Singular or non-Fermi liquids(16)

Singular or non-Fermi liquids

282C.M.Varmaetal./PhysicsReports361(2002)

267–417

Fig.8.(a)Restrictiononallowedparticle–holeexcitationsinaFermiseaduetokinematics.Theplasmonmodehasbeendrawnforthecased=3;(b)theabsorptivepartoftheparticle–holesusceptibility(inthecharge,currentandspinchannels)for!¡qvFintheFermigas.

for!→0.Theimaginarypartinthislimitisproportionalto4!:

!

Im (q;!)=g2 N(0)for!¡qvF;

F

(21)

whileIm (q;!)=0for!¿vFq.ThisbehaviorissketchedinFig.8(b).Anexplicitevaluationfortherealpartyields

!! qvF 2 ;Re (q;!)=gN(0)1+ln (22)FF whichgivesaconstant(leadingtoaÿnitecompressibilityandspinsusceptibility)at!small

comparedtoqvF.Fordiagram7(b),wegetaterm! (Ep1 p2+k+q Ek)inthedenominator.Thistermisalwaysÿniteforgeneralmomentap1andp2,andhencethecontributionfromthisdiagramcanalwaysbeneglectedrelativetotheonefrom7(a).Alongsimilarlines,oneÿndsthatdiagram7(c),whichdescribesscatteringintheparticle–particlechannel,isirrelevantexceptwhenp1= p2,whenitdivergesasln!.

Ofcourse,ndaunoticedthissingularitybutignoreditsimplication.5Indeed,aslongasthee ectiveinteractionsdonotfavorsuperconductivityoraslongasweareattemperaturesmuchhigherthanthesupercon-ductingtransitiontemperature,itisnotimportantforFermi-liquidtheory.

Letusnowlookfurtherattheabsorptivespectrumofparticle–holeexcitationsintwoandthreedimensions,i.e.,weexaminetheimaginarypartofEq.(19).Whenthetotalenergy!ofthepairissmall,boththeparticleandtheholehavetoliveclosetotheFermisurface.In

Thisbehaviorimpliesthatthisscatteringcontributionisamarginaltermintherenormalizationgroupsense,whichmeansthatita ectsthenumericalfactors,butnotthequalitativebehavior.5

Attractiveinteractionsinanyangularmomentumchannel(leadingtosuperconductivity)arethereforemarginallyrelevantoperators.

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