Critical Spectral Statistics at the Metal-Insulator Transiti

The spectral properties of a disordered system with few interacting three-dimensional spinless fermions are investigated. We show the existence of a critical spacings distribution which is invariant upon increase of the system size, but strongly depends on

CriticalSpectralStatisticsattheMetal-InsulatorTransitioninInteractingFermionic

Systems

PhilippeJacquod

DepartmentofAppliedPhysics,P.O.Box208284,YaleUniversity,NewHaven,CT06520-8284

(june29,1998)

arXiv:cond-mat/9802226v2 [cond-mat.mes-hall] 9 Jul 1998

Thespectralpropertiesofadisorderedsystemwithfewinteractingthree-dimensionalspinlessfermionsareinvestigated.Weshowtheexistenceofacriticalspacingsdistributionwhichisin-variantuponincreaseofthesystemsize,butstronglydependsonthenumberofparticles.Atthecriticalpoint,wereportasubstantialdecreaseofthedegreeoflevelrepulsionasthenumberofparticlesincreasesindicatingadecreaseofnearestlevelcorrelationsassociatedwiththesparsityoftheHamiltonianmatrix.

PACSnumbers:05.45.+b,72.15.Rn,71.30.+h

Thepresenceandstrengthofadisorderpotentialstronglymodifythein uenceofelectronicinteractionsonlocalization.Whileincleansystemsrepulsiveinterac-tionsalwayslocalize,theire ectcanbereversedbythepresenceofastrongdisorder.There,interaction-inducedhoppingsbetweenlocalizedstatesmayhelpelectronstoovercometherandompotentialthusreducinglocalization[1–3].Inthispaperwewillshowthatrepulsiveinterac-tionscanindeedtriggerametal-insulatortransitioninafew-particlestronglydisorderedsystem,therebyturningaone-bodyinsulatorintoamany-bodymetal.

Thestatisticalpropertiesofspectraofdisorderedone-particlesystemsareknowntobecloselyrelatedtothelocalizationpropertiesofthecorrespondingeigenstates[4–6].Inthelocalizedphase,stateswhicharecloseinen-ergyhaveanexponentiallysmalloverlap.ConsequentlythelevelsEiareuncorrelatedandthecorrespondingnor-malizedspacingssi≡(Ei+1 Ei)/ Ei+1 Ei aredis-tributedaccordingtothePoissondistributionPP(s)=exp( s).Ontheotherhand,thelargeoverlapofdelo-calizedeigenstatesofneighboringenergyinducescorrela-tionsinthespectrumandleadstolevelrepulsion,sothatinthetime-reversalsymmetriccase,thedistributionofenergyspacingscorrespondstotheWigner-Dysondistri-butionPWD(s)=

π

The spectral properties of a disordered system with few interacting three-dimensional spinless fermions are investigated. We show the existence of a critical spacings distribution which is invariant upon increase of the system size, but strongly depends on

ingone-particlestateslocatedinthefrozenFermiseaisadvantageousintworespects:(i)itallowstoreducesig-ni cantlytheHilbertspacevolumeandconsequentlytostudylargesystemsoflinearsizesuptoL=10inthreedimensions(Presumably,itistheimpossibilityofstudy-ingsystemswithlargeenoughsizesthathinderedinves-tigationssimilarto[5]ininteractingsystemssofar.)and(ii)itprojectsoutone-bodytailstateswhichareknowntobestronglylocalizedandhencemuchlesssensitivetotheinteraction-induceddelocalizinge ect.Forthetwo-quasiparticlecase,thismodelwasoriginallyproposedin[2]andfurtherstudiedin[13].

Morespeci cally,weconsiderspinlessfermionsinadisorderedAnderson-likemodelwitharepulsivenearestneighborH=Pb

interactionofstrengthUwhoseHamiltonianis

Wia iai+ (Ua iajajai ;j

taiaj) Pb(1)

i

iHere,PbprojectsoutSlaterdeterminantscontainingone-particlestatesofenergylowerthanthethresholdenergyEb. i;j restrictsthesumtonearest-neighborsitesonathree-dimensionalcubiclatticeandWiistheon-sitedisorderrandomlydistributedbetween W/2andW/2.Inournumericalinvestigationswe xedEbrespondingtoa llingfactorofν=1/3in≈the 3disorder.3tcor-regimewewillconsiderW=18.WepointoutthatthischoiceisarbitraryandcheckednumericallythatthephysicalpicturepresentedheredoesnotdependonthisparticularchoiceofEb.ForasystemoflinearsizeLandnquasiparticles,theground-stateenergyisapprox-imately g≈dnEb+n(n+1) 1/2+3Un(n 1) 1/B1. 1≈B1/Listheone-particleaveragespacingsandB1de ne≈2thedt+many-bodyWistheone-particleexcitationenergybandwidth.asδ ≡We also g.AstheintroductionofthefrozenFermiseamayappearsomehowarti cial,westressthatforlowmany-bodyen-ergyexcitationsδ andmoderateinteractionUenergylayerofsizeT～

√ B1,onlyparticlesinanδ / 1sothatfor

neff<nourmodelisfullyjusti ed.

Thenumericalinvestigationswerecarriedoutmuchinthesamewayasinreference[5,10].WecomputedthedistributionP(s)forthemany-bodylevelspacingssi≡( i+1 i)/ nforn=2 6quasiparticles,at xedexcitationenergiesanddisorder,fordi erentin-teractionstrengthsandlinearsystemsize.Asthein-teractionstrengthincreases,moreandmorestatesarecoupledandweexpectquantumchaostosetin,i.e.thelevelspacingsdistributionundergoesacrossoverfromPoissontoWigner-Dysondistribution.Toquantitativelystudythiscrossoverwecomputedvalueoftheeterη= s0

0[P(s) PWD(s)]ds/the thesparam-00[PP(s) PWD(s)]ds.Here,s0≈0.473is1smallest(P(s)=rootPP(ofs))PP(s)=PWD(s)andηvariesbetweenand0(P(s)=

PWD(s)).Allinvestigationswerecarriedoutat xed

disorderW=18.slightlyabovethecriticaldisorderWcdisorder≈16.[5,14].5oftheAtAndersonthisdisorder,modelone-particlewithboxeddistributedstatesstillhavealargeenoughlocalizationlengthl1sothattheshort-rangeinteractioncouplesalargenumberofstates.Weaveragedoverupto10000disorderrealisations,soastogetatleastN=50000levelsinanenergyinterval[δ /2;δ + /2], < 1.

1.00.8

n=3n=4

η

0.60.40.21.00.8

n=5n=6

η

0.60.40.20.0

0.20.4

U

0.60.81.00.00.20.4

U

0.60.81.0

FIG.1.Evolutionofthecrossoverparameterηforn=3,δ =0.35,n=4,δ =0.5,n=5,δ =0.65andn=6,δ =0.8anddisorderW=18.(Fromlefttorightandtoptobottom).ThelinearsystemsizesareL=6(circles),L=8(squares)andL=10(diamonds).energiessatisfyneff≈2

Foreachn,excitation

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