Topological Insulators and Topological Superconductors

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TOPOLOGICAL INSULATORS AND TOPOLOGICAL SUPER CONDUCTORS

Copyright © 2013 by Princeton University Press
     ISBN-13: 978-0-691-15175-5 (hardback)
     QC61 l .95.B465 2013 530.4'1-dc23
     LCCN 2012035384

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 Spin less Fermions
     4.2 Time Reversal for Spinful Particles
     4.3 Kramers' Theorem
     4.4 Time-Reversal Symmetry in Crystals for Half-Integer Spin Particles
     4.5 Vanishing of Hall Conductance for T-lnvariant 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 C 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 a Correspond to Real Spin
          8.1.2 When the Matrices u Correspond to lsospin
     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 > 0 Phase and M < -4 Phase
          8.3.2 The -2 < M < 0 Phase
          8.3.3 The -4 < M < - 2 Phase
          8.3.4 Back to the Trivial State for M < -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 t 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 rtaffian 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 'la = 'lb
          11.2.2 T/a = -T/b
     11.3 Mercury Telluride Hamiltonian
     11.4 Problems

12  Time-Reversal Topological Insulators withI nversion 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
               16.3.1.1 Non-Abelian Statistics of Vortices in Chiral p-Wave Superconductors
     16.4 Problems

17  Time-Reversal-Invariant Topological Superconductors
     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

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