Slave Particle Studies of the Electron Momentum Distribution(2)

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

σ

(3)(6)

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

′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)

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

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

Gσ(k,iwn)=

1

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)=

1

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)

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

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

1

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

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

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