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arXiv:cond-mat/0012043v1[cond-mat.stat-mech]4Dec2000NonuniversalEffectsintheHomogeneousBoseGasEricBraatena,∗,H.-W.Hammera,†,andShawnHermansb,‡aDepartmentofPhysics,TheOhioStateUniversity,Columbus,OH43210,USAbSt.John’sUniversity,Collegeville,MN56321(December4,2000)AbstractEffectivefieldtheorypredictsthattheleadingnonuniversaleffectsinthehomogeneousBosegasarisefromtheeffectiverangeforS-wavescatteringandfromaneffectivethree-bodycontactinteraction.Wecalculatethelead-ingnonuniversalcontributionstotheenergydensityandcondensatefractionandcomparethepredictionswithresultsfromdiffusionMonteCarlocalcula-tionsbyGiorgini,Boronat,andCasulleras.Wegiveacrudedeterminationofthestrengthofthethree-bodycontactinteractionforvariousmodelpoten-tials.AccuratedeterminationscouldbeobtainedfromdiffusionMonteCarlocalculationsoftheenergydensitywithhigherstatistics.∗braaten@mps.ohio-state.edu†hammer@mps.ohio-state.edu‡schermans@csbsju.eduI.INTRODUCTIONTheBose-EinsteincondensationoftrappedatomsallowstheexperimentalstudyofcoldBosegaseswithhighprecision.Itiswell-knownthatthedominanteffectoftheinteractionsbetweentheatomscanbecapturedbyasingleconstantacalledtheS-wavescatteringlength.Thispropertyisoftencalleduniversality.However,sufficientlyaccuratemeasurementswillrevealthesensitivitytoaspectsoftheinteratomicinteractionsotherthanthescatteringlength.Thesearecallednonuniversaleffects.TheoreticalinvestigationsofthehomogeneousBosegasinthe1950’sshowedthatitspropertiescouldbecalculatedusingalow-densityexpansioninpowersof√na3,wherenisthenumberdensity.Forexample,theenergyperparticlehastheexpansionEN=2π¯h2man(1+3215π√16πna3+4π−3√36πlog(na3)+c′#(16πna3)+...).(1)Theleadingtermistheuniversalmean-fieldcontributionfirstdeterminedbyBogoliubov[1].The√na3correctionwasfirstcalculatedbyLee,Huang,andYangforbosonsinteractingthroughahard-spherepotential[2].Inthena3correction,thecoefficientoflog(na3)wasfirstcalculatedbyWuforahard-spherepotential[3].Theseleadingtermsintheexpansionwereshowntobeuniversal[4],applyingtobosonsinteractingthroughanyshort-rangepotentialwithscatteringlengtha.Ontheotherhand,HugenholtzandPinesshowedthattheconstantc′in(1)aswellashigher-ordercorrectionsarenotuniversal[5].TheydependonpropertiesoftheinteractionsbetweenthebosonsotherthantheS-wavescatteringlength,althoughthespecificpropertieswerenotidentified.Giorgini,Boronat,andCasullerashavestudiedthepropertiesofthegroundstateofhomogeneousBosegasesnumericallyusingadiffusionMonteCarlomethod[6].Theycon-sidered4differenttwo-bodypotentialsasmodelsfortheinteractionbetweenthebosons,andcalculatedtheenergyperparticleandcondensatefractionforthehomogeneousgas.Inthecaseoftheenergyperparticle,theuniversal√na3correctionin(1)accountsformostofthedeviationsfromtheuniversalmean-fieldpredictionatthedensitiesstudied,whichrangedfromna3=10−6tona3=0.244.However,theyalsoobservedsmalldifferencesbetweenthemodelpotentials,i.e.nonuniversaleffects.EffectivefieldtheoryprovidesapowerfulmethodforanalyzingthenonuniversaleffectsinthehomogeneousBosegas.Itidentifiespreciselywhichaspectsoftheinteractionsbetweenlow-energybosonsenterateachorderintheexpansionin√na3.Forexample,thecoefficientc′inthena3correctionin(1)isdeterminedbyaneffectivethree-bodycontactinteractionbetweenlow-energybosons[7].The(na3)3/2correctioniscompletelydeterminedbyc′andbytheeffectiverangeforS-wavescattering.Thenonuniversalcorrectionsinthelow-densityexpansionoftheFermigasatzerotem-peraturehavebeenstudiedpreviously[8–11].Thelow-densityexpansioncanbeexpressedasanexpansioninkF,wherekFistheFermiwavenumber.Forfermionswithasinglespinstate,theleadingcorrectiontotheenergyofanidealFermigasisproportionaltok3Fa3p,whereapistheP-wavescatteringlength[8].Forfermionswithg1degeneratespinstates,theuniversalcorrectionstotheenergycanbeexpandedinpowersofkFa,whereaistheS-wavescatteringlength[8].Theleadingnonuniversalcorrectionsarethek3Fa3ptermanda2k3Fa2rsterm,wherersistheeffectiverangeforS-wavescattering.Atorderk4Fa4,thereisalsoanonuniversalcorrectionfromtheeffectivethree-bodycontactinteraction[9].ThesecorrectionswererecentlydiscussedwithintheeffectivefieldtheoryframeworkbyHammerandFurnstahl[11].Inthispaper,weuseeffectivefieldtheorytocalculatetheleadingandnext-to-leadingnonuniversalcorrectionstotheenergydensityandcondensatefractionforahomogeneousBosegas.WecomparethecalculationswiththediffusionMonteCarlodatafromRef.[6]andattempttodeterminethecoefficientofthethree-bodycontactinteractionforeachoftheinteractionpotentialsthatwereconsidered.InSectionII,wedescribethegeneralphilosophyofeffectivetheoriesfordealingwiththelow-energybehaviorofphysicalsystems.WeuseaneffectivequantumfieldtheorytoidentifythoseparametersthatgivetheleadingnonuniversalcontributionstothepropertiesofahomogeneousBosegas.Thecalculationsoftheleadingandnext-to-leadingnonuniversalcontributionstotheenergydensityandthecondensatefractionarepresentedinSectionIII.InSectionIV,wedescribethemodelinteractionpotentialsusedbyGiorgini,Boronat,andCasulleras[6]andreviewtheirdiffusionMonteCarloresults.InSectionV,weanalyzethediffusionMonteCarloresultsofRef.[6]andgiveacrudedeterminationofth
本文标题:Nonuniversal Effects in the Homogeneous Bose Gas
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