A modern approach to critical phenomena

224 Pages·2007·1.211 MB·English

by  Igor Herbut

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A MODERN APPROACH
TO CRITICAL PHENOMENA
The study of critical phenomena is one of the most exciting areas of mod-
ern physics. This book provides a thorough but economic introduction into
the principles and techniques of the theory of critical phenomena and the
renormalization group, from the perspective of modern condensed matter
physics. Assuming basic knowledge of quantum and statistical mechanics,
the book discusses phase transitions in magnets, superfluids, superconduc-
tors, and gauge field theories. Particular attention is given to modern topics
such as gauge field fluctuations in superconductors, the Kosterlitz-Thouless
transition,dualitytransformations,andquantumphasetransitions,allofwhich
areattheforefrontoftoday’sphysicsresearch.
AModernApproachtoCriticalPhenomenacontainsnumerousproblemsof
varyingdegreesofdifficulty,withsolutions.Theseproblemsprovidereaders
withawealthofmaterialtotesttheirunderstandingofthesubject.Itisidealfor
graduatestudentsandmoreexperiencedresearchersinthefieldsofcondensed
matterphysics,statisticalphysics,andmany-bodyphysics.
Igor Herbut isProfessorofPhysicsatSimonFraserUniversityinBurn-
aby,BritishColumbia.HehasheldvisitingappointmentsattheMaxPlanck
Institute, the Kavli Institute for Theoretical Physics, and the Tokyo Institute
of Technology. Professor Herbut has authored a number of research papers
on quantum phase transitions, disordered systems, gauge field theories, and
high-temperaturesuperconductivity.

A MODERN APPROACH
TO CRITICAL PHENOMENA
IGOR HERBUT
SimonFraserUniversity
CAMBRIDGEUNIVERSITYPRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationo nthi stitle :www.cambri dge.org/9780521854528
©I.Herbut2007
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-13 978-0-511-24940-2 eBook (EBL)
ISBN-10 0-511-24940-3 eBook (EBL)
ISBN-13 978-0-521-85452-8 hardback
ISBN-10 0-521-85452-0 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.
Dedicatedtomyparents,DivnaandFedorHerbut

Contents
Preface page ix
1 Introduction 1
1.1 Phasetransitionsandorderparameters 1
1.2 Models:Ising,XY,Heisenberg 3
1.3 Universalityandcriticalexponents 10
1.4 Scalingoffreeenergy 12
1.5 Correlationsandhyperscaling 19
2 Ginzburg–Landau–Wilsontheory 23
2.1 Partitionfunctionforinteractingbosons 23
2.2 Bose–Einsteincondensation 27
2.3 Hartreeapproximation 30
2.4 Landau’smean-fieldtheory 34
2.5 Uppercriticaldimension 40
3 Renormalizationgroup 43
3.1 Idea 43
3.2 Momentum-shelltransformation 46
3.3 (cid:2)-expansion 53
3.4 Dangerouslyirrelevantcoupling 58
3.5 Correctionstoscaling 59
3.6 Field-theoreticperspective 61
3.7 Computationofanomalousdimension 65
3.8 Summary 69
vii
viii Contents
4 Superconductingtransition 77
4.1 Meissnereffect 77
4.2 Fluctuation-inducedfirst-ordertransition 81
4.3 Type-IIsuperconductorsnearfourdimensions 85
4.4 Anomalousdimensionforthegaugefield 92
4.5 Widthofthecriticalregion 93
5 Nearlowercriticaldimension 97
5.1 Goldstonemodes 97
5.2 Mermin–Wagner–Hohenbergtheorem 99
5.3 Non-linearσ-model 102
5.4 Low-temperatureexpansion 104
5.5 Discussion 106
6 Kosterlitz–Thoulesstransition 115
6.1 Vorticesandspinwaves 115
6.2 Mean-fieldtheory 118
6.3 Dualityandthesine-Gordontheory 123
6.4 Renormalizationofthesine-Gordonmodel 131
6.5 Universaljumpofsuperfluiddensity 138
6.6 Heisenbergmodel 140
7 Dualityinhigherdimensions 147
7.1 Frozenlatticesuperconductor 147
7.2 Confinementofmagneticmonopoles 152
7.3 Magneticfieldcorrelations 154
7.4 Compactelectrodynamics 156
8 Quantumphasetransitions 165
8.1 Dynamicalcriticalexponent 165
8.2 Quantumcriticalpointin(cid:4)4-theory 168
8.3 Bose–Hubbardmodel 174
8.4 Quantumfluctuationsandthesuperfluiddensity 181
8.5 Universalconductivityintwodimensions 185
AppendixA Hubbard–Stratonovichtransformation 195
AppendixB Linked-clustertheorem 197
AppendixC Gaugefixingforlong-rangeorder 199
Selectbibliography 203
Index 209