Topological Orders with Spins and Fermions (eBook)

Supervisor’s Foreword 6
Abstract 9
Outline of the Thesis 12
Contents 14
Introduction 17
1 Topology in Condensed Matter 18
1.1 SET Phases: Symmetry Fractionalisation 20
1.2 SPT Phases: Edge States 21
References 22
2 Topology in Quantum Information 26
2.1 Topological Quantum Computation with Ground State 27
2.2 Topological Quantum Computation with Excitations 28
References 29
Spin Systems 31
3 The Bilayer Double Semion Model 32
3.1 Introduction 32
3.2 Summary of Main Results 34
3.3 The Double Semion Model 34
3.4 The String-Flux Mechanism 38
3.5 The Bilayer DS Model 42
3.5.1 Invariance Under Global Symmetry 48
3.6 New Paramagnetic Phase 50
3.6.1 The Ground State 54
3.6.2 Edge States 57
3.7 Outlook 59
References 61
4 Double Semion Model as a Quantum Memory 62
4.1 Introduction 62
4.2 Summary of Main Results 63
4.3 Off-Shell Double Semion: Microscopic Model 64
4.3.1 Double Semion Model on the Vertex-Free Subspace 64
4.3.2 Exactly Solvable Model in the Whole Hilbert Space 66
4.4 Constructing Open-String Operators 68
4.4.1 Algorithm to Generate Strings 69
4.4.2 Example of an Open-String Operator 73
4.4.3 Completeness of String Operators 74
4.5 Constructing Closed Strings: Logical Operators 76
4.6 Error Correction with the Semion Code 79
4.7 Outlook 80
References 80
Fermionic Systems 82
5 Topological Insulators 83
5.1 Introduction 83
5.2 Summary of Main Results 85
5.3 Generic Helical Edge States 86
5.3.1 Phenomenology of the Generic Helical Edge States 86
5.3.2 Realistic GHES: The BHZ Model with Rashba SOC 88
5.4 Outlook 102
References 102
6 Topological Superconductors 105
6.1 Introduction 105
6.2 Summary of Main Results 108
6.3 Microscopic Characterisation 109
6.3.1 Disorder Analysis 113
6.3.2 Topological Phase Diagrams 115
6.4 Topological Edge States 117
6.4.1 Edge States from d-wave Superconductor 118
6.4.2 Edge States from Nodeless Pairing 123
6.5 Outlook 126
References 127
Conclusions 131
7 Summary of Results 132
8 Outlook 135
References 137
A Technical Details on the Bilayer DS Model 138
A.1 Remarks on the DS Model 138
A.2 Proofs of Commutativity of the Bilayer Double Semion 142
B Technical Details on the Semion Code 146
B.1 Commutation of the Plaquette Operators on the Vertex-Free Subspace 146
B.2 Properties of the Generalised Plaquette Operators 147
B.2.1 Hermiticity and Unitarity of the Plaquette Operators 147
B.2.2 Commutation of the Plaquette Operators 148
B.2.3 The Product of All Plaquette Operators Gives the Identity 152
B.3 Proofs Regarding Algorithm 1 155
B.3.1 Useful Technical Lemmas 156
B.3.2 Proof of Theorem 1 161
B.4 Properties of Open Crossing Strings 161
C Technical Details on How to Calculate GHESs 165
D Technical Details on Derivation of Majorana States 175
References 176