Random Fields and Spin Glasses: A Field Theory Approach

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Overview

In this book the authors deal with the theory of magnetic systems in the presence of frozen disorder, and in particular paradigmatic and well-known spin models such as the Random Field Ising Model and the Ising Spin Glass. They describe some of the most successful approaches to the physics of disordered systems, such as the replica method and Langevin dynamics, together with lesser known results in finite dimension. This is a unified presentation using a field theory language which covers mean field theory, dynamics and perturbation expansion within the same theoretical framework. Particular emphasis is given to the connections between different approaches such as statics vs. dynamics, microscopic vs. phenomenological models. The book introduces some useful and little known techniques in statistical mechanics and field theory including multiple Legendre transforms, supersymmetry, Fourier transforms on a tree, infinitesimal permutations and Ward Takahashi identities. This book will be of great interest to graduate students and researchers in statistical physics and basic field theory.

About the Author:
Cirano De Dominicis is affiliated with the Service de Physique Theorique, Centre d'Etudes de Saclay, France

About the Author:
Irene Giardina is a researcher at the Istituto Nazionale di Fisica della Materia, Consiglio Nazionale delle Ricerche (CNR), and is also affiliated with the Department of Physics at the University of Rome La Sapienza, Italy

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Editorial Reviews

From the Publisher
"All in all the book will take an important place on the desk of everyone working in this exciting field of research. In particular, it should be highly recommended to mathematicians and mathematical physicists who want to get thoroughly acquainted with the methods and results of theoretical physics in this field."
Anton Bovier, Mathematical Reviews

"It is written in masterly and elegant fashion by two remarkable researchers who have themselves, in particular prof. De Dominicis, fundamentally contributed to the subject. The authors masterly wield the various methods of the field theory in order to "clarify and unify" via its mathematical language some of, in their opinion, more cryptic and pointed details of the theory of disordered systems."
Rudi Podgornik for the Journal of Statistical Physics

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Product Details

  • ISBN-13: 9780521143554
  • Publisher: Cambridge University Press
  • Publication date: 6/10/2010
  • Edition description: Reissue
  • Pages: 232
  • Product dimensions: 6.69 (w) x 9.61 (h) x 0.47 (d)

Meet the Author

Cirano De Dominicis is affiliated with the Service de Physique Théoretique, Centre Commissariat ... l'Energie Atomique (CEA), Saclay, France. He has won several prizes including the Langevin prize and Ricard prize from the French Physical Society, the Ampere prize from the French Academy of Sciences, and the Silver Medal of the National Center of Scientific Research.

Irene Giardina is a researcher at the Istituto Nazionale de Fisica della Materia, Consiglio Nazionale delle Ricerche (CNR), and is also affiliated with the Department of Physics at the University of Rome La Sapienza, Italy.

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Table of Contents


Preface     xi
List of abbreviations     xv
A brief introduction     1
Quenched and annealed averages     2
The replica method     4
The generating functional     5
General comments     6
The Random Field Ising Model     9
The model     10
The replicated field theory     11
Perturbation expansion     13
Bare propagators     13
Vertices     14
Perturbation expansion: the free energy     14
Perturbation expansion: two-point and four-point functions     16
Most divergent graphs and dimensional reduction     17
Dimensional reduction and supersymmetry     18
Generalized couplings     19
Renormalization at one loop: single replica     21
Renormalization at one loop: multi-replicas     22
General structure of the [Gamma][superscript (4)] functions     23
IR divergences     24
RG flow equations and stability on the Curie line     25
A vitreous transition     26
Legendre Transforms     26
The eigenvalues of the Jacobian     29
A naive estimate     30
Summary     32
The dynamical approach     35
Langevin dynamics     36
The bare average values     37
Perturbation expansion     39
Martin-Siggia-Rose formulation     41
Perturbation expansion from MSR     43
Role of the Jacobian     45
Fluctuation-Dissipation Theorem     46
Static limit     47
RFIM dynamics     49
Dynamics/replicas relationship     52
Summary     54
The p = 2 spherical model     57
The model     58
Statics     58
The Wigner distribution     59
A disguised ferromagnet     61
Statics with a random field     62
Langevin dynamics     63
Equilibrium initial conditions     64
Random-like initial conditions     64
Correlation function and weak-ergodicity breaking     66
Response function and generalized FDT     69
Connection with domain coarsening     70
Dynamics with a random field     71
What do we get from the replica way?     73
Comments and summary     76
Mean field spin glasses: one-step RSB      79
The Random Energy Model (REM)     80
Statistics of extremes     83
The REM via replicas     86
The p-spin model     88
Relationship with the REM     88
The replica approach     89
The RS solution     90
Overlap distribution and Replica Symmetry Breaking     91
The one-step RSB solution     93
Metastability, complexity and glassy behaviour     96
Summary     98
The Sherrington-Kirkpatrick Model     101
The model     101
The RS ansatz     104
Stability around the RS saddle point     105
The Replica Fourier Transform     107
Stability for the truncated model     111
Parisi Replica Symmetry Breaking     113
The one-step ansatz     113
Multiple breaking     114
Stationarity equations for the truncated model     118
Overlap distribution     120
Overlaps and susceptibilities     121
Summary     124
Mean field via TAP equations     127
The TAP free energy and mean field equations     127
The TAP free energy for the SK model     128
The paramagnetic phase and the spin glass transition     129
Number of solutions and complexity     130
The BRST supersymmetry     132
Thermodynamic properties and equivalence to the replica method     134
Metastable states and supersymmetry breaking     136
BRST supersymmetry and fluctuation-dissipation relations     136
Supersymmetry violation and fragility of the structure of states     137
Summary     138
Spin glass above D = 6     141
The spin glass Lagrangian     142
RS propagators     143
RSB propagators     146
Parametrization     147
The unitarity equation in the near infrared     148
One-loop correction to the equation of state     150
Physical meaning of the RSB propagators     153
The correlation overlaps     153
Correlation functions     155
Summary     157
Propagators, mostly replicon     159
RFT diagonalization on a tree     159
Coordinates on a tree     160
Toy model diagonalization     160
Properties of the RFT on a tree     163
Equation of state via RFT     164
The propagators in the replicon sector     165
Parametrization of the replicon sector     165
The solution via RFT     166
The replicon propagators in RPT     168
The standard replicon propagators     170
A hint at the lower critical dimension     172
What about the other sectors?     174
The generalized susceptibility     178
Summary     180
Ward-Takahashi Identities and Goldstone modes     183
The Legendre Transform and invariance properties     183
The case of the spin glass     185
Defining a small permutation     186
The simplest Ward-Takahashi Identity     187
A very special Goldstone mode     189
Summary     190
Alternative approaches and conclusions     193
The droplet picture     193
The droplet model     193
Scaling theory and the behaviour of space correlations     194
General predictions     196
Zero temperature excitations and the TNT picture     197
Conclusions     198
Renormalization at one loop: [phiv][superscript 4] theory (pure Ising)     201
Renormalization at one loop: tr[phiv][superscript 3] theory (spin glass)      205
Index     209
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