Introduction to Statistical Field Theory

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1.1 The classical canonical ensemble......Page 13
1.1.2 The quantum canonical ensemble......Page 14
1.2.1 The thermodynamic limit......Page 15
1.3 Gaussian integrals and Wick\'s theorem......Page 16
1.5 d-dimensional integrals......Page 18
Additional references......Page 20
2.1 Can statistical mechanics be used to describe phase transitions?......Page 21
2.2 The order–disorder competition......Page 22
2.3 Order parameter, symmetry and broken symmetry......Page 24
2.4 More general symmetries......Page 28
2.5 Characterization of a phase transition through correlations......Page 30
2.6 Phase coexistence, critical points, critical exponents......Page 31
3.1 Heisenberg\'s exchange forces......Page 34
3.2 Heisenberg and Ising Hamiltonians......Page 36
3.3 Lattice gas......Page 38
3.4 More examples......Page 40
3.5 A first connection with field theory......Page 41
4.1 One-dimensional Ising model: transfer matrix......Page 44
4.2 One-dimensional Ising model: correlation functions......Page 47
4.3 Absence of phase transition in one dimension......Page 49
4.5 Proof of broken symmetry in two dimensions (and more)......Page 50
4.6 Correlation inequalities......Page 54
4.7 Lower critical dimension: heuristic approach......Page 56
Discrete symmetries......Page 57
Continuous symmetries......Page 58
4.8 Digression: Feynman path integrals, the transfer matrix and the Schrödinger equation......Page 59
Appendix Derivation of the GKS inequalities......Page 61
5.1 High-temperature expansion for the Ising model......Page 64
5.1.1 Continuous symmetry......Page 67
5.2 Low-temperature expansion......Page 68
5.2.1 Kramers–Wannier duality......Page 69
5.3 Low-temperature expansion for a continuous symmetry group......Page 70
6.1 Polymers and self-avoiding walks......Page 72
6.2 Potts model and percolation......Page 76
7.1 Landau theory......Page 80
7.2 Landau theory near the critical point: homogeneous case......Page 83
Spontaneous magnetization......Page 84
Susceptibility......Page 85
Critical isotherm......Page 86
7.3 Landau theory and spatial correlations......Page 87
7.4 Transitions without symmetry breaking: the liquid–gas transition......Page 90
7.5 Thermodynamic meaning of gamma{m}......Page 91
7.6 Universality......Page 92
7.7 Scaling laws......Page 94
Appendix Large-distance behaviour of the free propagator......Page 96
8.1 Weiss `molecular field\'......Page 97
8.2 Mean field theory: the variational method......Page 99
8.3 A simpler alternative approach......Page 104
9.1 The first correction to the mean-field free energy......Page 107
9.2 Physical consequences......Page 109
10 Introduction to the renormalization group......Page 112
10.2 Kadanoff block spins......Page 113
One dimension......Page 115
More than one dimension......Page 116
Percolation......Page 118
10.4 Structure of the renormalization group equations......Page 121
Summary......Page 124
11 Renormalization group for the phi4 theory......Page 125
Susceptibility......Page 126
Renormalization group flow......Page 127
11.2 Study of the renormalization group flow in dimension four......Page 128
11.3 Critical behaviour of the susceptibility in dimension four......Page 130
11.4 Multi-component order parameters......Page 132
11.5 Epsilon expansion......Page 134
11.6 An exercise on the renormalization group: the cubic fixed point......Page 137
Conclusion......Page 139
12.1 The meaning of renormalizability......Page 140
The coefficient of m6 in the free energy......Page 142
The shift of g0......Page 143
12.2 Renormalization of the massless theory......Page 144
12.3 The renormalized critical free energy (at one-loop order)......Page 146
12.4 Away from Tc......Page 148
13.1 Broken symmetries and massless modes......Page 150
13.2 Linear and non-linear O(n) sigma models......Page 154
13.3.1 Regularization......Page 156
Lattice regularization......Page 157
Dimensional regularization......Page 158
13.3.2 Perturbation expansion and renormalization......Page 160
13.4 Renormalization group equations for the O(n) non-linear sigma model and the (d-2) expansion......Page 162
13.4.1 Integration of RG equations and scaling......Page 163
Sigma model on an arbitrary Riemannian manifold......Page 165
Localization by a disordered potential......Page 166
14.1 The linear O(n) model......Page 168
Broken symmetry......Page 170
14.2 O(n) sigma model......Page 173
Index......Page 177