Quantum Scaling in Many-Body Systems: An Approach to Quantum Phase Transitions

Contents......Page 8
Preface......Page 11
1.1 Quantum Phase Transitions......Page 14
1.2 Renormalisation Group and Scaling Relations......Page 17
1.3 The Critical Exponents......Page 18
1.4 Scaling Properties Close to a Zero-Temperature Fixed Point......Page 19
1.5 Extension to Finite Temperatures......Page 25
1.7 Generalised Scaling......Page 31
1.8 Conclusions......Page 36
2.2 Landau Theory of Phase Transitions......Page 38
2.3 Gaussian Approximation (T > Tc)......Page 41
2.4 Gaussian Approximation (T < Tc)......Page 45
2.5 Goldstone Mode......Page 47
2.6 Ising Model in a Transverse Field – Mean-Field Approximation......Page 49
3.1 Introduction......Page 52
3.2 The Ising Model in a Transverse Field......Page 53
3.3 Recursion Relations and Fixed Points......Page 55
3.4 Conclusions......Page 57
4.1 The Landau–Wilson Functional......Page 59
4.2 The Renormalisation Group in Momentum Space......Page 60
4.3 Fixed Points......Page 66
4.4 Renormalisation Group Flows and Critical Exponents......Page 67
4.5 Conclusions......Page 69
5.1 Effective Action for a Nearly Ferromagnetic Metal......Page 70
5.2 The Quantum Paramagnetic-to-Ferromagnetic Transition......Page 73
5.3 Extension to Finite Temperatures......Page 80
5.4 Effective Action Close to a Spin-Density Wave Instability......Page 83
5.5 Gaussian Effective Actions and Magnetic Instabilities in Metallic Systems......Page 84
5.6 Field-Dependent Free Energy......Page 86
5.7 Gaussian versus Mean Field at T ≠ 0......Page 87
5.8 Critique of Hertz Approach......Page 88
6.1 Introduction......Page 90
6.2 Scaling Analysis......Page 95
6.3 Conclusions......Page 99
7.1 The Model......Page 100
7.2 Local Quantum Criticality......Page 102
7.3 Critical Regime......Page 108
7.4 Generalised Scaling and the Non-Fermi Liquid Regime......Page 109
7.5 Local Regime near the QCP......Page 111
7.6 Quantum Lifshitz Point......Page 112
7.7 Conclusions......Page 113
8.1 Conductivity and Charge Stiffness......Page 115
8.2 Scaling Properties Close to a Metal–Insulator Transition......Page 120
8.3 Different Types of Metal–Insulator Transitions......Page 121
8.4 Disorder-Driven Superfluid–Insulator Transition......Page 123
9.1 The Simplest Density-Driven Transition......Page 128
9.2 Renormalisation Group Approach......Page 130
9.3 Metal–Insulator Transition in Divalent Metals......Page 133
9.5 The Effect of Electron–Electron Interactions......Page 137
9.6 The Density-Driven MI Transition in the d = 1 Hubbard Model......Page 138
9.7 Effects of Disorder......Page 140
10.1 Introduction......Page 142
10.2 Gutzwiller Approach......Page 144
10.3 Density-Driven Transition......Page 154
10.4 Scaling Analysis......Page 155
10.5 Conclusions......Page 157
11.1 Introduction......Page 159
11.2 Transverse Fluctuations......Page 160
11.3 The Quantum Non-Linear Sigma Model......Page 164
11.4 Some Notable β-Functions......Page 166
12.1 Introduction......Page 171
12.2 Non-Uniform Superconductor......Page 175
12.3 Criterion for Superconductivity......Page 177
12.4 Normal-to-FFLO Quantum Phase Transition in Three Dimensions......Page 178
12.5 The Universality Class of the T=0 d=3 FFLO Quantum Phase Transition......Page 180
12.6 The Two-Dimensional Problem......Page 181
12.7 Disorder-Induced SQCP......Page 188
13.2 Topological Quantum Phase Transitions......Page 190
13.3 The Kitaev Model......Page 191
13.4 Renormalisation Group Approach to the Kitaev Model......Page 196
13.5 The Simplest Topological Insulator: the sp-Chain......Page 200
13.6 Weyl Fermions in Superconductors......Page 205
14.1 Introduction......Page 209
14.2 Goldstone Modes and Anderson–Higgs Mechanism......Page 211
14.3 The Effective Potential......Page 212
14.4 At the Quantum Critical Point......Page 214
14.5 The Nature of the Transition......Page 216
14.6 The Neutral Superfluid......Page 219
14.7 The Charged Superfluid......Page 220
14.8 Quantum First-Order Transitions in Systems with Competing Order Parameters......Page 222
14.9 Superconducting Transition......Page 225
14.10 Antiferromagnetic Transition......Page 226
14.11 One-Loop Effective Potentials and Renormalisation Group......Page 227
14.12 Conclusions......Page 228
15.1 Scaling Theory of First-Order Quantum Phase Transitions......Page 230
15.2 The Charged Superfluid and the Coleman–Weinberg Potential: Scaling Approach......Page 231
15.3 Conclusions......Page 236
A.1 Green’s Functions......Page 237
References......Page 240
Index......Page 247