Random Fields and Spin Glasses: A Field Theory Approach

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Abbreviations......Page 17
1 A brief introduction......Page 19
1.1 Quenched and annealed averages......Page 20
1.2 The replica method......Page 22
1.3 The generating functional......Page 23
1.4 General comments......Page 24
References......Page 25
2 The Random Field Ising Model......Page 27
2.1 The model......Page 28
2.2 The replicated field theory......Page 29
2.3.1 Bare propagators......Page 31
2.3.3 Perturbation expansion: the free energy......Page 32
2.3.4 Perturbation expansion: two-point and four-point functions......Page 34
2.4 Most divergent graphs and dimensional reduction......Page 35
2.4.1 Dimensional reduction and supersymmetry......Page 36
2.5 Generalized couplings......Page 37
2.6 Renormalization at one loop: single replica......Page 39
2.7 Renormalization at one loop: multi-replicas......Page 40
2.7.1 General structure of the Gamma(4) functions......Page 41
2.7.2 IR divergences......Page 42
2.7.3 RG flow equations and stability on the Curie line......Page 43
2.8.1 Legendre Transforms......Page 44
2.8.2 The eigenvalues of the Jacobian......Page 47
2.8.3 A naive estimate......Page 48
References......Page 50
3 The dynamical approach......Page 53
3.1 Langevin dynamics......Page 54
3.1.1 The bare average values......Page 55
3.1.2 Perturbation expansion......Page 57
3.2 Martin–Siggia–Rose formulation......Page 59
3.2.1 Perturbation expansion from MSR......Page 61
3.2.2 Role of the Jacobian......Page 63
3.2.3 Fluctuation–Dissipation Theorem......Page 64
Two-point functions......Page 65
Multipoint functions......Page 66
3.3 RFIM dynamics......Page 67
Response......Page 68
Correlation......Page 69
Generalized vertices......Page 70
Singularities at exceptional momenta......Page 71
3.5 Summary......Page 72
References......Page 73
The p = 2 spherical model......Page 75
4.2 Statics......Page 76
4.3 The Wigner distribution......Page 77
4.4 A disguised ferromagnet......Page 79
4.5 Statics with a random field......Page 80
4.6 Langevin dynamics......Page 81
4.6.2 Random-like initial conditions......Page 82
4.6.3 Correlation function and weak-ergodicity breaking......Page 84
(ii) Aging regime: asymptotic but widely separated times......Page 85
4.6.4 Response function and generalized FDT......Page 87
4.7 Connection with domain coarsening......Page 88
4.8 Dynamics with a random field......Page 89
4.9 What do we get from the replica way?......Page 91
4.10 Comments and summary......Page 94
References......Page 96
5 Mean field spin glasses: one-step RSB......Page 97
5.1 The Random Energy Model (REM)......Page 98
5.1.1 Statistics of extremes......Page 101
RS solution......Page 104
5.2.1 Relationship with the REM......Page 106
5.2.2 The replica approach......Page 107
5.2.3 The RS solution......Page 108
5.2.4 Overlap distribution and Replica Symmetry Breaking......Page 109
5.2.5 The one-step RSB solution......Page 111
5.2.6 Metastability, complexity and glassy behaviour......Page 114
5.3 Summary......Page 116
References......Page 117
6.1 The model......Page 119
6.2 The RS ansatz......Page 122
6.3 Stability around the RS saddle point......Page 123
6.3.1 The Replica Fourier Transform......Page 125
6.3.2 Stability for the truncated model......Page 129
6.4.1 The one-step ansatz......Page 131
6.4.2 Multiple breaking......Page 132
6.4.3 Stationarity equations for the truncated model......Page 136
6.4.4 Overlap distribution......Page 138
6.4.5 Overlaps and susceptibilities......Page 139
6.5 Summary......Page 142
References......Page 143
7.1 The TAP free energy and mean field equations......Page 145
7.1.1 The TAP free energy for the SK model......Page 146
7.1.2 The paramagnetic phase and the spin glass transition......Page 147
7.2 Number of solutions and complexity......Page 148
7.3 The BRST supersymmetry......Page 150
7.4 Thermodynamic properties and equivalence to the replica method......Page 152
7.5.1 BRST supersymmetry and fluctuation–dissipation relations......Page 154
7.5.2 Supersymmetry violation and fragility of the structure of states......Page 155
7.6 Summary......Page 156
References......Page 157
8 Spin glass above D = 6......Page 159
8.1 The spin glass Lagrangian......Page 160
8.2 RS propagators......Page 161
8.3 RSB propagators......Page 164
8.3.1 Parametrization......Page 165
8.4 The unitarity equation in the near infrared......Page 166
8.5 One-loop correction to the equation of state......Page 168
8.6.1 The correlation overlaps......Page 171
8.6.2 Correlation functions......Page 173
8.7 Summary......Page 175
References......Page 176
9.1 RFT diagonalization on a tree......Page 177
9.1.2 Toy model diagonalization......Page 178
9.1.3 Properties of the RFT on a tree......Page 181
9.1.4 Equation of state via RFT......Page 182
9.2.1 Parametrization of the replicon sector......Page 183
9.2.2 The solution via RFT......Page 184
9.2.3 The replicon propagators in RFT......Page 186
9.2.4 The standard replicon propagators......Page 188
9.3 A hint at the lower critical dimension......Page 190
9.4 What about the other sectors?......Page 192
9.5 The generalized susceptibility......Page 196
References......Page 198
10.1 The Legendre Transform and invariance properties......Page 201
10.2 The case of the spin glass......Page 203
10.3 Defining a small permutation......Page 204
10.4 The simplest Ward–Takahashi Identity......Page 205
10.5 A very special Goldstone mode......Page 207
10.6 Summary......Page 208
References......Page 209
11.1.1 The droplet model......Page 211
11.1.2 Scaling theory and the behaviour of space correlations......Page 212
11.1.3 General predictions......Page 214
11.1.4 Zero temperature excitations and the TNT picture......Page 215
11.2 Conclusions......Page 216
References......Page 217
Appendix A Renormalization at one loop: phi4 theory (pure Ising)......Page 219
References......Page 222
Appendix B Renormalization at one loop: tr phi3 theory (spin glass)......Page 223
References......Page 226
Index......Page 227