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Topological Insulators

Panagiotis Kotetes

Description

This book provides an introduction to topological matter, with a focus on insulating bulk systems. A number of prerequisite concepts and tools are first laid out, including the notion of symmetry transformations, the band theory of semiconductors and aspects of electronic transport. The main part of the book discusses realistic models for both time-reversal-preserving and -violating topological insulators, as well as their characteristic responses to external perturbations. Special emphasis is given to the study of the anomalous electric, thermal and thermoelectric transport properties, the theory of orbital magnetisation, and the polar Kerr effect. The topological models studied throughout this book become unified and generalised by means of the tenfold topological-classification framework and the respective systematic construction of topological invariants. This approach is further extended to topological superconductors and topological semimetals. This book covers a wide range of topics and aims for a transparent presentation of the technical aspects involved. For this purpose, homework problems are also provided in dedicated hands-on sections. Given its structure, and the required background level of the reader, this book is particularly recommended for graduate students or researchers who are new to the field.

About Editors

Panagiotis Kotetes is currently a postdoctoral researcher at the Niels Bohr Institute in the University of Copenhagen, Denmark. He carried out his diploma, MS and PhD studies at the National Technical University of Athens, Greece. In 2010, he moved to the Karlsruhe Institute of Technology, Germany, where he worked for five years as a postdoctoral researcher. His research interests and activity covers topics such as topological phases of matter and quantum computing, strongly correlated systems, and the interplay of superconductivity and exotic magnetic phases.

Table of Contents

1 Symmetries and effective Hamiltonians 1

1.1 Crash course on symmetry transformations . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Unitary symmetry transformations . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Action of symmetry transformations on operators . . . . . . . . . . . . . . . 3

1.1.3 Antiunitary symmetry transformations: Time-reversal . . . . . . . . . . . . 4

1.1.4 Symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.5 Translations, Bloch's theorem and space groups . . . . . . . . . . . . . . . . 8

1.2 Effective Hamiltonians for bulk III-V semiconductors . . . . . . . . . . . . . . . . . 9

1.2.1 Effective Hamiltonian about the G-point: Plain vanilla model . . . . . . . . . 9

1.2.2 Cubic crystalline effects and double covering groups . . . . . . . . . . . . . 11

1.2.3 Bulk inversion asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.4 Confinement and structural inversion asymmetry . . . . . . . . . . . . . . . 13

1.3 Hands-on: Symmetry analysis of a triple quantum-dot . . . . . . . . . . . . . . . . . 16

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Coupling to external fields and transport theory 19

2.1 Electromagnetic potentials, fields and currents . . . . . . . . . . . . . . . . . . . . . 19

2.2 Minimal-coupling and electric charge conservation law . . . . . . . . . . . . . . . . 20

2.3 Charge current in lattice systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Linear response and current-current correlation functions . . . . . . . . . . . . . . . 23

2.5 Matsubara technique and thermal Green functions . . . . . . . . . . . . . . . . . . . 24

2.6 Matsubara formulation of linear response . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Charge conductivity of an electron gas . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Thermoelectric and thermal transport . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8.1 Energy conservation and heat current . . . . . . . . . . . . . . . . . . . . . 31

2.8.2 Luttinger's gravitational field approach . . . . . . . . . . . . . . . . . . . . 32

2.8.3 Nature of the gravitational field . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9 Hands-on: Magnetoconductivity of a triple-quantum-dot . . . . . . . . . . . . . . . 34

2.10 Hands-on: Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . 36

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Jackiw-Rebbi model and Goldstone-Wilczek formula 39

3.1 Helical electrons in nanowires: Emergent Jackiw-Rebbi model . . . . . . . . . . . . 39

3.2 Zero-energy solutions in the Jackiw-Rebbi model . . . . . . . . . . . . . . . . . . . 41

3.3 The Jackiw-Rebbi model in condensed matter physics . . . . . . . . . . . . . . . . . 44

3.3.1 Polyacetylene and Su-Schrieffer-Heeger model . . . . . . . . . . . . . . . . 44

3.3.2 One-dimensional conductors and sliding charge density waves . . . . . . . . 45

3.4 Goldstone-Wilczek formula and dissipationless current . . . . . . . . . . . . . . . . 46

3.4.1 Connection to Dirac physics and chiral anomaly . . . . . . . . . . . . . . . 47

3.4.2 Fractional electric charge at solitons and electric charge pumping . . . . . . 48

3.5 Hands-on: Derivation of the Goldstone-Wilczek formula . . . . . . . . . . . . . . . 49

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Topological insulators in 1+1 dimensions 55

4.1 Topological insulator in 1+1 dimensions . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 Model Hamiltonian and zero-energy edge states . . . . . . . . . . . . . . . . 55

4.1.2 Topological invariant - Winding number . . . . . . . . . . . . . . . . . . . . 58

4.1.3 Homotopy mapping and winding number . . . . . . . . . . . . . . . . . . . 58

4.1.4 Topological invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.5 Generalised winding number . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Lattice topological insulator models . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Adiabatic transport: Thouless pump and Berry curvature . . . . . . . . . . . . . . . 62

4.3.1 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.2 Relation between Chern and winding numbers . . . . . . . . . . . . . . . . 65

4.3.3 Lattice model and electric polarisation . . . . . . . . . . . . . . . . . . . . . 66

4.4 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Hands-on: Winding number in a 3+1d model . . . . . . . . . . . . . . . . . . . . . 69

4.6 Hands-on: Current and electric polarisation formula . . . . . . . . . . . . . . . . . . 69

4.7 Hands-on: Violation of chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Chern insulators - Fundamentals 73

5.1 Jackiw-Rebbi model and Dirac physics in 2+1d . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Electric charge and current responses of the chiral edge modes . . . . . . . . 74

5.1.2 Chiral edge modes in the quantum Hall effect: Laughlin's argument . . . . . 75

5.1.3 Connection to Dirac physics and parity anomaly . . . . . . . . . . . . . . . 77

5.1.4 Maxwell-Chern-Simons action and topological Meissner effect . . . . . . . . 78

5.2 Chern insulator in 2+1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 Lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Quantised Hall conductivity - Bulk approach . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1 Bulk eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.2 Adiabatic Hall transport and Berry curvature . . . . . . . . . . . . . . . . . 81

5.3.3 Homotopy mapping and Chern number . . . . . . . . . . . . . . . . . . . . 82

5.4 Chern insulators in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Chern insulator model in 4+1d . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Second Chern number and non-Abelian Berry gauge potentials . . . . . . . . 84

5.4.3 4+1d Chern-Simons action and four-dimensional quantum Hall effect . . . . 85

5.4.4 Generalisation to arbitrary dimensions . . . . . . . . . . . . . . . . . . . . . 85

5.5 Dimensional reduction: Chiral anomaly . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6 Hands-on: Chern-Simons action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.7 Hands-on: Chern number for interacting systems . . . . . . . . . . . . . . . . . . . 88

5.8 Hands-on: Second Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography . . .

6 Chern insulators - Applications 91

6.1 Dynamical anomalous Hall response and polar Kerr effect . . . . . . . . . . . . . . 91

6.1.1 Dynamical anomalous Hall conductivity . . . . . . . . . . . . . . . . . . . . 91

6.1.2 Polar Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.1.3 Dielectric tensor and circular polarisation birefringence . . . . . . . . . . . . 92

6.1.4 Kerr-angle formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.5 Polar Kerr effect for a 2+1d Chern insulator . . . . . . . . . . . . . . . . . . 95

6.2 Chern insulators in an external magnetic field . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 High-field limit and the formation of Landau levels . . . . . . . . . . . . . . 95

6.2.2 Theory of orbital magnetisation - a Green function method . . . . . . . . . . 97

6.3 Anomalous thermoelectric and thermal Hall transport . . . . . . . . . . . . . . . . . 99

6.3.1 Thermoelectric conductivity tensor . . . . . . . . . . . . . . . . . . . . . . 99

6.3.2 Thermal conductivity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.3 Diathermal contributions to the conductivities and transport current . . . . . 101

6.4 Hands-on: Magnetic-field-induced Chern systems . . . . . . . . . . . . . . . . . . . 102

6.5 Hands-on: Thermoelectricity in the Haldane model . . . . . . . . . . . . . . . . . . 103

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Z2 topological insulators 105

7.1 Z2 topological insulators in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . . 105

7.1.1 Bottom-up construction based on Chern insulators: BHZ model . . . . . . . 106

7.1.2 Violation of chiral symmetry and Z2 topological invariant . . . . . . . . . . 107

7.2 Z2 topological insulators in 3+1 dimensions . . . . . . . . . . . . . . . . . . . . . . 110

7.2.1 Crystal structure and model Hamiltonian . . . . . . . . . . . . . . . . . . . 111

7.2.2 Surface states for negligible warping . . . . . . . . . . . . . . . . . . . . . . 111

7.2.3 Consequences of warping and p-Berry phase . . . . . . . . . . . . . . . . . 112

7.2.4 Magnetoelectric polarisation and Z2 topological invariants in 3+1d . . . . . . 114

7.3 Dimensional reduction and magnetoelectric coupling . . . . . . . . . . . . . . . . . 116

7.3.1 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.3.2 Magnetoelectric polarisation domain wall and quantum anomalous Hall effect 118

7.4 Hands-on: Quasiparticle interference . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.5 Hands-on: Topological Kondo insulator . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 Topological classification of insulators and beyond 121

8.1 Generalised antinunitary symmetries and symmetry classes . . . . . . . . . . . . . . 121

8.2 The art of topological classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2.1 Complex symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.2 Real symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.3 Z2 classification and relative Chern and winding numbers . . . . . . . . . . 125

8.2.4 Weak topological invariants and flat bands . . . . . . . . . . . . . . . . . . . 128

8.2.5 Topological classification with unitary symmetries . . . . . . . . . . . . . . 129

8.2.6 Crystalline topological insulators . . . . . . . . . . . . . . . . . . . . . . . 130

8.3 Topological classification of gapless systems . . . . . . . . . . . . . . . . . . . . . . 132

8.3.1 2+1d semimetals - Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.3.2 Weyl semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.4 Topological classification of insulators and defects . . . . . . . . . . . . . . . . . . 137

8.5 Topological superconductors and Majorana fermions . . . . . . . . . . . . . . . . . 138

8.6 Further topics and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.7 Hands-on: Berry magnetic monopoles in semiconductors . . . . . . . . . . . . . . . 140

8.8 Hands-on: Floquet topological insulator . . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Bibliographic

Paperback ISBN: 9780750327503

Ebook ISBN: 9781681745183

DOI: 10.1088/978-1-68174-517-6

Publisher: Morgan & Claypool Publishers

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