Singular or non-Fermi liquids

280C.M.Varmaetal./PhysicsReports361(2002)267–417

Heresubscripts0refertothequantitiesofthenon-interactingreferencesystem,andmisthemassofthefermions.ForaGalileaninvariantsystem(like3He),thereisasimplerelation

s,andthereisnorenormalizationbetweenthemassenhancementandtheLandauparameterF1

oftheparticlecurrentj;however,thereisarenormalizationofthevelocity:onehas

s Fm

j=k=m;v=k=m ;(18)=1+:

Thetransportpropertiesarecalculatedbydeÿningadistributionfunction n(k ;r;t)whichis

slowlyvaryinginspaceandtimeandwritingaBoltzmannequationforit[208,37].

ItisadelightfulconceitoftheLandautheorythattheexpressionsofthelow-energyproper-tiesintermsofthequasiparticlesinnoplaceinvolvethequasiparticleamplitudeZk.Infactinatranslationallyinvariantproblemsuchasliquid3He;Zkcannotbemeasuredbyanythermo-dynamicortransportmeasurements.AmasterlyuseofconservationlawsensuresthatZ’scanceloutinallphysicalproperties(onecanextractZfrommeasurementofthemomentumdistri-bution.Byneutronscatteringmeasurements,itisfoundthatZ≈1=4[112]forHe3nearthemeltingline).Thisisnolongertrueonalattice,intheelectron–phononinteractionproblem[212]orinheavyfermions[265]orevenmoregenerallyinanysituationwheretheinter-actingproblemcontainsmorethanonetypeofparticlewithdi erentcharacteristicfrequencyscales.

2.3.UnderstandingmicroscopicallywhyFermi-liquidtheoryworks

LetustrytounderstandfromamoremicroscopicapproachwhytheLandautheoryworkssowell.Wepresentaqualitativediscussioninthissubsectionandoutlinetheprincipalfeaturesoftheformalderivationinthenextsubsection.

Aswealreadyremarked,acrucialelementintheapproachistochoosethepropernon-interactingreferencesystem.ThatthisispossibleatallisduetothefactthatthenumberofstatestowhichanaddedparticlecanscatterduetointeractionsisseverelylimitedduetothePauliprinciple.Asaresult,non-interactingfermionsareagoodstablesystemtoperturbabout;theyhaveaÿnitecompressibilityandsusceptibilityinthegroundstate,andsocollectivemodesandthermodynamicquantitieschangesmoothlywhentheinteractionsareturnedon.Thisisnottruefornon-interactingbosonswhichdonotsupportcollectivemodeslikesoundwaves.Soonecannotperturbaboutthenon-interactingbosonsasareferencesystem.

LandaualsolaidthefoundationsfortheformaljustiÿcationofFermiliquidtheoryintwoandthreedimensions.The urryofactivityinthisÿeldfollowingthediscoveryofhigh-TcphenomenahasledtonewwaysofjustifyingFermi-liquidtheory(andunderstandingwhytheone-dimensionalproblemisdi erent).However,theprincipalphysicalreason,whichwenowdiscuss,remainsthephase-spacerestrictionsduetokinematicalconstraints.

WelearnedinSection2.2thatinordertodeÿnequasiparticles,itwasnecessarytohaveaÿniteZkF,whichinturnneededaself-energyfunction (kF;!)whichissmoothnearthechemicalpotential,i.e.,at!=0.LetusÿrstseewhyaFermigashassuchpropertieswheninteractionsareintroducedperturbatively.