Density matrix renormalization group study of the

arXiv:cond-mat/9710058v16Oct1997HEP/123-qedDensityMatrixRenormalizationGroupstudyofthepolaronproblemintheHolsteinmodelEricJeckelmannandStevenR.WhiteDepartmentofPhysicsandAstronomy,UniversityofCalifornia,Irvine,CA92697.(February1,2008)AbstractWeproposeanewdensitymatrixrenormalizationgroup(DMRG)approachtostudylatticesincludingbosons.Thekeytothenewapproachisanexactmappingofabosonsitecontaining2NstatestoNpseudo-sites,eachwith2states.Thepseudo-sitescanbeviewedasthebinarydigitsofabosonlevel.Weapplythepseudo-siteDMRGmethodtothepolaronproblemintheone-andtwo-dimensionalHolsteinmodels.Groundstateresultsarepresentedforawiderangeofelectron-phononcouplingstrengthsandphononfrequenciesonlatticeslargeenough(upto80sitesinonedimensionandupto20×20sitesintwodimensions)toeliminatefinitesizeeffects,withupto128phononstatesperphononmode.Wefindasmoothbutquiteabruptcrossoverfromaquasi-freeelectrongroundstatewithaslightlyrenormalizedmassatweakelectron-phononcouplingtoapolaronicgroundstatewithalargeeffectivemassatstrongcoupling,inagreementwithpreviousstudies.71.38.+iTypesetusingREVTEX1I.INTRODUCTIONThedensitymatrixrenormalizationgroup(DMRG)method1,2hasprovedtobeaverysuccessfulnumericaltechniqueforstudyingspinandfermionlatticemodelswithshort-rangeinteractionsinlowdimensions.AlthoughtheDMRGalgorithmcaneasilybegeneralizedtotreatsystemsincludingbosons,calculationsareoftennotpractical.Asforexactdiag-onalizations,thisisduetothedifficultyindealingwiththelarge(inprinciple,infinite)dimensionoftheHilbertspaceforbosons.AlthoughtheproblemislesssevereinDMRGthaninexactdiagonalizations,applicationsofDMRGtobosonsystemshavebeenrestrictedtoproblemsforwhichoneneedstoconsideratmostaboutadozenstatesforeachboson3–5.Inthispaper,wepresentanewapproachfordealingwithlargebosonicHilbertspaceswithDMRG.Thebasicideaistotransformeachbosonsiteintoseveralartificialinteract-ing2-statesites(pseudo-sites)andthentouseDMRGtechniquestotreatthisinteractingsystem.DMRGismuchbetterabletohandleseveral2-statesitesratherthanonemany-statesite.AlthoughthisprocedureintroducessomecomplicationsinaDMRGprogram,thepseudo-siteapproachismoreefficientandallowsustokeepmanymorestatesineachbosonicHilbertspacethantheapproachusedinearlierworks3–5.Totestourmethod,wehavestudiedthepolaronproblem,theself-trappingofanelec-tronbyalocalizedlatticedeformation,intheHolsteinmodel6inoneandtwodimensions.Weconsiderasingleelectrononalatticewithoscillatorsoffrequencyωateachsiterep-resentingdispersionlessopticalphononmodesandacouplingbetweentheelectrondensityandoscillatordisplacementsqℓ=b†ℓ+bℓ,whereb†ℓandbℓaretheusualbosoncreationandannihilationoperators.TheHamiltonianisgivenbyH=ωXℓb†ℓbℓ−gωXℓb†ℓ+bℓnℓ−tXhℓ,mic†mcℓ+c†ℓcm,(1.1)wherec†ℓandcℓareelectroncreationandannihilationoperators,nℓ=c†ℓcℓandtisthehoppingintegral.gisadimensionlesselectron-phononcouplingconstant.(Whencomparingresultsreadersshouldbeawarethatnotationsformodelparameters,especiallyg,differin2otherpapers.)Asummationoverℓorhℓ,mimeansasumoverallsitesoroverallbondsbetweennearest-neighborsitesinachainoflengthLorasquarelatticeofsizeL×L.OnlyopensystemshavebeenconsideredbecausetheDMRGmethodusuallyperformsmuchbetterinthiscasethanforperiodicboundaryconditions.Thepolaronproblemhasbeenextensivelystudiedusingvariationalmethods7,quantumMonteCarlosimulations8,9,exactdiagonalizations10–14andperturbationtheory12,14,15.Itisknownthatarathersharpcrossoveroccursbetweenaquasi-free-electrongroundstatewithaslightlyrenormalizedmassatweakelectron-phononcoupling,andapolaronicgroundstatewithanarrowband-widthatstrongcoupling.However,despitetheseconsiderabletheo-reticalefforts,thephysicsofthisself-trappingtransitionisnotfullyunderstood.Previousstudieshavebeenlimitedeithertosmallsystemsortoaparticularregimeofparametersgandω/torbyaseverelytruncatedphononicHilbertspaceorbyuncontrolledapproxima-tions.WiththeDMRGmethod,wehavebeenabletostudytheone-electrongroundstateoftheHolsteinmodelforallregimesofparametersω/tandgonlargelatticesandwithgreataccuracy.Inthisworkwereportanddiscusssomegroundstateresultswhichshowstheself-trappingcrossover,suchaselectron-latticedisplacementcorrelationfunctions,electronickineticenergy,andeffectivemass.Thispaperisorganizedasfollows:inthenextsection,wepresentournewpseudo-sitemethodforbosons.InSection3wedescribehowweapplythismethodtotheHolsteinmodel.MostresultsforthepolaronproblemarepresentedanddiscussedinSection4.InSection5weexplainhowwehavecomputedtheeffectivemassofelectronsandpolaronsandpresenttheseresults.Finally,Section6containsourconclusions.II.DMRGFORBOSONSYSTEMSIntheDMRGmethod,thelatticeisbrokenupintoblocksmadeofoneorseveralsitesandHilbertspacesrepresentingblocksaretruncated(formoredetails,seeRefs.1and2).Ineachblockonekeepsonlythemmostimportantstatesforformingthegroundstate(or3low-energyeigenstates)ofthefullsystem.AstepoftheDMRGalgorithmistheprocessofforminganewblockbyaddingasitetoablockobtainedinapreviousstep.Tofindthemoptimalstatesofthenewenlargedblock,onehastofindthegroundstateofaneffectiveHamiltonianinasuperblockmadeoftwoblocksandtwositesandthentodiagonalizeadens