Singular or non-Fermi liquids

C.M.Varmaetal./PhysicsReports361(2002)267–417273

ndau’sFermiliquid

2.1.EssentialsofLandauFermiliquids

ThebasicideaunderlyingLandau’sFermi-liquidtheory[152,153,208,37]isthatofanalyt-icity,i.e.,thatstateswiththesamesymmetrycanbeadiabaticallyconnected.Simplyput,thismeansthatwhetherornotwecanactuallycarryoutthecalculationweknowthattheeigen-statesofthefullHamiltonianofthesamesymmetrycanbeobtainedperturbativelyfromthoseofasimplerHamiltonian.Atthesametimestatesofdi erentsymmetrycannotbeobtainedby“continuation”fromthesamestate.Thissuggeststhatgivenatoughproblemwhichisim-possibletosolve,wemayguessarightsimpleproblem.Thelowenergyandlongwavelengthexcitations,aswellasthecorrelationandtheresponsefunctionsoftheimpossibleproblembearaone-to-onecorrespondencewiththesimplerproblemintheiranalyticproperties.Thisleavesÿxingonlynumericalvalues.Thesearetobedeterminedbyparameters,theminimumnumberofwhichisÿxedbythesymmetries.Experimentsoftenprovideanintuitionastowhattherightsimpleproblemmaybe:fortheinteractingelectrons,inthemetallicrangeofdensities,itistheproblemofkineticenergyofparticleswithFermistatistics.(Ifonehadstartedwiththeoppositelimit,justthepotentialenergyalone,thestartinggroundstateistheWignercrystal—abadplacetostartthinkingaboutametal!)Ifwestartwithnon-interactingfermions,andthenturnontheinteractions,thequalitativebehaviorofthesystemdoesnotchangeaslongasthesystemdoesnotgothrough(oriscloseto)aphasetransition.Owingtotheanalyticity,wecanevenconsiderstronglyinteractingsystems—thelow-energyexcitationsinthesehavestronglyrenormalizedvaluesoftheirparameterscomparedtothenon-interactingproblem,buttheirqualitativebehavioristhesameasthatofthesimplerproblem.

TheheavyfermionproblemprovidesanextremeexampleofthedomainofvalidityoftheLandauapproach.ThisisillustratedinFig.2,whichshowsthespeciÿcheatoftheheavyfermioncompoundCeAl3.AsintheSommerfeldmodel,thespeciÿcheatislinearinthetemperatureatlowT,butifwewriteCv≈ Tatlowtemperatures,thevalueof isaboutathousandtimesaslargeasonewouldestimatefromthedensityofstatesofatypicalmetal,usingthefreeelectronmass.ForaFermigas,thedensityofstatesN(0)attheFermienergyisproportionaltoane ectivemassm :

m kF

N(0)=;

(1)

withkFtheFermiwavenumber.Thenthefactthatthedensityofstatesatthechemicalpotentialisathousandtimeslargerthanfornormalmetalscanbeexpressedbythestatementthatthee ectivemassm ofthequasiparticlesisathousandtimeslargerthanthefreeelectronmassm.Likewise,asFig.3shows,theresistivityofCeAl3atlowtemperaturesincreasesasT2.ThisalsoisacharacteristicsignofaFermiliquid,inwhichthequasiparticlelifetime attheFermisurface,determinedbyelectron–electroninteractions,behavesas ～1=T2.2However,justas

Inheavyfermions,atleastintheobservedrangeoftemperatures,thetransportlifetimedeterminingthetemper-aturedependenceofresistivityisproportionaltothesingle-particlelifetime.

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