fff Thermodynamics of a Weakly Interacting Bose-Ei

fThermodynamicsofaWeaklyInteractingBose-EinsteinGasT.Haugset,H.HaugerudandF.RavndalDepartmentofPhysicsUniversityofOsloN-0316Oslo,NorwayAbstract:Theone-loopeectivepotentialfornon-relativisticbosonswithadeltafunctionrepulsivepotentialiscalculatedforagivenchemicalpotentialusingfunctionalmethods.Afterrenormalizationandatzerotemperatureitreproducesthestandardgroundstateenergyandpressureasfunctionoftheparticledensity.Atnitetemperaturesitisfoundnecessarytoincluderingcorrectionstotheone-loopresultinordertosatisfytheGoldstonetheorem.Itisnaturaltointroduceaneectivechemicalpotentialdirectlyrelatedtotheorderparameterandwhichuniformlydecreaseswithincreasingtemperatures.Thisisincontrasttothetheordinarychemicalpotentialwhichpeaksatthecriticaltemperature.TheresultingthermodynamicsinthecondensedphaseatverylowtemperaturesisfoundtobethesameasintheBogoliubovapproximationwherethedegreesoffreedomaregivenbytheGoldstonebosons.Athighertemperaturestheringcorrectionsdominateandresultinacriticaltemperatureunaectedbytheinteraction.1IntroductionBose-Einsteincondensationiscentraltomuchofourunderstandingofphenomenaincondensedmatterphysics[1].Itisoneofthesimplestprocesseswherequan-tumeectsmanifestthemselvesonthemacroscopiclevelwhenanitefractionofthenon-interactingbosonsinasystemstarttooccupythelowestenergylevel.Al-thoughalltheparticleswillthenbeinthesamequantumstateatzerotemperature,thiscondensatedoesnothavereallong-rangeorderanddoesnottrulyrepresentadierentphase.ItwasBogoliubov[2]whorstshowedthatashort-rangere-pulsionbetweentheparticlesisnecessaryinordertohavearealcondensatewiththeparticlesinanewphysicalphasewhichissuperuid.Hisdescriptionofthecondensationofinteractingbosonshassincethenformedthebasisforamuchmoredetailedunderstandingoftheseimportantphenomena[3].UntilveryrecentlytheonlyphysicalBose-Einsteinsystemofnon-relativisticparticlesexhibitingaphasetransitionatlowtemperatures,wasliquidHe4.Butheretheparticledensityissohighthatitisastronglyinteractingsystemsothatperturbationtheoryaroundthefreesystemdoesnotwork[3].However,withtherecentexperimentalprogressmadeinconnectionwithmagneticallytrappedbosonsinthegasphase[4],thesituationhasradicallychangedandsystemsofweaklyinteractingbosonscannowbestudied.TheseweretheoreticallyinvestigatedinaseriesofpapersbyLee,Yangandtheircollaboratorsfortyyearsagousingmethodsfromstatisticalmechanics[5,6,7,8].Theirmanyresultsstillrepresenttoalargedegreethestateoftheoreticalunderstandingofweaklyinteractingbosongases.Inthenormalphaseonecanassignadenitenumberofparticlestothesystemwhilethisisimpossibleafterthecondensatehasformed[9].Bose-Einsteinconden-sationofinteractingparticlesisthereforetheoldestandstillprobablythesimplestexampleofspontaneoussymmetrybreakdownwhichtodayliesattheveryheartofmodernelementaryparticletheory[10].Sincethesetheoriesarerelativistic,itisnotobvioushowBogoliubov’smethodcanbeusedinthiscase.InsteadonehasdevelopedverypowerfulmethodsbasedonFeynman’spathintegralformulationofquantumeldtheories[11]whichallowthecalculationofthecorrespondingeectivepotentialsinaverysystematicway[12,13,14].Thisapproachtospontaneoussymmetrybreakdownbaseduponfunctionalme-thodshasnotyetbeenusedtothesameextentinthestudyofBose-Einsteincon-densationofnon-relativisticsystemsalthoughthebasicelementsarealreadyinamoderntextbook[15].IntwodimensionsithasbeenusedbyLozano[16].ItisnotclearifthisapproachisequivalenttotheBogoliubovmethodornot.Thisiswhatwehavesetouttoinvestigatehere.ArststepinthisdirectionwastakenseveralyearsagobyKapusta[17]whoconsideredinteractingsystemsofrelativisticbosonsatnon-zerochemicalpotentialandtheircondensationatlowtemperatures.Morerecently,BernsteinandDodelson[18]andBenson,BernsteinandDodelson1[19]extendedtheserelativisticcalculationsandalsoconsideredthenon-relativisticlimit.Wewillhereshowthattheirnite-temperatureresultsareincompleteinthattheyhavenotincludedthecontributionsfromtheringordaisydiagramswhichareknowntobeessentialatnon-zerotemperatures[13,14,20,21].Sincethen,thenon-relativisticBosegashasalsobeenstudiedbyStoofandBijlsma[22].Theuseoffunctionalmethodsinquantumstatisticalphysicsofnon-relativisticsystemshasnotyetaswidespreadasforrelativisticsystems.OneofthebestintroductionshavebeengivenbyPopov[23].Ourapproachisdierentandmorealongthelinesusedinrelativisticquantumeldtheories[24],butwewilltoalargedegreereproducehisresults.Thenecessaryformalismisestablishedinthenextsectionwherewewillderivethethermodynamicsofagasoffreebosonsinthislanguage.Wewillworkwiththetworealcomponentsoftheeldinsteadofthecomplexelditselfanditsconjugatewhichisusuallydoneincondensedmatterphysics[25,26].WendthischoiceofvariablesespeciallyadvantageousinthecaseofinteractingparticlesatzerotemperatureconsideredinSection3.Wecalculatetheeectivepotentialandfreeenergyatnon-zerochemicalpotentialintheone-loopapproximationwhereweincludethequantumeectsoftheuctuationsaroundtheclassicalsolution.Afterremovingthedivergencesinthetheorybyrenormalizationofthecouplingconstantandthechemicalpotential,wendthegroundstateenergyofthehard-corebos