Singular or non-Fermi liquids

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Fig.9.Thesingle-particleself-energydiagraminsecondorder.

thislimit,wecanmakeanyexcitationwithmomentumq62kF.Forÿxedbutsmallvaluesofq,themaximumexcitationenergyis!≈qvF;thisoccurswhenqisinthesamedirectionasthemainmomentumkofeachquasiparticle.Forqnear2kF,themaximumpossibleenergyis!=vF|q 2kF|.Combiningtheseresults,weobtainthesketchinFig.8(a),inwhichtheshadedareainthe!–qspaceistheregionofallowedparticle–holeexcitations.6Fromthisspectrum,onecancalculatethepolarizability,orthemagneticsusceptibility.

Thebehaviorsketchedaboveisvalidgenerallyintwoandthreedimensions(butaswewillseeinSection4,notinonedimension).Theimportantpointtorememberisthatthedensityofparticle–holeexcitationsdecreaseslinearlywith!for!smallcomparedtoqvF.WeshallseelaterthatonewayofundoingFermi-liquidtheoryistohave!～k2intwodimensionsor!～k3inthreedimensions.

WecannowuseIm (q;!)tocalculatethesingle-particleself-energytosecondorderintheinteractions.ThisisshowninFig.9wherethewigglylinedenotes (q; )whichinthepresentapproximationisjustgivenbythediagramofFig.7(a).

Fortheperturbativeevaluationofthisprocess,theintermediateparticlewithenergy–momentum(!+ );(k+q)isafreeparticle.Second-orderperturbationtheorythenyieldsanimaginarypart,oradecayrate,

2

1!

Im (k;!)=(23)=g2N(0)

Finthreedimensionsfork≈kF.Intwodimensions,thesameprocessyieldsIm (kF;!)～

!2ln(EF=!).

The!2decayrateisintimatelyrelatedtotheanalyticresult(22)forIm (q;!)exhibitedinFig.(8).Asmaybefoundintextbooks,thesamecalculationforelectron–phononinteractionsorforinteractionwithspinwavesinanantiferromagneticmetalgivesIm (kF;!)～(!=!c)3,where!cisthephononDebyefrequencyintheformerandthecharacteristiczone-boundaryspin-wavefrequencyinthelatter.

Therealpartoftheself-energymaybeobtaineddirectlyorbyKramers–Kronigtransforma-tionof(23).Itisproportionalto!.Therefore,ifthequasiparticleamplitudeZkFisevaluated

Inthepresenceoflong-rangeCoulombinteractions,inadditiontotheparticle–holeexcitationspectrumassociatedwiththescreened(andhencee ectivelyshort-ranged)interactionsonegetsacollectivemodewithaÿniteplasma

√

frequencyasq→0ind=3anda!～behaviorind=2.Theplasmamodeisahigh-frequencymodeinwhichthemotionofthelightelectronscannotbefollowedbytheheavyions:screeningisabsentinthisregimeandthelong-rangeCoulombinteractionsthengiverisetoaÿniteplasmafrequencyind=3.

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