Thermodynamic Properties of Generalized Exclusion

arXiv:cond-mat/9606165v216Sep1998HYUPT-96/3,SNUTP96-070,cond-mat/9606165ThermodynamicPropertiesofGeneralizedExclusionStatisticsHyunSeokYang,Bum-HoonLeeandChanyongParkDepartmentofPhysics,HanyangUniversity,Seoul133-791,KoreaAbstractWeanalyticallycalculatesomethermodynamicquantitiesofanidealg-ongasobeyinggeneralizedexclusionstatistics.Weshowthatthespecificheatofag-ongas(g6=0)vanisheslinearlyinanydimensionasT→0whentheparticlenumberisconservedandexhibitsaninterestingdualsymmetrythatrelatestheparticle-statisticsatgtothehole-statisticsat1/gatlowtemper-atures.Wederivethecompletesolutionfortheclustercoefficientsbl(g)asafunctionofHaldane’sstatisticalinteractionginDdimensions.Wealsofindthattheclustercoefficientsbl(g)andthevirialcoefficientsal(g)areex-actlymirrorsymmetric(l=odd)orantisymmetric(l=even)aboutg=1/2.Intwodimensions,wecompletelydeterminetheclosedformsabouttheclusterandthevirialcoefficientsofthegeneralizedexclusionstatistics,whichexactlyagreewiththevirialcoefficientsofananyongasoflinearenergies.Weshowthattheg-ongaswithzerochemicalpotentialshowsthermodynamicproper-tiessimilartothephotonstatistics.Wediscusssomephysicalimplicationsofourresults.PACSnumbers:05.30.-d,05.70.Ce,64.10.+hTypesetusingREVTEX1I.INTRODUCTIONRecently,therehasbeenextensiveinterestingeneralizedexclusionstatistics(GES)[1–14]initiatedbyHaldane.Ithasbeenrealized[2,5,7–12]thatthereareseveralmodelsobeyingGESinwhichinterparticleinteractionscanberegardedaspurelystatisticalinteractionsbygeneralizedexclusionprinciple.GESdefinedbyHaldanegeneralizesthePauli’sexclusionprinciplethroughthelineardifferentialrelation[1]Δdα=−XβgαβΔNβ,(1.1)whereΔdαisthechangeoftheavailableone-particleHilbert-spacedimensionwhenthenumberofaddedparticlesamountstoΔNβandαandβindicatedifferentparticlespecies.Thisdefinitionofstatisticsisindependentofthespacedimensionandobviouslyinterpolatesbetweenboson(gαβ=0)andfermion(gαβ=δαβ)statisticscontinuously.Notegαβcanhavearbitrarynonnegativevalues.Anearlierformoffractionalstatistics-anyons[15]isrelatedtobraidingpropertiesofparticletrajectories,whichisapeculiarpropertyoftwodimensionsunlikeGES.GESsu-perficiallyseemstohavelittletodowiththebraid-groupnotionof2Dstatistics.However,severalpeoplehaveshown[1,2,7–9]thatananyongasinthelowestLandaulevel(LLL)satisfiestheGESgivenbythestatisticalinteractiong=α(braidingstatisticsparameter).Moreover,MurthyandShankarhaveargued[5]thatanyonsobeytheGESbyrelatingthestatisticalinteractiongtothehigh-temperaturelimitofthesecondvirialcoefficient(VC)oftheanyongaswhichshowsanontrivialdependenceon2Dbraidstatistics[16].AsdiscussedbyNayakandWilczek[3],freeanyonsarenotidealg-onsbutinteractingg-ons.ItwillbeaninterestingquestionwhethertheidealGESprovidesgoodapproximationsoftheanyonmodelinwhichtheChern-Simonsfluxisattachedtochargedparticles[15].Inthisrespect,itwillbeshownthatidealg-onstatisticsprovidesagoodleadingapproximationofanyonstatistics,i.e.,theanyonstatisticshasacorrectlimitofGES,inthefirst-orderperturbation-clusterexpansion[17,18]andinthelinearenergiesinaregulatingharmonicpotential[19].2Inthispaper,weanalyticallycalculatesomethermodynamicquantitiesofanidealg-ongaswithgαβ=gδαβ.Forthesingle-particleenergygivenbyǫ(p)=apn,weverifythatexclusionstatisticssatisfythethirdlawofthermodynamicsasconjecturedbyNayakandWilczek[3],i.e.,whentheparticlenumberisconserved,thespecificheatofag-ongas(g6=0)vanisheslinearlyinanydimensionasT→0andshowthatthespecificheatatlowtemperatureexhibitsaninterestingdualsymmetrythatrelatestheparticle-statisticsatgtothehole-statisticsat1/g.Wederivethecompletesolutionfortheclustercoefficientsbl(g)andthusdeterminetheVCsal(g)asafunctionofHaldane’sstatisticalinteractionginDdimensions.Wealsodemonstratethattheclustercoefficient(CC)bl(g)andtheVCal(g)areexactlymirrorsymmetric(l=odd)orantisymmetric(l=even)aboutg=1/2,sothatthesemion(g=1/2)whichisthecaseof1DspinonasdiscussedbyHaldane[1,20]mustexhibitapeculiarproperty,a2l(12)=b2l(12)=0for∀l∈N.InthecaseofD=n,wecompletelydeterminetheclusterandthevirialcoefficientsofGESinclosedforms,andtheresultsexactlyagreewiththeVCsofanyongasobtainedbySen[17]andDasni`eresdeVeigy[19].Thisresultimpliesthatafreeanyongaswithasmallαorwithlinearenergiescanbewelldescribedbyanidealg-ongas.Wealsoshowthattheg-ongaswithzerochemicalpotentialshowsthermodynamicpropertiessimilartothephotonstatisticsandallthethermodynamicquantitiesforagivenvolumedependonlyonthetemperatureandtheirdependencescanbedeterminedbysimpledimensionalarguments.Finally,wediscusssomephysicalimplicationsofourresults.II.THERMODYNAMICSOFGENERALIZEDEXCLUSIONSTATISTICSConsideraone-particleenergyspectrumdividedintoalargenumberofenergycellsǫα,eachofwhichcontainsGαindependentlevelsandNαidenticalparticles.Inthelimitofadiscreteenergyspectrum,Gαwouldbethedegeneracyfactor.IfdNαisthedimensionoftheone-particleHilbertspaceintheαthcellwiththecoordinatesofNα−1particlesheldfixed,Haldane’sdefinitionofexclusionprinciple,Eq.(1.1),leadsto3dNα=Gα−g(Nα−1).LetW({Nα})bethenumberofconfigurationsofthesystemcorrespondingtothesetofoccupationnumbers{Nα}.Thenanelementarycombinatorialargumentgi