Low energy transition in spectral statistics of 2D interactingfermions

Max-Planck-Institutf¨urKernphysik,Postfach103980,69029Heidelberg,Germany

LaboratoiredePhysiqueQuantique,UMR5626duCNRS,Universit´ePaulSabatier,F-31062ToulouseCedex4,France

(April16,1999;revisedSeptember7,1999)

WestudythelevelspacingstatisticsP(s)andeigenstatepropertiesofspinlessfermionswithCoulombinteractiononatwodimensionallatticeatconstantfillingfactorandvariousdisorderstrength.Inthelimitoflargelatticesize,P(s)undergoesatransitionfromthePoissontotheWigner-Dysondistributionatacriticaltotalenergyindependentofthenumberoffermions.ThisimpliestheemergenceofquantumergodicityinducedbyinteractionanddelocalizationintheHilbertspaceatzerotemperature.

(a)

PilHunSong(a)andDimaL.Shepelyansky(b)

(b)

arXiv:cond-mat/9904229v2 7 Sep 1999PACSnumbers:71.30.+h,72.15.Rn,05.45.Mt

Theexperimentalobservationofthemetal-insulatortransitionintwodimensions(2D)byKravchenkoetal.[1]hasattractedagreatinteresttointeractingfermionsinadisorderedpotential.Indeed,accordingtothewell-establishedresult[2],allstatesarelocalizedfornonin-teractingparticlesin2D.Therefore,intheviewoftheexperimentalresult[1],anewtheoryshouldbedevel-opedtounderstandtheinteractioneffectsbetweenthelocalizedfermionicstates.However,inspiteofvarioustheoreticalattempts,acoherenttheoryforsuchsystemsisstillnotavailable.Whileforhighlyexcitedstates,ithasbeenshownthattherepulsive/attractiveinteractioncaninduceadelocalizationoftwointeractingparticles[3–6],thepropertiesoflowenergystatesarenotunder-stoodyet.Recently,inadditiontoexperimentalandthe-oreticalinvestigations,anumberofattemptshavebeenmadetostudythesemanyfermionicsystemsthroughnumericalsimulations[7–9].Eventhoughseveralinter-estingfeatureshavebeenreported,thesystemsstudiedtherearenotlargeenoughtoobserveinteractioninduceddelocalization.

Inthispaperweuseanothernumericalapproachbasedontheanalysisofspectralpropertiesofmulti-particlefermionicsystems.Indeed,Shklovskiietal.haveshownthatthelevelspacingstatisticsisapowerfultooltoan-alyzetheAndersontransitionindisorderedsystems[10].Whenthestatesarelocalized,thelevelsarenotcorre-latedandthestatisticsisgivenbythePoissondistri-bution,P(s)=PP(s)=exp(−s),whileinthemetallicphase,thestatesareergodicandthestatisticsisclosetotheWignersurmise,PW(s)=(πs/2)exp(−πs2/4).Thecriticaltransitionpointischaracterizedbyanintermedi-atestatisticswhichdependsontheboundaryconditions[11]andthespatialdimensionofthesystem[12].Thisapproachhasalsobeenusedtodeterminethequantumchaosborderandtheinteractioninducedthermalizationinfinitefermionicsystems[13]andtodetectAndersontransitionfortwoelectronswiththeCoulombinterac-tionon2Ddisorderedlattice[14,15].Alltheseresultsdemonstratethattheapproachdevelopedin[10]allows

toinvestigateefficientlythetransitionfromnonergodic(localized)toergodiceigenstates.

Hereweusetheabovemethodtostudythechangeofthespectralstatistics,P(s),withexcitationenergyEinamodelofspinlessfermionswithCoulombinteractionon2Ddisorderedlattice.TheHamiltonianreads

󰀁ninj󰀁󰀁†

wini+Uaiaj+H=V

i

i>j

πν),where

ν=Np/ListhefillingfactorandǫF=4πνV.Themajorityofourdatahavebeenobtainedforν≈1/32(nearestrationalvalue)andforU/V=2,whichcorre-spondstors=3.22.

TostudythelevelspacingstatisticsP(s),wegeneral-izetheapproachusedin[14]formanyparticles.First,oneparticleeigenstates(orbitals)atU=0areobtainedandtheHamiltonian(1)isrewritteninthisbasisus-ingthetwo-bodyinteractionmatrixelements.Wecon-siderthefirstMorbitalsfromthelowestenergyandtheHamiltonianmulti-particlematrix,constructedonlyfromtheseorbitals,isdiagonalizedatthefinalstage.Tothemaximum,Np=20particleswithuptoM=42or-bitalshasbeenconsideredandtheresultingmatrixsizeisNm≈5000.ThissizeissignificantlysmallerthanMCNp

󰀂Np−1󰀂Np

mi≤i=1i+Mappliesforsincetheconditioni=1

2

1

theone-particleorbitalindexmi.Suchapyramidruleallowsonetouseefficientlyonlylowenergymulti-particlestatesandtomakeastrikingreductionoftheresultingtotalmatrixsizewithoutanyseriousmodificationofthelowenergystates.WecheckedthatourlevelstatisticsresultsatlowenergyarenotsensitivetothevariationofMandNm(seeforexampleinsertinFig.1).TocomputethespectralstatisticsP(s)atagiventotalexcitationen-ergyE,whichiscountedfromthegroundstateenergy,disorderaveragehasbeenperformedoverND=5000(forlowenergy)andND=1000(forhigherenergy)con-figurations.Inthisway,thetotalstatisticsforP(s)forsmallenergyintervalvariesfromNS=104atlowEtoNS=3×105athighE.

1.0Wenotethatforgivenvaluesofdisordertheone-particleinverseparticipationratio(IPR),ξ1(thenumberofsitescontributingaone-particleeigenstate)ismuchsmallerthanthetotalnumberofsitesL2forthecaseofFig.1:

1.0η0.84ξs/Np2↓0.600.00.2E/B0.40.4P(s)0.8η0.40.30.20.2(a)0.00.000.20.40.60.60.100.5E/B1.0E/B0.4η0.20.6302010ξs/Np0.00.0↓0.21.02.0s3.00.400.0E/B0.4FIG.1.LevelstatisticsP(s)for10particlesatL=18,rs=3.22intheenergyinterval0.250.2(b)0.00.00.20.40.6E/BAchangeofP(s)withW,atagivenEandallotherparametersfixed,isshowninFig.1forNp=10.AsWdecreases,P(s)evolvesfromthePoissoniantotheWigner-Dysondistribution.TomeasuretheproximityofP(s)tooneofthesetwolimitingcases,itisconvenientto󰀃s0

usetheparameterηwhichisdefinedasη=0(P(s)−

󰀃s

PW(s))ds/00(PP(s)−PW(s))ds,wheres0=0.4729...isthesmallerintersectionpointofPP(s)andPW(s).Inthiswayη=1correspondstoPP(s)andη=0toPW(s).AccordingtoFig.1,ηchangesalmostbyanorderofmagnitudewhenWdecreasesbyfactoroftwo.Thisshowsthatforstrongdisorder,W/V=15,themulti-particlestatesatgivenEarenotergodic(localized)whileforweakerdisorder,W/V=7,theybecomeergodicandcharacterizedbytherandommatrixspectralstatistics.

2

FIG.2.DependenceofηontherescaledtotalenergyE/BforvariousnumberofparticlesNp=2(△),3(fulltriangleup),4(2),6(fulldiamond),8(⋄),10(•),14(◦)and20(fulltriangledown);(a)W/V=10and(b)W/V=7;fillingfactorν≈1/32andrs=3.22,8≤L≤25.InsertsshowthedependenceofξS/NponE/Bbysamesymbols;arrowsmarkthecriticalenergyEcfromthemainfigure.

ξ1=3.4and4.2(W/V=15);5.2and11.6(W/V=10);8.2and36.7(W/V=7),wheretheformersareforthegroundstateandthelattersforthecenteroftheband.Thismeansthatthetransitiontoergodicityisin-ducedbytheinteractionwhichbecomeseffectivelymorestrongwhentheone-particlelocalizationlengthincreases(similartothetwointeractingparticlecase[3–6]).Atthe

sametime,thecomparisonwiththedataforNp=2[14]atthesameE(η≈0.92,0.82and0.51forW/V=15,10and7,respectively)showsthatthedelocalizationef-fectduetomulti-particleinteractionisstrongerforweakdisorderwhileforstrongdisorderitdoesnotaffectlocal-ization.

10.8P(s)0.60.40.20-0.2ofFig.2.ThisIPRξSiscomputedinthebasisofnonin-teractingSlaterdeterminantsanddetermineshowmanyofsuchstatesarerequiredtoconstructaneigenstate.Theresultsareobtainedbyaveragingover100disorderrealizationssothatthestatisticalerrorislessthan5%.ForEEcitgrowsrapidlywithNp.Forexample,thegroundstateaveragevalueisξS/Np=0.35;0.40;0.46forW/V=10;7;5(thelatterisnotshown)withonly±21;13;18%variationwhenNpchangesfrom3to10;14;20,respectivelyforeachW/V.Onthecontrary,forE=0.4B>EctheratioξS/Npgrowsin2.7;9;20timeswhenNpchangesfrom3to8forW/V=10;7;5re-spectively.TheseresultsshowthatthedelocalizationintheHilbertspaceofSlaterdeterminantstakesplaceforE>EcwithEcindependentofnumberofparticles.AcrossoverfromlocalizedtoextendedstatesintheHilbertspacewasextensivelydiscussedrecentlyforthemetallicphasewithextendedone-particlestates[16,13].OurdatainFig.2representthefirstevidenceforadelocalizationtransitionintheHilbertspacefortheinsulatingphasewithlocalizedone-particlestates.

0.0η1.0-0.40123FIG.3.LevelstatisticsP(s)atthecrossingpointsofFig.2(withthesamesymbols).TheupperdataareforW/V=10,0.225sm0.8log(εη /Β)0.50.2−1.00.02.04.06.0rsInFig.2wepresentthevariationofηwiththeto-talenergyEforvariousnumberofparticlesatfixed

fillingfactorandtwovaluesofdisorder.ForNp>2,allcurveshaveapproximatelythesamecrossingpoint:Ec≈0.25B(W/V=10),Ec≈0.15B(W/V=7)andEc≈0.1B(ηc≈0.19,W/V=5,notshown).AsNpincreases,thestatisticsapproachesthePoissoniancaseforEEc.Thetransitionpoint,Ec,ischaracterizedbyaninterme-diatestatistics,whichisindependentofthesystemsizeasshowninFig.3.TheenergyEcgrowswiththein-creaseofWsothatforW/V=15thecrossingpointisnotfoundclearlywithinourcomputations(inthiscaseη≈0.8-1.0for02).TheabovedatagiveastrongevidencethatthegroundstateandtheexcitedstateswithEEcthestatisticsisclosetoPW(s)thatimpliestheergodicityofeigenstatesandtheirspacedelocalization.ToshowthattheeigenstatepropertiesarequalitativelydifferentbelowandaboveEcwedeterminedthevaria-tionofIPRξSwithNpandEaspresentedintheinserts

3

−2.00.00.51.01.5log NpFIG.4.Dependenceofǫη/BonthenumberofparticlesNp,obtainedfromFig.2:W/V=10withη(Eη)=0.4(fulldiamond)andW/V=7withη(Eη)=0.2(•),whereǫη=Eη/Np.ThestraightlineshowstheslopewhenEη=const.InsertgivesthedependenceofmaximalηonrsforW/V=7andNp=6:U/V=2,8≤L≤28(fulldiamond)andL=14,0.25≤U/V≤2(3).

Anunusualpropertyoftheabovetransitionisthatittakesplaceatthefinitetotalenergy.Thismeansthattheenergyperparticleǫ=E/Np,orthetemperature,isequaltozeroatthetransitionpointasNp→∞.Inthis

sensethistransitioncanbeconsideredasaquantumzerotemperaturetransition.Toensurethatinournumericalcomputationsǫwassufficientlylow,wepresentvariationofǫη,definedatthefixedηlevel,withthenumberofparticlesinFig.4.Accordingtothesedataǫηdropsmorethanbyanorderofmagnitudeandbecomesap-proximatelybyanorderofmagnitudesmallerthantheFermienergyǫFDependenceof≈ηm0.on1Br.

sisfoundintheinsertofFig.4.Thevalueofrsisvariedintwodifferentways:i)bychangingthecellsizeLorii)bychangingU/V.Thefirstcasei)allowstoobtainlargersvalueandshowsthatthestatisticsapproachesthePoissoniancase.Thisresultisinaqualitativeagreementwiththeexperimentaldata[1],wherethelocalized(insulating)phaseappearsforlargers,andwiththetheoreticalargumentgivenin[14],accordingtowhichthetwo-bodyinteractiondropsasthefillingfactordecreases.Thesameargumentsexplainwhyηgoesto1whentheinteractionstrengthUdecreasesasinthesecondcaseii).OnecanexpectthatthevariationofηcwithrsatlargeNpandfixedνwillbequalitativelysimilartothatoneofηminFig.4.

ThecomparisonoftheenergyEc,atwhichthetransi-tionatfixedνtakesplace(seeFig.2),withthetransitionenergyfortwoparticles(E2c/B=1.2forW/V=10andE2c/B=0.56forW/V=7[14])showsthatEcissignif-icantlysmallerthanE2c(Ec/E2cthatthemulti-particleinteractionis≈more0.2).efficientThissignifiesforde-localizationthanthetwo-bodyinteractionforonlytwoparticles.Ourqualitativeunderstandingforthetransi-tiontothequantumergodicityatfixedνandfixedtotalenergyEcisbasedonthefollowingscenario.Intheanal-ogywiththeapproachdevelopedin[14]forNp=2,thedynamicsofthreeorfourparticlesatfiniteenergyEcanbeassumedtobeequivalenttooneparticledynamicsinadisorderedsystemwitheffectivespatialdimension3Inconclusion,ourresultsshowthat2Dfermions,whicharestronglylocalizedbydisorderwithoutinteraction,be-comeergodicduetoCoulombinteraction.Thetransitiontoergodicity,inlatticeandHilbertspaces,andtheran-

dommatrixstatisticstakesplaceatthefinitetotalenergyorzerotemperature.Theseresultsareinfavoroftheex-perimentallyobservedmetal-insulatortransition[1],evenifourapproachdoesnotallowtoinvestigatethecon-ductancedependenceontemperaturewhichisthemainexperimentalmethodtodetectthetransition.Further-more,theyshowthatthevariablerangehoppingtrans-portcanbeinducedbyelectron-electroninteractioninagreementwithrecentexperiments[18].

WeacknowledgetheIDRISatOrsayforallocationoftheCPUtimeonthesupercomputers.WealsothanktheMax-Planck-Institutf¨urPhysikKomplexerSystemeatDresdenforthehospitalityatthefinalstageofthisworkduringtheworkshopDynamicsofComplexSystems.

neff∼νL2T/ǫF.Thenthespacingbetweendirectlycou-pledmulti-particlestatesis∆c∼∆2/n2effwhereatlowenergythetwo-particlespacingis∆2∼V/ξ1,sinceeachparticlecanjumponlyinsideξ1sites[4,14].Thetransi-tiontoη=0takesplaceforUs>∆c[13]thatgivesthecriticalenergyEc∼Tneff∼(Np/ν)1/2V2/Uξ1.How-ever,thisborderishigherthanthenumericallyfoundone.Thisindicatesthatthedelocalizationofsmallgroupofparticlesisdominating.

[18]S.I.Khondaker,I.S.Shlimak,J.T.Nicholls,M.Pepper,

andD.A.Ritchie,Phys.Rev.B,594580(1999).

5