Singular or non-Fermi liquids(10)

Singular or non-Fermi liquids

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267–417

Fig.4.Bare-particledistributionatT=0foragivenspindirectioninatranslationallyinvariantFermisystemwithinteractions(fullline)andwithoutinteractions(dashedline).Notethatthepositionofthediscontinuity,i.e.,theFermiwavenumberkF,isnotrenormalizedby

interactions.

Fig.5.(a).Thenon-interactingspectralfunctionA(k;!)atÿxedkasafunctionof!;(b)thespectralfunctionofsingle-electronexcitationsinaFermiliquidatÿxedkasafunctionof!.If(1= )A(k;!)isnormalizedto1,signifyingonebareparticle,theweightundertheLorentzian,i.e.,thequasiparticlepart,isZ.Asexplainedinthetext,atthesametimeZisthediscontinuityinFig.4.

thesingle-particleGreen’sfunctionG(k;!)isdeÿnedintermsofthecorrelationfunctionofparticlecreationandannihilationoperatorsinstandardtextbooks[195,4,222,168].Forourpresentpurpose,itissu cienttonotethatitisrelatedtothespectralfunctionA(k;!),whichhasaclearphysicalmeaningandwhichcanbededucedthrough-angleresolvedphotoemissionexperiments

∞

A(k;x)

G(k;!)=dx:(5)

∞A(k;!)thusisthespectralrepresentationofthecomplexfunctionG(k;!).Herewehavedeÿned

theso-calledretardedGreen’sfunctionwhichisespeciallyusefulsinceitsrealandimaginary