Preface ........................................................ xi
1 Scaling Theory of Quantum Critical Phenomena ................. 1
1.1 Quantum Phase Transitions ............................... 1
1.2 Renormalisation Group and Scaling Relations ............. 4
1.3 The Critical Exponents .................................. 5
1.4 Scaling Properties Close to a Zero-Temperature Fixed
Point ................................................... 6
1.5 Extension to Finite Temperatures ....................... 12
1.6 Temperature-Dependent Behaviour near a Quantum
Critical Point ......................................... 18
1.7 Generalised Scaling .................................... 18
1.8 Conclusions ............................................ 23
2 Landau and Gaussian Theories ................................ 25
2.1 Introduction ........................................... 25
2.2 Landau Theory of Phase Transitions ..................... 25
2.3 Gaussian Approximation (T > Tc) ........................ 28
2.4 Gaussian Approximation (T < Tc) ........................ 32
2.5 Goldstone Mode ......................................... 34
2.6 Ising Model in a Transverse Field - Mean-Field
Approximation .......................................... 36
3 Real Space Renormalisation Group Approach ................... 39
3.1 Introduction ........................................... 39
3.2 The Ising Model in a Transverse Field .................. 40
3.3 Recursion Relations and Fixed Points ................... 42
3.4 Conclusions ............................................ 44
4 Renormalisation Group: the ϵ-Expansion ...................... 46
4.1 The Landau-Wilson Functional ........................... 46
4.2 The Renormalisation Group in Momentum Space ............ 47
4.3 Fixed Points ........................................... 53
4.4 Renormalisation Group Flows and Critical Exponents ..... 54
4.5 Conclusions ............................................ 56
5 Quantum Phase Transitions ................................... 57
5.1 Effective Action for a Nearly Ferromagnetic Metal ...... 57
5.2 The Quantum Paramagnetic-to-Ferromagnetic Transition ... 60
5.3 Extension to Finite Temperatures ....................... 67
5.4 Effective Action Close to a Spin-Density Wave
Instability ............................................ 70
5.5 Gaussian Effective Actions and Magnetic
Instabilities in Metallic Systems ...................... 71
5.6 Field-Dependent Free Energy ............................ 73
5.7 Gaussian versus Mean Field at T ≠ 0 .................... 74
5.8 Critique of Hertz Approach ............................. 75
6 Heavy Fermions .............................................. 77
6.1 Introduction ........................................... 77
6.2 Scaling Analysis ....................................... 82
6.3 Conclusions ............................................ 86
7 A Microscopic Model for Heavy Fermions ...................... 87
7.1 The Model .............................................. 87
7.2 Local Quantum Criticality .............................. 89
7.3 Critical Regime ........................................ 95
7.4 Generalised Scaling and the Non-Fermi Liquid Regime .... 96
7.5 Local Regime near the QCP .............................. 98
7.6 Quantum Lifshitz Point ................................. 99
7.7 Conclusions ........................................... 100
8 Metal and Superfluid-Insulator Transitions ................. 102
8.1 Conductivity and Charge Stiffness ..................... 102
8.2 Scaling Properties Close to a Metal-Insulator
Transition ............................................ 107
8.3 Different Types of Metal-Insulator Transitions ........ 108
8.4 Disorder-Driven Superfluid-Insulator Transition ....... 110
9 Density-Driven Metal-Insulator Transitions ................. 115
9.1 The Simplest Density-Driven Transition ................ 115
9.2 Renormalisation Group Approach ........................ 117
9.3 Metal-Insulator Transition in Divalent Metals ......... 120
9.4 The Excitonic Transition .............................. 124
9.5 The Effect of Electron-Electron Interactions .......... 124
9.6 The Density-Driven MI Transition in the d = I
Hubbard Model ......................................... 125
9.7 Effects of Disorder ................................... 127
10 Mott Transitions ........................................... 129
10.1 Introduction .......................................... 129
10.2 Gutzwiller Approach ................................... 131
10.3 Density-Driven Transition ............................. 141
10.4 Scaling Analysis ...................................... 142
10.5 Conclusions ........................................... 144
11 The Non-Linear Sigma Model ................................. 146
11.1 Introduction .......................................... 146
11.2 Transverse Fluctuations ............................... 147
11.3 The Quantum Non-Linear Sigma Model .................... 151
11.4 Some Notable β-Functions .............................. 153
12 Superconductor Quantum Critical Points ..................... 158
12.1 Introduction .......................................... 158
12.2 Non-Uniform Superconductor ............................ 162
12.3 Criterion for Superconductivity ....................... 164
12.4 Normal-to-FFLO Quantum Phase Transition in Three
Dimensions ............................................ 165
12.5 The Universality Class of the T=0 d=3 FFLO Quantum
Phase Transition ...................................... 167
12.6 The Two-Dimensional Problem ........................... 168
12.7 Disorder-Induced SQCP ................................. 175
13 Topological Quantum Phase Transitions ...................... 177
13.1 The Landau Paradigm ................................... 177
13.2 Topological Quantum Phase Transitions ................. 177
13.3 The Kitaev Model ...................................... 178
13.4 Renormalisation Group Approach to the Kitaev Model .... 183
13.5 The Simplest Topological Insulator: the sp-Chain ...... 187
13.6 Weyl Fermions in Superconductors ...................... 192
14 Fluctuation-Induced Quantum Phase Transitions .............. 196
14.1 Introduction .......................................... 196
14.2 Goldstone Modes and Anderson-Higgs Mechanism .......... 198
14.3 The Effective Potential ............................... 199
14.4 At the Quantum Critical Point ......................... 201
14.5 The Nature of the Transition .......................... 203
14.6 The Neutral Superfluid ................................ 206
14.7 The Charged Superfluid ................................ 207
14.8 Quantum First-Order Transitions in Systems with
Competing Order Parameters ............................ 209
14.9 Superconducting Transition ............................ 212
14.10 Antiferromagnetic Transition ......................... 213
14.11 One-Loop Effective Potentials and Renormalisation
Group ................................................ 214
14.12 Conclusions .......................................... 215
15 Scaling Theory of First-Order Quantum Phase Transitions .... 217
15.1 Scaling Theory of First-Order Quantum Phase
Transitions ........................................... 217
15.2 The Charged Superfluid and the Coleman-Weinberg
Potential: Scaling Approach ........................... 218
15.3 Conclusions ........................................... 223
Appendix ...................................................... 224
A.1 Green's Functions ..................................... 224
References .................................................... 227
Index ......................................................... 234
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