Slave Particle Studies of the Electron Momentum Distribution in the Low Dimensional t-J M

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The electron momentum distribution function in the $t-J$ model is studied in the framework of slave particle approach. Within the decoupling scheme used in the gauge field and related theories, we treat formally phase and amplitude fluctuations as well as

arXiv:cond-mat/9304011v1 8 Apr 1993SlaveParticleStudiesoftheElectronMomentumDistributionintheLowDimensionalt JModelShipingFeng ,1J.B.Wu,2Z.B.Su,1,2andL.Yu1,21International2InstituteCentreforTheoreticalPhysics,P.O.Box586,34100Trieste,ItalyofTheoreticalPhysics,ChineseAcademyofSciences,Beijing100080,ChinaTheelectronmomentumdistributionfunctioninthet Jmodelisstudiedintheframeworkofslaveparticleapproach.Withinthedecouplingschemeusedinthegauge eldandrelatedtheories,wetreatformallyphaseandamplitude uctuationsaswellasconstraintswithoutfurtherapproximations.Ourresultindicatesthatthe

electronFermisurfaceobservedinthehigh-resolutionangle-resolvedphotoemissionandinversephotoemissionexperimentscannotbeexplainedwithinthisframework,andthesumruleforthephysicalelectronisnotobeyed.Acorrectscalingbehavioroftheelectronmomentumdistributionfunctionneark～kFandk～3kFinonedimensioncanbereproducedbyconsideringthenonlocalstring elds[Z.Y.Wengetal.,Phys.Rev.B45,7850(1992)],buttheoverallmomentumdistributionisstillnotcorrect,atleastatthemean eldlevel.

PACSnumbers:71.45.-d,75.10.Jm

I.INTRODUCTION

Thet Jmodelisoneofthesimplestmodelscontainingtheessenceofstrongcorrelationsanditsimplicationsforoxidesuperconductivity[1,2]stillremainanoutstandingproblem.Thet Jmodelwasoriginallyintroducedasane ectiveHamiltonianoftheHubbardmodelinthestrongcouplingregime,wheretheon-siteCoulombrepulsionUisverylargeascomparedwiththeelectronhoppingenergyt,andthereforetheelectronsbecomestronglycorrelatedtoavoidthedoubleoccupancy.Inthiscase,theelectron’sHilbertspaceisseverelyrestrictedduetothisconstraint σ CiσCiσ≤1.Anderson[1]andlaterZhangandRice[2]havearguedstronglythatthebasicphysicsofoxidesuperconductorscanbedescribedbythet Jmodel.

Thenormalstatepropertiesofoxidesuperconductorsexhibitanumberofanomalouspropertiesinthesensethattheydonot tintheconventionalFermiliquidtheory[3,4].SomepropertiescanbeinterpretedonlyintermsofadopedMottinsulator[3,4].Acentralquestioninthetheoryofthesestronglycorrelatedsystemsconcernsthenatureoftheelec-tronFermisurface(EFS)[4].Thehigh-resolutionangle-resolvedphotoemissionandinversephotoemissionexperiments[5]demonstratetheexistenceofalargeEFS,withanareacon-sistentwiththebandstructurecalculations.SincethebandtheoryisconsistentwiththeLuttingertheorem[6],thismeansthattheEFSareacontains1-δelectronpersite,whereδistheholedopingconcentration.AlthoughthetopologyofEFSisingeneralagreementwithone-electron-bandcalculations,theFermivelocityisquitedi erent.Thisindicatesthatelectroncorrelationsrenormalizeconsiderablytheresultsobtainedintheframeworkofasingle-particletreatment.Ithasrecentlybeenshownbysmallclusterdiagonalizationthatinatwo-dimensional(2D)squarelatticetheEFSwithinthet JmodelisconsistentwithLuttinger’stheorem[7].MonteCarlosimulationsforanearlyhalf- lled2DHubbardmodelalsosupportthisresult[8].Moreover,theelectronmomentumdistributionfunctionofa2Dt Jmodelwasstudied[9]byusingtheLuttinger-Jastrow-Gutzwillervariational

Fermiedge.Forone-dimensional(1D)largeUlimitHubbardmodelwhichisequivalenttothet Jmodel,OgataandShiba[10]obtainedtheelectronmomentumdistributionfunc-tionbyusingLieb-Wu’sexactwavefunction[11].TheirresultalsoshowstheexistenceofEFSaswellasthesingularbehavioratk～kFandk～3kFinthemomentumdistributionfunction.Furthermore,YokoyamaandOgata[12]studiedthe1Dt Jmodelbyusingexactdiagonalizationofsmallsystemsandfoundthepower-lawsingularityappearingatkFinthemomentumdistributionfunction.

Sofar,strongcorrelatione ectscanbeproperlytakenintoaccountonlybynumericalmethods[7–10,12],suchasvariationalMonte-Carlotechnique[13],exactclusterdiagonaliza-tion[14],andvariousrealizationsofquantumMonte-Carlomethod[15].Apartfromthesenumericaltechniques,ananalyticalapproachtothet Jmodelreceivingagreatdealofattentionistheslaveparticletheory[4,16],wheretheelectronoperatorCiσispresentedas

Ciσ=a ifiσwithaiastheslavebosonandfiσasthefermion,orviceversa.Thiswaythe

non-holonomicconstraint

σ CiσCiσ≤1(1)

isconvertedintoaholonomicone

a iai+ σ fiσfiσ=1,(2)

whichmeansagivensitecannotbeoccupiedbymorethanoneparticle.Anewgaugedegreeoffreedommustbeintroducedtoincorporatetheconstraint,whichmeansthattheslaveparticlerepresentationshouldbeinvariantunderalocalgaugetransformationai→aieiθ(ri,t),fiσ→fiσeiθ(ri,t),andallphysicalquantitiesshouldbeinvariantwithrespecttothistransformation.Wecallsuchslaveparticleapproachas”conventional”.Theadvantageofthisformalismisthatthechargeandspindegreesoffreedomofelectronmaybeseparatedatthemean eldlevel,wheretheelementarychargeandspinexcitationsarecalledholons

uidsofspinonsandholonsrepresentthesamesetofelectrons,andtheymust,onaverage, owtogether.ThedecouplingofchargeandspindegreesoffreedominthelargeUlimitHubbardmodelisundoubtedlycorrectin1D[10],wherethechargedegreesoffreedomofthegroundstateareexpressedasaSlaterdeterminantofspinlessfermions,whileitsspindegreesoffreedomareequivalenttothe1DS=1

followingRef.20,toshowthatthescalingbehavioroftheelectronmomentumdistributionfunctionneark～kFandk～3kFin1Dcanbereproducedbyconsideringthenonlocalstring eldsinthemean eldapproximation,butthereisstillnoEFS,andthesumrulefortheelectronnumberisstillviolated.IfLuttingertheoremisobeyed,theFermivolumeisinvariantunderinteractions.Thusourmean eldresultseemstoindicatethatthecorrectscalingbehavioroftheelectronmomentumdistributiondoesnotguaranteetheexistenceofEFS(inthesenseofglobaldistribution)inthesametheoreticalframework.SectionIVisdevotedtoasummaryanddiscussionsonrelatedproblems.

II.FORMALSTUDYOFELECTRONMOMENTUMDISTRIBUTION

Theslaveparticletheorycanbetheslavebosonortheslavefermiontheoriesaccordingtostatisticsassignedtospinonsandholons.Theslavebosonformulation[21]isoneofthepopularmethodsoftreatingthet Jmodel.Thismethod,however,doesnotgiveagoodenergyofthegroundstate[22].Forexample,inthehalf- lledcase,wherethet JmodelreducestotheantiferromagneticHeisenbergmodel,thelowestenergystateobtainedbythismethodfailstoshowtheexpectedlong-rangeN´eelorder,andtheenergyisconsiderablyhigherthannumericalestimates[13].Also,itdoesnotsatisfytheMarshallsignrule[23].However,thistheoryprovidesaspinonFermisurface[17]eveninthemean eldapproximation.Alternatively,theslavefermionapprochnaturallygivesanorderedN´eelstateathalf- lling[24],whichobeystheMarshall[23]signrule.Thegroundstateenergyobtainedinthiscaseismuchbetterthantheslavebosoncase.Atthemean eldlevel,theyarequitedi erentalthoughinprincpletheyshouldbeequivalenttoeachother.Thedi erencesmaybereducedbygoingbeyondthemean eld[25]approximation.Inthefollowing,weusebothapproachestodiscusstheelectronmomentumdistributionfunctionandthesumruleobeyedbyit,andadirectcomparisonoftheobtainedresultsismade.Firstconsidertheslavefermionrepresentationinwhichtheelectronoperatorcanbe

withtheconstraint σbiσbiσ+e iei=1.Inthiscase,theLagrangianLsfandthepartitionfunctionZsfofthet Jmodelintheimaginarytimeτcanbewrittenas

Lsf= b iσ τbiσ+biσ 1),

iσ e i τei+H+λi(e iei+i i b iσσ

H= teie b J

jiσbjσ+

<ij>σ

ZDλe

sf DeDeDbDbieie dτLsf(τ),

1 δ= <b iσbiσ>=1

(3)(6)

′Dσ(Ri Rj,τ τ′)= <Tτbiσ(τ)b jσ(τ)>

= 1

Zsf DeDeDbDb ′′ Dλe i(τ)ej(τ)biσ(τ)bjσ(τ)e dτLsf(τ).(10)

Thespinon-holonscatteringcontainedineq.(10)isafourparticleprocess,thereforethespinonandholonarestronglycoupled.Atthemean eldlevel,theholonsandspinonsareseparatedcompletely.However,inmanytheoreticalframeworks,suchastheusualgaugetheoriesdiscussedbymanyauthors[17,26],wherethevertexcorrectionsareignored,buttheRPAbubblesareincluded,thespinonsandholonsarestillstronglycoupledbyphase uctuations(gauge elds)[17],amplitude uctuations[25]andothere ects.Nevertheless,inallthesecasestheelectronGreen’sfunctionGσ(k,iwn)canbepresentedasaconvolutionofthefermionGreen’sfunctiong(k,iwn)andbosonGreen’sfunctionDσ(k,iwn)

Gσ(k,iwn)=1

β wmg(q,iwm)Dσ(q+k,iwm+iwn).(11)

Thecouplingofthegauge eldtotheseparticlescanbestrongandapartialresummationofdigramshasbeencarriedout[17],butthevertexcorrectionswereneglected,beingtheessenceofthedecouplingapproximation.Thisisanimportantbuttheonlyapproximationapartfromthesumrulesforslaveparticles(6)-(7)inourformalstudy.Inwhatfollows,onewill ndthatmanydi cultiesmightappearduetothisapproximation.ThefermionandbosonGreen’sfunctionscanbeexpressedasfrequencyintegralsoffermionandbosonspectralfunctionsas

g(k,iwn)= ∞dw

iwn w ∞,(12)

respectively[27].Substitutingeqs.(12)and(13)intoeq.(11),weobtaintheelectronGreen’sfunctionbysummingovertheMatsubarafrequencyiwm,

Gσ(k,iwn)=

2π ∞ ∞dw′′iwn+w′ w′′,(14)

wherenF(w′)andnB(w′′)aretheFermiandBosedistributionfunctions,respectively.ThespectralfunctionsAe(q,w′)andAbσ(k,w′′)obeythesumrulescomingfromthecommutationrelations,

∞dw

∞

2πAbσ(k,w)=2.(16)

Theelectron’sHilbertspacehasbeenseverelyrestricted,butthefermionandbosonthem-selvesarenotrestricted.Byananalyticcontinuationiwn→w+iηintheelectronGreen’sfunction(14),theelectronspectralfunctionAcσ(k,w)= 2ImGcσ(k,w)canbeobtainedas

Acσ(k,w)=

2πAe(q,w′)Abσ(q+k,w+w′)[nF(w′)+nB(w+w′)].(17)

Therefore,theelectronspectralfunctionAcσ(k,w)obeysthesumrule

σ∞dw ∞

2πnB(w)Abσ(k,w),(19)

ThisisbecausenB(w)Abσ(k,w)andnF(w)Ae(k,w)canbeinterpretedastheprobabilityfunctionsofstatekwithenergywforbosonandfermion,respectevely.AsimilarinterpretionisalsovalidfortheelectronspectralfunctionAcσ(k,w).Thusthenumberofelectronsinstatekisobtainedbysummingoverallenergiesw,weightedbytheelectronspectralfunction

nc(k)= σ∞dwN ∞ qσnbσ(k+q)ne(q).(21)

Ineq.(21),the rsttermoftherighthandside1-δisindependentofk,andtherforeitistrueforallkstatesoftheentireBrillouinzone.Thevalueofthesecondtermoftherighthandsideisoftheorderofδ,andhenceitisnotenoughtorestoretheEFS,i.e.,thedistributionoutsidetheshould-beEFSisstilloftheorder1.Fig.1showsthemean eldelectronmomentumdistributionnc(k)fordopingδ=0.125in1D.Beyondthemean eldapproximation,butstillwithinthedecouplingscheme(11),therearenoimportantcorrectionsfortheglobalfeaturesoftheelectronmomentumdistributionandEFS,buttheFermivelocitywillbemodi ed.Thisisbecausetheessentialglobalfeaturesoftheelectronmomentumdistributionaredominatedbythe rsttermoftherighthandside,1 δineq.

(21),andthesecondtermoftherighthandsideineq.(21),whichisoftheorderofδ,isnotenoughtocancelout1 δateachkoutsidethekFstate,i.e.,kF<k<π,fork>0,and π<k< kF,fork<0.Sincenbσ(k+q)≥0andne(q)≥0,andaminussignappearsbetweenthe rstandthesecondtermsoftherighthandsideineq.(21),thenitisimpossibletoshifttheweight1 δinstatekoutsidetheFermipointsintostatek′insidetheFermipoints,i.e.,0<k′<kF,fork′>0,and kF<k′<0,fork′<0.Theabovediscussionsarealsotrueforthe2Dcase.Therefore,thereisnoEFSinthestandardsensewithinthedecouplingscheme(11)intheconventionalslavefermionapproach.Ifδholesareintroducedintothehalf- lledsystem,onemightexpectthatthetotalelectronnumberpersitewouldbe1-δ.However,asurprisingresultis

Alternatively,intheslavebosonrepresentation,theelectronoperatorcanbeexpressedas

Ciσ=fiσb i,wherefiσisfermionandbiisslaveboson,withthecontraint

Inthiscase,theLagrangianLsbandthepartitionfunctionZsbofthet Jmodelmaybewrittenas

Lsb= iσ fiσ τfiσ σ fiσfiσ+b ibi=1.+ ib i τbi

+H+ iλi(b ibi

iσ+ σ fiσfiσ 1),(23)H= t

+J bib jfiσfjσ µ fiσfiσ<ij>σ

Zsb

1 δ= σ fiσ>=<fiσ DbDbDfDf Dλb ibie dτLsb(τ),(26)1

2π

nc(k)=1 δ+

Acσ(k,w)=1+δ,1(28)

Incomparisonwitheq.(21),thesecondtermoftherighthandsideofeq.(29)changessignasanessentialdi erenceoftheelectronmomentumdistributionfunctionbetweentheslavebosonandslavefermionrepresentations.However,thevalueofthesecondtermoftherighthandsideofeq.(29)isalsooftheorderofδ,anditisalsonotenoughtorestoretheEFS.Fig.2showsthemean eldelectronmomentumdistributionnc(k)fordopingδ=0.125in1D.Beyondthemean eldapproximation,butstillwithinthedecouplingscheme,therearenoimportantcorrectionsfortheglobalfeaturesoftheelectronmomentumdistribution.Thereasonisalmostthesameasintheconventionalslavefermionapproach.Theessentialglobalfeaturesoftheelectronmomentumdistributionaredominatedbythe rsttermoftherighthandside1 δineq.(29).Sincenfσ(k+q)≥0andnb(q)≥0,andaplussignappearsbetweenthe rstandthesecondtermsoftherighthandsideineq.(29),itisimpossibletoremovethose1 δstatesbeyondtheFermipoints.Thebestsituationisthatanamountoforderofδisaddedintoeachk′statewithintheFermipoints.Theabovediscussionisalsovalidforthe2Dcase.Therefore,thereisnoEFSwithinthedecouplingschemeintheconventionalslavebosonapproach.Inthiscase,thesumruleoftheelectronnumberisalsoviolatedasintheslavefermioncase.Thedi erencebetweenthesetwoapproachesisthattheelectronnumberismorethantheexpectedvalueintheslavebosonrepresentation,butlessthanitintheslavefermionrepresentation.

TheseresultsindicatethatthereisnorealEFSfortheelectronmomentumdistributionfunctionnc(k)neark～kFwithinthedecouplingschemeintheconventionalslaveparticleapproach.Thesumruleofthephysicalelectronnumberisviolated.ThescalingbehavioroftheelectronmomentumdistributionnearkFisalsonotcorrectlydescribedbythisscheme.Thetheorycanbeappliedtoboth1Dand2Dsystems.Itwasproposed[17,26]thatthelowenergyphysicsofthet Jmodelcanbedescribedbyatheoryoffermionsandbosonscoupledbyagauge eld.Ourresultsalsoindicatethatthegauge elds(phase uctuations)discussedbymanyauthors[17,26,28]arenotstrongenoughtorestoretheEFSunderthedecoupling

experiments[5]thatoxidesuperconductingmaterialsexhibitanEFS,andtheEFSobeysLuttingertheorem[6],whichmeanstheFermivolumeisinvariantunderinteraction.InthissensetheEFSofstrongcorrelatedsystemsshouldalsobedescribedbyanadequatetheoryeveninthemean eldapproximation.Infact,therearrangementsofspincon guationsintheelectronhoppingprocess[10],whicharenonlocale ectsandbeyondtheconventionalslaveparticleapproach,playanessentialroletorestoreEFSforasystemofdecoupledchargeandspindegreesoffreedom.Thustheelectronisnotacompositeofholonandspinononly,anditshouldincludeother eldswhichdescribethenonlocale ects.InthenextSection,wewillseethatthescalingbehaviorofnc(k)neark～kFandk～3kFcanbeobtainedin1Dbyconsideringsomenonlocale ects.BeforegoingtothenextSection,wewouldliketoemphasizethatthe”FermiSurface”obtainedpreviously[29]fromtheslavebosontheoryisaspinonFermisurface,nottherealEFS.Inordertointerpretthehigh-resolutionangle-resolvedphotoemissionandinversephotoemissionexperiments[5],thespinonFermisurfaceisnotenoughbecauseexperimentshaveshowntheEFSforrealelectrons.

III.THESCALINGBEHAVIOROFTHEELECTRONMOMENTUM

DISTRIBUTION

FUNCTIONANDTHEEFFECTSOFNONLOCALSTRINGFIELDIN1DInteracting1DelectronsystemsgenerallybehavelikeLuttingerliquids[18]inwhichthecorrelationfunctionshavepower-lawdecaywithexponentswhichdependontheinteractionstrength.Forthe1DHubbardmodel,anexactsolutionwasexplicitlyobtainedbyLiebandWu[11].InthelimitofJ→0,thet JmodelisequivalenttothelargeUlimitHubbardmodel[1,2,10].Thusthe1Dt Jmodelprovidesagoodtestforvariousapproaches.TheasymptoticformsofsomecorrelationfunctionsaswellasthesingleelectronGreen’sfunctionhavebeenobtainedbymanyauthorsusingdi erentapproximations[20,30].Inparticular,OgataandShiba[10]obtainedtheelectronmomentumdistributionfunctionnc(k)byusing

andthecorrectscalingbehaviorofnc(k)atk～kFandk～3kF.ItiswellestablishedthattheLandauFermiliquidtheorybreaksdownin1D,namely,(1)thereisno nitejumpofthemomentumdistributionattheFermisurface;(2)thereisnoquasi-particlepropagationand(3)thespinandchargeareseparated.Ontheotherhand,thereisstillawellde nedFermisurfaceatkF,asonewouldexpectfromtheLuttingertheorem.Itisremarkablethattheexactsolutionofthe1DHubbardmodeldemonstratesexplicitlythesetwoaspectsatthesametime.Inthissenseitisimportanttocheckwhetherbothaspectsremaininanyappropriateapproximatetreatmentofthe1Dmodel.Toourknowledge,suchglobalfeaturesoftheelectronmomentumdistributionfunctionevenin1Dwereobtainedonlybyusingnumericaltechniqes.InthisSection,wetrytostudyanalyticallythisproblem.ACP1boson-solitonapproachincludingthee ectsofthenonlocalstring eldtostudythelargeUHubbardmodelwasrecentlydevelopedbyWengetal.[20].Theyhaveshownthattheelectronisacompositeparticleofholon,spinon,togetherwithanonlocalstring eld.Wewilldrawheavilyontheirresults,butwetrytomakethepresentationself-contained.Thecorrectscalingformoftheelectronmomentumdistributionfunctioncanbeobtainedeveninthemean eldapproximationifoneconsidersthenonlocalstring eld.

Forconvenience,webeginwiththet Jmodel,butconsiderthelimitJ→0+whichisa xedpointdi erentfromJ=0.Inthiscase,thet JHamiltonianmaybewrittenas

H= t CiσCjσ+h.c.(31)

<ij>σ

AsintheCP1boson-solitonapproach[20],theelectronoperatorCiσcanbeexpressedintheslavefermionrepresentationincludingthee ectsofnonlocalstring eldsas

iCiσ=e ieπ

2 l<inlisthestring

thatbiσannihilatesthespinσandisanonlocalphaseshiftGi=eiπ

2Xjhj,(34)

andtheHamiltonianmayberewrittenas

H= t iσ [b iσbi+1σhi+1hi+h.c] µ

+ iλi(h ihi+ σ ih ihi(35)b iσbiσ 1).

FollowingWengetal.

[20],wecanobtaintheasymptoticsingularbehavioroftheelectronmomentumdistributionbyasimilarcalculation,andtheresultisinagreementwiththeirs.

factorwillcontributeanadditionalpower-lawdecayintheasymptoticsingleelectronGreen’sfunction,andonemaynegelectitforsimplicityifonlyinterestedintheleadingcontribution.Themaine ectduetothenonlocalstringisalreadypresent.IntheMFA,oursituationisverysimilar.Eq.(32)holdsexactlyintheN´eellimitofthespincon guration,andthereforethequantummany-bodye ectsareneglected.Thisdoesnota ecttheleadinglong-wavelengthbehavior.Inthispaper,however,wearemainlyinterestedintheglobalfeaturesoftheelectronmomentumdistribution.IfLuttingertheorem[6]isobeyed,theFermivolumeisinvariantunderinteractionandastronglyinteractingsystemshouldalsoshowalargeEFSeveninthemean eldapproximation.Thusamean eldtreatmentisausefultestforthepresentapproach.

Themean eldapproximationtotheHamiltoian(35)amountstotreatingλiasaconstant,independentofpositionandtodecouplingthespinon-holoninteractioninaHartree-likeapproximationbyintroducingtheorderparamerters

χ= σ B<bAiσbi+1σ>,(36)

Bφ=<hAihi+1>,(37)

wherewehaveconsideredtwosublatticesAandBwithi∈A,i+1∈B.Theself-consistentequationsaboutλ,χ,φ,andµcanbeobtainedbyminimizingthefreeenergy.

UnderthesameapproximationastheoneusedtoobtaintheasymptoticsingleelectronGreen’sfunction(i.e.,Gifactorineq.(32)hasbeendropped),thesingleparticleelectronGreen’sfunctionGcσ(k,iwn)inthepresentmean eldapproxmationcanbeobtainedas

Gcσ(k,iwn)=

where

wk=λ tφγk,εk=λ µ tχγk,γk=2cos(k).(39)12(1+δ))(1+δ) wq2,(38)

Acσ(k,w)=π

2(1+δ))]δ(w+εq k±π

2π

1Acσ(k,w)=1+δ,(41)nc(k)=1 δ 2(1+δ)),(42)

2(1+δ)

duetothepresenceofnonlocalstring elds.Fig.3showstheelectronmomentumdistribu-tionnc(k)inthemean eldapproximation,includingthenonlocalstring eldsfordopingδ=0.125in1D.ThesingularbehavioroftheelectronmomentumdistributionfunctionatkFand3kFisinqualitativeagreementwiththenumericalresults[10],butthesumruleforthetotalelectronnumberisstillviolatedandEFSstilldoesnotexistinthepresentframeworkundertheaboveapproximations.Beyondtheseapproximations,thesituationisnotclearyet.IfLuttingertheorem[6]isobeyed,thentherearenoimportantcorrectionstotheglobalfeaturesoftheelectronmomentumdistributionfunctionandnoEFSbeyondthemean eldapproximationinthesametheoreticalframework.Sincetherearrangementsofquantumspincon gurationsinthehoppingprocessplayanessentialroletorestoretheEFS[10],theneq.(32)whichholdsexactly[20]intheN´eellimitofthespincon guration,isperhapsnotenoughtodescribeallofthesequantumspincon gurations.Thusourfeelingisthatinthistheoreticalframework,thecorrectsingularbehaviorofelectronmomentumdistributionmaybeobtained,butitdoesnotguaranteetheexistenceofEFS.Wealsonotethatonlytheasymptoticformswerediscussedinthepreviousworksbymanyauthors[30],whiletheglobalfeaturesofthemomentumdistributionfunctionhavenotbeenconsideredbytheirtheoreticalmethods.

IV.SUMMARYANDDISCUSSIONS

Wehavediscussedtheelectronmomentumdistributionfunctionandthesumruleoftheelectronnumberinthet Jmodelbyusingtheslaveparticleapproach.Underthedecouplingscheme(11)usedinthegauge eld[17]andrelatedtheories,wehaveformallyprovedthatEFScannotberestoredandthesumruleofelectronnumberisviolatedintheframeworkofconventionalslaveparticleapproach.For1Dt Jmodel,thecorrectsingularityformscanbereproducedbyconsideringthenonlocalstring eldsinthespecialmean eldapproximation,butthereisstillnoEFSandthesumruleisstillnotsatis ed.

Inthepresentslaveparticleapproachtostudyt Jmodel,wehavepushedouttheupperHubbardband.Thismeanswehaveneglectedthedoublyoccupiedsitesintheoriginalslaveparticleapproach[21]:Ciσ=a ifiσ+σdifi σ,withaconstraintaiai+

wherea iistheslaveboson,f f σiσfiσ+d idi=1,iσisfermion,anddiisbosonwhichdescribesthedoublyoccupiedsites,orviceversa.Infactintheoriginalslaveparticleapproachtheelectronspectralfunctionobeysthesumruleofconventionalelectron,i.e., σ ∞ ∞dw

vertexcorrectionsareneglected,i.e.,thedecouplingschemeisadopted,thedi cultywillremain.Ontheotherhand,itisdi cultytointroducevertexcorrectionsinthepresentformofthegauge eldtheory,becausetheinfrareddivergencehasnotyetbeenproperlyhandled.Anotherpossibilitytoavoidthisdi cultyisthatoneshouldmakeanewinterpretationoftheconstraint(2)andthephysicalmeaningofspinonsandholons.Infacttheconstraint

(2)isanoperatoridentity,andonereplacestheconstraint(2)byeqs.(6)and(7)intheslavefermionapproach,oreqs.(26)and(27)intheslavebosoncase.Itisnotclearthatthisisacorrectwayofimposingtheconstraint,i.e.,thechargeisrepresentedbyafermion,whilethespinisrepresentedbyabosonintheslavefermionapproachandviceversaintheslavebosonversion.Thecrucialpointistoimplementthelocalconstraintrigorously[34].Finally,accordingtotheimplicationsofnumericalsolutions[10],theelectronisperhapsnotacompositeofholonandspinononly,butitrathershouldbeacompositeofholon,spinontogetherwithsomethingelse.ThisisbecausetheBethe-ansatzLieb-Wu’sexactwavefunction[10]for1DHubbardmodelatlargeUlimitmaybewrittenas

Ψ(x1,···,xN)=( 1)Qdet[exp(ikixQi)]Φ(y1,···,yM),(44)

wherethedeterminantdependsonlyonthecoordinatesofparticles(xQ1<···<xQN)but

notontheirspins.ThusitisthesameastheSlaterdeterminantofspinlessfermionswithmomentakj’s.Thespinwavefunctionφ(y1,···,yM)isthesameastheBethe’sexactsolutionof1DHeisenbergspinsystem.Inthesecondquantizationrepresentation,aspinlessfermionoperatormayberesponsiblefortheSlaterdeterminantofspinlessfermions,butasimplefermionorbosonoperatormaybenotenoughtodescribethespinwavefunction.Perhapsitshouldbeafermionorbosonoperatortogetherwithsomethingelse,suchastheJordan-Wigner’sform[35]in1D,oramapofthethreespinoperatorsoftheSU(2)algebraontoacoupleofcanonicalbosonoperators,liketheHolstein-Primako form[36],areresponsibleforthespinwavefunction.Webelievethatasuccessfultheoreticalframeworkmustinclude

Tosummarize,wearefacingadilemma:ifwewouldliketosplitoneelectronintotwoparticles,onekeepingtrackofthecharge,theotherkeepingtrackofthespin,andimposecorrespondingsumrulesonthem[eqs.(6)-(7)oreqs.(26)-(27)]andthenusethedecouplingschemefortheirexpectationvalues,theresultingelectrondistributiondoesnotsatisfythesumruleanddoesnotshowanEFS.Inthissense,wehaveprovedinthispaperaNo-Gotheorem.Thealternativewouldbetogiveuptheattractivelysimpleinterpretationofspinonsandholonsandtolookformorecomplicatedchargeandspincollectiveexcitations.Finallywenotethatthepresentslaveparticleapproachisdi erentfromthose rstproposedbyBarnes[37]andrediscoveredandextendedbyColeman[38],ReadandNewns

[39],andKotliarandRuckenstein[40]intheirworksonthemixedvalenceproblemandtheheavyfermionsystems.Intheirformulation,thespinandchargedegreesoffreedomarenotdecoupled,wheretheauxiliarybosonskeeptrackonlyoftheenvironmentbymeasuringtheoccupationnumbersineachofthepossiblestatesforelectronhopping.TheirtheorydescribesthepropertiesofaFermiliquid,andisanotherstory.

ACKNOWLEDGMENTS

TheauthorswouldliketothankDr.Z.Y.Wengforhelpfuldiscussions.S.P.FengandZ.B.SuwouldliketothanktheICTPforthehospitality.

REFERENCES

OnleavefromtheDepartmentofPhysics,BeijingNormalUniversity,Beijing100875,

China.

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