第五章關聯5.1單電子近似的理論基礎5.2費米液體理論5.3強關聯體系多電子體系(AfterBorn-Oppenheimer絕熱近似):5.1單電子近似的理論基礎關聯：電子－電子相互作用弱：單電子近似，電子平均場1.Hartree方程(1928)連乘積形式：按變分原理，的選取E達到極小正交歸一條件單電子方程Hartree方程中的勢：第二項是全部電子在r處形成的勢，與相抵消第三項是須扣除的自作用，與j有關，但如取r為計算原點：所以對凝膠模型，Hartree方程：相互作用→沒有相互作用電子＋正電荷背景→自由電子氣3.Hartree-Fock方程(1930)Hartree方程不滿足Pauli不相容原理電子：費米子單電子波函數f：→N電子體系的總波函數：

不涉及自旋－軌道耦合時：N電子體系能量期待值：1.第二項j,j'可以相等，自相互作用2.自相互作用嚴格相消（通過第二，三項）3.第三項為交換項，同自旋電子通過變分:么正變換:單電子方程：與Hartree方程的差別：第三項對全體電子，第四項新增，交換作用項。求和只涉及與j態自旋平行的j’態，是電子服從Fermi統計的反映。4.Koopmann定理（1934)單電子軌道能量等于N電子體系從第j個軌道上取走一個電子并保持N－1個電子狀態不不變的總能變化值。定性討論：假設Fermihole:與某電子自旋相同的其余鄰近電子在圍繞該電子形成總量為1的密度虧欠域energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理論(Densityfunctionaltheory)

(1)Thomas-Fermi-DiracModel(2)TheHohenberg-KohnTheorem

propertiesareuniquelydeterminedbytheground-stateelectron

In1964,HohenbergandKohnprovedthatmolecularenergy,wavefunction

andallothermolecularelectronic

probabilitydensity

namely,Phys.Rev.136,13864(1964)

.”Densityfunctionaltheory(DFT)attemptstoandotherground-statemolecularproperties

fromtheground-stateelectrondensity

“Formoleculeswitha

nondegenerate

groundstate,theground-state

calculate

Nowweneedtoprovethattheground-stateelectronprobabilitydensitythenumberofelectrons.

theexternalpotential(exceptforanarbitraryadditiveconstant)

a)Sincedeterminesthenumberofelectrons.b)Toseethatdeterminestheexternalpotential,wesupposethatthisisfalseandthattherearetwoexternalpotentialsand(differingbymorethanaconstant)thateachgiveriseto

thesameground-stateelectrondensity.determinestheexactground-statewavefunctionandenergyoftheexactground-statewavefunctionandenergyofLetSinceanddifferbymorethanaconstant,andmustbe

differentfunctions.Proof:Assumethusthuswhichcontradictsthegiveninformation.function,theexactground-statewavefunction

stateenergy

foragivenHamiltonianIfthegroundstateisnondegenerate,thenthereisonlyonenormalizedthatgivestheexactgroundLetbeafunctionofthespatialcoordinatesofelectroni,thenUsingtheaboveresult,wegetSimilarly,ifwegothroughthesamereasoning

withaandbinterchanged,wegetByhypothesis,thetwodifferentwavefunctionsgivethesameelectron.Puttingandaddingtheabovetwoinequalitiesdensity:

yieldpotentialscouldproducethesameground-stateelectrondensitymustbefalse.

energy)

andalsodeterminesthenumberofelectrons.Thisresultisfalse,soourinitialassumptionthattwodifferentexternalpotential(towithinanadditiveconstantthat

simplyaffectsthezerolevel

ofHence,the

ground-stateelectronprobabilitydensity

determinestheexternalprobabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

fordifferentmolecules.“Forsystemswithanondegenerategroundstate,theground-stateelectrondeterminestheground-statewavefunctionandenergy,,whichHowever,thefunctionalsareunknown.isalsowrittenasThefunctionalindependentoftheexternalonispotential.withHamiltonian.AccordingtothevariationtheoremLetususethewavefunctionasatrialvariationfunctionforthe

moleculeSincethelefthandsideofthisinequalitycanberewrittenasOnegetsstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground(4)TheKohn-Shammethod

Ifweknowtheground-stateelectrondensity

molecularpropertiesfromfunction.,theHohenberg-Kohntheoremtellsusthatitispossibleinprincipletocalculatealltheground-state,withouthavingtofindthemolecularwave

1965,KohnandShamdevisedapracticalmethodforfinding

andforfinding

from.[Phys.Rev.,140,A1133(1965)].Theirmethod

iscapable,inprinciple,ofyieldingexactresults,butbecausetheequationsof

theKohn-Sham(KS)methodcontainanunknownfunctionalthatmustbeapproximated,theKSformationofDFTyield

approximateresults.沈呂九electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

RememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributionsThusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it(6)Variousapproximatefunctionals

DFcalculations.Thefunctionalandacorrelation-energyfunctionalAmongvariousCommonlyusedandPW91(PerdewandWang’s1991functional)Lee-Yang-Parr(LYP)functionalareusedinmolecularapproximations,gradient-corrected

exchangeandcorrelationenergyfunctionalsarethemostaccurate.PW86(PerdewandWang’s1986functional)B88(Becke’s1988functional)P86(the

Perdew1986correlationfunctional)

(7)NowadaysKSDFTmethodsaregenerallybelievedtobebetterthantheHFmethod,andinmostcasestheyareevenbetterthanMP2

iswrittenasthesumofanexchange-energyfunctional

XLocalexchangeApproximatedensityfunctionaltheoriesforexchangeandcorrelationX:

LocalexchangefunctionalofthehomogeneouselectrongasLDALocalexchange+localcorrelationGGALocalexchange+localcorrelation+gradientcorrections3rdGenerationoffunctionalsLDA:Localexchangefunctional+localcorrelationfunctionalofthehomogeneouselectrongasGGA:SameasLDA+“non-local”gradientcorrectionstoexchangeandcorrelation3rdGenerationoffunctionals:SameasGGA+instilationof“exact-exchange”and+2ndderivativesofthedensitycorrectionsTermsinDensityFunctionalsr Localdensityrs Seitzradius=(3/4pr)1/3kF Fermiwavenumber=(3p2r)1/3t Densitygradient=|gradr|/2fksrz Spinpolarization=(rup-rdown)/rf Spinscalingfactor=[(1+z)2/3+(1-z)2/3]/2ks

Thomas-Fermiscreeningwavenumber =(4kF/pa0)1/2s Anotherdensitygradient=|gradr|/2kFrJ.Chem.Phys.,100,1290(1994);PRL77,3865(1996).LocalDensityApproximationLocalSpinDensityApproximationLocalSpinDensityCorrelationFunctionalNotforthefaintofheart:GeneralizedGradientApproximationFunctionalsTheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)5.2費米液體理論費米體系費米溫度：均勻的無相互作用的三維系統，費米溫度：費米簡并系統：費米子系統的溫度通常運運低于費米溫度

室溫下金屬中的傳導電子費米溫度給出了系統中元激發存在與否的標度在費米溫度以下，系統的性質由數目有限的低激發態決定。有相互作用和無相互作用的簡并費米子系統中，低激發態的性質具有較強的對應性。2.費米液體金屬中電子通常是可遷移的，稱為電子氣，電子動能：電子勢能：在高密度下，電子動能為主，自由電子氣模型是較好的近似。在低密度下，電子之間的勢能或關聯變得越來越重要，電子可能由于這種關聯作用進入液相甚至晶相。較強關聯下，電子系統被稱為電子液體或費米液體或Luttinger液體(1D)相互作用：(1)單電子能級分布變化(勢的變化);(2)電子散射導致某一態上有限壽命(馳豫時間)3.朗道費米液體理論單電子圖象不是一個正確的出發點，但只要把電子改成準粒子或準電子，就能描述費米液體。準粒子遵從費米統計，準粒子數守恒，因而費米面包含的體積不發生變化。假設激發態用動量表示朗道費米液體理論的適用條件：(1).必須有可明確定義的費米面存在(2).準粒子有足夠長的壽命FermiLiquidTheorySimplePictureforFermiLiquid朗道費米液體理論是處理相互作用費米子體系的唯象理論。在相互作用不是很強時，理論對三維液體正確。二維情況下，多大程度上成立不知道。一維情況下，不成立。luttinger液體一維：低能激發為自旋為1/2的電中性自旋子和無自旋荷電為的波色子的激發。非費米液體行為：與費米液體理論預言相偏離的性質THEPHYSICS

OFLUTTINGERLIQUIDSFERMISURFACEHASONLYTWOPOINTSfailureofLandau′sFermiliquidpictureELECTRONSFORMAHARMONICCHAINATLOWENERGIES

Coulomb+PauliinteractionTHELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIEScollectiveexcitationsarevibrationalmodesREMARKABLEPROPERTIESAbsenceofelectron-likequasi-particles(onlycollectivebosonicexcitations)Spin-chargeseparation(spinandchargearedecoupledandpropagatewithdifferentvelocities)AbsenceofjumpdiscontinuityinthemomentumdistributionatPower-lawbehaviorofvariouscorrelationfunctionsandtransportquantities.Theexponentdependsontheelectron-electroninteractionOUTLINEWhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1DTheTomonaga-LuttingermodelTheTL-HamiltoniananditsbosonizationDiagonalizationBosonicfieldsandelectronoperatorsLocaldensityofstatesTunnelingintoaLuttingerliquidLuttingerliquidwithasingleimpurityPhysicalrealizationsofLuttingerliquidsLITERATURE

K.FlensbergLecturenotesontheone-dimensionalelectrongasandthetheoryofLuttingerliquids

J.vonDelftandH.SchoellerBosonizationforbeginnersrefermionizationforexperts,cond-mat/9805275J.VoitOne-dimensionalFermiliquids,Rep.Prog.Phys.58,977(1995)H.J.Schulz,G.CunibertiandP.PieriFermiliquidsandLuttingerliquids,cond-mat/9807366SHORTLYABOUTFERMILIQUIDSLandau1957-1959Alsocollectiveexcitationsoccur(e.g.zerosound)atfiniteenergiesLowenergyexcitationsofasystemofinteractingparticlesdescribedintermsof``quasi-particles``(single-particleexcitations)Keypoint:quasi-particleshavesamequantumnumbersasthecorrespondingnon-interactingsystem(adiabaticcontinuity)StartfromappropriatenoninteractingsystemRenormalizationofasetofparameters(e.g.effectivemass)FERMILIQUIDSIIPauliexclusionprinciple

onlystateswithinkTaroundFermisphereavailablequasiparticlestatesnearFermispherescatteronlyweaklyQUASI-PARTICLEPICTUREISAPPLICABLEIN3DEffectofCoulombinteractionistoinduceafinitelife-timet3DFERMILIQUIDSIIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSIONOFEXCITATIONSIN3D0nointeractingT=0FinitejumpinmomentumdistributionZZquasi-particleweightLIFETIMEOF``QUASI-PARTICLES′′scatteringoutofstatekscatteringintostatekspinscreenedCoulombinteractionenergyconservationIn3Danintegrationoverangulardependencetakescareofd-functionFermi′sgoldenruleyieldsforthelifetimetT=0LIFETIMEOF``QUASI-PARTICLES′′IIIn1Dk,k′arescalars.Integrationoverk′yieldsWhataboutthelifetimetin1D?formally,itdivergesatsmallqbutwecaninsertasmallcut-offAtsmallTi.e.,thisratiocannotbemadearbitrarilysmallasin3DBREAKDOWNOFLANDAUTHEORYIN1D12340132DISPERSIONOFEXCITATIONSIN1D

collectiveexcitationsareplasmonswith(RPA)singleparticlegaplessplasmon

COLLECTIVEAND

SINGLE-PARTICLEEXCITATIONNONDISTINCT

nolongerdivergesat(noangularintegrationoverdirectionofasin3D)THETOMONAGA-LUTTINGERMODELEXACTLYSOLVABLEMODELFORINTERACTING1DELECTRONSATLOWENERGIESDispersionrelationislinearizednear(bothcollectiveandsingle-particleexcitationshavelineardispersion)ModelbecomesexactwhenlinearizedbranchesextendfromAssumptions:OnlysmallmomentaexchangesareincludedTOMONAGA-LUTTINGERHAMILTONIANFreepart

freepartinteraction

fermionicannihilation/creationoperatorsIntroducerightmoving

k>0,andleftmovingk<0electronsTLHAMILTONIANIIInteractions

freepartinteractionbackscatteringforwardumklappforwardBOSONIZATIONBOSONIZATION:EXPRESSFERMIONICHAMILTONIANINTERMSOFBOSONICOPERATORSconstructbosonicHamiltonianwiththesamespectrun(a)(b)(c)(d)(a)and(b)havesamespectrumbutdifferentgroundstateEXCITEDSTATECANBEWRITTENINTERMSOFCHARGEEXCITATIONS,ORBOSONICELECTRON-HOLEEXCITATIONSSTEP1WHICHOPERATORSDOTHEJOB?Introducethedensityoperators(createexcitationofmomentumq)andconsidertheircommutationrelations

nearlybosonic

commutationrelationsSTEP1:PROOFConsidere.g.algebraoffermionicoperatorsoccupationoperatorSTEP2ExaminenowBOSONIZEDHAMILTONIANSTATESCREATEDBYAREEIGENSTATESOFWITHENERGY

andinteractionsSTEP2:PROOFExample:STEP3IntroducethebosonicoperatorsyieldingDIAGONALIZATIONSPIN-CHARGESEPARATIONandinteraction(satisfyingSU2symmetry)Ifweincludespin,itgetsslightlymorecomplicated...andinterestingIntroducethespinandchargedensitiesHamiltoniandecoupleintwoindependentspinandchargeparts,withexcitationspropagatingwithvelocitiesSPACEREPRESENTATIONLongwavelengthlimit(interactions)AppropriatelinearcombinationsP,qofthefieldr(x)canbedefined.ThenonefindswhereLuttingerparameterg<1repulsiveinteractionBOSONICREPRESENTATIONOFYFermionicoperatorWheree.g.Expressyintheformofabosonicdisplacementoperator

B

from

decreasesthenumberofelectronsbyonedisplacesthebosonconfigurationforthatstateBOSONIZATIONIDENTITYifac-numberUladderoperator,qbosonicLOCALDENSITYOFSTATESi)Localdensityofstatesatx=0ndensityofstatesofnon-interactingsystematT=0ii)LocaldensityofstatesattheendofaLuttingerliquidatT=0cut-offenergyG

gammafunctionMEASURINGTHELDOS

Measurementofthelocaldensityofstatessystem1system2couplingIVbytunnelingSeee.g.carbonnanotubeexperimentbyBockrathetal.Nature,397,598(1999)MEASURINGTHELDOSIItunnelingrateitojTunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenruleconstant

LLtoLLLLtometalSINGLEIMPURITYAgaintunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenrule

endtoendWeaklinkx=0However,nowistunnelingfromtheendofaLLChargedensitywaveispinnedattheimpurityPHYSICALREALIZATIONS

SemiconductingquantumwiresEdgestatesinfractionalquantumHalleffectSingle-walledmetalliccarbonnanotubesEFEnergymetallic1Dconductorwith

2linearbandsk5.3強關聯體系窄能帶現象金屬與絕緣體之分：(1)能帶框架下的區分：導帶導帶價帶價帶(2)無序引起的Anderson轉變：局域態擴展態局域態局域態局域態擴展態EFEF(3)電子間關聯導致的Mott金屬－絕緣體轉變(a).MnO:5個3d未滿3d帶；O2-2p是滿帶不與3d能帶重疊能帶論MnO的3d帶將具有金屬導電性實際上，MnO是絕緣體！(b).ReO3:能帶論絕緣體。實際上是金屬。(c).一些過渡金屬氧化物當溫度升高時會從絕緣體金屬f電子或d電子波函數的分布范圍是否和近鄰產生重疊，是電子離域還是局域化的基本判據l殼層體積與Winger-Seitz元胞體積的比值：4f最小，5f次之，3d,4d,5d…多電子態的局域化強度的順序：4f>5f>3d>4d>5d______________能帶寬度上升另外，從左往右穿過周期表，部分填充殼層的半徑逐步降低，關聯重要性增加。4f，5f元素和3d,4d,5d元素的殼層體積與Winger-Seitz元胞體積的比值YScSmith和Kmetko準周期表窄帶區域重費米子強鐵磁性超導體離域性局域性另一類窄帶現象：來自能帶中的近自由電子與溶在晶格中具有3d,5f或4f殼層電子的溶質原子相互作用

Friedel與Anderson稀土元素或過渡金屬化合物中的能隙不可能僅用“電荷轉移能”、“雜化能隙”、“有效庫侖相關能”三者之一來描述，而應該說三者同時發揮作用。稀土化合物部分存在混價“mixedvalence”。混價的作用導致在Fermi面附近存在非常窄的能帶(部分填充f能帶或f能級），電子可以在4f能級和離域化能帶之間轉移，對固體基態性質產生顯著影響。2.窄能帶現象的理論模型選擇經驗參數的模型Hamilton量方法Hubbard模型和Anderson模型TheHubbardModelFromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroductionHistoricalfactsHubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJosé,USA.Picturetakenfrom:PhysicsToday,Vol.34,No4,1981What,ingeneral,istheHM?

Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheoryQuantummechanicsBasics:Schr?dingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotationBasistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

BlochwaveSingleparticleequationsDispersionrelationforfreeelectrons(dashedline):DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesat arenolongerdegenerated.Twoeigenenergiesatthosepoints.GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagSingleparticleequationsWannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:ComparisonbetweenthetwonewwavefunctionsBlochwavefunctionWannierwavefunction(w-part)GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagGraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagWavefunctionformanyparticlesWavefunctionisnotsimplytheproductofallsingleparticlewavefunctions;ParticlescannotbedifferedFermionsmustobeyPauliprincipleAnsatz:SlaterdeterminanteSecondQuantizationforFermionsCreationanddistructionoperatorscreateordestroystates:SecondQuantizationTheoperatorsfulfillthecommutatorrelation:Thisisamust,otherwiseonewoulddisturbclosurerelationandorthonormalityofwavefunctionsdescribedbysecondquantizationHamiltonianformanyparticlesSummationoverallsingleparticlesHamiltonians+interactionHamiltonian:interactionpotentialuistherepulsiveCoulombinteractionOperatorsinsecondquantizationOperatorsinsecondquantizationHamiltonianinsecondquantizationIstransformedliketheone-particleoperatorA(1)andthetwo-particleoperatorA(2)HamiltonianinsecondquantizationNow:Matrixelement mustbedetermined.Herefore,awavefunctionhastobechosen.Example:Bloch-waveComingclosertoHubbard...EvaluationofmatrixelementswithWannierwavefunctions:FinalAssumptionsNow:onlydirectneighborinteractions,restrictiontooneband.Meaningofmatrixelementst:singleparticlehoppingU:Hubbard-U,describesonsite-CoulombinteractionV:Nearest-neighbor(density)interactionX:conditionalhoppinginteractionTheHubbardModelssimpleHubbardmodelextendedHubbardmodelandanycombinationofmatrixelements...Mott-Hubbardtransition,insulating(Mott)phaseCase1:Strongcoupling,U/t>>1:Mottinsulating

stateforahalf-filledsystem.Thedensityofstates(availablestatesforaddingorremovingparticle)consitsof