Topological Quantum Matter - A Field Theoretical Perspective

Supervisor’s Foreword

Acknowledgements

Contents

1 Introduction

1.1 Entanglement

1.2 Topological Interactions

1.3 Topological Response

1.4 Outline

1.5 Notations and Conventions for Topological Matter

References

2 Anyons and Topological Order

2.1 Bosons, Fermions and Abelian Anyons

2.1.1 A Single Particle on An Annulus

2.1.2 Indistinguishable Particles and the Braid Group

2.1.3 Mediating Topological Interactions with a Gauge Boson

2.2 Topological Order in Two Dimensions

2.2.1 Torus Degeneracy

2.3 Abelian Chern-Simons Theory

References

3 Response Theory and Symmetry Protected Topological Phases

3.1 The Quantization of the Hall Conductance

3.2 U(1) Response Theory

3.2.1 Functional Bosonization

3.3 The Wen-Zee Term and Chiral Superconductors

3.3.1 Chiral Superconductors

References

4 Topological Geometric Response and Topological Order of Super Conductors

4.1 A Superconducting Bosonic Model

4.2 Adding Fermions to Get Topological Order

4.2.1 The Chiral p-Wave SC

4.3 Non-Abelian Statistics in the Adiabatic Limit

4.4 The Majorino Conundrum

References

A Conventions and Notation Concerning Differential Forms and U(1) Connections

A.1 Differential Forms

A.2 Normalization

Appendix B Definitions

Appendix C Vector Bundles and Chern Numbers in Quantum Mechanics

C.1 The Berry Connection

C.2 The Berry Field Strength

C.3 Chern Numbers and Chern-Simons Invariants

Reference