Singular or non-Fermi liquids(9)

Singular or non-Fermi liquids

C.M.Varmaetal./PhysicsReports361(2002)267–417275

Itshouldberemarkedthattherightsimpleproblemisnotalwayseasytoguess.Therightsimpleproblemforliquid4Heisnotthenon-interactingBosegasbuttheweaklyinteractingBosegas(i.e.,theBogoliubovproblem[45,154]).TherightsimpleproblemfortheKondoproblem(alow-temperaturelocalFermiliquid)wasguessed[197]onlyafterthenumericalrenormalizationgroupsolutionwasobtainedbyWilson[289].Therightsimpleproblemfortwo-dimensionalinteractingdisorderedelectronsinthe“metallic”rangeofdensities(Section8inthispaper)isatpresentunknown.

ForSFLs,theproblemisdi erent:usuallyoneisinaregimeofparameterswherenosimpleproblemisastartingpoint—insomecasesthe uctuationsbetweensolutionstodi erentsimpleproblemsdeterminesthephysicalproperties,ndauFermiliquidandthewavefunctionrenormalizationZ

Landautheoryistheforerunnerofourmodernwayofthinkingaboutlow-energye ectiveHamiltoniansincomplicatedproblemsandoftherenormalizationgroup.TheformalstatementsofLandautheoryintheiroriginalformareoftensomewhatcrypticandmysterious—thisre ectsbothLandau’sstyleandhisingenuity.Weshalltakeamorepedestrianapproach.

Letusconsidertheessentialdi erencebetweennon-interactingfermionsandaninteractingFermiliquidfromasimplemicroscopicperspective.Forfreefermions,themomentumstates|k arealsoeigenstatesoftheHamiltonianwitheigenvalue

k=

2k2

:(2)

Moreover,thethermaldistributionofparticlesn0k ,isgivenbytheFermi–Diracfunctionwhere denotesthespinlabel.AtT=0,thedistributionjumpsfrom1(allstatesoccupiedwithintheFermisphere)tozero(nostatesoccupiedwithintheFermisphere)at|k|=kFandenergyequaltothechemicalpotential .ThisisillustratedinFig.4.

Agoodwayofprobingasystemistoinvestigatethespectralfunction;thespectralfunctionA(k;!)givesthedistributionofenergies!inthesystemwhenaparticlewithmomentumkisaddedorremovedfromit(rememberthatremovingaparticleexcitationbelowtheFermienergymeansthatweaddaholeexcitation).AssketchedinFig.5(a),forthenon-interactingsystem,A0(k;!)issimplya -functionpeakattheenergy k,becauseallmomentumstatesarealsoenergyeigenstates

A0(k;!)= (! ( k ))

for!¿ ;

(3)(4)

111= Im= ImG0(k;!):

kHere, issmallandpositive;itre ectsthatparticlesorholesareintroducedadiabatically,and

itistakentozeroattheendofthecalculationforthepurenon-interactingproblem.Theÿrststepofthesecondlineisjustasimplemathematicalrewritingofthedeltafunction.InthesecondlinetheGreen’sfunctionG0fornon-interactingelectronsisintroduced.Moregenerally