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.
B. Andrei Bernevig is the Eugene and Mary Wigner Assistant Professor of Theoretical Physics at Princeton University. Taylor L. Hughes is an assistant professor in the condensed matter theory group at the University of Illinois, Urbana-Champaign.
Author(s): B. Andrei Bernevig, Taylor L. Hughes
Publisher: Princeton University Press
Year: 2013
Language: English
Pages: C, xii, 247, B
Cover
S Title
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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