Topological Insulators and Topological Superconductors

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...

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Bibliographic Details
Main Authors:	Bernevig, B. Andrei, Hughes, Taylor L
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2013
Edition:	STU - Student edition
Subjects:	Adiabatic theorem Aharonov–Bohm effect Canonical commutation relation Chern–Simons theory Chiral anomaly Correlation function (quantum field theory) Creation and annihilation operators Density matrix Dimensional reduction Dirac delta function Dirac equation Dirac fermion Dirac spectrum Domain wall (magnetism) Domain wall (string theory) Eigenvalues and eigenvectors Electron magnetic moment Electronic band structure Energy-band theory of solids Expectation value (quantum mechanics) Fermi arc Fermi energy Fermi level Fermi surface Fermion Fermion doubling Fractional quantum Hall effect Gauge theory General Topics for Engineers Geometric phase Ginzburg–Landau theory Graphene Hamiltonian matrix Isospin Lagrangian (field theory) Landau quantization Lattice model (physics) Magnetic coupling Magnetic field Magnetic flux Magnetic translation Magnetization Majorana fermion Mathematics Mirror symmetry (string theory) Perturbation theory (quantum mechanics) Phase transition Physics Quantum critical point Quantum decoherence Quantum entanglement Quantum Hall effect Quantum mechanics Quantum number Quantum well Quasiparticle SCIENCE SCIENCE / Physics / Condensed Matter SCIENCE / Physics / Electromagnetism SCIENCE / Physics / General Slater determinant Solid state physics Solid state physics > Mathematics Soliton Spin (physics) Spin structure Spin–orbit interaction Stokes' theorem Superconductivity Superconductors Superconductors > Mathematics Superfluidity Surface states Thermodynamic limit Topological insulator Topological order Topological quantum computer Topological quantum number Translation operator (quantum mechanics) Two-dimensional space Unitarity (physics) Weyl semimetal Quasiparticle Density matrix Soliton Mirror symmetry (string theory) Dirac delta function Topological insulator Slater determinant Isospin Lagrangian (field theory) Stokes' theorem Majorana fermion Topological quantum computer Chern–Simons theory Eigenvalues and eigenvectors Thermodynamic limit Dimensional reduction Creation and annihilation operators Fermi surface Two-dimensional space Magnetic field Spin structure Ginzburg–Landau theory Aharonov–Bohm effect Geometric phase Quantum entanglement Fermi energy Electronic band structure Fermion Quantum critical point Topological order Magnetic translation Graphene Magnetic coupling Domain wall (magnetism) Dirac fermion Phase transition Quantum number Dirac spectrum Magnetic flux Quantum decoherence Quantum Hall effect Surface states Electron magnetic moment Weyl semimetal Landau quantization Fractional quantum Hall effect Adiabatic theorem Perturbation theory (quantum mechanics) Superfluidity Quantum well Magnetization Spin (physics) Superconductivity Topological quantum number Fermi arc Fermion doubling Domain wall (string theory) Spin–orbit interaction Unitarity (physics) Chiral anomaly Correlation function (quantum field theory) Lattice model (physics) Gauge theory Fermi level Canonical commutation relation Quantum mechanics Dirac equation Hamiltonian matrix Translation operator (quantum mechanics) Expectation value (quantum mechanics)
ISBN:	9780691151755, 069115175X, 9781400846733, 1400846730
Online Access:	Get full text
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Table of Contents:

Front Matter Table of Contents 1: Introduction 2: Berry Phase 3: Hall Conductance and Chern Numbers 4: Time-Reversal Symmetry 5: Magnetic Field on the Square Lattice 6: Hall Conductance and Edge Modes: 7: Graphene 8: Simple Models for the Chern Insulator 9: Time-Reversal-Invariant Topological Insulators 10: Z₂ Invariants 11: Crossings in Different Dimensions 12: Time-Reversal Topological Insulators with Inversion Symmetry 13: Quantum Hall Effect and Chern Insulators in Higher Dimensions 14: Dimensional Reduction of 4-D Chern Insulators to 3-D Time-Reversal Insulators 15: Experimental Consequences of the Z₂ Topological Invariant 16: Topological Superconductors in One and Two Dimensions 17: Time-Reversal-Invariant Topological Superconductors 18: Superconductivity and Magnetism in Proximity to Topological Insulator Surfaces APPENDIX: References Index
Cover -- Title -- Copyright -- Contents -- 1 Introduction -- 2 Berry Phase -- 2.1 General Formalism -- 2.2 Gauge-Independent Computation of the Berry Phase -- 2.3 Degeneracies and Level Crossing -- 2.3.1 Two-Level System Using the Berry Curvature -- 2.3.2 Two-Level System Using the Hamiltonian Approach -- 2.4 Spin in a Magnetic Field -- 2.5 Can the Berry Phase Be Measured? -- 2.6 Problems -- 3 Hall Conductance and Chern Numbers -- 3.1 Current Operators -- 3.1.1 Current Operators from the Continuity Equation -- 3.1.2 Current Operators from Peierls Substitution -- 3.2 Linear Response to an Applied External Electric Field -- 3.2.1 The Fluctuation Dissipation Theorem -- 3.2.2 Finite-Temperature Green's Function -- 3.3 Current-Current Correlation Function and Electrical Conductivity -- 3.4 Computing the Hall Conductance -- 3.4.1 Diagonalizing the Hamiltonian and the Flat-Band Basis -- 3.5 Alternative Form of the Hall Response -- 3.6 Chern Number as an Obstruction to Stokes' Theorem over the Whole BZ -- 3.7 Problems -- 4 Time-Reversal Symmetry -- 4.1 Time Reversal for Spinless Particles -- 4.1.1 Time Reversal in Crystals for Spinless Particles -- 4.1.2 Vanishing of Hall Conductance for T-Invariant Spinless Fermions -- 4.2 Time Reversal for Spinful Particles -- 4.3 Kramers' Theorem -- 4.4 Time-Reversal Symmetry in Crystals for Half-Integer Spin Par -- 4.5 Vanishing of Hall Conductance for T-Invariant Half-Integer Spin Particles -- 4.6 Problems -- 5 Magnetic Field on the Square Lattice -- 5.1 Hamiltonian and Lattice Translations -- 5.2 Diagonalization of the Hamiltonian of a 2-D Lattice in a Magnetic Field -- 5.2.1 Dependence on ky -- 5.2.2 Dirac Fermions in the Magnetic Field on the Lattice -- 5.3 Hall Conductance -- 5.3.1 Diophantine Equation and Streda Formula Method -- 5.4 Explicit Calculation of the Hall Conductance -- 5.5 Problems
6 Hall Conductance and Edge Modes: The Bulk-Edge Correspondence -- 6.1 Laughlin's Gauge Argument -- 6.2 The Transfer Matrix Method -- 6.3 Edge Modes -- 6.4 Bulk Bands -- 6.5 Problems -- 7 Graphene -- 7.1 Hexagonal Lattices -- 7.2 Dirac Fermions -- 7.3 Symmetries of a Graphene Sheet -- 7.3.1 Time Reversal -- 7.3.2 Inversion Symmetry -- 7.3.3 Local Stability of Dirac Points with Inversion and Time Reversal -- 7.4 Global Stability of Dirac Points -- 7.4.1 C3 Symmetry and the Position of the Dirac Nodes -- 7.4.2 Breaking of C3 Symmetry -- 7.5 Edge Modes of the Graphene Layer -- 7.5.1 Chains with Even Number of Sites -- 7.5.2 Chains with Odd Number of Sites -- 7.5.3 Influence of Different Mass Terms on the Graphene Edge Modes -- 7.6 Problems -- 8 Simple Models for the Chern Insulator -- 8.1 Dirac Fermions and the Breaking of Time-Reversal Symmetry -- 8.1.1 When the Matrices r Correspond to Real Spin -- 8.1.2 When the Matrices r Correspond to Isospin -- 8.2 Explicit Berry Potential of a Two-Level System -- 8.2.1 Berry Phase of a Continuum Dirac Hamiltonian -- 8.2.2 The Berry Phase for a Generic Dirac Hamiltonian in Two Dimensions -- 8.2.3 Hall Conductivity of a Dirac Fermion in the Continuum -- 8.3 Skyrmion Number and the Lattice Chern Insulator -- 8.3.1 M &gt -- 0 Phase and M &lt -- −4 Phase -- 8.3.2 The −2 &lt -- M &lt -- 0 Phase -- 8.3.3 The −4 &lt -- M &lt -- −2 Phase -- 8.3.4 Back to the Trivial State for M &lt -- −4 -- 8.4 Determinant Formula for the Hall Conductance of a Generic Dirac Hamiltonian -- 8.5 Behavior of the Vector Potential on the Lattice -- 8.6 The Problem of Choosing a Consistent Gauge in the Chern Insulator -- 8.7 Chern Insulator in a Magnetic Field -- 8.8 Edge Modes and the Dirac Equation -- 8.9 Haldane's Graphene Model -- 8.9.1 Symmetry Properties of the Haldane Hamiltonian -- 8.9.2 Phase Diagram of the Haldane Hamiltonian
8.10 Problems -- 9 Time-Reversal-Invariant Topological Insulators -- 9.1 The Kane and Mele Model: Continuum Version -- 9.1.1 Adding Spin -- 9.1.2 Spin ↑ and Spin ↓ -- 9.1.3 Rashba Term -- 9.2 The Kane and Mele Model: Lattice Version -- 9.3 First Topological Insulator: Mercury Telluride Quantum Wells -- 9.3.1 Inverted Quantum Wells -- 9.4 Experimental Detection of the Quantum Spin Hall State -- 9.5 Problems -- 10 Z2 Invariants -- 10.1 Z2 Invariant as Zeros of the Pfaffian -- 10.1.1 Pfaffian in the Even Subspace -- 10.1.2 The Odd Subspace -- 10.1.3 Example of an Odd Subspace: da = 0 Subspace -- 10.1.4 Zeros of the Pfaffian -- 10.1.5 Explicit Example for the Kane and Mele Model -- 10.2 Theory of Charge Polarization in One Dimension -- 10.3 Time-Reversal Polarization -- 10.3.1 Non-Abelian Berry Potentials at k, −k -- 10.3.2 Proof of the Unitarity of the Sewing Matrix B -- 10.3.3 A New Pfaffian Z2 Index -- 10.4 Z2 Index for 3-D Topological Insulators -- 10.5 Z2 Number as an Obstruction -- 10.6 Equivalence between Topological Insulator Descriptions -- 10.7 Problems -- 11 Crossings in Different Dimensions -- 11.1 Inversion-Asymmetric Systems -- 11.1.1 Two Dimensions -- 11.1.2 Three Dimensions -- 11.2 Inversion-Symmetric Systems -- 11.2.1 na = nb -- 11.2.2 na = −nb -- 11.3 Mercury Telluride Hamiltonian -- 11.4 Problems -- 12 Time-Reversal Topological Insulators with Inversion Symmetry -- 12.1 Both Inversion and Time-Reversal Invariance -- 12.2 Role of Spin-Orbit Coupling -- 12.3 Problems -- 13 Quantum Hall Effect and Chern Insulators in Higher Dimensions -- 13.1 Chern Insulator in Four Dimensions -- 13.2 Proof That the Second Chern Number Is Topological -- 13.3 Evaluation of the Second Chern Number: From a Green's Function Expression to the Non-Abelian Berry Curvature -- 13.4 Physical Consequences of the Transport Law of the 4-D Chern Insulator
13.5 Simple Example of Time-Reversal-Invariant Topological Insulators with Time-Reversal and Inversion Symmetry Based on Lattice Dirac Models -- 13.6 Problems -- 14 Dimensional Reduction of 4-D Chern Insulators to 3-D Time-Reversal Insulators -- 14.1 Low-Energy Effective Action of (3 + 1)-D Insulators and the Magnetoelectric Polarization -- 14.2 Magnetoelectric Polarization for a 3-D Insulator with Time-Reversal Symmetry -- 14.3 Magnetoelectric Polarization for a 3-D Insulator with Inversion Symmetry -- 14.4 3-D Hamiltonians with Time-Reversal Symmetry and/or Inversion Symmetry as Dimensional Reductions of 4-D Time-Reversal-Invariant Chern Insulators -- 14.5 Problems -- 15 Experimental Consequences of the Z2 Topological Invariant -- 15.1 Quantum Hall Effect on the Surface of a Topological Insulator -- 15.2 Physical Properties of Time-Reversal Z2-Nontrivial Insulators -- 15.3 Half-Quantized Hall Conductance at the Surface of Topological Insulators with Ferromagnetic Hard Boundary -- 15.4 Experimental Setup for Indirect Measurement of the Half-Quantized Hall Conductance on the Surface of a Topological Insulator -- 15.5 Topological Magnetoelectric Effect -- 15.6 Problems -- 16 Topological Superconductors in One and Two Dimensions -- 16.1 Introducing the Bogoliubov-de-Gennes (BdG) Formalism for s-Wave Superconductors -- 16.2 p-Wave Superconductors in One Dimension -- 16.2.1 1-D p-Wave Wire -- 16.2.2 Lattice p-Wave Wire and Majorana Fermions -- 16.3 2-D Chiral p-Wave Superconductor -- 16.3.1 Bound States on Vortices in 2-D Chiral p-wave Superconductors -- 16.4 Problems -- 17 Time-Reversal-Invariant Topological Superconductors by Taylor L. Hughes -- 17.1 Superconducting Pairing with Spin -- 17.2 Time-Reversal-Invariant Superconductors in Two Dimensions -- 17.2.1 Vortices in 2-D Time-Reversal-Invariant Superconductors
17.3 Time-Reversal-Invariant Superconductors in Three Dimensions -- 17.4 Finishing the Classification of Time-Reversal-Invariant Superconductors -- 17.5 Problems -- 18 Superconductivity and Magnetism in Proximity to Topological Insulator Surfaces -- 18.1 Generating 1-D Topological Insulators and Superconductors on the Edge of the Quantum-Spin Hall Effect -- 18.2 Constructing Topological States from Interfaces on the Boundary of Topological Insulators -- 18.3 Problems -- APPENDIX: 3-D Topological Insulator in a Magnetic Field -- References -- Index
References --
Contents --
3-D Topological Insulator in a Magnetic Field --
2. Berry Phase --
16. Topological Superconductors in One and Two Dimensions
1. Introduction --
Index
3. Hall Conductance and Chern Numbers --
Taylor L. Hughes --
11. Crossings in Different Dimensions --
13. Quantum Hall Effect and Chern Insulators in Higher Dimensions --
15. Experimental Consequences of the Z2 Topological Invariant --
8. Simple Models for the Chern Insulator --
6. Hall Conductance and Edge Modes: The Bulk-Edge Correspondence --
5. Magnetic Field on the Square Lattice --
7. Graphene --
10. Z2 Invariants --
4. Time-Reversal Symmetry --
17. Time-Reversal-Invariant Topological Superconductors
APPENDIX --
9. Time-Reversal-Invariant Topological Insulators --
14. Dimensional Reduction of 4-D Chern Insulators to 3-D Time-Reversal Insulators --
Frontmatter --
12. Time-Reversal Topological Insulators with Inversion Symmetry --
18. Superconductivity and Magnetism in Proximity to Topological Insulator Surfaces