Slave Particle Studies of the Electron Momentum Distribution(3)

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

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

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

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

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

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.

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

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.