Topological Quantum Matter (eBook)

Supervisor’s Foreword 6
Acknowledgements 8
Contents 10
1 Introduction 12
1.1 Entanglement 13
1.2 Topological Interactions 14
1.3 Topological Response 16
1.4 Outline 20
1.5 Notations and Conventions for Topological Matter 20
References 22
2 Anyons and Topological Order 23
2.1 Bosons, Fermions and Abelian Anyons 23
2.1.1 A Single Particle on An Annulus 24
2.1.2 Indistinguishable Particles and the Braid Group 26
2.1.3 Mediating Topological Interactions with a Gauge Boson 30
2.2 Topological Order in Two Dimensions 32
2.2.1 Torus Degeneracy 37
2.3 Abelian Chern-Simons Theory 39
References 42
3 Response Theory and Symmetry Protected Topological Phases 44
3.1 The Quantization of the Hall Conductance 45
3.2 U(1) Response Theory 49
3.2.1 Functional Bosonization 52
3.3 The Wen-Zee Term and Chiral Superconductors 55
3.3.1 Chiral Superconductors 56
References 58
4 Topological Geometric Response and Topological Order of Super Conductors 60
4.1 A Superconducting Bosonic Model 61
4.2 Adding Fermions to Get Topological Order 64
4.2.1 The Chiral p-Wave SC 65
4.3 Non-Abelian Statistics in the Adiabatic Limit 70
4.4 The Majorino Conundrum 74
References 76
A Conventions and Notation Concerning Differential Forms and U(1) Connections 77
A.1 Differential Forms 77
A.2 Normalization 78
Appendix B Definitions 79
Appendix C Vector Bundles and Chern Numbers in Quantum Mechanics 85
C.1 The Berry Connection 85
C.2 The Berry Field Strength 86
C.3 Chern Numbers and Chern-Simons Invariants 88
Reference 89