On the scaling spectrum of the Anderson impurity m

arXiv:cond-mat/0110056v1[cond-mat.str-el]2Oct2001OnthescalingspectrumoftheAndersonimpuritymodel.NigelL.DickensandDavidE.LoganUniversityofOxford,PhysicalandTheoreticalChemistryLaboratory,SouthParksRd,OxfordOX13QZ,UKAbstract.WeconsidertheuniversalscalingbehaviouroftheKondoresonanceinthestrongcouplinglimitofthesymmetricAndersonimpuritymodel,usingarecentlydevelopedlocalmomentapproach.Theresultantscalingspectrumisobtainedinclosedform,andisdominatedbylongtailsthatincontrasttopreviousworkarefoundtoexhibitaslowlogarithmicdecayratherthanpower-lawform,crossingovertocharacteristicFermiliquidbehaviouronthelowestenergyscales.Theresultanttheory,whilenaturallyapproximate,isfoundtogiveverygoodagreementforessentiallyallfrequencieswithnumericalrenormalizationgroupcalculationsofboththesingle-particlescalingspectrumandtheself-energy.PACSnumbers:71.27.+a,72.15.Qm,75.20.HrSubmittedto:J.Phys.:Condens.Matter1.IntroductionAsaparadigmfortheeffectsofstrong,localcoulombinteractions,theAndersonimpuritymodel(AIM)[1]remainshighlytopicalsomefortyyearsafteritsinception(foracomprehensivereview,see[2]).Itsessentialphysicsinthestrongcouplingregimeoflargeon-siteinteractionstrength(U)isthatoftheKondoeffect,characterizedbyalow-energyscaleωK;andmanifestfamouslyinthemany-bodyKondoorAbrikosov-Suhlresonanceappearinginthesingle-particlespectrumD(ω).AlthoughωKitselfnaturallydependsupontheinteractionstrength,thefactthatitisthesolelow-energyscalemeansthattheKondoresonanceexhibitsuniversal(U-independent)scalingintermsofω/ωKalone.Anobviousquestionis:whatistheformoftheuniversalscalingspectrum?Itisthisweconsiderinthepresentpaper,inpossiblythesimplestcontextoftheparticle-holesymmetricAIM.Perhapsthefirstpointtomakeisthatwedonotbelievetheanswertothisquestionisknown,andthatthisreflectsinpartthewellknowndifficultiesinconstructingapproximatetheoriesfordynamicalpropertiesoftheAIM.Theproblemisofcoursewellunderstoodatlowfrequencies,whereD(ω)−D(0)∝−(ω/ωK)2satisfiesthedictatesofFermiliquidtheory,asarisesdirectlyfromasimplelow-frequencyOnthescalingspectrumoftheAndersonimpuritymodel.2expansionoftheimpuritysingle-particleGreenfunction[2].Suchbehaviourishoweverconfinedtothelowestoffrequencies|ω|/ωK≪1.NaiveextrapolationofitleadstoarathertrivialLorentzianscalingspectrum,asindeedarisesinavarietyoftheoreticalapproaches;forexample[2]microscopicFermiliquidtheorytoleadingorder,theslavebosonmean-fieldapproximation,orD(ω)approximatedbythespinonspectrumthatmaybeobtainedviatheBetheansatz.Numericalapproachesbycontrast,suchasthenumericalrenormalizationgroup(NRG)[3,4]orquantumMonteCarlo(QMC)[5,6]calculations,revealaverydifferentbehaviourintheformoflong,slowlyvaryingtailsthatentirelydominatethescalingspectrumfor|ω|/ωK&1,andareknowntobeimportantexperimentally[7].Thesearecurrentlybelieved[3-6]tobeso-calledDoniach-˘Sunji´c(DS)[8]tails,ofasymptoticformD(ω)∝(|ω|/ωK)−12,reflectingphysicallyanincipientorthogonalitycatastrophe.SuchbehaviourhashoweverbeeninferredbydirectcomparisonofthenumericalresultstoanempiricalDSform[3-6];andwhilethereisnodoubtthattheyaretherebyquitewelldescribed,wedonotknowofatheoreticalapproachthat(a)explicitlyyieldssuchbehaviour(ifcorrect),and(b)simultaneouslyrecoverstherequisiteFermiliquidformas|ω|/ωK→0.Weconsidertheseissueswithintheframeworkofthelocalmomentapproach(LMA)thathasrecentlybeendeveloped[9-11]tohandleinparticulardynamicalpropertiesofAIMs[9,10].Theaimsofthepaperarestraightforward,andthreefold.(i)ToobtaininclosedformthestrongcouplingLMAscalingspectrumD(ω)forthesymmetricAIM.Thishashithertobeendeterminednumericallyinreference[9],andshowntogiveverygoodagreementwithNRGresults[4].Butitnaturallyprecludedanexplicitdeterminationoftheformofthedominantspectraltails,which(ii)isoursecondaim.WefindthatthesearenotofDSform,butratherexhibitamuchmoreslowlyvaryinglogarithmicdecay.ThisbehaviourisshowntobeinlargepartindependentofthedetailsoftheLMA,andwegivefurtherqualitativeargumentsinsupportofit.(iii)InviewofthiswereassesscomparisonbetweenLMAandNRGresults[4]forthescalingspectrum,concludinginparticularthattheslowlogarithmictailsareverywellsupportedbyNRGdata.Weconsiderinadditionthescalingbehaviouroftheconventionalinteractionself-energyΣ(ω),whichtheLMAcorrespondinglypredictstodivergelogarithmicallyfor|ω|/ωK≫1.SincerecentNRGadvances[12]nowpermitanaccuratenumericaldeterminationoftheself-energyitself,thistoomaybecomparedtoLMApredictions;verygoodagreementisagainfound.§2ofthepapergivesabriefintroductiontotheLMA,andthebackgroundrequiredfortheremainderofthework;fulldetailsmaybefoundinreferences[9,10].GeneralconsiderationofthescalingspectrumwithintheLMAframeworkisgivenin§3;andin§4theasymptoticbehaviourofthespectraltailsisdeducedexplicitlyandcompareddirectlytoNRGresults[4].In§5boththeLMAscalingspectrumandresultantsingleself-energyareconsideredonallfrequencyscales,andlikewisecomparedtoresultsfromNRGcalculations.Thepaperconcludeswithabriefsummary.OnthescalingspectrumoftheAndersonimpuritymodel.32.BackgroundTheHamiltonianfortheAIM[1]isgiveninconventionalnotationbyˆH=Xk,σǫkˆnkσ+Xσǫi+U2ˆni−σˆniσ+Xk,σVikc†iσc