凝聚態(tài)物理現(xiàn)象理論與計(jì)算I 周森

凝聚態(tài)物理現(xiàn)象理論與計(jì)算I周森第1頁/共35頁Correlation:motionofoneinfluencestheothersUncorrelated:Lighttraffic=“idealgas”CorrelationsandemergenceCorrelations:jammed(Emergence)

Controlledcorrelations:FastandefficientCorrelationsandemergenceareeverywhere.Correlatedparticles:electrons,atoms,molecules,grains,biologicalstructures,cars…第2頁/共35頁BiologicalcorrelationsThehumanheartisdevelopmentallyprogrammedtooccurinthesamepositionagainandagain.第3頁/共35頁Atomic/molecularcorrelationsSelf-assemblingofnano-metalsinasolutionSinglecrystal:verystrongcorrelationsPackingofionsinmetallicglasses第4頁/共35頁Electroniccorrelations1cm3ofmatter~1023atoms,electronsWeakcorrelationsconventionalmaterial,semiconductorStrongcorrelationsuniquematerials/deviceproperties

Superconductivity

Magnetism

Spin-chargecoupling,e.g.multiferroics

Spin-orbitalcoupling,e.g.topologicalinsulator

Largethermopower

Controlledmany-electroncoherenceinnanostructures……Overlap:tCoulomb:UCorrelationstrength:t/U第5頁/共35頁ComparisonConventionalMaterialsCorrelatedElectronMaterialsLargeoverlapofs+porbitalsgivesvery

extendedwavefunctionsLocalizationofd+forbitalsenhancesCoulombinteractionHighqualityandflexiblefabricationMaterialschemistrychallenging!Sensitivityduetoweakdonor/acceptorbindingSensitivityduetocompetingorderedstatesNointrinsicmagnetismorothercorrelationsDiversemagneticandothercorrelationsIntrinsiclengthscale=largeeffectiveBohrradiusa0IntrinsiclengthscalesasshortasatomicsizeWeakcorrelationandlargea0enablesimpleandaccuratemodelingStrongcorrelations,verychallengingtoexistingtheoreticaltoolsPotentialgain:newmultifunctionalmaterialsanddevices,whichdomoreanddoitbetterthansemiconductorsdo.Challenges:Understandingphenomena,controllingmaterialsandinterfaces第6頁/共35頁ImportantlandmarkinsuperconductivitySm(O1-xFx)FeAsMar.(2008)

ConventionalSC:1911–onwards:

Electron-phononsuperconductors1957–BCSTheory

HighestTc:MgB2(2001)UnconventionalSC:1977:HeavyFermionSC1986:High–Tccuprates(manyTc>77K)

LaSrCuO,YBaCuO,BiSrCaCuO…1995:RuthenatesSrRuO,CaSrRuO2003:CobaltatesNaCoO22008:Fe-PnictidesLaOFeAs,BaFe2As2…SC:perfectconductor+perfectdiamagnetism第7頁/共35頁SCisnotdrivenbyelectron-phononinteraction–easytoargue(e-phinteractionscanstillbeimportant)Electron-electroninteractionisthedrivingforce–DifficulttoproveUnconventionalsuperconductivity“Newclues”:unconventionalSCsareoftenfoundinthevicinityofelectronicorderedphasesinducedbyinteractionsMagnetism:Cuprates,HeavyFermions,Fe-pnictidesChargeorder:Organics,transition-metalchalcogenides,CuxTiSe2,Ba1-xKxBaO3ElectronicPhaseseparation/Nematic:Cuprates,Fe-pnictides.第8頁/共35頁WhatarethemostbasicpropertiesofadopedMottinsulator?ExperimentalevidencefromthecupratesHowtoconstructtheoreticalmodelsforcuprates:fromCud-electronsandOp-electronstotheeffectivet-JmodelWhatarethemostessentialideasofResonanceValenceBond?Short-rangeRVBfromthet-JmodelSlave-bosonformulationHigh-TcCuprates第9頁/共35頁Y-123CuO2CuO2CuO2CuO2CuOYBa2Cu3O7La-214Cuprates:crystalstructureTwo-dimensionallayeredstructureMostimportantphysicsinthecommonCuO2planeBi-2212第10頁/共35頁

OctahedralcrystalfieldsplittingCu2+[Ar]3d9JTdistortionO2-[Be]2p6Undopedcuprateshave1e/unitcell→half-filled,lowspinS=1/2Cuprates:electronicconfigurationpx,pypz第11頁/共35頁CopperOxygen3delectrons2pelectronsCupratesarep-dchargetransfersystemsTransitionmetalsCu,Fe,Co,Ru(4d)…Oxygen,Pnictigen,ChalcogenO,As,P,Se,Te…..Two-dimensionallayeredstructureMostimportantphysicsinthecommonCuO2planeCuprates:p-dchargetransfer第12頁/共35頁Cuprates:p-dchargetransferinsulators

ElectronPicture:as1electronon3d8U=7~10eVEnergycostforchargetransfer?pd=U-εp=1~2eVIncontrast,Fe-pnictidesarechargetransfermetalsHowtodescribeCu2+=3d9?p-delectrontransferUεp3d83d10UU-εp3d83d10p-dholetransferCuCuOOHolePicture:as1holeon3d10第13頁/共35頁Cuprates:dopedAFMottinsulatorEverythingstartsfromdopingahalf-filledAFMottinsulatorThisphasediagramisonerealizationofdopingachargetransferMottinsulatorHigh-Tcisastrongcorrelationproblem.Noweak-couplinganalog!第14頁/共35頁Cuprates:dopedAFMottinsulatorMostessentialingredientsofadopedMottinsulator:

Particlenumber=1-xMobilecarrierdensity/coherencefactor=xExperimentalevidenceLargeFermisurfacewithLuttingervolumeproportionalto1-x(ARPES+QuantumOscillations)第15頁/共35頁Cuprates:dopedAFMottinsulatorQuantitieshavingtodowithcoherentmotionofholesscalewithdopingx

(Opticalspectroscopyandtransport)Padillaetal,PRB(2005)第16頁/共35頁CoherentweightZACuprates:dopedAFMottinsulatorLowtemperatureQPcoherenceweightscaleswithx(ARPES)Dinget.al.,PRL87,227001(2001)第17頁/共35頁TheseessentialpropertiesdonotcomefromweaklyinteractingelectronsStrongcorrelationphysicsisrequiredtounderstandcupratesConstructionoftheoreticalmodelsforthestronglycorrelatedCuO2plane第18頁/共35頁StronglycorrelatedelectronsonCuO2planeMinimal3-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.第19頁/共35頁ZhangandRice,PRB37,3759(1988).DopingintroducesnewstatesinCTgap:Keepingthelowestone-holeandthelowesttwo-holeZRsingletOCuOperatorA

dd

A2-holeboundstatehopping→effectivesingle-holehopping.Single-bandt-Jmodel:

Undopedcase:Charge-transferinsulator→CopperspinsinteractviasuperexchangeSpin-1/2HeisenbergmodelEffective1-bandmodelviaformationofZRsinglet第20頁/共35頁Systematicderivationusingcellperturbation

Construct“molecular”orbitalscenteredonCuDecomposeintoequivalent(overlapping)CuO4cells:H=H0(non-interactingCells)+Hcc(cell-cellinteractions)

Solveone-cellproblem,ConstructeffectiveHamiltonian(Rayleigh-Schr?dingerexpansion)P0:projectionoperatorfor|ofh0ExpressHinthebasisoftheone-cellstates:Jefferson,Eskes&Feiner,PRB45,7959(1992);Feiner,Jefferson&Raimondi,PRB53,8751(1996);…第21頁/共35頁SimplestdescriptionofthecupratesUndopedCuO2planeAFHeisenbergmodelDopeholestJt

3J,t’

-0.3tt’DopedCuO2planeOne-bandt-JmodelNochargefluctuationsPG:projectionoperatorNodoubleoccupationConnectiontoHubbardmodelinthelarge-Ulimit第22頁/共35頁BasicphysicsofstrongcorrelationBand(Pauli)Insulator=Evennumberofe-persiteMottInsulator=Interactiondriveninsulatorhalf-filledcase=onee-persiteHubbardModelWhyisitdifficulttosolve?Single-siteHilbertspace:Totalnumberofstates:latticewithNssites:

4Ns第23頁/共35頁Two-siteHubbardmodelConsidersubspacewithn=2,Sz=0:4statesEigenvalues:Totalnumberofstates=42=16,Hisa1616matrix.Duetosymmetries,electronnumbernandSzandagoodquantumnumber,HblockdiagonalfordifferentnandSz第24頁/共35頁UpperandlowerHubbardbandsEDOSLHBUHBWtdoubleocc.0-JUU+JUMott-HubbardGapLowestenergystate:S=0SpinsingletbondTwosites第25頁/共35頁LargeU:projectionofupperHubbardbandVirtualhoppingfavorsAFMcorrelationsForlargeU>>t,projectoutdoublyoccupiedsitesCanonicaltransformation→t-JmodelinprojectedHilbertspace第26頁/共35頁Manytheoriesforhigh-Tc,butthetheoryis…RVBResonatingValencebond共振共價(jià)鍵第27頁/共35頁WhydopinganAFMottinsulatorSCRVBstateforHeisenbergmodelontriangularlattice(1973)Andersonresonatingvalencebond(RVB)ideaSuperpositionofspin-singletpairsratherthanNeelstateBetteraccountforthequantumfluctuations:spinliquidstateRVBpictureforhigh-Tccuprates(1987)Electronsformspin-singletpairs,mobilizedupondopingandcondenseintoaSCstate第28頁/共35頁Butweknowcuprateshaveantiferromagneticorderatx=0Holedopingx0Frustration…AFT=0SpinLiquidSCRVBCupratesHoledopingx0AFHiddenSpinLiquidSCTShort-rangeRVB第29頁/共35頁InRVBpicture:Thereisnotapairingmechanism.

Spinsingletpairsalreadyexist.SCcomesfromcoherenceof