Electron correlation in metals

257 Pages·2004·2.501 MB·English

by  K. Yamada

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ELECTRON CORRELATION IN METALS
Since the discovery of high T superconductivity, the role of electron correlation
c
insuperconductivityhasbeenanimportantnewissueincondensedmatterphysics.
Heretheroleofelectroncorrelationinmetalsisexplainedindetailonthebasisof
theFermiliquidtheory.Thebookdiscussesthefollowingissues:enhancementsof
electronicspecificheatandmagneticsusceptibility;effectsofelectroncorrelation
ontransportphenomenasuchaselectricresistivityandHallcoefficient;magnetism;
Motttransitionandunconventionalsuperconductivity.Theseoriginatecommonly
from the coulomb repulsion between electrons. In particular, superconductivity
in strongly correlated electron systems is discussed with a unified point of view.
Thisbookiswrittentoexplaininterestingphysicsinmetalsforundergraduateand
graduatestudentsandresearchersincondensedmatterphysics.
Kosaku Yamada has been a professor in the Department of Physics at Kyoto
Universitysince1995.

ELECTRON CORRELATION
IN METALS
K. YAMADA
DepartmentofPhysics,KyotoUniversity,Japan
cambridge university press
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© K. Yamada 1993
English translation Cambridge University Press 2004
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 2004
isbn-13 978-0-511-22958-9 eBook (EBL)
isbn-10 0-511-22958-5
eBook (EBL)
isbn-13 978-0-521-57232-3 hardback
isbn-10 0-521-57232-0 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
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Contents
Preface page vii
1 Fermigas 1
1.1 Metals 1
1.2 FreeFermigas 2
1.3 ElectronicspecificheatandPaulisusceptibility 4
1.4 Many-bodyeffectofelectrongas 7
1.5 Exchangeenergy 10
1.6 Screeningeffect 13
1.7 Plasmaoscillation 16
1.8 Groundstateenergy 18
1.9 Wignercrystal 20
2 Fermiliquidtheory 22
2.1 Principleofcontinuity 22
2.2 Landau’sFermiliquidtheory 25
3 Anderson’sorthogonalitytheorem 29
3.1 Friedelsumrule 29
3.2 Orthogonalitytheoremonlocalperturbation 33
3.3 Photoemissioninmetalsandtheorthogonalitytheorem 36
3.4 Diffusionofchargedparticlesinmetals 41
4 s–d HamiltonianandKondoeffect 49
4.1 Spinsusceptibilityofconductionelectrons 49
4.2 s–d Exchangeinteractionandspinpolarization 51
4.3 Kondoeffect 54
4.4 Groundstateofdilutemagneticalloysystem 60
4.5 Scalinglawofthes–d system 63
4.6 Wilson’stheory 67
4.7 FermiliquidtheorybyNozie`res 68
v
vi Contents
5 AndersonHamiltonian 72
5.1 Hartree–Fockapproximation 72
5.2 Perturbationexpansionwithrespectto V 76
kd
5.3 Green’sfunction 79
5.4 PerturbationexpansionwithrespecttoU 87
5.5 ExactsolutionoftheAndersonHamiltonian 99
6 HubbardHamiltonian 103
6.1 Basicproperties 103
6.2 Theoryofelectroncorrelation 106
6.3 Infinite-dimensionalHubbardHamiltonian 112
6.4 Motttransition 117
6.5 One-dimensionalHubbardmodel 119
6.6 Ferromagnetismoftransitionmetals 120
6.7 SuperconductivityintheHubbardHamiltonian 121
7 Fermiliquidtheoryofstronglycorrelatedelectronsystems 122
7.1 Heavyfermionsystems 122
7.2 Heavyfermions 123
7.3 Kondotemperatureincrystallinefields 124
7.4 Fermiliquidtheoryonheavyfermionsystems 127
7.5 SpinfluctuationandFermiLiquid 141
8 TransporttheorybasedonFermiliquidtheory 147
8.1 Conductivity 147
8.2 Opticalconductivity 148
8.3 Hallconductivity 152
8.4 Cyclotronresonance 153
8.5 Magnetoresistance 157
8.6 Concludingremarks 158
9 Superconductivityinstronglycorrelatedelectronsystems 159
9.1 Cupratehigh-temperaturesuperconductors 159
9.2 Anisotropicsuperconductivityduetocoulombrepulsion 172
9.3 Electron-dopedcuprates 206
9.4 TripletsuperconductorSr RuO 210
2 4
9.5 Conclusion 218
AppendixA.Feynmanrelation 223
AppendixB.Secondquantization 224
AppendixC.Interactionrepresentationandthermal
Green’sfunction 226
AppendixD.Linearresponsetheory 233
AppendixE.TransportequationderivedbyE´liashberg 238
Index 243
Preface
P.W.Andersonhasachievedmanybrillianttheoriesinthewidefieldofcondensed
matter physics. His book titled Basic Notions of the Condensed Matter Physics
waspublishedin1984.InthisbookAndersonstressestwobasicprinciplesofcon-
densedmatterphysics.Oneoftheprinciplesis‘brokensymmetry’.Thismeansthat
condensedmattersystemsundergophasetransitiontotakeastatepossessinglower
symmetrythanthatoftheHamiltonian.Thisstatementcorrespondstotheappear-
anceofaferromagneticstateandasuperconductingstate,etc.atlowtemperatures.
Thisprinciplemanifestsdiscontinuouschange.
Anotherbasicprincipleistheprincipleof‘adiabaticcontinuity’.Thisprinciple
tells us that when we study a generally complicated physical system we can refer
to a simple system that contains the essential nature of the real system and un-
derstand thecomplicated system onthe basis ofknowledge of thesimple system.
Anderson stresses that the most beautiful and appropriate example showing the
importanceofthecontinuityprincipleisLandau’sFermiliquidtheory.Following
the continuity principle, we start from a non-interacting Fermi gas and introduce
interactions among particles gradually. There exists a one-to-one correspondence
betweenthefreeparticlesystembeforetheintroductionoftheinteractionsandthe
Fermiliquidaftertheintroduction.ItisthebasiccharacteroftheFermiliquidatlow
temperaturesthatwecanintroduceinteractionsasslowlyaspossibleowingtothe
longlifetimeofquasi-particles.Eventhoughmany-bodyinteractionsexistamong
particles, by considering quasi-particles renormalized by the interactions we can
treatthemasiftheyarefreeparticles.BythisprocedurestronglyinteractingFermi
systems are much simplified. Strictly speaking, however, the systems cannot be
completelyfreeparticlesystemsevenafterrenormalization;thereremaindamping
effects giving a finite lifetime and weak renormalized interactions among quasi-
particles. In particular, since attractive forces make the Fermi surface unstable, it
is only the repulsive force that can be continuously renormalized on the basis of
theFermiliquidtheory.Thisfactplaysanimportantroleinmany-bodyproblems.
vii
viii Preface
Afterreducingsystemstoweaklyinteractingquasi-particlesystemsfollowingthe
Fermiliquidtheory,wecanthendiscussphasetransitionssuchassuperconductivity,
inducedbytherenormalizedinteractionsamongquasi-particles.
Fermi liquid theory, which was introduced by Landau for liquid 3He, has been
applied to electrons in metals and developed. Along with progress in the electron
theory of solids, strongly interacting electron systems have been studied actively
for various metals. By this development, the Fermi liquid theory has been made
profound and based on microscopic foundations. After completion of the many-
bodytheoryofelectrongasandtheBCStheoryofsuperconductivity,inthe1960s
magnetism of transition metals and Mott transitions of their oxides were studied
asmainissuesofelectroncorrelation.Then,thediscoveryoftheKondoeffectand
the study of the Anderson Hamiltonian relating to the appearance and disappear-
ance of localized moments in metals followed. Recently, we have studied heavy
electrons whose masses are thousands of times as large as the free electron mass.
Theseelectronsarenothingbutthequasi-particlesintheFermiliquidtheory.Our
presentsubjectinelectroncorrelationisthestudyofcopper-oxidesuperconductors
as metallic systems near the Mott insulator. These are called strongly correlated
electron systems. The normal state of strongly correlated electron systems can be
describedastheFermiliquid.InrecentyearsithasbeenmadeclearthattheFermi
liquid theory contains various fruitful physical properties. At present the Fermi
liquidtheoryisinthecourseofdevelopment.
The Hubbard Hamiltonian contains a wealth of physics, such as metal–
insulatortransitionsandmagnetictransitions.AlthoughthetheoryoftheHubbard
Hamiltonianhasbeengreatlydeveloped,asseeninthestudyoftheMotttransition
usinganinfinite-dimensionalmodel,wearestillfarfromacompleteunderstanding.
Recently, the new problem of whether the Hubbard system can exhibit supercon-
ductivity has been added, as an important problem to be solved. In this book we
showthatsuperconductivityisactuallyrealizedowingtoelectroncorrelationinthe
HubbardHamiltonian.
ThepurposeofthisbookistointroducetheFermiliquidtheoryandtodescribe
the physics of strongly correlated electron systems. I intend to introduce unique
and rich physical phenomena in each system mentioned above; I would like to
describe the universality of the Fermi liquid and the basic principles common to
various systems. At this point, I believe there exists a powerful theory that is not
changed by details of experimental data. Nature is much more complicated than
wecanimagine,butitisnotcapricious.
To accomplish the purpose of this book, I intend to introduce the microscopic
Fermi liquid theory developed by Luttinger et al. and describe the concepts con-
firmedthroughitsdevelopment,althoughIamnotsurewhetherIhavesucceeded
inthistaskornot.