高等固体物理中科大5关联

高等固体物理中科大5关联第1页/共155页1.Hartree方程(1928)连乘积形式：按变分原理，的选取E达到极小正交归一条件单电子方程第2页/共155页动能原子核对电子形成的势能其余N-1个电子对j电子的库仑作用能自洽求解，H2,He计算与实验相符。26个电子的Fe原子，运算要涉及1076个数，对称简化1053个整个太阳系没有足够物质打印这个数据表！2.凝胶模型(jelliummodel)为突出探讨相互作用电子系统的哪些特征是区别于不计其相互作用者，可人为地简化假定电子是沉浸在空间密度持恒的正电荷背景之中(不考虑离子的周期性)。正电荷的作用体现于在相互作用电子体系的Hamiltonian中出现一个维持系统聚集的附加项金属体系，设电子波函数：第3页/共155页Hartree方程中的势：第二项是全部电子在r处形成的势，与相抵消第三项是须扣除的自作用，与j有关，但如取r为计算原点：所以对凝胶模型，Hartree方程：相互作用→没有相互作用电子＋正电荷背景→自由电子气第4页/共155页3.Hartree-Fock方程(1930)Hartree方程不满足Pauli不相容原理电子：费米子单电子波函数f：→N电子体系的总波函数：

不涉及自旋－轨道耦合时：N电子体系能量期待值：1.第二项j,j'可以相等，自相互作用2.自相互作用严格相消（通过第二，三项）3.第三项为交换项，同自旋电子第5页/共155页通过变分:么正变换:单电子方程：与Hartree方程的差别：第三项对全体电子，第四项新增，交换作用项。求和只涉及与j态自旋平行的j’态，是电子服从Fermi统计的反映。4.Koopmann定理（1934)单电子轨道能量等于N电子体系从第j个轨道上取走一个电子并保持N－1个电子状态不不变的总能变化值。第6页/共155页推广：系统中一个电子由状态j转移到态i而引起系统能量的变化5.交换空穴(Fermihole)

将H-F方程改写为：其中：第7页/共155页定性讨论：假设Fermihole:与某电子自旋相同的其余邻近电子在围绕该电子形成总量为1的密度亏欠域第8页/共155页energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理论(Densityfunctionaltheory)

(1)Thomas-Fermi-DiracModel第9页/共155页(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

第10页/共155页Proof:TheelectronicHamiltonianisitisproduced

bychargesexternaltothesystemofelectrons.InDFT,

iscalledtheexternalpotentialactingonelectroni,sinceOncetheexternalpotential

theelectronicwavefunctionsandallowedenergiesofthemoleculeare

andthenumberofelectronsnarespecified,

determinedasthesolutionsoftheelectronicSchrödingerequation.

第11页/共155页Nowweneedtoprovethattheground-stateelectronprobabilitydensitythenumberofelectrons.

theexternalpotential(exceptforanarbitraryadditiveconstant)

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

thesameground-stateelectrondensity.determines第12页/共155页theexactground-statewavefunctionandenergyoftheexactground-statewavefunctionandenergyofLetSinceanddifferbymorethanaconstant,andmustbe

differentfunctions.第13页/共155页Proof:Assumethusthuswhichcontradictsthegiveninformation.function,theexactground-statewavefunction

stateenergy

foragivenHamiltonianIfthegroundstateisnondegenerate,thenthereisonlyonenormalizedthatgivestheexactground第14页/共155页Accordingtothevariationtheorem,supposethatIfthenisanynormalizedwell-behavedtrialvariationfunction.

NowuseasatrialfunctionwiththeHamiltonianthenSubstitutinggives第15页/共155页Letbeafunctionofthespatialcoordinatesofelectroni,thenUsingtheaboveresult,wegetSimilarly,ifwegothroughthesamereasoning

withaandbinterchanged,weget第16页/共155页Byhypothesis,thetwodifferentwavefunctionsgivethesameelectron.Puttingandaddingtheabovetwoinequalitiesdensity:

yieldpotentialscouldproducethesameground-stateelectrondensitymustbefalse.

energy)

andalsodeterminesthenumberofelectrons.Thisresultisfalse,soourinitialassumptionthattwodifferentexternalpotential(towithinanadditiveconstantthat

simplyaffectsthezerolevel

ofHence,the

ground-stateelectronprobabilitydensity

determinestheexternal第17页/共155页probabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

fordifferentmolecules.“Forsystemswithanondegenerategroundstate,theground-stateelectrondeterminestheground-statewavefunctionandenergy,,whichHowever,thefunctionalsareunknown.isalsowrittenasThefunctionalindependentoftheexternalonispotential.第18页/共155页(3)TheHohenberg-kohnvariationaltheorem“Foreverytrialdensityfunctionthatsatisfiesandforall,thefollowinginequalityholds:,isthetrueground–stateenergy.”Proof:LetsatisfythatandHohenberg-Kohntheorem,determinestheexternalpotential

andthisinturndeterminesthewavefunctiondensity

.Bythe,thatcorrespondstothe

.where第19页/共155页withHamiltonian.AccordingtothevariationtheoremLetususethewavefunctionasatrialvariationfunctionforthe

moleculeSincethelefthandsideofthisinequalitycanberewrittenasOnegetsstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground第20页/共155页(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.沈吕九第21页/共155页electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

第22页/共155页Thus,theground-statewavefunctionofthereferencesystemis:isaspinfunctionorbitalenergies.areKohn-ShamForconvenience,thezerosubscriptonisomittedhereafter.Defineasfollows:ground-state

electronickineticenergysystemofnoninteractingelectrons.(either)isthedifferenceintheaveragebetweenthemoleculeand

thereference

Thequantityrepulsionenergy.units)

for

theelectrostaticinterelectronicistheclassicalexpression(inatomic第23页/共155页RememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributions第24页/共155页Thusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it第25页/共155页ground-stateenergybyvaryingtominimizethefunctional

canvarytheKSorbitals

minimizetheaboveenergyexpressionsubjecttotheorthonormalityconstraint:TheHohenberg-Kohnvariationaltheoremtellusthatwecanfindthe

soas.Equivalently,insteadofvaryingweThus,theKohn-Shamorbitalsarethosethatwiththeexchange-correlationpotential

definedby(Ifisknown,itsfunctionalderivative

isalsoknown.)第26页/共155页CommentsontheDFTmethods:(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequations.(2)ThecomputationtimerequiredforaDFTcalculationformallyscalesthe

third

power

ofthenumberofbasisfunctions.(3)ThereisnoDFmolecularwavefunction.(4)TheKSorbitalscanbeusedinqualitativeMOdiscussions,liketheHF

orbitals.(5)Koopmans’theorem

doesn’t

holdhere,exceptTheKSoperatorexchangeoperatorsintheHFoperatorarereplacedbytheeffectsofbothexchangeandelectroncorrelation.isthesameastheHFoperator

exceptthatthe,whichhandles第27页/共155页(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

第28页/共155页XLocalexchangeApproximatedensityfunctionaltheoriesforexchangeandcorrelationX:

LocalexchangefunctionalofthehomogeneouselectrongasLDALocalexchange+localcorrelationGGALocalexchange+localcorrelation+gradientcorrections3rdGenerationoffunctionalsLDA:Localexchangefunctional+localcorrelationfunctionalofthehomogeneouselectrongasGGA:SameasLDA+“non-local”gradientcorrectionstoexchangeandcorrelation3rdGenerationoffunctionals:SameasGGA+instilationof“exact-exchange”and+2ndderivativesofthedensitycorrections第29页/共155页TermsinDensityFunctionalsr 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).第30页/共155页LocalDensityApproximation第31页/共155页LocalSpinDensityApproximation第32页/共155页LocalSpinDensityCorrelationFunctionalNotforthefaintofheart:第33页/共155页GeneralizedGradientApproximationFunctionals第34页/共155页第35页/共155页第36页/共155页第37页/共155页TheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)第38页/共155页第39页/共155页第40页/共155页第41页/共155页第42页/共155页第43页/共155页第44页/共155页第45页/共155页第46页/共155页第47页/共155页第48页/共155页第49页/共155页第50页/共155页第51页/共155页第52页/共155页第53页/共155页第54页/共155页第55页/共155页第56页/共155页第57页/共155页第58页/共155页第59页/共155页第60页/共155页第61页/共155页第62页/共155页第63页/共155页第64页/共155页第65页/共155页第66页/共155页第67页/共155页第68页/共155页第69页/共155页第70页/共155页第71页/共155页第72页/共155页第73页/共155页第74页/共155页第75页/共155页5.2费米液体理论费米体系费米温度：均匀的无相互作用的三维系统，费米温度：费米简并系统：费米子系统的温度通常运运低于费米温度

室温下金属中的传导电子费米温度给出了系统中元激发存在与否的标度在费米温度以下，系统的性质由数目有限的低激发态决定。有相互作用和无相互作用的简并费米子系统中，低激发态的性质具有较强的对应性。第76页/共155页2.费米液体金属中电子通常是可迁移的，称为电子气，电子动能：电子势能：在高密度下，电子动能为主，自由电子气模型是较好的近似。在低密度下，电子之间的势能或关联变得越来越重要，电子可能由于这种关联作用进入液相甚至晶相。较强关联下，电子系统被称为电子液体或费米液体或Luttinger液体(1D)第77页/共155页相互作用：(1)单电子能级分布变化(势的变化);(2)电子散射导致某一态上有限寿命(驰豫时间)3.朗道费米液体理论单电子图象不是一个正确的出发点，但只要把电子改成准粒子或准电子，就能描述费米液体。准粒子遵从费米统计，准粒子数守恒，因而费米面包含的体积不发生变化。假设激发态用动量表示第78页/共155页朗道费米液体理论的适用条件：(1).必须有可明确定义的费米面存在(2).准粒子有足够长的寿命第79页/共155页FermiLiquidTheory第80页/共155页第81页/共155页第82页/共155页第83页/共155页第84页/共155页第85页/共155页第86页/共155页SimplePictureforFermiLiquid第87页/共155页第88页/共155页朗道费米液体理论是处理相互作用费米子体系的唯象理论。在相互作用不是很强时，理论对三维液体正确。二维情况下，多大程度上成立不知道。一维情况下，不成立。luttinger液体一维：低能激发为自旋为1/2的电中性自旋子和无自旋荷电为的波色子的激发。非费米液体行为：与费米液体理论预言相偏离的性质第89页/共155页THEPHYSICS

OFLUTTINGERLIQUIDSFERMISURFACEHASONLYTWOPOINTSfailureofLandau´sFermiliquidpictureELECTRONSFORMAHARMONICCHAINATLOWENERGIES

Coulomb+PauliinteractionTHELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIEScollectiveexcitationsarevibrationalmodes第90页/共155页REMARKABLEPROPERTIESAbsenceofelectron-likequasi-particles(onlycollectivebosonicexcitations)Spin-chargeseparation(spinandchargearedecoupledandpropagatewithdifferentvelocities)AbsenceofjumpdiscontinuityinthemomentumdistributionatPower-lawbehaviorofvariouscorrelationfunctionsandtransportquantities.Theexponentdependsontheelectron-electroninteraction第91页/共155页OUTLINEWhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1DTheTomonaga-LuttingermodelTheTL-HamiltoniananditsbosonizationDiagonalizationBosonicfieldsandelectronoperatorsLocaldensityofstatesTunnelingintoaLuttingerliquidLuttingerliquidwithasingleimpurityPhysicalrealizationsofLuttingerliquids第92页/共155页LITERATURE

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/9807366第93页/共155页SHORTLYABOUTFERMILIQUIDSLandau1957-1959Alsocollectiveexcitationsoccur(e.g.zerosound)atfiniteenergiesLowenergyexcitationsofasystemofinteractingparticlesdescribedintermsof``quasi-particles``(single-particleexcitations)Keypoint:quasi-particleshavesamequantumnumbersasthecorrespondingnon-interactingsystem(adiabaticcontinuity)StartfromappropriatenoninteractingsystemRenormalizationofasetofparameters(e.g.effectivemass)第94页/共155页FERMILIQUIDSIIPauliexclusionprinciple

onlystateswithinkTaroundFermisphereavailablequasiparticlestatesnearFermispherescatteronlyweaklyQUASI-PARTICLEPICTUREISAPPLICABLEIN3DEffectofCoulombinteractionistoinduceafinitelife-timet3D第95页/共155页FERMILIQUIDSIIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSIONOFEXCITATIONSIN3D0nointeractingT=0FinitejumpinmomentumdistributionZZquasi-particleweight第96页/共155页LIFETIMEOF``QUASI-PARTICLES´´scatteringoutofstatekscatteringintostatekspinscreenedCoulombinteractionenergyconservationIn3Danintegrationoverangulardependencetakescareofd-functionFermi´sgoldenruleyieldsforthelifetimetT=0第97页/共155页LIFETIMEOF``QUASI-PARTICLES´´IIIn1Dk,k´arescalars.Integrationoverk´yieldsWhataboutthelifetimetin1D?formally,itdivergesatsmallqbutwecaninsertasmallcut-offAtsmallTi.e.,thisratiocannotbemadearbitrarilysmallasin3D第98页/共155页BREAKDOWNOFLANDAUTHEORYIN1D12340132DISPERSIONOFEXCITATIONSIN1D

collectiveexcitationsareplasmonswith(RPA)singleparticlegaplessplasmon

COLLECTIVEAND

SINGLE-PARTICLEEXCITATIONNONDISTINCT

nolongerdivergesat(noangularintegrationoverdirectionofasin3D)第99页/共155页THETOMONAGA-LUTTINGERMODELEXACTLYSOLVABLEMODELFORINTERACTING1DELECTRONSATLOWENERGIESDispersionrelationislinearizednear(bothcollectiveandsingle-particleexcitationshavelineardispersion)ModelbecomesexactwhenlinearizedbranchesextendfromAssumptions:Onlysmallmomentaexchangesareincluded第100页/共155页TOMONAGA-LUTTINGERHAMILTONIANFreepart

freepartinteraction

fermionicannihilation/creationoperatorsIntroducerightmoving

k>0,andleftmovingk<0electrons第101页/共155页TLHAMILTONIANIIInteractions

freepartinteractionbackscatteringforwardumklappforward第102页/共155页BOSONIZATIONBOSONIZATION:EXPRESSFERMIONICHAMILTONIANINTERMSOFBOSONICOPERATORSconstructbosonicHamiltonianwiththesamespectrun(a)(b)(c)(d)(a)and(b)havesamespectrumbutdifferentgroundstateEXCITEDSTATECANBEWRITTENINTERMSOFCHARGEEXCITATIONS,ORBOSONICELECTRON-HOLEEXCITATIONS第103页/共155页STEP1WHICHOPERATORSDOTHEJOB?Introducethedensityoperators(createexcitationofmomentumq)andconsidertheircommutationrelations

nearlybosonic

commutationrelations第104页/共155页STEP1:PROOFConsidere.g.algebraoffermionicoperatorsoccupationoperator第105页/共155页STEP2ExaminenowBOSONIZEDHAMILTONIANSTATESCREATEDBYAREEIGENSTATESOFWITHENERGY

andinteractions第106页/共155页STEP2:PROOFExample:第107页/共155页STEP3IntroducethebosonicoperatorsyieldingDIAGONALIZATION第108页/共155页SPIN-CHARGESEPARATIONandinteraction(satisfyingSU2symmetry)Ifweincludespin,itgetsslightlymorecomplicated...andinterestingIntroducethespinandchargedensitiesHamiltoniandecoupleintwoindependentspinandchargeparts,withexcitationspropagatingwithvelocities第109页/共155页SPACEREPRESENTATIONLongwavelengthlimit(interactions)AppropriatelinearcombinationsP,qofthefieldr(x)canbedefined.ThenonefindswhereLuttingerparameterg<1repulsiveinteraction第110页/共155页BOSONICREPRESENTATIONOFYFermionicoperatorWheree.g.Expressyintheformofabosonicdisplacementoperator

B

from

decreasesthenumberofelectronsbyonedisplacesthebosonconfigurationforthatstateBOSONIZATIONIDENTITYifac-numberUladderoperator,qbosonic第111页/共155页LOCALDENSITYOFSTATESi)Localdensityofstatesatx=0ndensityofstatesofnon-interactingsystematT=0ii)LocaldensityofstatesattheendofaLuttingerliquidatT=0cut-offenergyG

gammafunction第112页/共155页MEASURINGTHELDOS

Measurementofthelocaldensityofstatessystem1system2couplingIVbytunnelingSeee.g.carbonnanotubeexperimentbyBockrathetal.Nature,397,598(1999)第113页/共155页MEASURINGTHELDOSIItunnelingrateitojTunnelingcurrentcanbeevaluatedbyuseofFermi´sgoldenruleconstant

LLtoLLLLtometal第114页/共155页SINGLEIMPURITYAgaintunnelingcurrentcanbeevaluatedbyuseofFermi´sgoldenrule

endtoendWeaklinkx=0However,nowistunnelingfromtheendofaLLChargedensitywaveispinnedattheimpurity第115页/共155页PHYSICALREALIZATIONS

SemiconductingquantumwiresEdgestatesinfractionalquantumHalleffectSingle-walledmetalliccarbonnanotubesEFEnergymetallic1Dconductorwith

2linearbandsk第116页/共155页5.3强关联体系窄能带现象金属与绝缘体之分：

(1)能带框架下的区分：导带导带价带价带(2)无序引起的Anderson转变：局域态扩展态局域态局域态局域态扩展态EFEF第117页/共155页(3)电子间关联导致的Mott金属－绝缘体转变

(a).MnO:5个3d未满3d带；O2-2p是满带不与3d能带重叠能带论MnO的3d带将具有金属导电性实际上，MnO是绝缘体！

(b).ReO3:能带论绝缘体。实际上是金属。

(c).一些过渡金属氧化物当温度升高时会从绝缘体金属f电子或d电子波函数的分布范围是否和近邻产生重叠，是电子离域还是局域化的基本判据l壳层体积与Winger-Seitz元胞体积的比值：4f最小，5f次之，3d,4d,5d…多电子态的局域化强度的顺序：4f>5f>3d>4d>5d______________能带宽度上升另外，从左往右穿过周期表，部分填充壳层的半径逐步降低，关联重要性增加。第118页/共155页4f，5f元素和3d,4d,5d元素的壳层体积与Winger-Seitz元胞体积的比值YSc第119页/共155页Smith和Kmetko准周期表窄带区域重费米子强铁磁性超导体离域性局域性第120页/共155页另一类窄带现象：来自能带中的近自由电子与溶在晶格中具有3d,5f或4f壳层电子的溶质原子相互作用

Friedel与Anderson稀土元素或过渡金属化合物中的能隙不可能仅用“电荷转移能”、“杂化能隙”、“有效库仑相关能”三者之一来描述，而应该说三者同时发挥作用。稀土化合物部分存在混价“mixedvalence”。混价的作用导致在Fermi面附近存在非常窄的能带(部分填充f能带或f能级），电子可以在4f能级和离域化能带之间转移，对固体基态性质产生显著影响。第121页/共155页2.窄能带现象的理论模型选择经验参数的模型Hamilton量方法Hubbard模型和Anderson模型第122页/共155页TheHubbardModelFromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroduction第123页/共155页HistoricalfactsHubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJosé,USA.Picturetakenfrom:PhysicsToday,Vol.34,No4,1981第124页/共155页What,ingeneral,istheHM?

Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheory第125页/共155页QuantummechanicsBasics:Schrödingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotation第126页/共155页Basistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:第127页/共155页SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

Blochwave第128页/共155页SingleparticleequationsDispersionrelationforfreeelectrons(dashedline):DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesat arenolongerdegenerated.Twoeigenenergiesatthosepoints.GraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,Vieweg-Verlag第129页/共155页SingleparticleequationsWannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:第130页/共155页ComparisonbetweenthetwonewwavefunctionsBlochwavefunctionWannierwavefunction(w-part)GraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,Vieweg-VerlagGraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,Vieweg-Verlag第131页/共155页WavefunctionformanyparticlesWavefunctionisnotsimplytheproductofallsingleparticlewavefunctions;ParticlescannotbedifferedFermionsmustobeyPauliprincipleAnsatz:Slaterdeterminante第132页/共155页SecondQuantizationforFermionsCreationanddistructionoperatorscreateordestroystates:第133页/共155页SecondQuantizationTheoperatorsfulfillthecommutatorrelation:Thisisamust,otherwiseonewoulddisturbclosurerelationandorthonormalityofwavefunctionsdescribedbysecondquantization第134页/共155页HamiltonianformanyparticlesSummationoverallsingleparticlesHamiltonians+interactionHamiltonian:interactionpotentialuistherepulsiveCoulombinteraction第135页/共155页Operatorsinsecondquantization第136页/共155页Operatorsinsecondquantization第137页/共155页HamiltonianinsecondquantizationIstransformedliketheone-particleoperatorA(1)andthetwo-particleoperatorA(2)第138页/共155页HamiltonianinsecondquantizationNow:Matrixelement mustbedetermined.Herefore,awavefunctionhastobechosen.Example:Bloch-wave第139页/共155页ComingclosertoHubbard...EvaluationofmatrixelementswithWannierwavefunctions:第140页/共155页FinalAssumptionsNow:onlydirectneighborinteractions,restrictiontooneband.第141页/共155页Meaningofmatrixelementst:singleparticlehoppingU:Hubbard-U,describesonsite-CoulombinteractionV:Nearest-neighbor(density)interactionX:conditionalhoppinginteraction第142页/共155页TheHubbardModels