Singular or non-Fermi liquids

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

287

Fig.13.DiagrammaticrepresentationofEq.(27).

where pistobeidentiÿedastheexcitationenergyofthequasiparticle,Zitsweight,andGinctheincoherentnon-singularpartofG.(ThelatterprovidesthesmoothbackgroundpartofthespectralfunctioninFig.5(b)andtheformerthesharppeak,whichisproportionaltothe functionfor p= p .)Itfollows[195,4]from(28)that

2i z2vq·k

G(q)G(q+k)= ( ) (|q| pF)+ (q)

Fq(29)

forsmallkand!,andwhere and( +!)arefrequenciesofthetwoGreen’sfunctions.NotethecrucialroleofkinematicsintheformoftheÿrsttermwhichcomesfromtheproductofthequasiparticlepartsofG; (q)comesfromthescatteringoftheincoherentpartwithitselfandwiththecoherentpartandisassumedsmoothandfeatureless(asitisindeed,giventhatGincissmoothandfeaturelessandthescatteringdoesnotproduceaninfraredsingularityatleastperturbativelyintheinteraction).Thevertex inregionsclosetok≈kFand!≈0isthereforedominatedbytheÿrstterm.ThederivationofFermi-liquidtheoryconsistsinprovingthatEqs.(27)forthevertexand(28)fortheGreen’sfunctionaremutuallyconsistent.Theproofproceedsbydeÿningaquantity !(p1;p2;k)through

d4q!(1)(1)!

(p1;p2;k)= (p1;p2;k) i (p1;q;k) (q) (q;p2;k):(30)

!containsrepeatedscatteringoftheincoherentpartoftheparticle–holepairsamongitselfandwiththecoherentpart,butnoscatteringofthecoherentpartwithitself.Then,providedtheirreduciblepartof (1)issmoothandnottoolarge, !issmoothinkbecause (q)isbyconstructionquitesmooth.

Usingthefactthattheÿrstpartof(29)vanishesforvF|k|=!→0,andcomparing(27)and(30)onecanwritetheforwardscatteringamplitude

limlim (p1;p2;k)= !(p1;p2):(31)

!→0k→0