Random fields and spin glasses: a field theory approach

231 Pages·2006·1.53 MB·English

by  Cirano De Dominicis, Irene Giardina

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RANDOM FIELDS AND SPIN GLASSES
AFieldTheoryApproach
Disorderedmagneticsystemsenjoynontrivialpropertieswhicharedifferentfrom
and richer than those observed in their pure, non-disordered counterparts. These
properties dramatically affect the thermodynamic behaviour and require specific
theoreticaltreatment.
Inthisbooktheauthorsdealwiththetheoryofmagneticsystemsinthepresence
offrozendisorder,andinparticularparadigmaticandwellknownspinmodelssuch
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
finitedimension.Thisisaunifiedpresentationusingafieldtheorylanguagewhich
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. phenomeno-
logical models. The book introduces some useful and little known techniques
in statistical mechanics and field theory including multiple Legendre transforms,
supersymmetry,Fouriertransformsonatree,infinitesimalpermutationsandWard
TakahashiIdentities.
Thisbookwillbeofgreatinteresttograduatestudentsandresearchersinstatis-
ticalphysicsandbasicfieldtheory.
Cirano De Dominicis is affiliated with the Service de Physique The´orique,
Centre d’Etudes de 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 Centre
NationaldelaRechercheScientifique(CNRS).
Irene Giardina is a researcher at the Istituto Nazionale di Fisica della
Materia, Consiglio Nazionale delle Ricerche (CNR), and is also affiliated with
theDepartmentofPhysicsattheUniversityofRomeLaSapienza,Italy.

RANDOM FIELDS AND SPIN GLASSES
A Field Theory Approach
CIRANO DE DOMINICIS
ServicedePhysiqueThe´orique,Saclay
Commissariata`l’EnergieAtomique,France
IRENE GIARDINA
IstitutoNazionalediFisicadellaMateria,Roma
ConsiglioNazionaledelleRicerche,Italy
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
TheEdinburghBuilding,Cambridgecb22ru,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521847834
© C. De Dominicis and I. Giardina 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-25644-8eBook (EBL)
isbn-10 0-511-25644-2 eBook (EBL)
isbn-13 978-0-521-84783-4hardback
isbn-10 0-521-84783-4 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
ToFlorence,Ariane, ToElsaandAndrea
MarionandBruno

Contents
Preface pagexi
Listofabbreviations xv
1 Abriefintroduction 1
1.1 Quenchedandannealedaverages 2
1.2 Thereplicamethod 4
1.3 Thegeneratingfunctional 5
1.4 Generalcomments 6
2 TheRandomFieldIsingModel 9
2.1 Themodel 10
2.2 Thereplicatedfieldtheory 11
2.3 Perturbationexpansion 13
2.3.1 Barepropagators 13
2.3.2 Vertices 14
2.3.3 Perturbationexpansion:thefreeenergy 14
2.3.4 Perturbationexpansion:two-pointandfour-pointfunctions 16
2.4 Mostdivergentgraphsanddimensionalreduction 17
2.4.1 Dimensionalreductionandsupersymmetry 18
2.5 Generalizedcouplings 19
2.6 Renormalizationatoneloop:singlereplica 21
2.7 Renormalizationatoneloop:multi-replicas 22
2.7.1 Generalstructureofthe(cid:2)(4) functions 23
2.7.2 IRdivergences 24
2.7.3 RGflowequationsandstabilityontheCurieline 25
2.8 Avitreoustransition 26
2.8.1 LegendreTransforms 26
2.8.2 TheeigenvaluesoftheJacobian 29
2.8.3 Anaiveestimate 30
2.9 Summary 32
vii
viii Contents
3 Thedynamicalapproach 35
3.1 Langevindynamics 36
3.1.1 Thebareaveragevalues 37
3.1.2 Perturbationexpansion 39
3.2 Martin–Siggia–Roseformulation 41
3.2.1 PerturbationexpansionfromMSR 43
3.2.2 RoleoftheJacobian 45
3.2.3 Fluctuation–DissipationTheorem 46
3.2.4 Staticlimit 47
3.3 RFIMdynamics 49
3.4 Dynamics/replicasrelationship 52
3.5 Summary 54
4 The p = 2sphericalmodel 57
4.1 Themodel 58
4.2 Statics 58
4.3 TheWignerdistribution 59
4.4 Adisguisedferromagnet 61
4.5 Staticswitharandomfield 62
4.6 Langevindynamics 63
4.6.1 Equilibriuminitialconditions 64
4.6.2 Random-likeinitialconditions 64
4.6.3 Correlationfunctionandweak-ergodicity
breaking 66
4.6.4 ResponsefunctionandgeneralizedFDT 69
4.7 Connectionwithdomaincoarsening 70
4.8 Dynamicswitharandomfield 71
4.9 Whatdowegetfromthereplicaway? 73
4.10 Commentsandsummary 76
5 Meanfieldspinglasses:one-stepRSB 79
5.1 TheRandomEnergyModel(REM) 80
5.1.1 Statisticsofextremes 83
5.1.2 TheREMviareplicas 86
5.2 The p-spinmodel 88
5.2.1 RelationshipwiththeREM 88
5.2.2 Thereplicaapproach 89
5.2.3 TheRSsolution 90
5.2.4 OverlapdistributionandReplicaSymmetryBreaking 91
5.2.5 Theone-stepRSBsolution 93
5.2.6 Metastability,complexityandglassybehaviour 96
5.3 Summary 98