Singular or non-Fermi liquids

C.M.Varmaetal./PhysicsReports361(2002)267–417279

Likewise,attheFermienergy1= kvarieswithtemperatureasT2.Fromthemicroscopicderiva-tionofthisresult,itfollowsthattheweightinthispeak,Zk,becomesequaltothejumpZinnk whenweapproachtheFermisurface:Zk→Zfork→kF.Forheavyfermions,aswealreadymentioned,Zcanbeoftheorderof10 3.However,aslongasZisnon-zero,onehasFermiliquidpropertiesfortemperatureslowerthanaboutZEF.Degeneracyise ectivelylostfortemperaturesmuchhigherthanZEFandclassicalstatisticalmechanicsprevails.3

Anadditionalresultfrommicroscopictheoryistheso-calledLuttingertheorem,whichstatesthatthevolumeenclosedbytheFermisurfacedoesnotchangeduetointeractions[195,4].ThemathematicsbehindthistheoremisthatwiththeassumptionsofFLT,thenumberofpolesintheinteractingGreen’sfunctionbelowthechemicalpotentialisthesameasthatforthenon-interactingGreen’sfunction.Recallthatthelatterisjustthenumberofparticlesinthesystem.

LandauactuallystartedhisdiscussionoftheFermiliquidbywritingtheequationforthedeviationofthe(Gibbs)freeenergyfromitsgroundstatevalueasafunctionalofthedevia-tionofthequasiparticledistributionfunctionn(k; )fromtheequilibriumdistributionfunctionn0(k; )

n(k; )=n(k; ) n0(k; )asfollows:

1 1

( k ) nk +f nk nk +···G=G0+ kk;

k;

kk;

(14)

(15)

Notethat( k )isitselfafunctionof n;sotheÿrsttermcontainsatleastacontributionof2order( n)whichmakesthesecondtermquitenecessary.Inprinciple,theunknownfunction

fkk ; dependsonspinandmomenta.However,spinrotationinvarianceallowsonetowritethespinpartintermsoftwoquantities,thesymmetricandantisymmetricpartsfsandfa.Moreover,forlowenergyandlong-wavelengthphenomenaonlymomentawithk≈kFplayarole;ifweconsiderthesimplecaseof3HewheretheFermisurfaceisspherical,rotationinvarianceimpliesthatformomentaneartheFermimomentumfcanonlydependontherelativeanglebetweenkandk ;thisallowsonetoexpandinLegendrepolynomialsPl(x)bywriting

k≈k ≈kFs;a

N(0)fkk ; →

∞ l=0

): ·kFls;aPl(k

(16)

Fromexpression(15)onecanthenrelatethelowestorderso-calledLandaucoe cientsF0

sandthee ectivemassm tothermodynamicquantitieslikethespeciÿcheatC,theandF1v

compressibilityÄ,andthesusceptibility :

m Cv=;v03

Äsm=(1+F0);

0 am=(1+F0):

0(17)

ItisanunfortunatecommonmistaketothinkofthepropertiesinthisregimeasSFLbehavior.