arXiv:cond-mat/0402471v1[cond-mat.dis-nn]18Feb2004 Amean-fieldtheoryofAndersonlocalization
V.JaniˇsandJ.KolorenˇcInstituteofPhysics,AcademyofSciencesoftheCzechRepublic, NaSlovance2,CZ-18221Praha8,CzechRepublic∗(Dated:November19,2018) Andersonmodelofnoninteractingdisorderedelectronsisstudiedinhighspatialdimensions.Wefindthatoff-diagonalone-andtwo-particlepropagatorsbehaveasgaussianrandomvariablesw.r.t.momentumsummations.Withthissimplificationandwiththeelectron-holesymmetrywereducetheparquetequationsfortwo-particleirreducibleverticestoasinglealgebraicequationforalocalvertex.Wefindadisorder-drivenbifurcationpointinthisequationsignallingvanishingofdiffusionandonsetofAndersonlocalization.Thereisnobifurcationind=1,2whereallstatesarelocalized.AnaturalorderparameterforAndersonlocalizationpopsupintheconstruction. PACSnumbers:72.10.Bg,72.15.Eb,72.15.Qm Understandingmobilityof(free)particlesinrandommediahasbeenachallengingtheoreticalproblemformanydecades.Itbecameclearfromtheearlydaysofthestudyofsystemswithrandomlydistributedscatterersthattheparticlemovementhasadiffusivecharacterdescribedatlongdistancesbyadiffusionequation.AbreakthroughintheconceptualperceptionofrandomsystemswasachievedinRef.[1].P.W.Andersondemonstratedthereonasimplemodelthatforasufficientlystrongdisordertheparticleremainsconfinedinafinitevolumeandfailstodiffusetolongdistances.Sincethen,thedisorder-inducedabsenceofdiffusion,calledAndersonlocalization,hasattractedalotofattentionofcondensedmattertheorists.Inspiteofyearsofintensivestudies,thephenomenonofAndersonlocalizationhasnotyetbeenfullyunderstood.Itismostlyduetomanyfacetsofthisdifficultproblem. Thereareplementarytools,numericalandanalytical,totackleAndersonlocalizationandthemetalinsulatortransition.Neitherofthemis,however,abletoanswerallquestionsaboutthedisorder-inducedvanishingofdiffusion.Whiletheformerdealswithfinitelatticesandmanyconfigurationsoftherandompotential[2],thelatterdealsmostlywiththethermodynamiclimitandconfigurationallyaveragedquantities[3].AlthoughtheexistenceofAndersonlocalizationwasprovedrigorouslytoexistinanyfinitedimensionifthedisorderissufficientlystrong[4],bothnumericalandanalyticalapproachesworkpreferablyinratherlowdimensions(d=2+ǫ).Asaconsequence,astandardmeanfield(high-dimensional)theoryofAndersonlocalizationismissing,exceptforspecialsolutionsonaBethelattice[5].Eventhoughaself-consistent,mean-field-typetheoryofAndersonlocalizationwasformulated[6],itwasderivedonlywithintheweak-scattering,low-dimensionallimit.Asystematicmean-fieldtheoryeoutoftheasymptoticlimittohighspatialdimensions[7]. TheaimofthisLetteristoemploythelimittohighspatialdimensionsfordevelopingamean-fieldtheoryfortheAndersonlocalizationtransition.Weusetheparquet approachsummingupsystematicallynonlocalvertexcor- rectionstothemean-field,d=∞,solution[8].Weshow howtheparquetequationsfortheirreducibletwo-particleverticescanbesimplifiedandsolvedintheasymptotic limittohighspatialdimensions.Weusetheasymptoticsolutionoftheparquetequationstobuildupaquantita- tivetheoryofthedisorder-drivenmetal-insulatortransi- tionwithamean-fieldcriticalbehavior,i.e.,independentofthespatialdimension. Theexistenceornonexistenceofdiffusioncanbede- terminedfromtheelectron-holecorrelationfunctiondefinedfromthetwo-particleresolventasΦ(z1,z2;q)=N−2kk′G(k2k)′(z1,z2;q),wherez1andz2plexenergies.Aspecificelementofthisfunctionwithenergiesz1=EF+ω+i0+andz2=EF−i0+,denotedΦAEFR(q,ω),isusedtodeterminethediffusionconstant[6,9] nFD=limω2∇2ΦAR(q,ω)
(1) ω→04πqEF q=
0 wherenFisthedensityofstatesattheFermienergyEF.VanishingofthediffusionconstantDindicatestheabsenceofdiffusioninthesystem. InsystematictheorieswedonotapproximatedirectlyeithertheGreenfunctionG(2)orthecorrelationfunctionΦAR,butratherthetwo-particlevertexΓdefinedfrom [10] G(k2k)′(q)=G+(k)G−(k+q)[δ(k−k′)+Γkk′(q)G+(k′)G−(k′+q)] whereG±(k)≡G(k,z±)areaveragedone-particleresolvents. Thesimplesttheoryforstronglydisorderedsystemsisthecoherent-potentialapproximation(CPA),beinganexactsolutionind=∞.Onlythediagonalpartoftheone-electronresolventG(z)=N−1kG(k,z)isrelevantintheCPA,sincepletelyneglectscoherencebetweenspatiallydistinctscatterings.Itisaconsistentmean-fieldtheoryonlyforone-electronfunctions.Nonlocalpartsoftwo-particlefunctionsinhighdimensionsdo notvanish.Theoff-diagonalelementsoftheone-electronpropagatorscannotbeneglectedandhavetobetakenexpliciltlyintoconsideration[7].Furtheron,theCPA,duetoitslocalcharacter,isdegenerateandcannotdistinguishbetweenscatteringsofelectronsandholes.Tobeabletoresolvevarioustypesoftwo-particlescatteringsinnoninteractingsystemswehavetogobeyondthemean-field,localapproximation. Asystematic(diagrammatic)expansionaroundthed=∞limitcanbeperformedbyusingtheoff-diagonalone-electronCPAresolventG¯(k,z)≡G(k,z)−G(z).ThreenonequivalentBethe-SalpeterequationswithnonlocalpropagatorscanbeconstructedforthefullvertexΓ[8].HerewewilluseonlytheBethe-Salpeterequationsfromtheelectron-holeandtheelectron-electronscatteringchannels[11].Theycanberepresentedas Γkk′(q)=Λ¯ekhk′(q)+N1Λ¯ekhk′′(q) k′′ (2a) ×G¯+(k′′)G¯−(k′′+q)Γk′′k′(q), Γkk′(q)=Λ¯ekek′(q)+N1Λ¯ekek′′(q+k′−k′′) k′′ ×G¯+(k′′)G¯−(Q−k′′)Γk′′k′(q+k−k′′), (2b) respectively.WeusedbarintheirreducibleverticesΛ¯ehandΛ¯eetoindicatethattheBethe-Salpeterequationsareconstructedwiththeoff-diagonalresolventsonly.Hence,intheinfinite-dimensionalcaseG¯±(k)=0andΛ¯eh,ee=γ,whereγisthefulllocalCPAvertex[8].ForsimplicityofnotationwedenotedQ≡q+k+k′.Noticethatqisthemomentumconservedforscatteringsintheelectron-holechannel,Eq.(2a),whileQisconservedintheelectron-electronchannel,Eq.(2b). Toreachthestrong-disorderlimitwithadiffusionlessregimewehavetodeterminetheirreducibleverticesΛ¯ehandΛ¯eeself-consistently.Theparquetconstructionprovidesasuitableframeworkforthispurpose.Itisbasedontheobservationthatreduciblediagramsinonechannelareirreducibleintheother,ologicallydistinctchannels.IfweapproximatethevertexirreducibleinallchannelsbythelocalCPAvertexγ,takeintoountonlytheehandeechannels,andrealizethatthefullvertexisasumofreducibleandirreducibleverticesinanychannel,weendupwithafundamentalparquetequation Γkk′(q)=Λ¯ekhk′(q)+Λ¯ekek′(q)−γ.
(3) TheminussignattheCPApensatesfortheidenticallocalpartinbothΛ¯ehandΛ¯ee.Acoupleof(nonlinear)parquetequationsdeterminingtheirreducibleverticesΛ¯ehandΛ¯eeasfunctionsofγandG¯±areobtainedbyreplacingthefullvertexΓinEqs.(2)byEq.
(3). PriortosolvingtheparquetequationsforΛ¯ehandΛ¯eeweutilizetheelectron-holesymmetryexpressedas
2 anidentityfortwo-particleverticesΓkk′(q)=Γkk′(−q−k−k′)andΛ¯ekek′(q)=Λ¯ekhk′(−q−k−k′).TheelectronholetransformationmapsEq.(2a)ontoEq.(2b).Thetwo-particleelectron-holesymmetryisaconsequenceofthetime-reversalinvariancetheone-electronresolventG¯(k,z)=G¯(−k,z)usedintheBethe-Salpeterequations
(2).ThissymmetryactuallyreducesthenumberofparquetequationstoasinglenonlinearintegralequationforΛ¯kk′(q)≡Λ¯ekek′(q)=Λ¯ekhk′(−q−k−k′). Generally,theparquetequationsareunsolvableduetomomentumconvolutionsintheBethe-Salpeterequations.Thelimittohighspatialdimensionsleadstosuppressionofspatialfluctuationsresultinginsimplificationsofmomentumconvolutions[12].Wetakeadvantageofthesesimplifications.Westartwiththeleadingasymptotictermintheoff-diagonalpropagatorG¯(k,z)thatonad-dimensionalhypercubiclatticewiththehoppingamplitudetreads G¯(k,z)=.√tddcos(kν) ν=
1 dǫρ(ǫ)G(z−Σ(z)−ǫ)2=tx(k)G(z)
2,
(4) whereρisthedensityofstatesandΣthelocal(CPA)selfenergy.Wereplacetheoff-diagonalone-electronpropagatorsintheparquetequationswiththisasymptoticrepresentation. Thesimplestandmostimportantconvolutionisatwo-particlebubbleχ¯(q)=N−1kG¯+(k)G¯−(k+q).Itsasymptoticbehaviorcanbefoundfromthefollowingformula
1 1 1d Nx(k)x(k+q)=2X(q)=2dcos(qν)(5a) k ν=
1 wherewedenotedX(q)atwo-particle(bosonic)dispersionfunction.Otherpossibleconvolutionsofthegenericfermionicandbosonicdispersionfunctionsare 1X(q′+q)x(q′+k)=1x(q−k),(5b)Nq′2d1X(q′+q1)X(q′+q2)=1X(q1−q2).(5c)Nq′2d Wecanseethatthefermionicandbosonicdispersionfunctionsformaclosedalgebrawithrespecttomomentumsummations.Theelementaryconvolutions(5)manifestthegenerationofthefactord−1duetomixingoftwo-particlepropagationsfromdifferentscatteringchannels.Thedispersionfunctionsthenbehaveintheleadingasymptoticorderofd−1asgaussianrandomvariableswhenmomentumsummationsareperformed. Tomakethecalculationsinhighspatialdimensionsmoremean-field-like,wereplacethebarefermionicandbosonicdispersionfunctionswiththerespectiveoffdiagonalone-andtwo-particlepropagators.Thatis,we useG¯insteadofxandχ¯insteadofX.Thesequan- titiesaredirectlyproportionalintheleadingasymp- toticorder.Withoutloosingtheasymptoticuracy wecanextendrelations(5)togenuinemean-fieldexpressionsN−1q′χ¯(q′+q)G¯±(q′+k)=WG¯±(q−k)/4d,N−1qχ¯(q+q1)χ¯(q+q2)=Wχ¯(q1−q2)/4d,whereweusedW=t2G2+G2−. ItisevidentfromEq.(4)andEq.(5)thatthetwo- particleverticescanberepresentedasfunctionsofthegenericoff-diagonalfermionicG¯(k)andbosonicχ¯(q) functions.Tofindtheleadinghigh-dimensionalasymptoticsofthesolutionoftheparquetequationforΛ¯kk′(q)wekeeponlytheleadingd−1termsforeachspecificmo- mentumdependenceofthevertex.Itiseasytodemon- stratethattheparquetequationthensimplifiesinthe leadingasymptoticlimitd→∞toanalgebraicequation Λ¯(q)=Λ¯(q)=γ+Λ¯Λ¯0χ¯(q) (6a) kk′ 01−Λ¯0χ¯(q) whereΛ¯0=N−1qΛ¯(q).Thehigh-dimensionalirreducibletwo-particlevertexpletelydeterminedfromasinglelocalparameterΛ¯0andthetwo-particlebubbleχ¯(q).SummingbothsidesofEq.(6a)overmo- mentaweobtain Λ¯=γ+Λ¯
1 Λ¯0χ¯(q). (6b)
0 0N1−Λ¯0χ¯(q) q Equations(6)werederivedfromtheleadinghighdimensionalasymptotics,butcanbeusedinanyfinitedimension.Itis,however,mandatorythattheproperd-dimensionalmomentumsummationisusedonaddimensionallattice.Wecannotdirectlyusethegaussianrulestoevaluatethesummationovermomentainddimensionsaswedidduringthederivationoftheseequations.Theintegrandwouldbesingular(nonintegrable)inthegaussianevaluation.Toassesstheasymptotichighdimensionalbehaviorofthetwo-particleirreduciblevertexwehavetorealizethatinderivingEq.(6)eachindependentone-dimensionalmomentumintegrationoverponentsqiwithi=1,2,...,dcancontainmaximallysquaresofthedispersiontoremainwithintheleadingasymptotics.Hence,onad-dimensionallatticewecanbuildmaximallydpairsofthetwo-particlebubblesχ¯.TheintegrandinEq.(6b)thereforecollapsestoapolynomialoforderd.Usingthegaussianintegrationrulesweexplicitlyobtain
1 Λ¯0χ¯(q) d(2n)!
n fd(a)=Nq1−Λ¯0χ¯(q)d≡n=1(2d)nn!
(a) where we denoted a =
W 2 Λ¯ 20 /
8 . Theasymptoticlimit ofthesumfd→∞(a)convergesfora<1/2,i.e.,forΛ¯20<4/W2.Thecriticalvalueoftherandomness
3 γ2=4/W2definesanultimateupperboundbeyondwhichperturbationtheoryaroundd=∞inpowersofd−1doesnotconvergeandesnonanalytic.Inrealisticmodels,however,suchanextremevaluecannotbereached,apartfromtinyregionsaroundbandedgesandinsatellitebands. Theone-electronfunctionsG,ΣandthelocaltwoparticlevertexγenteringEqs.(6)wereassumedtobetakenfromtheCPA.Therewehaveγ=λ/(1−λG+G−)andλ=(Σ+−Σ−)/(G+−G−).IfweanalogouslydefineΛ¯0=Λ0/(1−Λ0G+G−)andχ(q)=χ¯(q)+G+G−wecanrepresenttheasymptoticformofthefulltwo-particlevertexasfollows Γkk′(q)=γ Λ¯0χ¯(q) Λ¯0χ¯(k+k′+q) +Λ01−Λ0χ(q)+1−Λ0χ(k+k′+q).
(7) ThefullnonlocalCPAvertexcanberecoveredfromtheaboveexpressionifweputΛ¯0=γandneglectthelasttermonther.h.s.ofEq.
(7).ThetermneglectedintheCPA,however,restorestheelectron-holesymmetryintheasymptoticvertexinhighdimensions. Uptonowwehaveanalyzedthetwo-particleasymptoticswiththeone-electronpropagatorsfixedbytheCPA.Toreproducediffusioninthisapproachwehavetomatchcorrectlytheirreduciblevertexcalculatedfromtheparquetequations(6)andtheone-electronself-energy.WehencehavetogobeyondtheCPAevenintheoneelectronpropagators.TodosoconsistentlyweusetheWardidentityanddeterminetheimaginarypartoftheself-energyfromthetwo-particleirreduciblevertexvia ℑΣ(E+i0+)=Λ0(E+i0+,E−i0+)ℑG(E+i0+).
(8) Therealpartoftheself-energyisdeterminedfromtheKramers-Kronigrelation[8].Equation(8)pletestheparquetequation(6b).BothequationstogetherwiththeKramers-Kronigrelationhavetobesolvedsimultaneouslytoachievefullself-consistencebetweenΛ0andGcalculatedfromΣviatheDysonequation. Theredefinitionoftheself-energyinEq.(8)isimportant,sinceonlywithitwerecoverthediffusionpoleinthevertexfunctions.Thatis,weobtainΛ0(E+i0+,E−i0+)χ
(0)=1,whenevertheparquetequation(6b)allowsforapositivesolution.TheexistenceofthediffusionpoleisessentialforthediffusionconstantfromEq.(1)tobenon-zero(positive).WecanimmediatelyconcludefromsimplepowercountinginthemomentumintegralofEq.(6b)thatthereisnopositivesolutionforΛ¯0(E+i0+,E−i0+)inlowdimensions,d=1,2,sincethediffusionpolewouldbenonintegrable.Consequently,nodiffusionpolecanexistandthediffusionconstantfromEq.(1)vanishesind=1,
2. Inhigherdimensionswecanexpandther.h.s.ofEq.(6b)inpowersofthelocalvertexΛ¯
0.Wethenobtain ℑΛ0(ω→0-)[a.u.] 1.0 D/D0 ℑΛ
0 0.5 D/D0 0.0024681012λ/w2 FIG.1:DiffusionconstantDandtheorderparameterinthelocalizedphaseℑΛ0calculatedfromEq.
(9).Weusedasemiellipticenergybandwiththebandwidth2w,theselfconsistentBornapproximationfortheself-energy,andsetCd=0.1W2. λ/w2 108 000111 6000111 4000111 20 0011 −6−4−20246 E/w FIG.2:PhasediagramforthesamesettingasinFig.1.Thehatchedareadenoteslocalizedstates. amean-field-likecubicequation Λ¯0=γ+CdΛ¯30
(9) withCd=limΛ¯0→0N−1qχ¯(q)2/(1−Λ¯0χ¯(q))
2.Thisconstantisgenerallyadecreasingfunctionofthespatial dimensiondandapproacheszerointhelimitd→∞viaCd∼W2/8d.DuetotheexistenceofthediffusionpoleinEq.(6b)theconstantCdesinfiniteind≤
4.Equation(9)derivedfromaTaylorexpansioninthelocalvertexΛ¯0doesnotsurviveinthisformtolowdimensions. Equation(9)hasgenerallythreesolutionsforΛ¯0(E+i0+,E−i0+).Forsufficientlysmalldisorderstrengths,γ<γc,allthreesolutionsarereal.Aperturbativesolutionisoforderγ,whiletwononperturbativesolutionsareoforder±1/Cd.Theperturbativesolutionincreasesandthemoduleofthenonperturbativeonesdecreases withincreasingthedisorderstrength.Atacriticalrandomness3CdΛ¯20=1,orequivalentlyγc=4/27Cd,the 4 twopositivesolutionsmergeandmoveintoplexplaneforγ>γc.DisappearanceofpositivesolutionsforΛ¯0(E+i0+,E−i0+)leadstosuppressionofthediffusionpoleandsimultaneouslytovanishingofthediffusionconstant.QuantityℑΛ¯0(E+i0+,E−i0+),emergingbeyondthecriticalpointinthelocalizedphase(γ>γc),playstheroleofanorderparameterforAndersonlocalization,seeFig.1.AtypicalphasediagramforlocalizedextendedstatescalculatedfromEq.(9)isplottedinFig.2.Althoughthemean-fieldequation(9)doesnotpredicttheprecisepositionofthemobilityedges,itdeterminesthemean-fielduniversalpropertiesurately. Toconclude,wederivedamean-fieldapproximationfortwo-particleirreducibleverticesmotivatedandjustifiedbytheasymptoticlimittohighdimensions.Weeededinderivinganalgebraicequationforthelocalirreduciblevertexwithabifurcationpointatwhichthediffusionconstantvanishesandarealirreduciblevertexplex.Afullyconsistentandcontrollablemeanfield-liketheoryofthedisorder-drivenvanishingofdiffusionandAndersonlocalizationwastherebyachieved.Itcorrectlyreproducesthelowandhigh-dimensionallimitsandallowsforfurthersystematicimprovements. ResearchonthisproblemwascarriedoutwithinaprojectAVOZ1-010-914oftheAcademyofSciencesoftheCzechRepublicandsupportedinpartbyGrantNo.202/01/0764oftheGrantAgencyoftheCzechRepublic. ∗Electronicaddress:janis@fzu.cz,kolorenc@fzu.cz[1]
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