Singular or non-Fermi liquids(12)
时间:2025-02-23
时间:2025-02-23
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.
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