Singular or non-Fermi liquids(12)

Singular or non-Fermi liquids

278C.M.Varmaetal./PhysicsReports361(2002)267–417

asthesumoftwoterms,A(k;!)=Acoh(k;!)+Aincoh(k;!).Thesingle-particleGreen’sfunc-tioncansimilarlybeexpressedasasumoftwocorrespondingterms,G(k;!)=Gcoh(k;!)+Gincoh(k;!).Then

Zk

Gcoh(k;!)=;(9)

kkwhichforlargelifetimes kgivesaLorentzianpeakinthespectraldensityatthequasiparticle

energy k≡ k .TheincoherentGreen’sfunctionissmoothandhenceforlarge kcorrespondstothesmoothbackgroundinthespectraldensity.

Theconditionfortheoccurrenceofthewell-deÿnedfeaturecanbeexpressedastheconditionthattheself-energy (k;!)hasananalyticexpansionabout!=0andk=kFandthatitsrealpartismuchlargerthanitsimaginarypart.Onecaneasilyseethatwereitnotso,thenexpression(9)forGcohcouldnotbeobtained.TheseconditionsarenecessaryforaLandauFermiliquid.Uponexpanding (k;!)in(12)forsmall!andsmalldeviationsofkfromkFandwritingitintheform(9),wemaketheidentiÿcations

k;1= ZkIm (kF;!=0); k= kZkZ(10)

k

where

19 9 1

=1+;Z:(11)Zk=1

!=0;k=kFF!=0;k=kFFromEq.(8),wehaveamorephysicaldeÿnitionofZk:Zkistheprojectionamplitudeof

N+1| k ontothestatewithonebareparticleaddedtothegroundstate,sinceallothertermsintheexpansionvanishinthethermodynamiclimitintheperturbativeexpressionembodiedby(8):

1=2N+1 NZk= k|ck| :

(12)

Inotherwords,ZkistheoverlapofthegroundstatewavefunctionofasystemofinteractingN±1fermionsoftotalmomentumkwiththewavefunctionofNinteractingparticlesandabareparticleofmomentumk.Zkiscalledthequasiparticleamplitude.

TheLandautheorytacitlyassumesthatZkisÿnite.Furthermore,itassertsthatforsmall!andkclosetokF,thephysicalpropertiescanbecalculatedfromquasiparticleswhichcarrythesamequantumnumbersastheparticles,i.e.,charge,spinandmomentumandwhichmaybedeÿnedsimplybythecreationoperator k; :

N+1N

| k = k; | :

(13)

ClosetokF,andforTsmallcomparedtotheFermienergy,thedistributionofthequasiparticlesisassumedtobetheFermi–Diracdistributionintermsoftherenormalizedquasiparticleen-ergies.Thebareparticledistributionisquitedi erent.AsisillustratedinFig.4,itisdepletedbelowkFandaugmentedabovekF,withadiscontinuityatT=0whosevalueisshowninmicroscopictheorytobeZk.AcentralresultofFermiliquidtheoryisthatclosetotheFermienergyatzerotemperature,thewidth1= kofthecoherentquasiparticlepeakisproportionalto( k )2sothatneartheFermienergythelifetimeislongandquasiparticlesarewell-deÿned.