Fractional Exclusion Statistics for the t-J Model

arXiv:cond-mat/9601132v127Jan1996typesetusingJPSJ.styver.0.7fFractionalExclusionStatisticsforthet-JModelwithLongRangeExchangeandHoppingYusukeKATO∗andYoshioKURAMOTO∗∗DepartmentofPhysics,TohokuUniversity,Sendai980-77(Received)Weconstructthermodynamicsoftheone-dimensionalsupersymmetrict-Jmodelwiththe1/sin2interactionandhopping.ThethermodynamicsisdescribedexactlyintermsoffreespinonsandholonsobeyingHaldane’sfractionalexclusionstatisticsatalltem-peratures.Moreover,atlowtemperaturesthesemionicspinonsandholonsdecoupleresultinginthespin-chargeseparationinthermodynamicproperties.Weobtainex-plicitresultsforthespinandchargesusceptibilitiesandspecificheat,andinterpretthemintermsofthefractionalexclusionstatistics.Extensiontothemulti-componentt-Jmodelshowsthattheexcitationsobeyeitherfractionalstatisticsforg-onswithpartialpolarizationofcomponents,ortheparafermioniconewithoutpolarization.KEYWORDS:t-Jmodel,Sutherlandmodel,thermodynamics,fractionalexclusionstatistics,g-on,parafermion∗e-mailaddress:[e-mail]∗∗e-mailaddress:[e-mail]1§1.IntroductionRecently,theCalogero-Sutherlandmodel1)hasattractedmuchinterestamongmanyphysi-cists.Thedynamicalsymmetryofthesystemenablesustocalculatetheenergyspectrum,explicitformofthewavefunction,andstaticanddynamicalcorrelationfunctions.Fromexplicitresultsonthesequantities,thesystemhasbeenidentifiedwithafree-particlesystemgovernedbythefractionalexclusionstatisticsproposedbyHaldane.2)AmongthefamilyoftheCalogero-Sutherlandmodel,thelatticemodelssuchastheHaldane-Shastrymodel3,4)andsupersymmetric(SUSY)t-Jmodel5)playimportantrolesinthecondensedmatterphysics.TheselatticemodelshavetheGutzwiller-Jastrowtypegroundstatewavefunction.5)Inthegroundstateenergyofthet-Jmodel,magneticandchargepartsdecouplefromeachother.Thishasbeeninterpretedasthecompletespinchargeseparation.5)InthestudyofthedynamicsatT=0,HaandHaldaneidentifiedholon,spinon,andantiholonaselementaryexcitations.6)Fromtheseresults,weregardthet-Jmodelasthefixed-pointmodelrealizingtheidealTomonaga-Luttingerliquid.Inthispaper,weclarifytherelationbetweenthet-Jmodelandthefractionalexclusionstatisticsfromtheviewpointofthermodynamics.Forthethermodynamicsoft-Jmodel,anapproachhasbeenproposedbyWang,Liu,andColeman7)andanotherbyHaandHaldane.8)Theseapproaches,however,relyonempiricalassumptionsonthestate-countingrule.Inthispaperweconstructthethermodynamicsofthet-JmodelexactlybyusingthemappingfromthecontinuumSutherlandmodelinthestrongcouplinglimit.9)Inthenextsection,weconstructthethermodynamicsoftheSU(K,1)t-JmodelbyusingthemethodofSutherlandandShastry.10)Inthismethod,wecanformulatethermodynamicsoftheSU(K,1)modelforarbitraryvalueofK.In§3-5,weconsidertheSU(2,1)model.In§3,weshowthatthet-Jmodelatalltemper-aturesisdescribedintermsoffreeparticlesobeyingthefractionalexclusionstatistics.2)In§4,wedealwiththelowtemperaturerangeandreducethedescriptionofthemulticompo-nentstatisticstoso-calledg-ons.In§5,wepresentexplicitresultsonthespinandchargesusceptibilitiesandthespecificheat.Theseresultsareinterpretedfromtheviewpointoftheg-onsobtainedin§4.In§6,weextendthelowtemperaturedescriptiontothegeneralSU(K,1)symmetry.Weshowthattheg-ondescriptionworkswellwhenallthechemicalpotentialsforKspeciesaredifferent.Ontheotherhand,ifallspeciessharethesamevalueofchemicalpotential,thethermodynamicsaredescribedintermsoffreeparafermionsinsteadofg-ons.Ashorteraccountofourresultshasbeenpresentedelsewhere.11)Inthispaper,weexplainthecalculationindetailandpresentadditionalanalyticalandnumericalresults.2§2.FormulationInthissection,weconstructthermodynamicsoftheSU(K,1)t-Jmodelwith1/sin2in-teraction.FirstweexplaintherelationbetweenthecontinuousSU(K,1)SutherlandmodelinthestrongcouplinglimitandtheSU(K,1)t-Jmodel.Nextweconstructthermody-namicsoftheSU(K,1)t-Jmodel,byusingtheknowledgeoftheenergyspectrumoftheSU(K,1)Sutherlandmodel.Lastwerewritethermodynamicquantitiesinasimplerform,byintroducinganewvariable.WestartfromtheSU(K,1)SutherlandmodelforNparticlesystems:Hλ=−12NXi=1∂2∂x2i+π2L2Xijλ(λ−˜Pij)sin2[π(xi−xj)/L],(2.1)wherexiisthecoordinateofthei-thparticle,Listhelineardimensionofthesystem,andλisacouplingparameter.˜PijisgivenintermsoftheexchangeoperatorPijfortheinternaldegreesoffreedomofparticlesby˜Pij≡−PijifbothiandjarefermionsPijotherwise.(2.2)TheenergyspectrumEλof(2.1)hasbeenobtained10,12,13)asfollows:Eλ=∞Xκ=−∞κ2ν(κ)2+π2λL2∞Xκ=∞∞Xκ′=−∞|κ−κ′|ν(κ)ν(κ′)+E0.(2.3)Hereκrunsoverintegersdescribingthemomemtum.ν(κ)isgivenbyνB(κ)+PKσ=1νσ(κ);νB(κ)(=0,1,2···)andνσ(κ)(=0,1)arethemomentumdistributionfunctionsofbosonsandfermionswithspeciesindexσ(henceforthreferredtoasspin),respectively.E0isgivenbyE0=π2λ2N(N2−1)6L2.(2.4)Letusconsiderthestrongcouplinglimitλ→∞inordertorelatetheSutherlandmodelwiththet-Jmodel.Inthislimit,particleslocalizewithalatticespacingL/N.UptoO(λ),therearetwokindsofdegreesoffreedom;oneisthevibrationaroundthelatticepointsandtheotheristheexchangeofparticlespeciesbetweenthetwolatticepoints.Theformercorrespondstoafreephononwhilethelatterdoestoahoppingofanelectronorexchangeof