2.2 Lattice p-WaveWire and Majorana Fermions 199
16.3 2-D Chiral p-Wave Superconductor 201
16.3.1 Bound States on Vortices in 2-D Chiral p-wave Superconductors 206
16.4 Problems 211
17 Time-Reversal-Invariant Topological Superconductors by Taylor L. Hughes 214
17.1 Superconducting Pairing with Spin 214
17.2 Time-Reversal-Invariant Superconductors in Two Dimensions 215
17.2.1 Vortices in 2-D Time-Reversal-Invariant Superconductors 218
17.3 Time-Reversal-Invariant Superconductors in Three Dimensions 219
17.4 Finishing the Classification of Time-Reversal-Invariant
Superconductors 222
17.5 Problems 224
18 Superconductivity and Magnetism in Proximity to Topological Insulator
Surfaces by Taylor L. Hughes 226
18.1 Generating 1-D Topological Insulators and Superconductors on the
Edge of the Quantum-Spin Hall Effect 226
18.2 Constructing Topological States from Interfaces on the Boundary of
Topological Insulators 228
18.3 Problems 234

APPENDIX: 3-D Topological Insulator in a Magnetic Field 237

References 241

Index