8.09_pm_凝聚态物理现象理论与计算_I _周森

强关联电子体系―现象、理论与计算略讲周森中国科学院理论物理研究所2012年8月9日2012年理论物理研究生暑期学校凝聚态物理―现象、理论与计算Correlation:motionofoneinfluencestheothersUncorrelated:Lighttraffic=“idealgas”CorrelationsandemergenceCorrelations:jammed(Emergence)Controlledcorrelations:FastandefficientCorrelationsandemergenceareeverywhere.Correlatedparticles:electrons,atoms,molecules,grains,biologicalstructures,cars…BiologicalcorrelationsThehumanheartisdevelopmentallyprogrammedtooccurinthesamepositionagainandagain.Atomic/molecularcorrelationsSelf-assemblingofnano-metalsinasolutionSinglecrystal:verystrongcorrelationsPackingofionsinmetallicglassesElectroniccorrelations1cm3ofmatter~1023atoms,electronsWeakcorrelationsconventionalmaterial,semiconductorStrongcorrelationsuniquematerials/devicepropertiesSuperconductivityMagnetismSpin-chargecoupling,e.g.multiferroicsSpin-orbitalcoupling,e.g.topologicalinsulatorLargethermopowerControlledmany-electroncoherenceinnanostructures……Overlap:tCoulomb:UCorrelationstrength:t/UComparisonConventionalMaterialsCorrelatedElectronMaterialsLargeoverlapofs+porbitalsgivesveryextendedwavefunctionsLocalizationofd+forbitalsenhancesCoulombinteractionHighqualityandflexiblefabricationMaterialschemistrychallenging!Sensitivityduetoweakdonor/acceptorbindingSensitivityduetocompetingorderedstatesNointrinsicmagnetismorothercorrelationsDiversemagneticandothercorrelationsIntrinsiclengthscale=largeeffectiveBohrradiusa0IntrinsiclengthscalesasshortasatomicsizeWeakcorrelationandlargea0enablesimpleandaccuratemodelingStrongcorrelations,verychallengingtoexistingtheoreticaltoolsPotentialgain:newmultifunctionalmaterialsanddevices,whichdomoreanddoitbetterthansemiconductorsdo.Challenges:Understandingphenomena,controllingmaterialsandinterfacesImportantlandmarkinsuperconductivitySm(O1-xFx)FeAsMar.(2008)ConventionalSC:1911–onwards:Electron-phononsuperconductors1957–BCSTheoryHighestTc:MgB2(2001)UnconventionalSC:1977:HeavyFermionSC1986:High–Tccuprates(manyTc>77K)LaSrCuO,YBaCuO,BiSrCaCuO…1995:RuthenatesSrRuO,CaSrRuO2003:CobaltatesNaCoO22008:Fe-PnictidesLaOFeAs,BaFe2As2…SC:perfectconductor+perfectdiamagnetismSCisnotdrivenbyelectron-phononinteraction–easytoargue(e-phinteractionscanstillbeimportant)Electron-electroninteractionisthedrivingforce–DifficulttoproveUnconventionalsuperconductivity“Newclues”:unconventionalSCsareoftenfoundinthevicinityofelectronicorderedphasesinducedbyinteractionsMagnetism:Cuprates,HeavyFermions,Fe-pnictidesChargeorder:Organics,transition-metalchalcogenides,CuxTiSe2,Ba1-xKxBaO3ElectronicPhaseseparation/Nematic:Cuprates,Fe-pnictides.WhatarethemostbasicpropertiesofadopedMottinsulator?ExperimentalevidencefromthecupratesHowtoconstructtheoreticalmodelsforcuprates:fromCud-electronsandOp-electronstotheeffectivet-JmodelWhatarethemostessentialideasofResonanceValenceBond?Short-rangeRVBfromthet-JmodelSlave-bosonformulationHigh-TcCupratesY-123CuO2CuO2CuO2CuO2CuOYBa2Cu3O7La-214Cuprates:crystalstructureTwo-dimensionallayeredstructureMostimportantphysicsinthecommonCuO2planeBi-2212OctahedralcrystalfieldsplittingCu2+[Ar]3d9JTdistortionO2-[Be]2p6Undopedcuprateshave1e/unitcell→half-filled,lowspinS=1/2Cuprates:electronicconfigurationpx,pypzCopperOxygen3delectrons2pelectronsCupratesarep-dchargetransfersystemsTransitionmetalsCu,Fe,Co,Ru(4d)…Oxygen,Pnictigen,ChalcogenO,As,P,Se,Te…..Two-dimensionallayeredstructureMostimportantphysicsinthecommonCuO2planeCuprates:p-dchargetransferCuprates:p-dchargetransferinsulatorsElectronPicture:as1electronon3d8U=7~10eVEnergycostforchargetransfer∆pd=U-εp=1~2eVIncontrast,Fe-pnictidesarechargetransfermetalsHowtodescribeCu2+=3d9?p-delectrontransferUεp3d83d10UU-εp3d83d10p-dholetransferCuCuOOHolePicture:as1holeon3d10Cuprates:dopedAFMottinsulatorEverythingstartsfromdopingahalf-filledAFMottinsulatorThisphasediagramisonerealizationofdopingachargetransferMottinsulatorHigh-Tcisastrongcorrelationproblem.Noweak-couplinganalog!Cuprates:dopedAFMottinsulatorMostessentialingredientsofadopedMottinsulator:Particlenumber=1-xMobilecarrierdensity/coherencefactor=xExperimentalevidenceLargeFermisurfacewithLuttingervolumeproportionalto1-x(ARPES+QuantumOscillations)Cuprates:dopedAFMottinsulatorQuantitieshavingtodowithcoherentmotionofholesscalewithdopingx(Opticalspectroscopyandtransport)Padillaetal,PRB(2005)CoherentweightZACuprates:dopedAFMottinsulatorLowtemperatureQPcoherenceweightscaleswithx(ARPES)Dinget.al.,PRL87,227001(2001)TheseessentialpropertiesdonotcomefromweaklyinteractingelectronsStrongcorrelationphysicsisrequiredtounderstandcupratesConstructionoftheoreticalmodelsforthestronglycorrelatedCuO2planeStronglycorrelatedelectronsonCuO2planeMinimal3-bandmodelinholepicture.Cu:3dx2-y2(d),PlanarO:2px,2pyExampleset:tpd=1eV,tpp=0.5eV,εp-εd0=6.5eV,U=10eVThisisanAndersonLatticeModel:Chargetransferinsulatorwhenundoped(oneholeperCu)iftpd<pd=U-(εp-εd)Chargefluctuationsquenchedbylarge-UonCu:onlyone-holestateallowed→localmoment(S=1/2)onCu.LargequantumspinfluctuationsDopedholesgotooxygen;newstatesintroducedinsidethechargetransfergap.ZhangandRice,PRB37,3759(1988).DopingintroducesnewstatesinCTgap:Keepingthelowestone-holeandthelowesttwo-holeZRsingletOCuOperatorAddA2-holeboundstatehopping→effectivesingle-holehopping.Single-bandt-Jmodel:Undopedcase:Charge-transferinsulator→CopperspinsinteractviasuperexchangeSpin-1/2HeisenbergmodelEffective1-bandmodelviaformationofZRsingletSystematicderivationusingcellperturbationConstruct“molecular”orbitalscenteredonCuDecomposeintoequivalent(overlapping)CuO4cells:H=H0(non-interactingCells)+Hcc(cell-cellinteractions)Solveone-cellproblem,ConstructeffectiveHamiltonian(Rayleigh-Schrödingerexpansion)P0:projectionoperatorfor|ofh0ExpressHinthebasisoftheone-cellstates:Jefferson,Eskes&Feiner,PRB45,7959(1992);Feiner,Jefferson&Raimondi,PRB53,8751(1996);…SimplestdescriptionofthecupratesUndopedCuO2planeAFHeisenbergmodelDopeholestJt3J,t’-0.3tt’DopedCuO2planeOne-bandt-JmodelNochargefluctuationsPG:projectionoperatorNodoubleoccupationConnectiontoHubbardmodelinthelarge-UlimitBasicphysicsofstrongcorrelationBand(Pauli)Insulator=Evennumberofe-persiteMottInsulator=Interactiondriveninsulatorhalf-filledcase=onee-persiteHubbardModelWhyisitdifficulttosolve?Single-siteHilbertspace:Totalnumberofstates:latticewithNssites:4NsTwo-siteHubbardmodelConsidersubspacewithn=2,Sz=0:4statesEigenvalues:Totalnumberofstates=42=16,Hisa1616matrix.Duetosymmetries,electronnumbernandSzandagoodquantumnumber,HblockdiagonalfordifferentnandSzUpperandlowerHubbardbandsEDOSLHBUHBWtdoubleocc.0-JUU+JUMott-HubbardGapLowestenergystate:S=0SpinsingletbondTwositesLargeU:projectionofupperHubbardbandVirtualhoppingfavorsAFMcorrelationsForlargeU>>t,projectoutdoublyoccupiedsitesCanonicaltransformation→t-JmodelinprojectedHilbertspaceManytheoriesforhigh-Tc,butthetheoryis…RVBResonatingValencebond共振共价键WhydopinganAFMottinsulatorSCRVBstateforHeisenbergmodelontriangularlattice(1973)Andersonresonatingvalencebond(RVB)ideaSuperpositionofspin-singletpairsratherthanNeelstateBetteraccountforthequantumfluctuations:spinliquidstateRVBpictureforhigh-Tccuprates(1987)Electronsformspin-singletpairs,mobilizedupondopingandcondenseintoaSCstateButweknowcuprateshaveantiferromagneticorderatx=0Holedopingx0Frustration…AFT=0SpinLiquidSCRVBCupratesHoledopingx0AFHiddenSpinLiquidSCTShort-rangeRVBInRVBpicture:Thereisnotapairingmechanism.Spinsingletpairsalreadyexist.SCcomesfromcoherenceofpairsSpinpairingthroughinstantaneoussuperexchangeinteraction.Howdoesthispicturematerializethrought-Jmodel?SpinliquidstateonaS=1/2KagomelatticeMostfrustrated2DlatticeSlave-bosonforprojectedHilbertspace:SpinlessbosonSpin-carryingfermion(spinon)EachsiteisandmustbeoccupiedbyeitherabosonorafermionConstraints→equality:EnforcedbyLagrangemultiplierscompletenessSlave-bosonformulationofthet-JmodelSlave-bosonmean-fieldtheoryRVBdecouplingofexchangeinteraction:ParamagneticvalencebondSpin-singletpairingUniformmean-fieldsolution:τij–symmetryofthepair:s-wave,d-wave,…NotinAFMDopingConcentrationxTd-SCSGincoherentmetalFermiliquidKotliarandLiu,PRB38,5142(1988).BosecondensationSpinonpairingantinodenodeSuperexchange→d-waveSCPG:spinpairinggapTcissetbyphasecoherencebelowoptimaldopingUniformmean-fieldphasediagram