Singular or non-Fermi liquids

Singular or non-Fermi liquids

PhysicsReports361(2002)

267–417

Singularornon-Fermiliquids

C.M.Varmaa;b;1,Z.Nussinovb,WimvanSaarloosb;

a

b

UniversiteitBellLaboratories,Leiden,Instituut-Lorentz,LucentTechnologies,PostbusMurray9506,2300Hill,RANJLeiden,07974,Netherlands

USA

ReceivedJune2001;editor:C:W:J:Beenakker

Contents

3.6.Aspinlessmodel1.Introduction

ndau’sOutline2723.8.Ainteractions

withÿniterange

Multichannelmodelformixed-valenceKondoproblem

impurities2.1.Fermiofthepaperliquid

paper2732.2.EssentialsLandauFermiofLandauFermiliquids2734.3.9.SFLThetwo-Kondo-impuritiesproblemonebehaviorforinteractingfermionsin2.3.Understandingwavefunctionrenormalizationliquidandthe

microscopicallyZwhy

2754.1.dimension

4.2.Theone-dimensionalelectrongas2.4.PrinciplesFermi-liquidtheoryworks2804.3.The4.4.Thermodynamics

Tomonaga–Luttingermodel2.5.derivationofofthemicroscopic

2864.5.One-particlespectralfunctions2904.6.Correlation3.2.6.LocalRoutesModerntoderivations

Landautheorybreakdownof2914.7.TheLuther–Emeryfunctions

model4.8.Spin–chargeSpin–chargeseparation

separationinmorethan

FermiFermiliquidsandlocalLandausingulartheory3.1.2953.2.Theliquids

Fermi-liquidKondoproblem

phenomenologyfor2964.9.oneRecoildimension?

catastropheandtheinoneorthogonality

dimensionand3.3.FerromagneticKondoproblem

the

2994.10.higher

3.4.anisotropicKondoproblemandthe

3004.11.CoupledExperimentalone-dimensionalchains3.5.OrthogonalityKondoX-rayedgesingularities

catastropheproblem3003015.Singulardimensionalgaugeÿelds

Fermi-liquidLuttingerobservationsbehaviorliquidofduebehavior

one-to

E-mailCorrespondingauthor.Tel.:+31-71-5275501;1

Presentaddress:andpermanentsaarloos@address:BellLaboratories,(W.vanfax:Saarloos).+31-71-5275511.LucentTechnologies,MurrayHill,NJ07974,USA.

0370-1573/02/$PII:S0370-1573(01)00060-6

-seefrontmatter

c2002ElsevierScienceB.V.Allrightsreserved.302305306310314315320321322324327328330332335335338

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268C.M.Varmaetal./PhysicsReports361(2002)267–417

338341343343344353359360361361364

7.3.Generalrequirementsinamicroscopic

theory

7.4.Microscopictheory

8.Themetallicstateintwodimensions8.1.Thetwo-dimensionalelectrongas8.2Non-interactingdisorderedelectrons:

scalingtheoryoflocalization

8.3.Interactionsindisorderedelectrons8.4.Finkelsteintheory

pressibility,screeninglengthand

amechanismformetal–insulatortransition8.6.Experiments

8.7.Discussionoftheexperimentsinlight

ofthetheoryofinteractingdisorderedelectrons

8.8.PhasediagramandconcludingremarksAcknowledgementsReferences

372373379380381385389391392400404406406

5.1.SFLbehaviorduetocouplingtothe

electromagneticÿeld

5.2.Generalizedgaugetheories

6.Quantumcriticalpointsinfermionicsystems

6.1.Quantumcriticalpointsin

ferromagnets,antiferromagnets,andchargedensitywaves6.2.Quantumcriticalscaling

6.3.ExperimentalexamplesofSFLdue

toquantumcriticality:opentheoreticalproblems

6.4.Specialcomplicationsinheavy

fermionphysics

6.5.E ectsofimpuritiesonquantum

criticalpoints

7.Thehigh-Tcprobleminthecopper-oxide-basedcompounds

7.1.Somebasicfeaturesofthehigh-Tc

materials

7.2.MarginalFermiliquidbehaviorofthe

normalstate

Abstract

Anintroductorysurveyofthetheoreticalideasandcalculationsandtheexperimentalresultswhichde-partfromLandauFermiliquidsispresented.ThecommonthemesandpossibleroutestothesingularitiesleadingtothebreakdownofLandauFermiliquidsarecategorizedfollowinganelementarydiscussionofthetheory.Solubleexamplesofsingularornon-Fermiliquidsincludemodelsofimpuritiesinmetalswithspecialsymmetriesandone-dimensionalinteractingfermions.AreviewoftheseisfollowedbyadiscussionofsingularFermiliquidsinawidevarietyofexperimentalsituationsandtheoreticalmodels.Theseincludethee ectsoflow-energycollective uctuations,gaugeÿeldsdueeithertosymmetriesintheHamiltonianorpossibledynamicallygeneratedsymmetries, uctuationsaroundquantumcriticalpoints,thenormalstateofhigh-temperaturesuperconductorsandthetwo-dimensionalmetallicstate.Forthelastthreesystems,theprincipalexperimentalresultsaresummarizedandtheoutstandingtheoretical

c2002ElsevierScienceB.V.Allrightsreserved.issuesarehighlighted.

PACS:7.10.Ay;71.10.Hf;71.10.Pm;71.27.+a

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C.M.Varmaetal./PhysicsReports361(2002)267–417269

1.Introduction

1.1.Aimandscopeofthispaper

Inthelasttwodecadesavarietyofmetalshavebeendiscoveredwhichdisplaythermodynamicandtransportpropertiesatlowtemperatureswhicharefundamentallydi erentfromthoseoftheusualmetallicsystemswhicharewelldescribedbytheLandauFermi-liquidtheory.TheyhaveoftenbeenreferredtoasNon-Fermiliquids.Afundamentalcharacteristicofsuchsystemsisthatthelow-energypropertiesinawiderangeoftheirphasediagramaredominatedbysingularitiesasafunctionofenergyandtemperature.Sincetheseproblemsstillrelatetoaliquidstateoffermionsandsinceitisnotagoodpracticetonamethingsafterwhattheyarenot,weprefertocallthemsingularFermiliquids(SFL).

ThebasicnotionsofFermi-liquidtheoryhaveactuallybeenwithusatanintuitivelevelsincethetimeofSommerfeld:Heshowedthatthelinearlow-temperaturespeciÿcheatbehaviorofmetalsaswellastheirasymptoticlow-temperatureresisitivityandopticalconductivitycouldbeunderstoodbyassumingthattheelectronsinametalcouldbethoughtofasagasofnon-interactingfermions,i.e.,intermsofquantummechanicalparticleswhichdonothaveanydirectinteractionbutwhichdoobeyFermistatistics.Meanwhile,Paulicalculatedthattheparamagneticsusceptibilityofnon-interactingelectronsisindependentoftemperature,alsoinaccordwithexperimentsinmetals.Atthesametimeitwasunderstood,atleastsincetheworkofBlochandWigner,thattheinteractionenergiesoftheelectronsinthemetallicrangeofdensitiesarenotsmallcomparedtothekineticenergy.Therationalizationforthequalitativesuccessofthenon-interactingmodelwasprovidedinamasterlypairofpapersbyLandau[152,153]whoinitiallywasconcernedwiththepropertiesofliquid3He.Thisworkepitomizedanewwayofthinkingaboutthepropertiesofinteractingsystemswhichisacornerstoneofourunderstandingofcondensedmatterphysics.Thenotionofquasiparticlesandelementaryexcitationsandthemethodologyofaskingusefulquestionsaboutthelow-energyexcitationsofthesystembasedonconceptsofsymmetry,withoutworryingaboutthemyriadunnecessarydetails,isepitomizedinLandau’sphenomenologicaltheoryofFermiliquids.Themicroscopicderivationofthetheorywasalsosoondeveloped.

OurperspectiveonFermiliquidshaschangedsigniÿcantlyinthelasttwodecadesorso.Thisisduebothtochangesinourtheoreticalperspective,andduetotheexperimentaldevel-opments:ontheexperimentalside,newmaterialshavebeenfoundwhichexhibitFermi-liquidbehaviorinthetemperaturedependenceoftheirlow-temperaturepropertieswiththecoe cientsoftenafactoroforder103di erentfromthenon-interactingelectronvalues.Theseobser-vationsdramaticallyillustratethepowerandrangeofvalidityoftheFermi-liquidideas.Ontheotherhand,newmaterialshavebeendiscoveredwhosepropertiesarequalitativelydif-ferentfromthepredictionsofFermi-liquidtheory(FLT).Themostprominentlydiscussedofthesematerialsarethenormalphaseofhigh-temperaturesuperconductingmaterialsforarangeofcompositionsneartheirhighestTc.Almosteveryideadiscussedinthisreviewhasbeenusedtounderstandthehigh-Tcproblem,butthereisnoconsensusyetonthesolution.

IthasofcoursebeenknownforalongtimethatFLTbreaksdowninthe uctuationregimeofclassicalphasetransitions.Thisbreakdownoccursinamoresubstantialregionofthephase

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Fig.1.Schematicphasediagramnearaquantumcriticalpoint.Theparameteralongthex-axiscanbequitegeneral,likethepressureoraratioofcouplingconstants.Wheneverthecriticaltemperaturevanishes,aQCP,indicatedwithadotintheÿgure,isencountered.Inthevicinityofsuchapointquantummechanical,zero-point uctuationsbecomeveryimportant.However,whenTcisÿnite,criticalslowingdownimpliesthattherelevantfrequencyscalegoesas!～|Tc T| z,dwarÿngquantume ects;thestandardclassicalcriticalmethodologythenapplies.AnexampleofaphasediagramofthistypeforMnSiisshowninFig.34below.

diagramaroundthequantumcriticalpoint(QCP)wherethetransitiontemperaturetendstozeroasafunctionofsomeparameter,seeFig.1.Thisphenomenonhasbeenextensivelyinvestigatedforawidevarietyofmagnetictransitionsinmetalswherethetransitiontemperaturecanbetunedthroughtheapplicationofpressureorbyvaryingtheelectronicdensitythroughalloying.Heavyfermions,withtheirclosecompetitionbetweenstatesofmagneticorderwithlocalizedmomentsanditinerantstatesduetoKondoe ects,appearparticularlypronetosuchQCPs.EquallyinterestingarequestionshavingtodowiththechangeinpropertiesduetoimpuritiesinsystemswhicharenearaQCPinthepurelimit.

Thedensity–densitycorrelationsofitinerantdisorderedelectronsatlongwavelengthsandlowenergiesmusthaveadi usiveform.Intwodimensionsthisleadstologarithmicsingularitiesinthee ectiveinteractionswhentheinteractionsaretreatedperturbatively.Theproblemofÿndingthegroundstateandlow-lyingexcitationsinthissituationisunsolved.Ontheexperimentalside,thediscoveryofthemetal–insulatortransitionintwodimensionsandtheunusualpropertiesobservedinthemetallicstatemakethisanimportantproblemtoresolve.

Theone-dimensionalelectrongasrevealslogarithmicsingularitiesinthee ectiveinteractionseveninasecond-orderperturbationcalculation.Avarietyofmathematicaltechniqueshavebeenusedtosolveawholeclassofinteractingone-dimensionalproblemsandonenowknowstheessentialsofthecorrelationfunctionseveninthemostgeneralcase.Animportantissueiswhetherandhowthisknowledgecanbeusedinhigherdimensions.

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ThesolutionoftheKondoproblemandtherealizationthatitslow-temperaturepropertiesmaybediscussedinthelanguageofFLThasledinturntotheformulationandsolutionofimpuritymodelswithsingularlow-energyproperties.SuchmodelshaveaQCPforaparticu-larrelationbetweenthecouplingconstants;insomeexamplestheyexhibitaquantumcriticalline.ThethermodynamicandtransportpropertiesaroundsuchcriticalpointsorlinesarethoseoflocalsingularFermiliquids.Althoughthedirectexperimentalrelevanceofsuchmodels(asofone-dimensionalmodels)toexperimentsisoftenquestionable,thesemodels,beingsol-uble,canbequiteinstructiveinhelpingtounderstandtheconditionsnecessaryforthebreak-downofFLTandassociatedquasiparticleconcepts.Theknowledgefromzero-dimensionalandone-dimensionalproblemsmustneverthelessbeextrapolatedwithcare.

AproblemwhichwedonotdiscussbutwhichbelongsinthestudyofSFLsisthequantumHalle ectproblem.Themassivedegeneracyoftwo-dimensionalelectronsinamagneticÿeldleadstospectacularnewpropertiesandinvolvesnewfractionalquantumnumbers.TheessentialsofthisproblemweresolvedfollowingLaughlin’sinspiredvariationalcalculation.Theprincipalreasonfortheomissionisÿrstlythatexcellentpapersreviewingthedevelopmentsareavailable[213,72,111]andsecondlythatthemethodologyusedinthisproblemisingeneraldistinctfromthosefordiscussingtheotherSFLswhichhaveacertainunity.WewillhoweverhaveoccasionstorefertoaspectsofthequantumHalle ectproblemoften.EspeciallyinterestingfromourpointofviewistheweaklysingularFermiliquidbehaviorpredictedinthe =quantumHalle ect[118].

Withlessjustiÿcation,wedonotdiscusstheproblemofsuperconductortoinsulatorand=ortometaltransitionsintwo-dimensionaldisorderedsystemsinthelimitofzerotemperaturewithandwithoutanappliedmagneticÿeld.Interestingnewdevelopmentsinthisproblemwithreferencestosubstantialearlierworkmaybefoundin[176,177,247].TheproblemoftransitionsinJosephsonarrays[247]isavariantofsuchproblems.

OneoftheprincipalaspectsthatwewanttobringtotheforegroundinthisreviewisthefactthatSFLsallhaveincommonsomefundamentalfeatureswhichcanbestatedusefullyinseveraldi erentways.(i)TheyhavedegenerategroundstatestowithinanenergyoforderkBT.Thisdegeneracyisnotduetostaticexternalpotentialsorconstraintsasin,forexamplethespin-glassproblem,butdegeneracieswhicharedynamicallygenerated.(ii)Suchdegeneraciesinevitablyleadtoabreakdownofperturbativecalculationsbecausetheygenerateinfra-redsingularitiesinthecorrelationfunctions.(iii)Ifabareparticleorholeisaddedtothesystem,itisattendedbyadivergentnumberoflow-energyparticle–holepairs,sothattheone-to-onecorrespondencebetweentheone-particleexcitationoftheinteractingproblemandthoseofthenon-interactingproblem,whichisthebasisforFLT,breaksdown.(iv)SinceSFLsareconcernedwithdynamicallygenerateddegeneracieswithinenergiesoforderofthemeasuringtemperature,theobservedpropertiesaredeterminedbyquantum-mechanicaltoclassicalcrossoversandinparticularbydissipationinsuchacrossover.

Onthetheoreticalside,onemaynowviewFermi-liquidtheoryasaforerunneroftherenor-malizationgroupideas.Therenormalizationgrouphasledtoasophisticatedunderstandingofsingularitiesinthecollectivebehaviorofmany-particlesystems.ItislikelythatthesemethodshaveanimportantroletoplayinunderstandingthebreakdownofFLT.

TheaimofthispaperistoprovideapedagogicalintroductiontoSFLs,focusedontheessentialconceptualideasandonissueswhicharesettledandwhichcanbeexpectedto

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survivefuturedevelopments.Therefore,wewillnotattempttogiveanexhaustivereviewoftheliteratureonthisproblemorofalltheexperimentalsystemswhichshowhintsofSFLbehavior.Theexperimentalexampleswediscusshavebeenselectedtoillustratebothwhatisessentiallyunderstoodandwhatisnotunderstoodeveninprinciple.Onthetheoreticalside,wewillshyawayfrompresentingindepththesophisticatedmethodsnecessaryforadetailedevaluationofcorrelationfunctionsnearQCP—forthiswerefertothebookbySachdev[225]—orforanexactsolutionoflocalimpuritymodels(see,e.g.[124,227,259]).Likewise,foradiscussionoftheapplicationofquantumcriticalscalingideastoJosephsonarraysorquantumHalle ects,werefertotheniceintroductionbySondhietal.[247].1.2.Outlineofthepaper

Theoutlineofthispaperisasfollows.WestartbysummarizinginSection2someofthekeyfeaturesofLandau’sFLT—indoingso,wewillnotattempttoretracealloftheingredientswhichcanbefoundinmanyoftheclassictextbooks[208,37];insteadourdiscussionwillbefocusedonthoseelementsofthetheoryandtherelationwithitsmicroscopicderivationthatallowustounderstandthepossibleroutesbywhichtheFLTcanbreakdown.ThisisfollowedinSection3bytheFermi-liquidformulationoftheKondoproblemandoftheSFLvariantsoftheKondoproblemandoftwointeractingKondoimpurities.TheintentionhereistoreinforcetheconceptsofFLTinadi erentcontextaswellastoprovideexamplesofSFLbehaviorwhicho erimportantinsightsbecausetheyarebothsimpleandsolvable.Wethendiscusstheproblemofonespatialdimension(d=1),presentingtheprincipalfeaturesofthesolutionsobtained.Wediscusswhyd=1isspecial,andtheproblemsencounteredinextendingthemethodsandthephysicstod¿1.WethenmovefromthecomfortsofsolvablemodelstotherealityofthediscussionofpossiblemechanismsforSFLbehaviorinhigherdimensions.FirstweanalyzeinSection5theparadigmaticcaseoflong-rangeinteractions.Coulombinteractionswillnotdointhisregard,sincetheyarealwaysscreenedinametal,buttransverseelectromagneticÿeldsdogiverisetolong-rangeinteractions.ThefactthatasaresultnometalisaFermiliquidforsu cientlylowtemperatureswasalreadyrealizedlongago[127]—fromapracticalpointofview,thismechanismisnotveryrelevant,sincethetemperatureswherethesee ectsbecomeimportantareoforder10 16K;nevertheless,conceptuallythisisimportantsinceitisasimpleexampleofagaugetheorygivingrisetoSFLbehavior.Gaugetheoriesonlatticeshavebeenintroducedtodiscussproblemsoffermionsmovingwiththeconstraintofonlyzeroorsingleoccupationpersite.WethendiscussinSection6thepropertiesnearaquantumcriticalpoint,takingÿrstanexampleinwhichtheferromagnetictransitiontemperaturegoestozeroasafunctionofsomeexternallychosensuitableparameter.Wereferinthissectiontoseveralexperimentsinheavyfermioncompoundswhichareonlypartiallyunderstoodornotunderstoodeveninprinciple.WethenturntoadiscussionofthemarginalFermiliquidphenomenologyfortheSFLstateofcopper-oxidehigh-Tcmaterialsanddiscusstherequirementsonamicroscopictheorythatthephenemenologyimposes.Asketchofamicroscopicderivationofthephenemenologyisalsogiven.WeclosethepaperinSection8withadiscussionofthemetallicstateind=2andthestateofthetheorytreatingthedi usivesingularitiesind=2anditsrelationtothemetal–insulatortransition.

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C.M.Varmaetal./PhysicsReports361(2002)267–417273

ndau’sFermiliquid

2.1.EssentialsofLandauFermiliquids

ThebasicideaunderlyingLandau’sFermi-liquidtheory[152,153,208,37]isthatofanalyt-icity,i.e.,thatstateswiththesamesymmetrycanbeadiabaticallyconnected.Simplyput,thismeansthatwhetherornotwecanactuallycarryoutthecalculationweknowthattheeigen-statesofthefullHamiltonianofthesamesymmetrycanbeobtainedperturbativelyfromthoseofasimplerHamiltonian.Atthesametimestatesofdi erentsymmetrycannotbeobtainedby“continuation”fromthesamestate.Thissuggeststhatgivenatoughproblemwhichisim-possibletosolve,wemayguessarightsimpleproblem.Thelowenergyandlongwavelengthexcitations,aswellasthecorrelationandtheresponsefunctionsoftheimpossibleproblembearaone-to-onecorrespondencewiththesimplerproblemintheiranalyticproperties.Thisleavesÿxingonlynumericalvalues.Thesearetobedeterminedbyparameters,theminimumnumberofwhichisÿxedbythesymmetries.Experimentsoftenprovideanintuitionastowhattherightsimpleproblemmaybe:fortheinteractingelectrons,inthemetallicrangeofdensities,itistheproblemofkineticenergyofparticleswithFermistatistics.(Ifonehadstartedwiththeoppositelimit,justthepotentialenergyalone,thestartinggroundstateistheWignercrystal—abadplacetostartthinkingaboutametal!)Ifwestartwithnon-interactingfermions,andthenturnontheinteractions,thequalitativebehaviorofthesystemdoesnotchangeaslongasthesystemdoesnotgothrough(oriscloseto)aphasetransition.Owingtotheanalyticity,wecanevenconsiderstronglyinteractingsystems—thelow-energyexcitationsinthesehavestronglyrenormalizedvaluesoftheirparameterscomparedtothenon-interactingproblem,buttheirqualitativebehavioristhesameasthatofthesimplerproblem.

TheheavyfermionproblemprovidesanextremeexampleofthedomainofvalidityoftheLandauapproach.ThisisillustratedinFig.2,whichshowsthespeciÿcheatoftheheavyfermioncompoundCeAl3.AsintheSommerfeldmodel,thespeciÿcheatislinearinthetemperatureatlowT,butifwewriteCv≈ Tatlowtemperatures,thevalueof isaboutathousandtimesaslargeasonewouldestimatefromthedensityofstatesofatypicalmetal,usingthefreeelectronmass.ForaFermigas,thedensityofstatesN(0)attheFermienergyisproportionaltoane ectivemassm :

m kF

N(0)=;

(1)

withkFtheFermiwavenumber.Thenthefactthatthedensityofstatesatthechemicalpotentialisathousandtimeslargerthanfornormalmetalscanbeexpressedbythestatementthatthee ectivemassm ofthequasiparticlesisathousandtimeslargerthanthefreeelectronmassm.Likewise,asFig.3shows,theresistivityofCeAl3atlowtemperaturesincreasesasT2.ThisalsoisacharacteristicsignofaFermiliquid,inwhichthequasiparticlelifetime attheFermisurface,determinedbyelectron–electroninteractions,behavesas ～1=T2.2However,justas

Inheavyfermions,atleastintheobservedrangeoftemperatures,thetransportlifetimedeterminingthetemper-aturedependenceofresistivityisproportionaltothesingle-particlelifetime.

2

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274C.M.Varmaetal./PhysicsReports361(2002)

267–417

Fig.2.SpeciÿcheatofCeAl3atlowtemperaturesfromAndresetal.[28].Theslopeofthelinearspeciÿcheatisabout3000timesthatofthelinearspeciÿcheatof,say,Cu.However,thehigh-temperaturecut-o ofthislineartermissmallerthanthatofCubyasimilaramount.Theriseofthespeciÿcheatinamagneticÿeldatlowtemperaturesisthenuclearcontribution,irrelevanttoour

discussion.

Fig.3.ElectricalresistivityofCeAl3below100mK,plottedagainstT2.FromAndresetal.[28].

theprefactor ofthespeciÿcheatisafactorthousandtimeslargerthanusual,theprefactoroftheT2termintheresistivityisafactor106larger—while scaleslinearlywiththee ectivemassratiom =m,theprefactoroftheT2termintheresistivityincreasesforthisclassofFermiliquidsas(m =m)2.

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Itshouldberemarkedthattherightsimpleproblemisnotalwayseasytoguess.Therightsimpleproblemforliquid4Heisnotthenon-interactingBosegasbuttheweaklyinteractingBosegas(i.e.,theBogoliubovproblem[45,154]).TherightsimpleproblemfortheKondoproblem(alow-temperaturelocalFermiliquid)wasguessed[197]onlyafterthenumericalrenormalizationgroupsolutionwasobtainedbyWilson[289].Therightsimpleproblemfortwo-dimensionalinteractingdisorderedelectronsinthe“metallic”rangeofdensities(Section8inthispaper)isatpresentunknown.

ForSFLs,theproblemisdi erent:usuallyoneisinaregimeofparameterswherenosimpleproblemisastartingpoint—insomecasesthe uctuationsbetweensolutionstodi erentsimpleproblemsdeterminesthephysicalproperties,ndauFermiliquidandthewavefunctionrenormalizationZ

Landautheoryistheforerunnerofourmodernwayofthinkingaboutlow-energye ectiveHamiltoniansincomplicatedproblemsandoftherenormalizationgroup.TheformalstatementsofLandautheoryintheiroriginalformareoftensomewhatcrypticandmysterious—thisre ectsbothLandau’sstyleandhisingenuity.Weshalltakeamorepedestrianapproach.

Letusconsidertheessentialdi erencebetweennon-interactingfermionsandaninteractingFermiliquidfromasimplemicroscopicperspective.Forfreefermions,themomentumstates|k arealsoeigenstatesoftheHamiltonianwitheigenvalue

k=

2k2

:(2)

Moreover,thethermaldistributionofparticlesn0k ,isgivenbytheFermi–Diracfunctionwhere denotesthespinlabel.AtT=0,thedistributionjumpsfrom1(allstatesoccupiedwithintheFermisphere)tozero(nostatesoccupiedwithintheFermisphere)at|k|=kFandenergyequaltothechemicalpotential .ThisisillustratedinFig.4.

Agoodwayofprobingasystemistoinvestigatethespectralfunction;thespectralfunctionA(k;!)givesthedistributionofenergies!inthesystemwhenaparticlewithmomentumkisaddedorremovedfromit(rememberthatremovingaparticleexcitationbelowtheFermienergymeansthatweaddaholeexcitation).AssketchedinFig.5(a),forthenon-interactingsystem,A0(k;!)issimplya -functionpeakattheenergy k,becauseallmomentumstatesarealsoenergyeigenstates

A0(k;!)= (! ( k ))

for!¿ ;

(3)(4)

111= Im= ImG0(k;!):

kHere, issmallandpositive;itre ectsthatparticlesorholesareintroducedadiabatically,and

itistakentozeroattheendofthecalculationforthepurenon-interactingproblem.Theÿrststepofthesecondlineisjustasimplemathematicalrewritingofthedeltafunction.InthesecondlinetheGreen’sfunctionG0fornon-interactingelectronsisintroduced.Moregenerally

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Fig.4.Bare-particledistributionatT=0foragivenspindirectioninatranslationallyinvariantFermisystemwithinteractions(fullline)andwithoutinteractions(dashedline).Notethatthepositionofthediscontinuity,i.e.,theFermiwavenumberkF,isnotrenormalizedby

interactions.

Fig.5.(a).Thenon-interactingspectralfunctionA(k;!)atÿxedkasafunctionof!;(b)thespectralfunctionofsingle-electronexcitationsinaFermiliquidatÿxedkasafunctionof!.If(1= )A(k;!)isnormalizedto1,signifyingonebareparticle,theweightundertheLorentzian,i.e.,thequasiparticlepart,isZ.Asexplainedinthetext,atthesametimeZisthediscontinuityinFig.4.

thesingle-particleGreen’sfunctionG(k;!)isdeÿnedintermsofthecorrelationfunctionofparticlecreationandannihilationoperatorsinstandardtextbooks[195,4,222,168].Forourpresentpurpose,itissu cienttonotethatitisrelatedtothespectralfunctionA(k;!),whichhasaclearphysicalmeaningandwhichcanbededucedthrough-angleresolvedphotoemissionexperiments

∞

A(k;x)

G(k;!)=dx:(5)

∞A(k;!)thusisthespectralrepresentationofthecomplexfunctionG(k;!).Herewehavedeÿned

theso-calledretardedGreen’sfunctionwhichisespeciallyusefulsinceitsrealandimaginary

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C.M.Varmaetal./PhysicsReports361(2002)267–417

277

Fig.6.Schematicillustrationoftheperturbativeexpansion(8)ofthechangeofwavefunctionasaresultoftheadditionofanelectrontotheFermiseaduetointeractionswiththeparticlesintheFermisea.

partsobeytheKramers–Kronigrelations.IntheproblemwithinteractionsG(k;!)willdif-ferfromG0(k;!).Thisdi erencecanbequitegenerallydeÿnedthroughthesingle-particleself-energyfunction (k;!):

(G(k;!)) 1=(G0(k;!)) 1 (k;!):

Eq.(5)ensurestherelationbetweenG(k;!)andA(k;!)

1

A(k;!)= ImG(k;!):

(7)(6)

Withthesepreliminariesoutoftheway,letusconsidertheformofA(k;!)whenweaddaparticletoaninteractingsystemoffermions.

Duetotheinteraction(assumedrepulsive)betweentheaddedparticleandthosealreadyintheFermisea,theaddedparticlewillkickparticlesfrombelowtheFermisurfacetoabove.Thepossibletermsinaperturbativedescriptionofthisprocessareconstrainedbytheconser-vationlawsofcharge,particlenumber,momentumandspin.ThosewhichareallowedbytheseconservationlawsareindicatedpictoriallyinFig.6,andleadtoanexpressionofthetype

1=2 N+1N

| k =Zkck |

+

1

k1;k2;k3 1; 2; 3

k1 1k2 2k3 3ckcck2k13

× k;k1 k2+k3 ( ; 1; 2; 3)| N +::::

(8)

’sandck’sarethebareparticlecreationandannihilationoperators,andthedotsindi-Heretheck

catehigher-orderterms,forwhichtwoormoreparticle–holepairsarecreatedand ( ; 1; 2; 3)expressesconservationofspinundervectoraddition.Themultiple-particle–holepairsforaÿxedtotalmomentumcanbecreatedwithacontinuumofmomentumsoftheindividualbareparticlesandholes.Therefore,anaddedparticlewithÿxedtotalmomentumhasawidedistributionofenergies.However,ifZkdeÿnedbyEq.(8)isÿnite,thereisawell-deÿnedfeatureinthisdis-tributionatsomeenergywhichisingeneraldi erentfromthenon-interactingvalue 2k2=(2m).ThespectralfunctioninsuchacasewillthenbeasillustratedinFig.5(b).Itisusefultoseparatethewell-deÿnedfeaturefromthebroadcontinuumbywritingthespectralfunction

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278C.M.Varmaetal./PhysicsReports361(2002)267–417

asthesumoftwoterms,A(k;!)=Acoh(k;!)+Aincoh(k;!).Thesingle-particleGreen’sfunc-tioncansimilarlybeexpressedasasumoftwocorrespondingterms,G(k;!)=Gcoh(k;!)+Gincoh(k;!).Then

Zk

Gcoh(k;!)=;(9)

kkwhichforlargelifetimes kgivesaLorentzianpeakinthespectraldensityatthequasiparticle

energy k≡ k .TheincoherentGreen’sfunctionissmoothandhenceforlarge kcorrespondstothesmoothbackgroundinthespectraldensity.

Theconditionfortheoccurrenceofthewell-deÿnedfeaturecanbeexpressedastheconditionthattheself-energy (k;!)hasananalyticexpansionabout!=0andk=kFandthatitsrealpartismuchlargerthanitsimaginarypart.Onecaneasilyseethatwereitnotso,thenexpression(9)forGcohcouldnotbeobtained.TheseconditionsarenecessaryforaLandauFermiliquid.Uponexpanding (k;!)in(12)forsmall!andsmalldeviationsofkfromkFandwritingitintheform(9),wemaketheidentiÿcations

k;1= ZkIm (kF;!=0); k= kZkZ(10)

k

where

19 9 1

=1+;Z:(11)Zk=1

!=0;k=kFF!=0;k=kFFromEq.(8),wehaveamorephysicaldeÿnitionofZk:Zkistheprojectionamplitudeof

N+1| k ontothestatewithonebareparticleaddedtothegroundstate,sinceallothertermsintheexpansionvanishinthethermodynamiclimitintheperturbativeexpressionembodiedby(8):

1=2N+1 NZk= k|ck| :

(12)

Inotherwords,ZkistheoverlapofthegroundstatewavefunctionofasystemofinteractingN±1fermionsoftotalmomentumkwiththewavefunctionofNinteractingparticlesandabareparticleofmomentumk.Zkiscalledthequasiparticleamplitude.

TheLandautheorytacitlyassumesthatZkisÿnite.Furthermore,itassertsthatforsmall!andkclosetokF,thephysicalpropertiescanbecalculatedfromquasiparticleswhichcarrythesamequantumnumbersastheparticles,i.e.,charge,spinandmomentumandwhichmaybedeÿnedsimplybythecreationoperator k; :

N+1N

| k = k; | :

(13)

ClosetokF,andforTsmallcomparedtotheFermienergy,thedistributionofthequasiparticlesisassumedtobetheFermi–Diracdistributionintermsoftherenormalizedquasiparticleen-ergies.Thebareparticledistributionisquitedi erent.AsisillustratedinFig.4,itisdepletedbelowkFandaugmentedabovekF,withadiscontinuityatT=0whosevalueisshowninmicroscopictheorytobeZk.AcentralresultofFermiliquidtheoryisthatclosetotheFermienergyatzerotemperature,thewidth1= kofthecoherentquasiparticlepeakisproportionalto( k )2sothatneartheFermienergythelifetimeislongandquasiparticlesarewell-deÿned.

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Likewise,attheFermienergy1= kvarieswithtemperatureasT2.Fromthemicroscopicderiva-tionofthisresult,itfollowsthattheweightinthispeak,Zk,becomesequaltothejumpZinnk whenweapproachtheFermisurface:Zk→Zfork→kF.Forheavyfermions,aswealreadymentioned,Zcanbeoftheorderof10 3.However,aslongasZisnon-zero,onehasFermiliquidpropertiesfortemperatureslowerthanaboutZEF.Degeneracyise ectivelylostfortemperaturesmuchhigherthanZEFandclassicalstatisticalmechanicsprevails.3

Anadditionalresultfrommicroscopictheoryistheso-calledLuttingertheorem,whichstatesthatthevolumeenclosedbytheFermisurfacedoesnotchangeduetointeractions[195,4].ThemathematicsbehindthistheoremisthatwiththeassumptionsofFLT,thenumberofpolesintheinteractingGreen’sfunctionbelowthechemicalpotentialisthesameasthatforthenon-interactingGreen’sfunction.Recallthatthelatterisjustthenumberofparticlesinthesystem.

LandauactuallystartedhisdiscussionoftheFermiliquidbywritingtheequationforthedeviationofthe(Gibbs)freeenergyfromitsgroundstatevalueasafunctionalofthedevia-tionofthequasiparticledistributionfunctionn(k; )fromtheequilibriumdistributionfunctionn0(k; )

n(k; )=n(k; ) n0(k; )asfollows:

1 1

( k ) nk +f nk nk +···G=G0+ kk;

k;

kk;

(14)

(15)

Notethat( k )isitselfafunctionof n;sotheÿrsttermcontainsatleastacontributionof2order( n)whichmakesthesecondtermquitenecessary.Inprinciple,theunknownfunction

fkk ; dependsonspinandmomenta.However,spinrotationinvarianceallowsonetowritethespinpartintermsoftwoquantities,thesymmetricandantisymmetricpartsfsandfa.Moreover,forlowenergyandlong-wavelengthphenomenaonlymomentawithk≈kFplayarole;ifweconsiderthesimplecaseof3HewheretheFermisurfaceisspherical,rotationinvarianceimpliesthatformomentaneartheFermimomentumfcanonlydependontherelativeanglebetweenkandk ;thisallowsonetoexpandinLegendrepolynomialsPl(x)bywriting

k≈k ≈kFs;a

N(0)fkk ; →

∞ l=0

): ·kFls;aPl(k

(16)

Fromexpression(15)onecanthenrelatethelowestorderso-calledLandaucoe cientsF0

sandthee ectivemassm tothermodynamicquantitieslikethespeciÿcheatC,theandF1v

compressibilityÄ,andthesusceptibility :

m Cv=;v03

Äsm=(1+F0);

0 am=(1+F0):

0(17)

ItisanunfortunatecommonmistaketothinkofthepropertiesinthisregimeasSFLbehavior.

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Heresubscripts0refertothequantitiesofthenon-interactingreferencesystem,andmisthemassofthefermions.ForaGalileaninvariantsystem(like3He),thereisasimplerelation

s,andthereisnorenormalizationbetweenthemassenhancementandtheLandauparameterF1

oftheparticlecurrentj;however,thereisarenormalizationofthevelocity:onehas

s Fm

j=k=m;v=k=m ;(18)=1+:

Thetransportpropertiesarecalculatedbydeÿningadistributionfunction n(k ;r;t)whichis

slowlyvaryinginspaceandtimeandwritingaBoltzmannequationforit[208,37].

ItisadelightfulconceitoftheLandautheorythattheexpressionsofthelow-energyproper-tiesintermsofthequasiparticlesinnoplaceinvolvethequasiparticleamplitudeZk.Infactinatranslationallyinvariantproblemsuchasliquid3He;Zkcannotbemeasuredbyanythermo-dynamicortransportmeasurements.AmasterlyuseofconservationlawsensuresthatZ’scanceloutinallphysicalproperties(onecanextractZfrommeasurementofthemomentumdistri-bution.Byneutronscatteringmeasurements,itisfoundthatZ≈1=4[112]forHe3nearthemeltingline).Thisisnolongertrueonalattice,intheelectron–phononinteractionproblem[212]orinheavyfermions[265]orevenmoregenerallyinanysituationwheretheinter-actingproblemcontainsmorethanonetypeofparticlewithdi erentcharacteristicfrequencyscales.

2.3.UnderstandingmicroscopicallywhyFermi-liquidtheoryworks

LetustrytounderstandfromamoremicroscopicapproachwhytheLandautheoryworkssowell.Wepresentaqualitativediscussioninthissubsectionandoutlinetheprincipalfeaturesoftheformalderivationinthenextsubsection.

Aswealreadyremarked,acrucialelementintheapproachistochoosethepropernon-interactingreferencesystem.ThatthisispossibleatallisduetothefactthatthenumberofstatestowhichanaddedparticlecanscatterduetointeractionsisseverelylimitedduetothePauliprinciple.Asaresult,non-interactingfermionsareagoodstablesystemtoperturbabout;theyhaveaÿnitecompressibilityandsusceptibilityinthegroundstate,andsocollectivemodesandthermodynamicquantitieschangesmoothlywhentheinteractionsareturnedon.Thisisnottruefornon-interactingbosonswhichdonotsupportcollectivemodeslikesoundwaves.Soonecannotperturbaboutthenon-interactingbosonsasareferencesystem.

LandaualsolaidthefoundationsfortheformaljustiÿcationofFermiliquidtheoryintwoandthreedimensions.The urryofactivityinthisÿeldfollowingthediscoveryofhigh-TcphenomenahasledtonewwaysofjustifyingFermi-liquidtheory(andunderstandingwhytheone-dimensionalproblemisdi erent).However,theprincipalphysicalreason,whichwenowdiscuss,remainsthephase-spacerestrictionsduetokinematicalconstraints.

WelearnedinSection2.2thatinordertodeÿnequasiparticles,itwasnecessarytohaveaÿniteZkF,whichinturnneededaself-energyfunction (kF;!)whichissmoothnearthechemicalpotential,i.e.,at!=0.LetusÿrstseewhyaFermigashassuchpropertieswheninteractionsareintroducedperturbatively.

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Fig.7.Thethreesecond-orderprocessesinaperturbativecalculationofthecorrectiontothebareinteractioninaFermiliquid.

Wewillexplicitlyconsideronlyshort-rangeinteractionsinthissection,sothattheycanbecharacterizedatallmomentumtransfersbyasingleparameter.Nevertheless,theessentialresultsofLandautheoryremainvalidinthepresenceofCoulombinteractionsbecausescreeningmakestheinteractionsessentiallyshort-ranged.Thecouplingconstantgbelowmaythenbeconsideredtoparametrizethescreenedinteraction.

InFig.7,weshowthethreepossibleprocessesthatariseinsecond-orderperturbationtheoryforthescatteringoftwoparticleswithÿxedinitialenergy!andmomentumq.Notethatintwoofthediagrams,Fig.7(a)and(b)theintermediatestatehasaparticleandaholewhiletheintermediatestateindiagram7(c)hasapairofparticles.

Wewillÿndthat,forourpresentpurpose,thecontributionofdiagram7(a)ismoreimportantthantheothertwo.Itgivesacontribution

g

2

k

fk+q fk

:

k+qk(19)

Here,gisameasureofthestrengthofthescatteringpotential(thevertexinthediagram)inthelimitofsmallq.Thedenominatorensuresthatthelargestcontributiontothescatteringcomesfromsmallscatteringmomentaq:forthesetheenergydi erenceislinearinq;Ek+q Ek≈q·vk,wherevkisavectoroflengthvFinthedirectionofk.Moreover,theterminthenumeratorisnon-zeroonlyintheareacontainedbetweentwocircles(ford=2)orspheres(ford=3)withtheircentersdisplacedbyq—herethephase-spacerestrictionisduetothePauliprinciple.Thisareaisalsoproportionaltoq·vk,andsointhesmallqapproximationfromdiagram7(a)wegetatermproportionalto

g2

q·vkdf

:

kk

(20)

Nowweseewhydiagram7(a)isspecial.Thereisasingularityat!=q·vkanditsvalueforsmall!andqdependsonwhichofthetwoissmaller.Thissingularityisresponsibleforthelow-energylong-wavelengthcollectivemodesoftheFermiliquidinLandautheory.Atlowtemperatures,df=d k= ( k ),sothesummationisrestrictedtotheFermisurface.Therealpartof(19)thereforevanishesinthelimitqvF=!→0,whileitapproachesaÿnitelimit

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Fig.8.(a)Restrictiononallowedparticle–holeexcitationsinaFermiseaduetokinematics.Theplasmonmodehasbeendrawnforthecased=3;(b)theabsorptivepartoftheparticle–holesusceptibility(inthecharge,currentandspinchannels)for!¡qvFintheFermigas.

for!→0.Theimaginarypartinthislimitisproportionalto4!:

!

Im (q;!)=g2 N(0)for!¡qvF;

F

(21)

whileIm (q;!)=0for!¿vFq.ThisbehaviorissketchedinFig.8(b).Anexplicitevaluationfortherealpartyields

!! qvF 2 ;Re (q;!)=gN(0)1+ln (22)FF whichgivesaconstant(leadingtoaÿnitecompressibilityandspinsusceptibility)at!small

comparedtoqvF.Fordiagram7(b),wegetaterm! (Ep1 p2+k+q Ek)inthedenominator.Thistermisalwaysÿniteforgeneralmomentap1andp2,andhencethecontributionfromthisdiagramcanalwaysbeneglectedrelativetotheonefrom7(a).Alongsimilarlines,oneÿndsthatdiagram7(c),whichdescribesscatteringintheparticle–particlechannel,isirrelevantexceptwhenp1= p2,whenitdivergesasln!.

Ofcourse,ndaunoticedthissingularitybutignoreditsimplication.5Indeed,aslongasthee ectiveinteractionsdonotfavorsuperconductivityoraslongasweareattemperaturesmuchhigherthanthesupercon-ductingtransitiontemperature,itisnotimportantforFermi-liquidtheory.

Letusnowlookfurtherattheabsorptivespectrumofparticle–holeexcitationsintwoandthreedimensions,i.e.,weexaminetheimaginarypartofEq.(19).Whenthetotalenergy!ofthepairissmall,boththeparticleandtheholehavetoliveclosetotheFermisurface.In

Thisbehaviorimpliesthatthisscatteringcontributionisamarginaltermintherenormalizationgroupsense,whichmeansthatita ectsthenumericalfactors,butnotthequalitativebehavior.5

Attractiveinteractionsinanyangularmomentumchannel(leadingtosuperconductivity)arethereforemarginallyrelevantoperators.

4

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Fig.9.Thesingle-particleself-energydiagraminsecondorder.

thislimit,wecanmakeanyexcitationwithmomentumq62kF.Forÿxedbutsmallvaluesofq,themaximumexcitationenergyis!≈qvF;thisoccurswhenqisinthesamedirectionasthemainmomentumkofeachquasiparticle.Forqnear2kF,themaximumpossibleenergyis!=vF|q 2kF|.Combiningtheseresults,weobtainthesketchinFig.8(a),inwhichtheshadedareainthe!–qspaceistheregionofallowedparticle–holeexcitations.6Fromthisspectrum,onecancalculatethepolarizability,orthemagneticsusceptibility.

Thebehaviorsketchedaboveisvalidgenerallyintwoandthreedimensions(butaswewillseeinSection4,notinonedimension).Theimportantpointtorememberisthatthedensityofparticle–holeexcitationsdecreaseslinearlywith!for!smallcomparedtoqvF.WeshallseelaterthatonewayofundoingFermi-liquidtheoryistohave!～k2intwodimensionsor!～k3inthreedimensions.

WecannowuseIm (q;!)tocalculatethesingle-particleself-energytosecondorderintheinteractions.ThisisshowninFig.9wherethewigglylinedenotes (q; )whichinthepresentapproximationisjustgivenbythediagramofFig.7(a).

Fortheperturbativeevaluationofthisprocess,theintermediateparticlewithenergy–momentum(!+ );(k+q)isafreeparticle.Second-orderperturbationtheorythenyieldsanimaginarypart,oradecayrate,

2

1!

Im (k;!)=(23)=g2N(0)

Finthreedimensionsfork≈kF.Intwodimensions,thesameprocessyieldsIm (kF;!)～

!2ln(EF=!).

The!2decayrateisintimatelyrelatedtotheanalyticresult(22)forIm (q;!)exhibitedinFig.(8).Asmaybefoundintextbooks,thesamecalculationforelectron–phononinteractionsorforinteractionwithspinwavesinanantiferromagneticmetalgivesIm (kF;!)～(!=!c)3,where!cisthephononDebyefrequencyintheformerandthecharacteristiczone-boundaryspin-wavefrequencyinthelatter.

Therealpartoftheself-energymaybeobtaineddirectlyorbyKramers–Kronigtransforma-tionof(23).Itisproportionalto!.Therefore,ifthequasiparticleamplitudeZkFisevaluated

Inthepresenceoflong-rangeCoulombinteractions,inadditiontotheparticle–holeexcitationspectrumassociatedwiththescreened(andhencee ectivelyshort-ranged)interactionsonegetsacollectivemodewithaÿniteplasma

√

frequencyasq→0ind=3anda!～behaviorind=2.Theplasmamodeisahigh-frequencymodeinwhichthemotionofthelightelectronscannotbefollowedbytheheavyions:screeningisabsentinthisregimeandthelong-rangeCoulombinteractionsthengiverisetoaÿniteplasmafrequencyind=3.

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Fig.10.Single-particleenergy kinonedimension,intheapproximationthatthedispersionrelationislinearizedaboutkF.NotethattheFermisurfaceconsistsofjusttwopoints.Thespectrumofparticle–holeexcitationsisgivenby!(q)= (k+q) (k)=kFq=m.Low-energyparticle–holeexcitationsareonlypossibleforqsmallorforqnear2kF.

Fig.11.Phasespaceforparticle–holeexcitationspectruminonedimensioncomparedwiththesameinhigherdimensions,Fig.8.Forlinearizedsingle-particlekineticenergy k=±vF(k kF),particle–holeexcitationsareonlypossibleonlinesgoingthroughk=0andk=2kF.

perturbatively7

ZkF≈1 2g2N(0)=EF:

(24)

Thusinaperturbativecalculationofthee ectofinteractionsthebasicanalyticstructureoftheGreen’sfunctionisleftthesameasfornon-interactingfermions.ThegeneralproofofthevalidityofLandautheoryconsistsinshowingthatwhatwehaveobtainedtosecondorderingremainsvalidtoallordersing.Theoriginalproofs[4]areself-consistencyarguments—wewillconsiderthembrie yinSection2.4.TheyassumeaÿniteZintheexactsingle-particleGreen’sfunctionsande ectivelyshowthattoanyorderinperturbationtheory,thepolarizabil-ityfunctionsretaintheanalyticstructureofthenon-interactingtheory,whichinturnensuresaÿniteZ.

Inonedimension,phase-spacerestrictionsonthepossibleexcitationsarecruciallydi erent.8HeretheFermisurfaceconsistsofjusttwopointsintheone-dimensionalspaceofmomenta—seeFig.10.Asaresult,whereasind=2and3acontinuumoflow-energyexcitationswithÿniteqispossible,inonedimensionatlow-energyonlyexcitationswithsmallkork≈2kFarepossible.ThesubsequentequivalentofFig.8fortheone-dimensionalcaseistheoneshowninFig.11.Uponintegratingoverthemomentumkwithacut-o ofO(kF)thecontributionfrom

ThisquantityhasbeenpreciselyevaluatedbyGalitski[104]forthemodelofadiluteFermigascharacterizedbyascatteringlength.8

Itmightappearsurprisingthattheyarenotdi erentinanyessentialwaybetweenhigherdimensions.

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Fig.12.(a)ThenestedFermisurfaceobtainedinatightbindingmodelonasquarelatticewithnearest-neighborhopping;(b)apartiallynestedFermisurfacewhichleadstocharge-densitywaveorantiferromagneticinstabilities.

thisparticle–holescatteringchanneltoRe (q;!)is

kF

1

dk～ln[(!+qvF)=EF]:F0

(25)

(Notethat(25)istrueforbothq kFand|q 2kF| kF.)Thisinturnleadstoasingle-particleself-energycalculatedbytheprocessinFig.9tobeRe (kF;!)～!ln!andso

Z～ln!givingahintoftrouble.TheCooper(particle–particle)channelhasthesamephase-spacerestrictions,andgivesacontributiontoRe (kF;!)proportionalto!ln!too.Thefactthatthesesingularcontributionsareofthesameorder,leadstoacompetitionbetweencharge=spin uctuationsandCooperpairing uctuations,andintheexactcalculationtopower-lawsingularities.Thefactthatinsteadofthecontinuumoflow-energyexcitationspresentinhigherdimensions,thewidthofthebandofallowedparticle–holeexcitationsvanishesas!→0,isthereasonthatthepropertiesofone-dimensionalinteractingmetalscanbeunderstoodintermsofbosonicmodes.Wewillpresentabriefsummaryoftheresultsforthesingle-particleGreen’sfunctionandcorrelationfunctionsinSection4.9.

Inspecialcasesofnestingintwoorthreedimensions,onecanhavesituationsthatresembletheone-dimensionalcase.Whenthenon-interactingFermisurfaceinatightbindingmodelhasthesquareshapesketchedinFig.12(a)(whichoccursforatight-bindingmodelwiththenearestneighborhoppingonasquarelatticeathalf-ÿlling)acontinuousrangeofmomentaonoppositesidesoftheFermisurfacecanbetransformedintoeachotherbyoneandthesamewavenumber.Thisso-callednestingleadstologandlog2singularitiesforacontinuousrangeofkintheperturbationtheoryfortheself-energy (k;!).Likewise,thepartiallynestedFermisurfaceofFig.12(b)leadstochargedensitywaveandantiferromagneticinstabilities.WewillcomebacktotheseissuesinSections2.6and6.

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2.4.PrinciplesofthemicroscopicderivationofLandautheory

Inthissection,wewillsketchhowtheconclusionsintheprevioussectionbasedonsecond-orderperturbationcalculationaregeneralizedtoallordersinperturbationtheory.Thissectionisslightlymoretechnicalthantherest;thereadermaychoosetoskiptoSection2.6.

WefollowthemicroscopicapproachwhosefoundationswerelaidbyLandauhimselfandwhichisdiscussedindetailinexcellenttextbooks[197,208,37,4].Formorerecentmethodswiththesameconclusions,see[237,128].OuremphasiswillbeonhighlightingtheassumptionsinthetheorysothatinthenextsectionwecansummarizetheroutesbywhichtheFermi-liquidtheorymaybreakdown.Theseassumptionsareusuallynotstatedexplicitly.

Thebasicideaisthatduetokinematicconstraints,anyperturbativeprocesswithnparticle–holepairsintheintermediatestateprovidescontributionstothepolarizabilityproportionalto(!=EF)n.Therefore,thelow-energypropertiescanbecalculatedwithprocesseswiththesame“skeletal”structureasthoseinFig.7,whichhaveonlyoneparticle–holepairintheintermediatestate.Soonemayconcentrateonthemodiÿcationofthefour-leggedverticesandthesingle-particlepropagatorsduetointeractionstoallorders.Accordingly,thetheoryisformulatedintermsofthesingle-particleGreen’sfunctionG(p)andthetwo-bodyscatteringvertex

(p1;p2;p1+k;p2 k)= (p1;p2;k):

(26)

Hereandbelowweuse,forthesakeofbrevity,p,etc.todenotetheenergy–momentumfourvector(p;!)andwesuppressthespinlabels.Theequationfor isexpandedinoneofthetwoparticle–holechannelsas9

d4q(1)(1)

(p1;p2;k)= (p1;p2;k) i (p1;q;k)G(q)G(q+k) (q;p2;k);(27)

where (1)istheirreduciblepartintheparticle–holechannelinwhichEq.(27)isexpressed.Inotherwords, (1)cannotbesplitintotwopartsbycuttingtwoGreen’sfunctionlineswithtotalmomentumk.So (1)includesthecompletevertexintheother(oftencalledcross-)particle–holechannel.ThediagrammaticrepresentationofEq.(27)isshowninFig.13.Inthesimplestapproximation (1)ndautheoryassumesthat (1)hasnosingularities.10AnassumptionisnowfurthermadethatG(p)doeshaveacoherentquasiparticlepartat|p| pFand! 0:

G(p)=

Z

+Ginc;

pp(28)

Tosecondorderintheinteractionsthecorrectiontothevertexinthetwopossibleparticle–holechannelshas

beenexhibitedintheÿrsttwopartsofFig.7.10

ThetheoryhasbeengeneralizedforCoulombinteractions[208,197,4].Thegeneralresultsremainunchangedbecauseascreenedshort-rangeinteractiontakestheplaceof (1).Thisisunlikelytobetrueinthecriticalregionofametal–insulatortransition,becauseontheinsulatingside,theCoulombinteractionisunscreened.

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Fig.13.DiagrammaticrepresentationofEq.(27).

where pistobeidentiÿedastheexcitationenergyofthequasiparticle,Zitsweight,andGinctheincoherentnon-singularpartofG.(ThelatterprovidesthesmoothbackgroundpartofthespectralfunctioninFig.5(b)andtheformerthesharppeak,whichisproportionaltothe functionfor p= p .)Itfollows[195,4]from(28)that

2i z2vq·k

G(q)G(q+k)= ( ) (|q| pF)+ (q)

Fq(29)

forsmallkand!,andwhere and( +!)arefrequenciesofthetwoGreen’sfunctions.NotethecrucialroleofkinematicsintheformoftheÿrsttermwhichcomesfromtheproductofthequasiparticlepartsofG; (q)comesfromthescatteringoftheincoherentpartwithitselfandwiththecoherentpartandisassumedsmoothandfeatureless(asitisindeed,giventhatGincissmoothandfeaturelessandthescatteringdoesnotproduceaninfraredsingularityatleastperturbativelyintheinteraction).Thevertex inregionsclosetok≈kFand!≈0isthereforedominatedbytheÿrstterm.ThederivationofFermi-liquidtheoryconsistsinprovingthatEqs.(27)forthevertexand(28)fortheGreen’sfunctionaremutuallyconsistent.Theproofproceedsbydeÿningaquantity !(p1;p2;k)through

d4q!(1)(1)!

(p1;p2;k)= (p1;p2;k) i (p1;q;k) (q) (q;p2;k):(30)

!containsrepeatedscatteringoftheincoherentpartoftheparticle–holepairsamongitselfandwiththecoherentpart,butnoscatteringofthecoherentpartwithitself.Then,providedtheirreduciblepartof (1)issmoothandnottoolarge, !issmoothinkbecause (q)isbyconstructionquitesmooth.

Usingthefactthattheÿrstpartof(29)vanishesforvF|k|=!→0,andcomparing(27)and(30)onecanwritetheforwardscatteringamplitude

limlim (p1;p2;k)= !(p1;p2):(31)

!→0k→0

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