Quantum Field Theory in Condensed Matter Physics

380 Pages·2007·2.61 MB·English

by  Alexei M. Tsvelik

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QuantumFieldTheoryinCondensedMatterPhysics
Thisbookisacourseinmodernquantumfieldtheoryasseenthroughtheeyesofatheorist
workingincondensedmatterphysics.Itcontainsagentleintroductiontothesubjectand
canthereforebeusedevenbygraduatestudents.Theintroductorypartsincludeaderiva-
tionofthepathintegralrepresentation,Feynmandiagramsandelementsofthetheoryof
metalsincludingadiscussionofLandauFermiliquidtheory.Inlaterchaptersthediscus-
sion gradually turns to more advanced methods used in the theory of strongly correlated
systems. The book contains a thorough exposition of such nonperturbative techniques as
1/N-expansion,bosonization(Abelianandnon-Abelian),conformalfieldtheoryandtheory
ofintegrablesystems.Thebookisintendedforgraduatestudents,postdoctoralassociates
andindependentresearchersworkingincondensedmatterphysics.
alexeitsvelikwasbornin1954inSamara,Russia,graduatedfromanelitemathematical
school and then from Moscow Physical Technical Institute (1977). He defended his PhD
intheoreticalphysicsin1980(thesubjectwasheavyfermionmetals).Hismostimportant
collaborativework(withWiegmannontheapplicationofBetheansatztomodelsofmagnetic
impurities) started in 1980. The summary of this work was published as a review article
in Advances in Physics in 1983. During the years 1983–89 Alexei Tsvelik worked at the
LandauInstituteforTheoreticalPhysics.Afterholdingseveraltemporaryappointmentsin
theUSAduringtheyears1989–92,hesettledinOxford,werehespentnineyears.Since2001
AlexeiTsvelikhasheldatenuredresearchappointmentatBrookhavenNationalLaboratory.
Themainareaofhisresearchisstronglycorrelatedsystems(withaviewofapplicationto
condensedmatterphysics).Heisanauthororco-authorofapproximately120papersand
twobooks.Hismostimportantpapersincludepapersontheintegrablemodelsofmagnetic
impurities,papersonlow-dimensionalspinliquidsandpapersonapplicationsofconformal
field theory to systems with disorder. Alexei Tsvelik has had nine graduate students of
whomsevenhaveremainedinphysics.

Quantum Field Theory in Condensed
Matter Physics
AlexeiM.Tsvelik
DepartmentofPhysics
BrookhavenNationalLaboratory
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521822848
© Alexei Tsvelik 2003
This book 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 2003
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Tomyfather

Contents
Prefacetothefirstedition pagexi
Prefacetothesecondedition xv
Acknowledgementsforthefirstedition xvii
Acknowledgementsforthesecondedition xviii
I Introductiontomethods
1 QFT:languageandgoals 3
2 Connectionbetweenquantumandclassical:pathintegrals 15
3 Definitionsofcorrelationfunctions:Wick’stheorem 25
4 Freebosonicfieldinanexternalfield 30
5 Perturbationtheory:Feynmandiagrams 41
6 Calculationmethodsfordiagramseries:divergencesandtheirelimination 48
7 Renormalizationgroupprocedures 56
8 O(N)-symmetricvectormodelbelowthetransitionpoint 66
9 Nonlinearsigmamodelsintwodimensions:renormalizationgroupand
1/N-expansion 74
10 O(3)nonlinearsigmamodelinthestrongcouplinglimit 82
II Fermions
11 PathintegralandWick’stheoremforfermions 89
12 Interactingelectrons:theFermiliquid 96
13 Electrodynamicsinmetals 103
viii Contents
14 Relativisticfermions:aspectsofquantumelectrodynamics 119
(1+1)-Dimensionalquantumelectrodynamics(Schwingermodel) 123
15 Aharonov–Bohmeffectandtransmutationofstatistics 129
Theindextheorem 135
QuantumHallferromagnet 137
III Stronglyfluctuatingspinsystems
Introduction 143
16 Schwinger–Wignerquantizationprocedure:nonlinearsigmamodels 148
Continuousfieldtheoryforaferromagnet 149
Continuousfieldtheoryforanantiferromagnet 150
17 O(3)nonlinearsigmamodelin(2+1)dimensions:thephasediagram 157
Topologicalexcitations:skyrmions 162
18 Orderfromdisorder 165
19 Jordan–WignertransformationforspinS =1/2modelsin D =1,2,3 172
20 MajoranarepresentationforspinS =1/2magnets:relationshipto Z
2
latticegaugetheories 179
21 Pathintegralrepresentationsforadopedantiferromagnet 184
IV Physicsintheworldofonespatialdimension
Introduction 197
22 Modelofthefreebosonicmasslessscalarfield 199
23 Relevantandirrelevantfields 206
24 Kosterlitz–Thoulesstransition 212
25 Conformalsymmetry 219
GaussianmodelintheHamiltonianformulation 222
26 Virasoroalgebra 226
Wardidentities 230
Subalgebrasl(2) 231
27 Differentialequationsforthecorrelationfunctions 233
Coulombgasconstructionfortheminimalmodels 239
28 Isingmodel 245
Isingmodelasaminimalmodel 245
QuantumIsingmodel 248
Orderanddisorderoperators 249