TRANSPORTANOMALIESOFTHESTRANGEMETAL:RESOLUTIONBY HIDDENFERMILIQUIDTHEORY PhilipWAnderson,PhilipACasey,PhysicsDept,PrincetonUniversity ABSTRACT Thestrangemetalphaseofoptimally andover dopedcuprates exhibitsanumberofanomaloustransportproperties unsaturatinglinear Tresistivity,distinctrelaxationtimesforHall angleandresistivity,temperature dependentanisotropic relaxationtimes,andacharacteristiccrossoverfromsupposed FermiLiquidtolinear Tbehavior.Allreceivenaturalexplanations andquantitativefitsintermsofthehiddenfermiliquidtheory. INTRODUCTION Fromtheveryfirstobservationsofthepropertiesofthecuprate high Tc superconductorsitwasseenthatthepropertiesofthe normal metalabovetcwereunusual.thereareactuallytwounusualregimes:at lowerdoping,theredevelopsa pseudogap regimewhichismost plausiblydescribed i, ii asastatewithbcspairingbutwithout superconductingorder.(atstilllowerdopingsvariouscomplexphases withinhomogeneitiesand/oralternativeorderingsshowupalso,butwe willconsideronlyhomogeneous,translationallysymmetricphases.) Nearoptimalandabove(andalsoforTabovethepseudogapregime) thereisnoevidenceofpairinginthenormalstatebutinsteada characteristic strange metallicbehaviorextendingtoveryhigh temperaturesandenergies.mostobviousisthenotorious lineart resistivity,sometimesextrapolatingatt=0to0orlessandpersistingin itslinearityoftentowellabovethemottlimit.acleanercharacterization, ifavailable,isthe Drude like tailofthemid infraredconductivity.this fallsoffasanon integerpoweroffrequencyconsiderablylessthanthe ω 2 ofthedrudetheory. iii Veryearlyaheuristicforthe strange behaviorwasdevelopedasthe marginalfermiliquid theory iv andthat isoftenusedasadescriptiveterm,butthisheuristicdoesnotdescribe theinfraredresultcorrectly,noranyofthefurtherregularities. ThisauthoracknowledgessupportfromanNSERCPGS DFellowship.
Oftenathirdregimeispostulated,thatthestatereturnstothesimple Fermiliquidwhenoverdopedbeyondthesuperconductingdome,andfor lowert,acrossoverlinebeingdrawnupandtotheright,startingatthe edgeofthedome.wewillseethatthisismerelyacrossoverinthe transportproperties,andthatfundamentalpropertiesliketheoneparticlegreen sfunctionremainanomalousaccordingtoourtheory.no transitiontoatruefermiliquidhasbeenobserved,inouropinion. AstrikinganomalyofthestrangephaseistheT dependenthalleffect.it isbestdescribedasthereexistingarelaxationrateforthelarmor precession1/τhdistinctfromthatfortheresistivityandmoreresembling thatforafermiliquid. v Thepurposeoftherestofthispaperistoshowhowallofthese anomaliesfollowfromthetheoryofthesimplestpossiblemodel,the HubbardmodelwithastronginteractionUandnothingelse. THEHIDDENFERMILIQUID ThehiddenFermiliquid(HFL,hereafter)theory vi dependsonthe assumptionthatthehubbardon siteinteractionuissufficientlystrong thatitmustberenormalizedtoinfinitybythegros Ricecanonical transformation,leavingbehindasuperexchangeinteractionandthe kineticenergyprojectedonthelowerhubbardband.thatis,the effectivehamiltonianis H = " J ij S i # S j + P[ " t ij c * i,$ c j,$ ]P i, j P = '(1% n i& n i( ) i i, j,$ [1]. This t JHamiltonian isnotsimplyaconvenientalternativetothe Hubbardmodel,itreflectsthephysicalfactthatthelow energystateslive withinasubspacewhichisovercompletelydescribedbyasinglefull bandofelectronstates,becauseanti boundstates(doublons)havebeen ejectedoutofthetopoftheband.noconvergentperturbativeroute existstoconnectthelowstatestotheoriginalbandofthehubbard model,sincetheyexistwithinhilbertspacesofdifferentdimensionality.
ItisassumedthatinthestrangemetalregionJistooweak,becauseof competitionwithkineticenergy vii orthermalfluctuations,tocausepair condensationandananomalousself energy,andthereforeitsmajor effectcanbelumpedinwiththatofphononsasarenormalizationofthe kineticenergy.itwillalsocontributeelectron electronscatteringbutwe donotexpectittobeaslargeasthatduetotheprojection.thereforethe problemreducestotheeffectofgutzwillerprojectiononthe renormalizedkineticenergy,representedbyasimplefermigas,thatis, tothesecondterminh,soweconsiderthehamiltonian H P = P# t ij c i" *c j" P = # t ij c ˆ i" * ˆ i, j," ˆ c i" = c i" (1$ n i,$" ) i, j," c j" [2] whereweintroducetheprojectivequasiparticleoperatorsc hatandc* hat,whichautomaticallyenforcetheprojection. TheHFLAnsatzisthattheprojectedHamiltonian[2]operatinginthe unprojectedhilbertspaceofmany electronwavefunctionsgivesonethe low energyspectrumofafermiliquid essentially,thatithasasharp Fermisurfacewiththeusualanalyticitypropertiesoftheself energiesof thequasiparticlescandc*.theansatzcanbethoughtofastheresultofa Shankar style viii renormalizationbutcanreallybejustifiedonlyby demonstratingitsself consistency,andbytestingtowhatextentitagrees withexperiment;inbothrespectsitseemssofartohavepassedmuster. ButthequasiparticlesinthisHilbertspacearenotthetruequasiparticles ofthephysicalsystem:thesearetheprojectedquasiparticleswhichwe designatewith hats.weshallhereafterinventthename pseudoparticle todescribetheobjectscandc*whichobeyfermiliquid rulesbecausetheyoperateinthefullhilbertspace. ThepseudoparticleshaverenormalizedFermivelocitieswhichcanbe estimatedwiththegutzwillerapproximation v F,ren = v F,0 g t with g t = 2x /(1+ x), x being the [3] doping percentage. Theycanbeexpectedtohaveratherlargeelectron electronscattering proportionalto(k kf) 2.Intheonecaseinwhichwehaveaccurate information,optimally dopedbissco ix,thecoefficientis
2 2! 3! 1 # ee = CvF ( k! k F ), C = 3.6" 10 ( mev) [4] Straightforwardphasespaceconsiderationswould,asobservedby Drew, x suggestthatthecoefficientshouldbeoftheorder1/w,wbeing thebandwidth,butinthehalleffectcaseofinteresttohimheobserved thatitwasconsiderablylarger,andwealsofindthis:wisoforderafew hundredmevratherthanafewthousand.alittlethoughtpersuadesus thatthisshouldbethecase.thegutzwillerprojectionslowsthe coherentfermivelocityforanelectronwithspinnearthefermisurface, butitdoesnotmuchaffecttheincoherentmotionsofbareholes,which arejustasrapidasintheunprojectedstate forinstance,thesecond momentoftheoverallspectrumisunaffected. xi Thequasiparticlesare broadenedbytheseincoherentmotionsproportionatelytothissecond moment,roughly,soonemightexpectthatthebroadeningwouldbe proportionaltog 2 oraboutanorderofmagnitudelargerthanthenaïve estimate.thusourhiddenfermiliquidwilltendnottobeaverygood one,inthesensethatthecoherenceofitspseudoparticleslastsonlyout to200 300mevfromtheFermisurface.Weshouldalsonotethatthere isnoreasontoexpectthisscatteringmechanismtobeanisotropic. Letusnowconsiderthetransportpropertiesofsuchasystem:firstthe resistivity.aspwadiscussedinmybookandinrelatedpapers xii,thisis complicatedbybeingatwo stepprocess.themomentumisdeliveredto thesystemviaacceleratingthetruequasiparticles,iebydisplacingtheir Fermisurface.Butthescatteringwhichtransfersmomentumtothe latticeisthet 2 umklappscatteringofthepseudoparticleswhichwehave justbeendiscussing.gutzwillerprojectionisperfectlytranslationinvariant,sothattheprocessofdecayoftruequasiparticlesinto pseudoparticlesismomentum conservingandcannotleadtoresisivity byitself.itacts,instead,asabottleneck,anecessarystepwhichmust takeplacebeforethetruescatteringeventscanoperate.(aspwanoted inref7,thisisactuallythesamephysicswhichisinvolvedinphonon drag,butithinkthe bottleneck descriptionisclearer.)itistheslower ofthetwoprocesseswhichwillcontroltherate:theydonotadd accordingtomatthiessen srulebutaccordingtoitsinverse. Inpreviouswork(ref7)PWAcalculatedthedissipationduetothe quasiparticledecayprocessbyapproximatingthetwo particlegreen s
functionwhichappearsintheresponsefunctionbythesimpleproductof twoone particlefunctions,sinceitshouldbeagoodapproximationfor thequasiparticlestodecayindependently.inref4andrelatedpapers xiii wehaveshownthattheformofthesingle particlegreen sfunctionat absolutezeroisthesimpleexpression " G(r,t) = G 0 (r,t)g * (t) 1 % # & $ g' G * (t) = t ( p ; p = (1( x) 2 /4 [5] Herethe1appliesontheholeside,thegontheelectron.(ForfiniteT, presumably,thejumpsingularityofthecoefficientbecomesafermi function.)g0isthepseudoparticlegreen sfunction.inreference[4]we showedhowtogeneralize[5]tofinitetemperature.g0followsthe conventionalrules,whileaswepointedoutthere,thepowerlawing* wasshownbyyuval xiv tofollowthegeneralruleofbeingantiperiodicin imaginarytimebybecoming # "T & G *(t,t) = % ( $ sinh"tt ' p ) e *"ptt fortt >>1 [6] Thisisthesourceoftheubiquitous linear T decay.notethatthe relaxationrateisisotropic,butthemeanfreepathandthereforethe conductivitywillhavetheanisotropyofthefermivelocity,sincethe Fermimomentumisfairlyisotropic. AthighfrequenciesandhightemperaturestheT 2,ω 2 decayimpliedby [4]maybeassumedtobemorerapidthan[6]anddissipativeprocesses willbedominatedbythepowerlawdecayofquasiparticlesinto pseudoparticles.themoststraightforwardsituationistheinfrared conductivitywhichhaslongbeenknowntoobeyafrequencypowerlaw xv, " ir (#) $(i#) %1+2 p,[7] whichcaneasilybederivedfrom[5]. Timusk xvi hasexperimentallyestimatedthedependenceofthepower2p ondoping,whichweshowinfigure1;theagreementastomagnitudeis good,thedependenceondopingabitslow.butourpredictioniswithin thescatterofthedata.
AsfarasDCresistivityisconcerned,[6]accountsfortheobservedlinear dependenceontnearoptimaldoping.thetrendwithdopingisin agreementwiththeexpected(1 x) 2 dependenceofp,thoughinorderto bequantitativeonewouldneedanestimateofthecarrierdensitywhich ishardtocomeby. Inthesameregimeweseethestrikingphenomenonfirstobservedby Ong xvii ofaqualitativedifferencebetweentherelaxationtimeτas estimatedfromthedcconductivityusingσ=ne 2 τ/m,asopposedtousing thehallangleformulaθh=ωcτh.thelattershowsaconventionalfermi liquidtemperaturedependence T 2,whiletheresistivityislinearinTas wehavejustbeendescribing.inthehfltheorythisdifferenceisvery natural:thehallangleobservedisthatoftheunderlyingpseudoparticles ofthehfl.thelarmorprecessionwhichiscausedbythemagneticfield doesnotchangerelativeoccupanciesandthereforedoesnotdisturbthe equilibriumbetweenquasiparticlesandpseudoparticles:effectively,the magneticfieldcommuteswithgutzwillerprojection.thusthehalleffect andothermagneticresponses suchasthedehaas vanalfveneffect willbeidenticallythoseofthehfl,withnobottleneckcausedbythe decayofthequasiparticles.wehaveestimatedthemagnitudeofthehall angleandfoundthatitisreasonablyaccountedforbyourestimatesof Drew sw. Theonlyeffectofthestronginteractionwillbequantitative.AsI remarkedabove,thet 2 relaxationratewillbeunexpectedlylarge.we haveasyetbeenunabletogetadirectcomparisonbetweenthe relaxationratesasmeasuredfromarpesdataandthosemeasuredvia thehalleffect,becausethesamplesarenotcomparable;butthegeneral observationofdrew,thatthet 2 ratesarehigh,seemstobeborneout.a moreaccuratenumericalfitwouldinvolveaverycompletestudyofthe FermisurfacecurvatureandtheanisotropyoftheFermivelocity. Thefinaltopictotakeupistheresistivityintheregioncompletely beyondthe dome whichisnormallydesignatedas thefermi Liquid. xviii Indeed,theresistivityatlowtemperaturesseemstoobeythe T 2 law;butweseenoreasontosupposethattheeffectsofthestrong interactiondieoutsosuddenly.actually,theresistivityinthisregion seemstobenicelyexplainedintermsofthebottleneckeffect,alongwith theanisotropyofthehflconductivityduetotheanisotropyofvf.
Thetemperaturedependenceoftheresistivity,then,isobtainedby combiningthetwoconductivities. " HFL = ne 2 # /m = e2 h 2 2 k F # h m h = e2 E F W h T 2 [8] Herewehaveignorednumericalfactorsoforder1,realizingthatthey maybesubsumedintheparameterw,theeffectivebandwidthdiscussed underequation[4].conductivitiesare2 dimensional,persingleplane, andtisinenergyunits.theeffectiveconductivitycorrespondingtothe decayprocess[6]is " decay = ne2 v F # = e2 (hk F v F ) mv F ht = e2 h E F T (v F /v F 0 ) [9] HerevF0isthemaximumFermivelocity,whichgivesusanestimateof theoverallbandwidthef;thenwemakeexplicitthedependenceon Fermivelocitywhichwillindeedvaryquitestronglyfromthediagonal directiontothezonecorners(andintherightdirectiontoaccountforthe anisotropyobservedbyhussey xix ). Firstwewouldliketocomparethegeneraltemperaturedependenceof theresistivityimpliedby[8]and[9]withrelativelyearlymeasurements onoverdopedcuprates,wheretherewasnoattempttodisentanglethe anisotropy(refs.18, xx ).Inthiscase,leavingouttheanisotropicFermi velocity,theresistivityistheuniversalexpression " = h T 2 e 2 E F T + W (+" ) res [10] d(ln(" # " res ) /d lnt =1+ W /(T + W ) (somesamplesshowasmallresidualresistancewhichwewouldexpect tobesimplyadditivealamatthiessen srule,playingnoroleinthe bottleneck.)thefitoftheform(10)tothedataisquitesatisfactory.for instance,inref20(the 92version)thereisaplotoftheeffective exponentvst,whichforlowt,wherethedataismostaccurate,follows thesecondequationof(10)accurately.reference18fitsthedataover theentirerangewithat 3/2 powerlaw,whichaccordingto[10]should onlybeapproximate;indeed,wegetasaccurateafit,exceptathight, wherethemeasurementisquestionablebecauseofthermalexpansion.
Fig2showsourfittothedataofref18,andFig3thevaluesofthe parametersin[8]and[9]obtainedfromthefit,asafunctionofdoping. Thex dependenceoftheparameterwisexperimentallyevenstronger thanx 2.Oneaspectwhichwehavenottakenintoaccountisthat[8]is nottheconventionalconductivityofthehflaswouldappeariftheefieldacteddirectlyonit;thethreepseudoparticlesmustrecohereintoa quasiparticletointeractwiththefield.surelythiseffectworksinthe rightdirection. Reference[19]providesanevenmoreexplicitconfirmationofour theory.hussey sequation[3]showsthatheisempiricallydriventothe necessityofaddingconductivities[8]and[9],ratherthanresistivities, butunfortunatelynotinquitethecorrectform[10].hisworkusing angle dependentmagnetoresistancemeasurements xxi hasshown experimentallythatintheoptimal tooverdopedregime,therearetwo scatteringmechanismsforeverymomentumonthefermisurface(not hot and cold spots)withdistincttemperatureandangledependences, andasipointedoutabovethetheoryprovidespreciselythose temperaturedependencesandthecorrectsignandmagnitudeforthe anisotropyofthelineartterm. Inaveryrecentpaper, xxii thesamegrouphaverevisitedthedopingrange ofreference18,butoveraveryrestrictedtemperaturerange.theyhave usedalargemagneticfieldtodestroysuperconductivitywhenpresentso havealowerminimumtemperature.theirfittingfunctionispurely empiricalandhasmoreparameterstoadjustthan[10],andinfactwe canachieveanequallevelofagreementovertheirlimitedtemperature range(seefig4andparametersinfig3). CONCLUSION TheHiddenFermiliquidmethodseemswellonthewaytoprovidinga completeresolutionoftheanomalouspropertiesofthe strangemetal phaseofthecupratesuperconductors.complex seemingastheyare, theseseemtofollowfromtheslightestpossiblegeneralizationofthe conventionalfermiliquidtheoryofmetals,namelytheinclusionofthe projectiveconstraintmadenecessarybytheexistenceofstrongon site electron electroninteractions.thissimplecase,farfrombeingan impenetrablemysteryasitissooftenpicturedtobe,shouldprovidethe
canonicalmodelformorecomplexexamplesofstronglyinteracting electronicsystems. Weshouldacknowledgeextensivediscussionoftheexperimentaldata withnpong. i PWAnderson,cond mat/0603726 ii YayuWang,PhDThesis,PrincetonUniversity,Dept.ofPhysics(2004). iii ZSchlesingerandRLCollins,PhysRevLett65,801(1990);DVanderMarel,HJA Molegraaf,etal,Nature425,271(2003);seealsorefs12,14,15 iv CMVarma,PBLittlewood,SSchmitt Rink,EAbrahams,andAERuckenstein, PhysRevLett63,1996(1989) v TRChien,ZZWang,andNPOng,physRevLett67,2088(1991);PWAnderson, PhysRevLett67,2092(1991);seealsoJMHarris,NPOngetal,PhysRevLett75, 1391(1995) vi PWAnderson,PhysRevB78,174505(2008);cond mat/0709.0656 vii PWAnderson,cond mat/0108522 viii RShankar,RevsModPhys66,129(1994) ix PACasey,JDKoraleketal,NaturePhys4,210(2008);cond mat/0707.3137 x ATZheleznyak,VYakovenko,HDennisDrew,PhysRevB57,3089(1998) xi WFBrinkmanandTMRice,PhysRevB2,1502(1970) xii PWAnderson, TheTheoryofSuperconductivityintheHighTcCuprates,Ch6, PrincetonUPress(1997);MOgataandPWAnderson,PhysLett70,3087(1993) xiii PWAnderson,PhysRevB78,175408(2008) xiv GYuvalandPWAnderson,PhysRevB1,1522(1970) xv AElAzrak,NBontemps,etal,JAlloys&Compds195,663(1993) xvi JHwang,TTimusk,GDGu,cond mat/0607653(2006);jphyscondmatt19, 125208(2007) xvii TRChien,ZZWang,andNPOng,PhysRevLett67,2088(1991) xviii butseehtakagi,bbatlogg,etal,physrevlett69,2975(1992) xix NEHussey,cond mat/0804.2984;subtojphyscondmat xx YShimakawa,JDJorgensen,TManako,YKubo,PhysRevB50,16033(1994);T Manako,YKubo,YShimakawa,PhysRevB46,11019(1992) xxi JMAbdel Jawad,NEHussey,etal,NaturePhys2,821(2006) xxii RACooper,NHussey,etal,Science323,603(2009) xxiii PWAnderson,NaturePhysics2,626 630(2006).
Figure 1. Infrared spectrum exponents for Bi2Sr2CaCu2O8+ δ. Data points fromref. 16 withlinearbestfitofref.16(redline) andpredicted value fromref.23(blueline).thepredictedexponentstemsfromσ(ω)=(iω) 2+γ withγ=1+2p,andpisgivenineq.[5].
Figure2.ComparisonofthepolycrystallineLa2 xsrxcuo4resistivity(data points) extracted from Ref. 18 with the bottleneck resistivity form of Eq.[10].Insetslowsthelowtemperatureregionindetail.
Figure3a. Figure3b.
Figure3c. Figure 3. Parameters of the bottleneck resistivity form of Eq.[10] for comparisons in Fig. 2 and Fig. 4. The three parameters are (a) the bandwidth,(b)apre factorforthefirsttermin[10],and(c)theresidual resistivity.
Figure 4. Comparison of the single crystal La2 xsrxcuo4 resistivity (data points) extracted from Ref. 22 with the bottleneck resistivity form of Eq.[10].FunctionalparameterscanbefoundinFig.3.Insetslowsthelow temperature region in detail. The low temperature resistivity data was determined by Hussey, et al. by suppressing superconductivity with a largemagneticfieldandthenextrapolatingthehighfieldresistivitydata tozerofield.