高等固體物理中科大5關(guān)聯(lián)

高等固體物理中科大5關(guān)聯(lián)第1頁(yè)/共155頁(yè)1.Hartree方程(1928)連乘積形式：按變分原理，的選取E達(dá)到極小正交歸一條件單電子方程第2頁(yè)/共155頁(yè)動(dòng)能原子核對(duì)電子形成的勢(shì)能其余N-1個(gè)電子對(duì)j電子的庫(kù)侖作用能自洽求解，H2,He計(jì)算與實(shí)驗(yàn)相符。26個(gè)電子的Fe原子，運(yùn)算要涉及1076個(gè)數(shù)，對(duì)稱簡(jiǎn)化1053個(gè)整個(gè)太陽(yáng)系沒(méi)有足夠物質(zhì)打印這個(gè)數(shù)據(jù)表！2.凝膠模型(jelliummodel)為突出探討相互作用電子系統(tǒng)的哪些特征是區(qū)別于不計(jì)其相互作用者，可人為地簡(jiǎn)化假定電子是沉浸在空間密度持恒的正電荷背景之中(不考慮離子的周期性)。正電荷的作用體現(xiàn)于在相互作用電子體系的Hamiltonian中出現(xiàn)一個(gè)維持系統(tǒng)聚集的附加項(xiàng)金屬體系，設(shè)電子波函數(shù)：第3頁(yè)/共155頁(yè)Hartree方程中的勢(shì)：第二項(xiàng)是全部電子在r處形成的勢(shì)，與相抵消第三項(xiàng)是須扣除的自作用，與j有關(guān)，但如取r為計(jì)算原點(diǎn)：所以對(duì)凝膠模型，Hartree方程：相互作用→沒(méi)有相互作用電子＋正電荷背景→自由電子氣第4頁(yè)/共155頁(yè)3.Hartree-Fock方程(1930)Hartree方程不滿足Pauli不相容原理電子：費(fèi)米子單電子波函數(shù)f：→N電子體系的總波函數(shù)：

不涉及自旋－軌道耦合時(shí)：N電子體系能量期待值：1.第二項(xiàng)j,j'可以相等，自相互作用2.自相互作用嚴(yán)格相消（通過(guò)第二，三項(xiàng)）3.第三項(xiàng)為交換項(xiàng)，同自旋電子第5頁(yè)/共155頁(yè)通過(guò)變分:么正變換:單電子方程：與Hartree方程的差別：第三項(xiàng)對(duì)全體電子，第四項(xiàng)新增，交換作用項(xiàng)。求和只涉及與j態(tài)自旋平行的j’態(tài)，是電子服從Fermi統(tǒng)計(jì)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(gè)軌道上取走一個(gè)電子并保持N－1個(gè)電子狀態(tài)不不變的總能變化值。第6頁(yè)/共155頁(yè)推廣：系統(tǒng)中一個(gè)電子由狀態(tài)j轉(zhuǎn)移到態(tài)i而引起系統(tǒng)能量的變化5.交換空穴(Fermihole)

將H-F方程改寫為：其中：第7頁(yè)/共155頁(yè)定性討論：假設(shè)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的密度虧欠域第8頁(yè)/共155頁(yè)energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理論(Densityfunctionaltheory)

(1)Thomas-Fermi-DiracModel第9頁(yè)/共155頁(yè)(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頁(yè)/共155頁(yè)P(yáng)roof:TheelectronicHamiltonianisitisproduced

bychargesexternaltothesystemofelectrons.InDFT,

iscalledtheexternalpotentialactingonelectroni,sinceOncetheexternalpotential

theelectronicwavefunctionsandallowedenergiesofthemoleculeare

andthenumberofelectronsnarespecified,

determinedasthesolutionsoftheelectronicSchr?dingerequation.

第11頁(yè)/共155頁(yè)Nowweneedtoprovethattheground-stateelectronprobabilitydensitythenumberofelectrons.

theexternalpotential(exceptforanarbitraryadditiveconstant)

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

thesameground-stateelectrondensity.determines第12頁(yè)/共155頁(yè)theexactground-statewavefunctionandenergyoftheexactground-statewavefunctionandenergyofLetSinceanddifferbymorethanaconstant,andmustbe

differentfunctions.第13頁(yè)/共155頁(yè)P(yáng)roof:Assumethusthuswhichcontradictsthegiveninformation.function,theexactground-statewavefunction

stateenergy

foragivenHamiltonianIfthegroundstateisnondegenerate,thenthereisonlyonenormalizedthatgivestheexactground第14頁(yè)/共155頁(yè)Accordingtothevariationtheorem,supposethatIfthenisanynormalizedwell-behavedtrialvariationfunction.

NowuseasatrialfunctionwiththeHamiltonianthenSubstitutinggives第15頁(yè)/共155頁(yè)Letbeafunctionofthespatialcoordinatesofelectroni,thenUsingtheaboveresult,wegetSimilarly,ifwegothroughthesamereasoning

withaandbinterchanged,weget第16頁(yè)/共155頁(yè)Byhypothesis,thetwodifferentwavefunctionsgivethesameelectron.Puttingandaddingtheabovetwoinequalitiesdensity:

yieldpotentialscouldproducethesameground-stateelectrondensitymustbefalse.

energy)

andalsodeterminesthenumberofelectrons.Thisresultisfalse,soourinitialassumptionthattwodifferentexternalpotential(towithinanadditiveconstantthat

simplyaffectsthezerolevel

ofHence,the

ground-stateelectronprobabilitydensity

determinestheexternal第17頁(yè)/共155頁(yè)probabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

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

andthisinturndeterminesthewavefunctiondensity

.Bythe,thatcorrespondstothe

.where第19頁(yè)/共155頁(yè)withHamiltonian.AccordingtothevariationtheoremLetususethewavefunctionasatrialvariationfunctionforthe

moleculeSincethelefthandsideofthisinequalitycanberewrittenasOnegetsstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground第20頁(yè)/共155頁(yè)(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.沈呂九第21頁(yè)/共155頁(yè)electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

第22頁(yè)/共155頁(yè)Thus,theground-statewavefunctionofthereferencesystemis:isaspinfunctionorbitalenergies.areKohn-ShamForconvenience,thezerosubscriptonisomittedhereafter.Defineasfollows:ground-state

electronickineticenergysystemofnoninteractingelectrons.(either)isthedifferenceintheaveragebetweenthemoleculeand

thereference

Thequantityrepulsionenergy.units)

for

theelectrostaticinterelectronicistheclassicalexpression(inatomic第23頁(yè)/共155頁(yè)RememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributions第24頁(yè)/共155頁(yè)Thusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it第25頁(yè)/共155頁(yè)ground-stateenergybyvaryingtominimizethefunctional

canvarytheKSorbitals

minimizetheaboveenergyexpressionsubjecttotheorthonormalityconstraint:TheHohenberg-Kohnvariationaltheoremtellusthatwecanfindthe

soas.Equivalently,insteadofvaryingweThus,theKohn-Shamorbitalsarethosethatwiththeexchange-correlationpotential

definedby(Ifisknown,itsfunctionalderivative

isalsoknown.)第26頁(yè)/共155頁(yè)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頁(yè)/共155頁(yè)(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頁(yè)/共155頁(yè)XLocalexchangeApproximatedensityfunctionaltheoriesforexchangeandcorrelationX:

LocalexchangefunctionalofthehomogeneouselectrongasLDALocalexchange+localcorrelationGGALocalexchange+localcorrelation+gradientcorrections3rdGenerationoffunctionalsLDA:Localexchangefunctional+localcorrelationfunctionalofthehomogeneouselectrongasGGA:SameasLDA+“non-local”gradientcorrectionstoexchangeandcorrelation3rdGenerationoffunctionals:SameasGGA+instilationof“exact-exchange”and+2ndderivativesofthedensitycorrections第29頁(yè)/共155頁(yè)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頁(yè)/共155頁(yè)LocalDensityApproximation第31頁(yè)/共155頁(yè)LocalSpinDensityApproximation第32頁(yè)/共155頁(yè)LocalSpinDensityCorrelationFunctionalNotforthefaintofheart:第33頁(yè)/共155頁(yè)GeneralizedGradientApproximationFunctionals第34頁(yè)/共155頁(yè)第35頁(yè)/共155頁(yè)第36頁(yè)/共155頁(yè)第37頁(yè)/共155頁(yè)TheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)第38頁(yè)/共155頁(yè)第39頁(yè)/共155頁(yè)第40頁(yè)/共155頁(yè)第41頁(yè)/共155頁(yè)第42頁(yè)/共155頁(yè)第43頁(yè)/共155頁(yè)第44頁(yè)/共155頁(yè)第45頁(yè)/共155頁(yè)第46頁(yè)/共155頁(yè)第47頁(yè)/共155頁(yè)第48頁(yè)/共155頁(yè)第49頁(yè)/共155頁(yè)第50頁(yè)/共155頁(yè)第51頁(yè)/共155頁(yè)第52頁(yè)/共155頁(yè)第53頁(yè)/共155頁(yè)第54頁(yè)/共155頁(yè)第55頁(yè)/共155頁(yè)第56頁(yè)/共155頁(yè)第57頁(yè)/共155頁(yè)第58頁(yè)/共155頁(yè)第59頁(yè)/共155頁(yè)第60頁(yè)/共155頁(yè)第61頁(yè)/共155頁(yè)第62頁(yè)/共155頁(yè)第63頁(yè)/共155頁(yè)第64頁(yè)/共155頁(yè)第65頁(yè)/共155頁(yè)第66頁(yè)/共155頁(yè)第67頁(yè)/共155頁(yè)第68頁(yè)/共155頁(yè)第69頁(yè)/共155頁(yè)第70頁(yè)/共155頁(yè)第71頁(yè)/共155頁(yè)第72頁(yè)/共155頁(yè)第73頁(yè)/共155頁(yè)第74頁(yè)/共155頁(yè)第75頁(yè)/共155頁(yè)5.2費(fèi)米液體理論費(fèi)米體系費(fèi)米溫度：均勻的無(wú)相互作用的三維系統(tǒng)，費(fèi)米溫度：費(fèi)米簡(jiǎn)并系統(tǒng)：費(fèi)米子系統(tǒng)的溫度通常運(yùn)運(yùn)低于費(fèi)米溫度

室溫下金屬中的傳導(dǎo)電子費(fèi)米溫度給出了系統(tǒng)中元激發(fā)存在與否的標(biāo)度在費(fèi)米溫度以下，系統(tǒng)的性質(zhì)由數(shù)目有限的低激發(fā)態(tài)決定。有相互作用和無(wú)相互作用的簡(jiǎn)并費(fèi)米子系統(tǒng)中，低激發(fā)態(tài)的性質(zhì)具有較強(qiáng)的對(duì)應(yīng)性。第76頁(yè)/共155頁(yè)2.費(fèi)米液體金屬中電子通常是可遷移的，稱為電子氣，電子動(dòng)能：電子勢(shì)能：在高密度下，電子動(dòng)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(shì)能或關(guān)聯(lián)變得越來(lái)越重要，電子可能由于這種關(guān)聯(lián)作用進(jìn)入液相甚至晶相。較強(qiáng)關(guān)聯(lián)下，電子系統(tǒng)被稱為電子液體或費(fèi)米液體或Luttinger液體(1D)第77頁(yè)/共155頁(yè)相互作用：(1)單電子能級(jí)分布變化(勢(shì)的變化);(2)電子散射導(dǎo)致某一態(tài)上有限壽命(馳豫時(shí)間)3.朗道費(fèi)米液體理論單電子圖象不是一個(gè)正確的出發(fā)點(diǎn)，但只要把電子改成準(zhǔn)粒子或準(zhǔn)電子，就能描述費(fèi)米液體。準(zhǔn)粒子遵從費(fèi)米統(tǒng)計(jì)，準(zhǔn)粒子數(shù)守恒，因而費(fèi)米面包含的體積不發(fā)生變化。假設(shè)激發(fā)態(tài)用動(dòng)量表示第78頁(yè)/共155頁(yè)朗道費(fèi)米液體理論的適用條件：(1).必須有可明確定義的費(fèi)米面存在(2).準(zhǔn)粒子有足夠長(zhǎng)的壽命第79頁(yè)/共155頁(yè)FermiLiquidTheory第80頁(yè)/共155頁(yè)第81頁(yè)/共155頁(yè)第82頁(yè)/共155頁(yè)第83頁(yè)/共155頁(yè)第84頁(yè)/共155頁(yè)第85頁(yè)/共155頁(yè)第86頁(yè)/共155頁(yè)SimplePictureforFermiLiquid第87頁(yè)/共155頁(yè)第88頁(yè)/共155頁(yè)朗道費(fèi)米液體理論是處理相互作用費(fèi)米子體系的唯象理論。在相互作用不是很強(qiáng)時(shí)，理論對(duì)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(fā)為自旋為1/2的電中性自旋子和無(wú)自旋荷電為的波色子的激發(fā)。非費(fèi)米液體行為：與費(fèi)米液體理論預(yù)言相偏離的性質(zhì)第89頁(yè)/共155頁(yè)THEPHYSICS

OFLUTTINGERLIQUIDSFERMISURFACEHASONLYTWOPOINTSfailureofLandau′sFermiliquidpictureELECTRONSFORMAHARMONICCHAINATLOWENERGIES

Coulomb+PauliinteractionTHELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIEScollectiveexcitationsarevibrationalmodes第90頁(yè)/共155頁(yè)REMARKABLEPROPERTIESAbsenceofelectron-likequasi-particles(onlycollectivebosonicexcitations)Spin-chargeseparation(spinandchargearedecoupledandpropagatewithdifferentvelocities)AbsenceofjumpdiscontinuityinthemomentumdistributionatPower-lawbehaviorofvariouscorrelationfunctionsandtransportquantities.Theexponentdependsontheelectron-electroninteraction第91頁(yè)/共155頁(yè)OUTLINEWhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1DTheTomonaga-LuttingermodelTheTL-HamiltoniananditsbosonizationDiagonalizationBosonicfieldsandelectronoperatorsLocaldensityofstatesTunnelingintoaLuttingerliquidLuttingerliquidwithasingleimpurityPhysicalrealizationsofLuttingerliquids第92頁(yè)/共155頁(yè)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頁(yè)/共155頁(yè)SHORTLYABOUTFERMILIQUIDSLandau1957-1959Alsocollectiveexcitationsoccur(e.g.zerosound)atfiniteenergiesLowenergyexcitationsofasystemofinteractingparticlesdescribedintermsof``quasi-particles``(single-particleexcitations)Keypoint:quasi-particleshavesamequantumnumbersasthecorrespondingnon-interactingsystem(adiabaticcontinuity)StartfromappropriatenoninteractingsystemRenormalizationofasetofparameters(e.g.effectivemass)第94頁(yè)/共155頁(yè)FERMILIQUIDSIIPauliexclusionprinciple

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

collectiveexcitationsareplasmonswith(RPA)singleparticlegaplessplasmon

COLLECTIVEAND

SINGLE-PARTICLEEXCITATIONNONDISTINCT

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

freepartinteraction

fermionicannihilation/creationoperatorsIntroducerightmoving

k>0,andleftmovingk<0electrons第101頁(yè)/共155頁(yè)TLHAMILTONIANIIInteractions

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

nearlybosonic

commutationrelations第104頁(yè)/共155頁(yè)STEP1:PROOFConsidere.g.algebraoffermionicoperatorsoccupationoperator第105頁(yè)/共155頁(yè)STEP2ExaminenowBOSONIZEDHAMILTONIANSTATESCREATEDBYAREEIGENSTATESOFWITHENERGY

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

B

from

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

gammafunction第112頁(yè)/共155頁(yè)MEASURINGTHELDOS

Measurementofthelocaldensityofstatessystem1system2couplingIVbytunnelingSeee.g.carbonnanotubeexperimentbyBockrathetal.Nature,397,598(1999)第113頁(yè)/共155頁(yè)MEASURINGTHELDOSIItunnelingrateitojTunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenruleconstant

LLtoLLLLtometal第114頁(yè)/共155頁(yè)SINGLEIMPURITYAgaintunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenrule

endtoendWeaklinkx=0However,nowistunnelingfromtheendofaLLChargedensitywaveispinnedattheimpurity第115頁(yè)/共155頁(yè)P(yáng)HYSICALREALIZATIONS

SemiconductingquantumwiresEdgestatesinfractionalquantumHalleffectSingle-walledmetalliccarbonnanotubesEFEnergymetallic1Dconductorwith

2linearbandsk第116頁(yè)/共155頁(yè)5.3強(qiáng)關(guān)聯(lián)體系窄能帶現(xiàn)象金屬與絕緣體之分：

(1)能帶框架下的區(qū)分：導(dǎo)帶導(dǎo)帶價(jià)帶價(jià)帶(2)無(wú)序引起的Anderson轉(zhuǎn)變：局域態(tài)擴(kuò)展態(tài)局域態(tài)局域態(tài)局域態(tài)擴(kuò)展態(tài)EFEF第117頁(yè)/共155頁(yè)(3)電子間關(guān)聯(lián)導(dǎo)致的Mott金屬－絕緣體轉(zhuǎn)變

(a).MnO:5個(gè)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(dǎo)電性實(shí)際上，MnO是絕緣體！

(b).ReO3:能帶論絕緣體。實(shí)際上是金屬。

(c).一些過(guò)渡金屬氧化物當(dāng)溫度升高時(shí)會(huì)從絕緣體金屬f電子或d電子波函數(shù)的分布范圍是否和近鄰產(chǎn)生重疊，是電子離域還是局域化的基本判據(jù)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(tài)的局域化強(qiáng)度的順序：4f>5f>3d>4d>5d______________能帶寬度上升另外，從左往右穿過(guò)周期表，部分填充殼層的半徑逐步降低，關(guān)聯(lián)重要性增加。第118頁(yè)/共155頁(yè)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YSc第119頁(yè)/共155頁(yè)Smith和Kmetko準(zhǔn)周期表窄帶區(qū)域重費(fèi)米子強(qiáng)鐵磁性超導(dǎo)體離域性局域性第120頁(yè)/共155頁(yè)另一類窄帶現(xiàn)象：來(lái)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(zhì)原子相互作用

Friedel與Anderson稀土元素或過(guò)渡金屬化合物中的能隙不可能僅用“電荷轉(zhuǎn)移能”、“雜化能隙”、“有效庫(kù)侖相關(guān)能”三者之一來(lái)描述，而應(yīng)該說(shuō)三者同時(shí)發(fā)揮作用。稀土化合物部分存在混價(jià)“mixedvalence”?；靸r(jià)的作用導(dǎo)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(jí)），電子可以在4f能級(jí)和離域化能帶之間轉(zhuǎn)移，對(duì)固體基態(tài)性質(zhì)產(chǎn)生顯著影響。第121頁(yè)/共155頁(yè)2.窄能帶現(xiàn)象的理論模型選擇經(jīng)驗(yàn)參數(shù)的模型Hamilton量方法Hubbard模型和Anderson模型第122頁(yè)/共155頁(yè)TheHubbardModelFromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroduction第123頁(yè)/共155頁(yè)HistoricalfactsHubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJosé,USA.Picturetakenfrom:PhysicsToday,Vol.34,No4,1981第124頁(yè)/共155頁(yè)What,ingeneral,istheHM?

Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheory第125頁(yè)/共155頁(yè)QuantummechanicsBasics:Schr?dingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotation第126頁(yè)/共155頁(yè)Basistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:第127頁(yè)/共155頁(yè)SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

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