Singular or non-Fermi liquids(19)

Singular or non-Fermi liquids

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

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Fig.12.(a)ThenestedFermisurfaceobtainedinatightbindingmodelonasquarelatticewithnearest-neighborhopping;(b)apartiallynestedFermisurfacewhichleadstocharge-densitywaveorantiferromagneticinstabilities.

thisparticle–holescatteringchanneltoRe (q;!)is

kF

1

dk～ln[(!+qvF)=EF]:F0

(25)

(Notethat(25)istrueforbothq kFand|q 2kF| kF.)Thisinturnleadstoasingle-particleself-energycalculatedbytheprocessinFig.9tobeRe (kF;!)～!ln!andso

Z～ln!givingahintoftrouble.TheCooper(particle–particle)channelhasthesamephase-spacerestrictions,andgivesacontributiontoRe (kF;!)proportionalto!ln!too.Thefactthatthesesingularcontributionsareofthesameorder,leadstoacompetitionbetweencharge=spin uctuationsandCooperpairing uctuations,andintheexactcalculationtopower-lawsingularities.Thefactthatinsteadofthecontinuumoflow-energyexcitationspresentinhigherdimensions,thewidthofthebandofallowedparticle–holeexcitationsvanishesas!→0,isthereasonthatthepropertiesofone-dimensionalinteractingmetalscanbeunderstoodintermsofbosonicmodes.Wewillpresentabriefsummaryoftheresultsforthesingle-particleGreen’sfunctionandcorrelationfunctionsinSection4.9.

Inspecialcasesofnestingintwoorthreedimensions,onecanhavesituationsthatresembletheone-dimensionalcase.Whenthenon-interactingFermisurfaceinatightbindingmodelhasthesquareshapesketchedinFig.12(a)(whichoccursforatight-bindingmodelwiththenearestneighborhoppingonasquarelatticeathalf-ÿlling)acontinuousrangeofmomentaonoppositesidesoftheFermisurfacecanbetransformedintoeachotherbyoneandthesamewavenumber.Thisso-callednestingleadstologandlog2singularitiesforacontinuousrangeofkintheperturbationtheoryfortheself-energy (k;!).Likewise,thepartiallynestedFermisurfaceofFig.12(b)leadstochargedensitywaveandantiferromagneticinstabilities.WewillcomebacktotheseissuesinSections2.6and6.