Density Matrix and Renormalization for Classical L

arXiv:cond-mat/9610107v1[cond-mat.stat-mech]14Oct1996DensityMatrixandRenromalizationforClassicalLatticeModelsDraftVer.1.0(Sep.9.1996)T.Nishino1andK.Okunishi21DepartmentofPhysics,GraduateSchoolofScience,KobeUniversity,Rokko-dai,Kobe657,Japan2DepartmentofPhysics,GraduateSchoolofScience,OsakaUniversity,Toyonaka,Osaka560,JapanAbstractThedensitymatrixrenormalizationgroupisavariationalapproxi-mationmethodthatmaximizesthepartitionfunction—orminimizethegroundstateenergy—ofquantumlatticesystems.ThevariationalrelationisexpressedasZ=Trρ≥Tr(˜1ρ),whereρisthedensitysub-matrixofthesystem,and˜1isaprojectionoperator.Inthisreportweapplythevariationalrelationtotwo-dimensional(2D)classicallatticemodels,wherethedensitysubmatrixρisobtainedasaproductofthecornertransfermatrices.Theobtainedrenormalizationgroupmethodfor2Dclassicallatticemodel,thecornertransfermatrixrenormaliza-tiongroupmethod,isappliedtotheq=2∼5Pottsmodels.Withthehelpofthefinitesizescaling,criticalexponents(q=2,3)andthelatentheat(q=5)arepreciselyobtained.Address:T.Nishino,DepartmentofPhysics,GraduateSchoolofScience,Kobeuniversity,Rokkodai,Kobe657,JAPANPhone:[phone],Fax:[phone]e-mail1:[e-mail]1IntroductionThebasicprocedureintherenormalizationgroup(RG)istokeeprelevantinformationofaphysicalsystem,andneglect(orintegrateout)irrelevantone.[1,2,3]Thedensitymatrixrenormalizationgroup(DMRG)introducedbyWhite[4]greatlyenhancestheapplicabilityofthenumericalRG,becausethemethodautomaticallykeepsafixednumbers(=m)oftherelevantbasis;DMRGpresentthebestapproximationwithinthelimitednumericalresourcethatwecanuse.TheDMRGhasbeenappliedtoanumberofone-dimensional(1D)quantumlatticesystems,suchasthespinchain,[5,6]ladder,[7,8]Bethelatticesystem,[9]stronglycorrelatedelectronsystems,[11,12,13,14]mod-elsinmomentumspace.[10]Notonlythenumericalsuperiority,butalsotheformulationofDMRGhaveattractedtheoreticalinterests.¨OstlundandRom-mer[15]haveshownthatDMRGisavariationalmethod,wherethegroundstateisexpressedasaproductof3-indextensors.[16,17]Mart`ın-DelgadoandSierrahaveinvestigatedtheanalyticformulationofDMRG,andhavefor-mulatedthecorrelatedblockRG.[18,19]RecentlyWhitehaverefinedthefinitesystemalgorithmofDMRG,andextendtheapplicabilityofDMRGto2Dquantumsystems.[20]QuiterecentlyXianghavereportedDMRGstudyof1Dquantumsystematfinitetemperature,[21]usingthequantumtransfermatrixformulation[22]andDMRGappliedtothetransfermatrix.[23]AnotherRGapproachhasbeendevelopmentfor2Dclassicallatticemodels:Baxter’smethodofthecornertransfermatrix.(CTM)[24]ThemethodisageneralizationofKramers-Wannierapproximation,[25,26]andthereforeBaxter’smethodisbasedonavariationalprincipleforthepartitionfunction.ItshouldbenotedthatBaxter’svariationalrelationisinprinciplethesameasthevariationalrelationinDMRG.[27]Thepurposeofthisreportistoexplain1howtheconceptofDMRGisappliedto2Dclassicallatticemodels.WestartfromashortreviewofthevariationalrelationinDMRGinthenextsection.ItisworthlookingatapracticaluseoftheRGmethodasthe2Dphotoimagecompression.[28,29]Aphotoimageinourcomputerisnormallycom-pressedbeforeitisstored,inordertodecreasethefilesize.ThecompressionalgorithmisrelatedtotheblockRGmethod,[1,3]sinceasmallregion—apixel—ina2Dpicturehasstrongcorrelationwithitsenvironment.(Com-puterscientistsmayinsistthattheirfindingsaboutthephotoimagecompres-sionareefficientforRGformulationinphysics.)Atpresent,compressionofmovies—TVpictures—areinprogressintheworldofcomputation;[30,31]thealgorithmmaybeagoodreferencefortheRGstudyof3Dclassicalsystemsand2Dquantumsystems.ThedevelopmentofRG,DMRG,Baxter’sCTMmethod,andthephotoimagecompressionissummarizedinFig.1.Originallythesemethodsarepro-posedindependently,however,nowitisapparentthattheirbackgroundisincommon.Itis,toapproximateasystem(ortheobjects)withinalimitednumberoffreedom.2VariationalPrincipleinDMRGWestartfromashortreviewofthevariationalprincipleinDMRG.WeconsidertheantiferromagneticS=1/2Heisenbergspinchainasanexampleof1Dquantumsystems.ThespinHamiltonianisH=JXiSi·Si+1,(1)whereSirepresentsthequantumspinati-site,andtheparameterJispositive.TheHamiltonianHisreal-symmetric,andsoisthedensitymatrixρ=e−βH.(Inthefollowingdiscussion,ρdoesnotalwayshavetobereal-symmetric,but2shouldbepositivedefinite.)Weconsideranopenspinchain,(Fig.2)whichconsistsofthelefthalf[L](=thelocalsystem)andtherighthalf[R](=thereserver).Theterms‘localsystem’and‘reserver’areratherformal,since[R]isnotalwayslongerthan[L].(InFig.2both[L]and[R]hasthesamesize.)TheHilbertspaceofthewholesystemisspannedbythereal-spacebasis|li|ri,where|liand|ricorrespondstothespinconfigurationfor[L]and[R],respectively.Thematrixelementofρisgivenbyρlr,l′r′=hl|hr|e−βH|r′i|l′i.(2)InthecontextofDMRG,whatiscalledthedensitymatrixisactuallythedensitysubmatrix(DSM)ρL=Xll′|liρLll′hl′|≡Xll′|liXrρlr,l′rhl′|(3)thatcontainstheinformationonlyaboutthelocalsystem[L].ThetraceofρLisequaltothepartitionfunction.Therelevantstateselection—renormalization—inDMRGisperformedthroughthediagonalizationoftheDSMOTρLQ=diag{λ1,λ2,...},(4)whereλ1≥λ2≥...≥0areeigenvaluesindecreasingorder,Q