Singular or non-Fermi liquids

286C.M.Varmaetal./PhysicsReports361(2002)267–417

2.4.PrinciplesofthemicroscopicderivationofLandautheory

Inthissection,wewillsketchhowtheconclusionsintheprevioussectionbasedonsecond-orderperturbationcalculationaregeneralizedtoallordersinperturbationtheory.Thissectionisslightlymoretechnicalthantherest;thereadermaychoosetoskiptoSection2.6.

WefollowthemicroscopicapproachwhosefoundationswerelaidbyLandauhimselfandwhichisdiscussedindetailinexcellenttextbooks[197,208,37,4].Formorerecentmethodswiththesameconclusions,see[237,128].OuremphasiswillbeonhighlightingtheassumptionsinthetheorysothatinthenextsectionwecansummarizetheroutesbywhichtheFermi-liquidtheorymaybreakdown.Theseassumptionsareusuallynotstatedexplicitly.

Thebasicideaisthatduetokinematicconstraints,anyperturbativeprocesswithnparticle–holepairsintheintermediatestateprovidescontributionstothepolarizabilityproportionalto(!=EF)n.Therefore,thelow-energypropertiescanbecalculatedwithprocesseswiththesame“skeletal”structureasthoseinFig.7,whichhaveonlyoneparticle–holepairintheintermediatestate.Soonemayconcentrateonthemodiÿcationofthefour-leggedverticesandthesingle-particlepropagatorsduetointeractionstoallorders.Accordingly,thetheoryisformulatedintermsofthesingle-particleGreen’sfunctionG(p)andthetwo-bodyscatteringvertex

(p1;p2;p1+k;p2 k)= (p1;p2;k):

(26)

Hereandbelowweuse,forthesakeofbrevity,p,etc.todenotetheenergy–momentumfourvector(p;!)andwesuppressthespinlabels.Theequationfor isexpandedinoneofthetwoparticle–holechannelsas9

d4q(1)(1)

(p1;p2;k)= (p1;p2;k) i (p1;q;k)G(q)G(q+k) (q;p2;k);(27)

where (1)istheirreduciblepartintheparticle–holechannelinwhichEq.(27)isexpressed.Inotherwords, (1)cannotbesplitintotwopartsbycuttingtwoGreen’sfunctionlineswithtotalmomentumk.So (1)includesthecompletevertexintheother(oftencalledcross-)particle–holechannel.ThediagrammaticrepresentationofEq.(27)isshowninFig.13.Inthesimplestapproximation (1)ndautheoryassumesthat (1)hasnosingularities.10AnassumptionisnowfurthermadethatG(p)doeshaveacoherentquasiparticlepartat|p| pFand! 0:

G(p)=

Z

+Ginc;

pp(28)

Tosecondorderintheinteractionsthecorrectiontothevertexinthetwopossibleparticle–holechannelshas

beenexhibitedintheÿrsttwopartsofFig.7.10

ThetheoryhasbeengeneralizedforCoulombinteractions[208,197,4].Thegeneralresultsremainunchangedbecauseascreenedshort-rangeinteractiontakestheplaceof (1).Thisisunlikelytobetrueinthecriticalregionofametal–insulatortransition,becauseontheinsulatingside,theCoulombinteractionisunscreened.

9