Singular or non-Fermi liquids(15)

Singular or non-Fermi liquids

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

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Fig.7.Thethreesecond-orderprocessesinaperturbativecalculationofthecorrectiontothebareinteractioninaFermiliquid.

Wewillexplicitlyconsideronlyshort-rangeinteractionsinthissection,sothattheycanbecharacterizedatallmomentumtransfersbyasingleparameter.Nevertheless,theessentialresultsofLandautheoryremainvalidinthepresenceofCoulombinteractionsbecausescreeningmakestheinteractionsessentiallyshort-ranged.Thecouplingconstantgbelowmaythenbeconsideredtoparametrizethescreenedinteraction.

InFig.7,weshowthethreepossibleprocessesthatariseinsecond-orderperturbationtheoryforthescatteringoftwoparticleswithÿxedinitialenergy!andmomentumq.Notethatintwoofthediagrams,Fig.7(a)and(b)theintermediatestatehasaparticleandaholewhiletheintermediatestateindiagram7(c)hasapairofparticles.

Wewillÿndthat,forourpresentpurpose,thecontributionofdiagram7(a)ismoreimportantthantheothertwo.Itgivesacontribution

g

2

k

fk+q fk

:

k+qk(19)

Here,gisameasureofthestrengthofthescatteringpotential(thevertexinthediagram)inthelimitofsmallq.Thedenominatorensuresthatthelargestcontributiontothescatteringcomesfromsmallscatteringmomentaq:forthesetheenergydi erenceislinearinq;Ek+q Ek≈q·vk,wherevkisavectoroflengthvFinthedirectionofk.Moreover,theterminthenumeratorisnon-zeroonlyintheareacontainedbetweentwocircles(ford=2)orspheres(ford=3)withtheircentersdisplacedbyq—herethephase-spacerestrictionisduetothePauliprinciple.Thisareaisalsoproportionaltoq·vk,andsointhesmallqapproximationfromdiagram7(a)wegetatermproportionalto

g2

q·vkdf

:

kk

(20)

Nowweseewhydiagram7(a)isspecial.Thereisasingularityat!=q·vkanditsvalueforsmall!andqdependsonwhichofthetwoissmaller.Thissingularityisresponsibleforthelow-energylong-wavelengthcollectivemodesoftheFermiliquidinLandautheory.Atlowtemperatures,df=d k= ( k ),sothesummationisrestrictedtotheFermisurface.Therealpartof(19)thereforevanishesinthelimitqvF=!→0,whileitapproachesaÿnitelimit