凝聚态物理-哈佛大学讲义

Hubbardmodel(s)EugeneDemlerHarvardUniversity$$NSF,AFOSR,MURI,DARPA,CollaborationswithexperimentalgroupsofI.Bloch,T.EsslingerCollaborationwithE.Altman(Weizmann),R.Barnett(Caltech),A.Imambekov(Yale),A.M.Rey(JILA),D.Pekker,R.Sensarma,M.Lukin,andmanyothersOutlineBoseHubbardmodel.SuperfluidandMottphasesExtendedHubbardmodel:CDWandSupersolidstatesTwocomponentBoseHubbardmodel:magneticsuperexchangeinteractionsintheMottstatesBoseHubbardmodelforF=1bosons:exoticspinstatesFermiHubbardmodel:competingordersHubbardmodelbeyondcondensedmatterparadigms:nonequilibriummany-bodyquantumdynamicsBoseHubbardmodelAtomsinopticallatticesTheory:Jakschetal.PRL(1998)Experiment:Kasevichetal.,Science(2001);Greineretal.,Nature(2001);Phillipsetal.,J.PhysicsB(2002)Esslingeretal.,PRL(2004);manymore…BoseHubbardmodeltunnelingofatomsbetweenneighboringwellsrepulsionofatomssittinginthesamewellUt4BoseHubbardmodel.Mean-fieldphasediagram1nU020M.P.A.Fisheretal.,PRB40:546(1989)MottN=1N=2N=3SuperfluidSuperfluidphaseMottinsulatorphaseWeakinteractionsStronginteractionsMottMottOpticallatticeandparabolicpotential41nU20N=1N=2N=3SFMIMIJakschetal.,PRL81:3108(1998)SuperfluidtoInsulatortransitionGreineretal.,Nature415:39(2002)U1nt/USuperfluidMottinsulatorShellstructureinopticallatticeS.Foellingetal.,PRL97:060403(2006)ObservationofspatialdistributionoflatticesitesusingspatiallyselectivemicrowavetransitionsandspinchangingcollisionssuperfluidregimeMottregimen=1n=2ExtendedHubbardmodelChargeDensityWaveandSupersolidphasesExtendedHubbardModel-onsiterepulsion-nearestneighborrepulsionCheckerboardphase:Crystalphaseofbosons.BreakstranslationalsymmetryExtendedHubbardmodel.MeanfieldphasediagramvanOtterloetal.,PRB52:16176(1995)Hardcorebosons.Supersolid–superfluidphasewithbrokentranslationalsymmetryExtendedHubbardmodel.QuantumMonteCarlostudySenguptaetal.,PRL94:207202(2005)Hebertetal.,PRB65:14513(2002)DipolarbosonsinopticallatticesGoraletal.,PRL88:170406(2002)TwocomponentBoseHubbardmodel.MagnetismttTwocomponentBosemixtureinopticallatticeExample:.Mandeletal.,Nature425:937(2003)TwocomponentBoseHubbardmodelQuantummagnetismofbosonsinopticallatticesDuanetal.,PRL(2003)•Ferromagnetic•AntiferromagneticKuklovandSvistunov,PRL(2003)ExchangeInteractionsinSolidsantibondingbondingKineticenergydominates:antiferromagneticstateCoulombenergydominates:ferromagneticstateTwocomponentBosemixtureinopticallattice.Meanfieldtheory+QuantumfluctuationsHysteresis1storderAltmanetal.,NJP5:113(2003)GroundstatehastopologicalorderExcitationsareAbelianornon-AbeliananyonsRealizationofspinliquidusingcoldatomsinanopticallatticeDuanetal.PRL91:94514(2003)H=-JxSsixsjx-JySsiysjy-JzSsizsjzKitaevmodelJJUsemagneticfieldgradienttoprepareastateObserveoscillationsbetweenandstatesObservationofsuperexchangeinadoublewellpotentialTheory:A.M.Reyetal.,PRL99:140601(2008)PreparationanddetectionofMottstatesofatomsinadoublewellpotentialComparisontotheHubbardmodelExperiments:S.Trotzkyetal.,Science319:295(2008)SpinF=1bosonsinopticallatticesSpinexchangeinteractions.Exoticspinorders(nematic,valencebondsolid)SpinorcondensatesinopticaltrapsSpinsymmetricinteractionofF=1atomsAntiferromagneticInteractionsforFerromagneticInteractionsforAntiferromagneticspinF=1atomsinopticallatticesHubbardHamiltonianSymmetryconstraintsDemler,Zhou,PRL(2003)NematicMottInsulatorSpinSingletMottInsulatorNematicinsulatingphaseforN=1EffectiveS=1spinmodelImambekovetal.,PRA(2003)Whenthegroundstateisnematicind=2,3.Onedimensionalsystemsaredimerized:Rizzietal.,PRL(2005)FermionicHubbardmodelUttP.W.Anderson(1950)J.Hubbard(1963)FermionicHubbardmodelPhenomenapredictedSuperexchangeandantiferromagnetism(P.W.Anderson)Itinerantferromagnetism.Stonerinstability(J.Hubbard)Incommensuratespinorder.Stripes(Schulz,Zaannen,Emery,Kivelson,White,Scalapino,Sachdev,…)Mottstatewithoutspinorder.DynamicalMeanFieldTheory(Kotliar,Georges,…)d-wavepairing(Scalapino,Pines,…)d-densitywave(Affleck,Marston,Chakravarty,Laughlin,…)SuperexchangeandantiferromagnetismintheHubbardmodel.LargeUlimitSingletstateallowsvirtualtunnelingandregainssomekineticenergyTripletstate:virtualtunnelingforbiddenbyPauliprincipleEffectiveHamiltonian:HeisenbergmodelHubbardmodelforsmallU.AntiferromagneticinstabilityathalffillingQ=(p,p)Fermisurfaceforn=1Analysisofspininstabilities.RandomPhaseApproximationNestingoftheFermisurfaceleadstosingularityBCS-typeinstabilityforweakinteractionHubbardmodelathalffillingUTNparamagneticMottphaseParamagneticMottphase:onefermionpersitechargefluctuationssuppressednospinorderBCS-typetheoryappliesHeisenbergmodelappliesDopedHubbardmodelAttractionbetweenholesintheHubbardmodelLossofsuperexchangeenergyfrom8bondsLossofsuperexchangeenergyfrom7bondsPairingofholesintheHubbardmodelNon-localpairingofholesLeadingistability:d-waveScalapinoetal,PRB(1986)k’k-k’-kspinfluctuationPairingofholesintheHubbardmodelQBCSequationforpairingamplitudek’k-k’-kspinfluctuationSystemsclosetoAFinstability:c(Q)islargeandpositiveDkshouldchangesignfork’=k+Q++--dx2-y2StripephasesintheHubbardmodelStripes:Antiferromagneticdomainssepara