Singular or non-Fermi liquids

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

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Fig.6.Schematicillustrationoftheperturbativeexpansion(8)ofthechangeofwavefunctionasaresultoftheadditionofanelectrontotheFermiseaduetointeractionswiththeparticlesintheFermisea.

partsobeytheKramers–Kronigrelations.IntheproblemwithinteractionsG(k;!)willdif-ferfromG0(k;!).Thisdi erencecanbequitegenerallydeÿnedthroughthesingle-particleself-energyfunction (k;!):

(G(k;!)) 1=(G0(k;!)) 1 (k;!):

Eq.(5)ensurestherelationbetweenG(k;!)andA(k;!)

1

A(k;!)= ImG(k;!):

(7)(6)

Withthesepreliminariesoutoftheway,letusconsidertheformofA(k;!)whenweaddaparticletoaninteractingsystemoffermions.

Duetotheinteraction(assumedrepulsive)betweentheaddedparticleandthosealreadyintheFermisea,theaddedparticlewillkickparticlesfrombelowtheFermisurfacetoabove.Thepossibletermsinaperturbativedescriptionofthisprocessareconstrainedbytheconser-vationlawsofcharge,particlenumber,momentumandspin.ThosewhichareallowedbytheseconservationlawsareindicatedpictoriallyinFig.6,andleadtoanexpressionofthetype

1=2 N+1N

| k =Zkck |

+

1

k1;k2;k3 1; 2; 3

k1 1k2 2k3 3ckcck2k13

× k;k1 k2+k3 ( ; 1; 2; 3)| N +::::

(8)

’sandck’sarethebareparticlecreationandannihilationoperators,andthedotsindi-Heretheck

catehigher-orderterms,forwhichtwoormoreparticle–holepairsarecreatedand ( ; 1; 2; 3)expressesconservationofspinundervectoraddition.Themultiple-particle–holepairsforaÿxedtotalmomentumcanbecreatedwithacontinuumofmomentumsoftheindividualbareparticlesandholes.Therefore,anaddedparticlewithÿxedtotalmomentumhasawidedistributionofenergies.However,ifZkdeÿnedbyEq.(8)isÿnite,thereisawell-deÿnedfeatureinthisdis-tributionatsomeenergywhichisingeneraldi erentfromthenon-interactingvalue 2k2=(2m).ThespectralfunctioninsuchacasewillthenbeasillustratedinFig.5(b).Itisusefultoseparatethewell-deÿnedfeaturefromthebroadcontinuumbywritingthespectralfunction