Application of the Density Matrix Renormalization

arXiv:cond-mat/0110420v1[cond-mat.str-el]19Oct2001ApplicationoftheDensityMatrixRenormalizationGroupinmomentumspaceSatoshiNishimoto,EricJeckelmann,andFlorianGebhardFachbereichPhysik,Philipps–Universit¨atMarburg,D–35032Marburg,GermanyReinhardM.NoackInstitutf¨urPhysik,Johannes–Gutenberg–Universit¨atMainz,D–55099Mainz,Germany(Dated:February1,2008)WeinvestigatetheapplicationoftheDensityMatrixRenormalizationGroup(DMRG)totheHubbardmodelinmomentum–space.Wetreattheone–dimensionalmodelswithdispersionrelationscorrespondingtonearest–neighborhoppingand1/rhoppingandthetwo–dimensionalmodelwithisotropicnearest–neighborhopping.Bycomparingwiththeexactsolutionsforbothone–dimensionalmodelsandwithexactdiagonalizationintwodimensions,wefirstinvestigatetheconvergenceoftheground–stateenergy.Wefindvariationalconvergenceoftheenergywiththenumberofstateskeptforallmodelsandparametersets.Incontrasttothereal–spacealgorithm,theaccuracybecomesrapidlyworsewithincreasinginteractionandisnotsignificantlybetterathalffilling.Wecomparetheresultsfordifferentdispersionrelationsatfixedinteractionstrengthoverbandwidthandfindthatextendingtherangeofthehoppinginonedimensionhaslittleeffect,butthatchangingthedimensionalityfromonetotwoleadstoloweraccuracyatweaktomoderateinteractionstrength.Intheone–dimensionalmodelsathalf–filling,wealsoinvestigatethebehaviorofthesingle–particlegap,thedispersionofspinonexcitations,andthemomentumdistributionfunction.Forthesingle–particlegap,wefindthatproperextrapolationinthenumberofstateskeptisimportant.Forthespinondispersion,wefindthatgoodagreementwiththeexactformscanbeachievedatweakcouplingifthelargemomentum–dependentfinite–sizeeffectsaretakenintoaccountfornearest–neighborhopping.Forthemomentumdistribution,wecomparewithvariousweak–couplingandstrong–couplingapproximationsanddiscusstheimportanceoffinite–sizeeffectsaswellastheaccuracyoftheDMRG.PACSnumbers:71.10.Fd,71.27.+aI.INTRODUCTIONManyrenormalizationschemesarecarriedoutinmo-mentumspaceandinvolveintegratingoutdegreesoffreedomusingamomentumcutoff.Forexample,Wil-son’snumericalRenormalizationGroup(RG)1imple-mentsthisprogramusingamappingofmomentumshellstoaneffectivelatticemodel.TherenormalizationprocessiscarriedoutbysuccessivenumericaldiagonalizationofafinitesystemandenergetictruncationoftheHilbertspace.Whilethislatticemodelcorrespondstosucces-sivemomentaor,equivalently,energyscales,itsformissimilartothatofastronglycorrelatedlatticemodel.Attemptsatapplyingareal–spaceversionoftheWil-sonproceduretoshort–rangequantumlatticemodelssuchastheHeisenbergortheHubbardmodelwerenotsuccessful,however,becausesuccessivelatticepointsdonotcorrespondtodifferentenergyscales.TheDensityMatrixRenormalizationGroup(DMRG)2,3overcomestheselimitationsbycarryingouttherenormalizationonasubsystem.Thetruncatedbasisisformedbypro-jectingthestateoftheentiresystemontothesubsys-temusingthereduceddensitymatrixratherthanse-lectingstatesenergetically.Thismethodhasbeenverysuccessfulattreatinglow–dimensionalquantumlatticemodelswithopenboundaryconditionsandshort–rangecouplings.However,forlonger–rangeoff–diagonalinter-actions,higherdimensionalsystemsorlatticeswithperi-odicboundaryconditions,thisreal–spaceformulationoftheDMRGismuchlesssuccessful.Inaddition,itlosessightofanenergyormomentum–basedclassificationoftherelevantdegreesoffreedom.Apotentialwayofovercomingthislimitationforitin-erantelectronsystemsistoapplytheDMRGideastothemomentum–spaceformulationoftheHamiltonian.Thisapproachhasanumberofpotentialadvantagesoverthereal–spaceapproach.First,sincethesingle–particlebasisinmomentumspaceisexplicitlytransla-tionallyinvariant,momentumisaconservedquantumnumber.Useofthismomentumquantumnumberre-ducesthesizeoftheHilbertspaceinthediagonaliza-tion.Second,momentum–dependentquantitiessuchasthemomentumdistributionorthedispersionofexcita-tionscanbedirectlycalculated.Third,thekineticen-ergytermisdiagonalsothatvaryingthedispersionby,forexample,changingtherangeofthehopping,iseasytodo.Attemptstoformulateanumericalrenormalizationgroupprocedureforquantumlatticesystemsinmomen-tumspace4predatingtheDMRGwerenotparticularlysuccessful–thiswasoneofWhite’smotivationsforturn-ingtorealspaceandformulatingtheDMRG.Shortlyaf-terthedevelopmentoftheDMRGinreal–space,WhiteattemptedtouseDMRGmethodsonthemomentum–spaceformulationoftheHubbardmodel.Hecalculatedtheground–stateenergyinoneandtwodimensionsatintermediatecouplings,butfoundthattheenergiesob-tainedwerenotsignificantlybetterthanthoseobtained2byothervariationalmethods.5Independently,XiangdevelopedasimilartechniqueandappliedittotheHubbardmodelinoneandtwodimensions.6Inthiswork,Xiangoutlinedanefficientim-plementationoftheDMRGinmomentumspace.Hede-velopedafactorizationoftheHubbardinteractionthatreducesthenumberoftermfromN3,whereNisthenumberofsingle–particleBlochwavefunctionsinthelat-tice,to6N.Healsopointedoutsomefeaturesofthealgorithmthatneedtobecarefullyconsideredinmo-mentumspace:Sincetheinteractionishighlynon–local,thereisnonaturalorderingofthesingle–particlestates;thechoiceoftheorderingcan,however,haveaneffectontheperformanceoftheDM