First Order Phase Transition in the 3-dimensional

arXiv:cond-mat/0507550v1[cond-mat.stat-mech]23Jul2005FirstOrderPhaseTransitioninthe3-dimensionalBlume-CapelModelonaCellularAutomatonN.Sefero˘glu∗,A.¨Ozkan∗andB.Kutlu∗∗∗Gazi¨Universitesi,FenBilimleriEnstit¨us¨u,FizikAnabilimDalı,Ankara,Turkey∗∗Gazi¨Universitesi,Fen-EdebiyatFak¨ultesi,FizikB¨ol¨um¨u,06500Teknikokullar,Ankara,Turkeye-mail:[e-mail]firstorderphasetransitionofthethree-dimensionalBlumeCapelareinvestigatedusingcoolingalgorithmwhichimprovedfromCreutzCellu-larAutomatonfortheD/J=2.9parametervalueinthefirstorderphasetransitionregion.Theanalysisofthedatausingthefinite-sizeeffectandthehistogramtechniqueindicatethatthemagneticsusceptibilitymaximaandthespecificheatmaximaincreasewiththesystemvolume(Ld)atD/J=2.9.Keywords:Blume-CapelModel;CreutzCellularAutomaton;FirstOrderPhaseTransition;Finite-SizeEffect;Histogram;SimpleCubicLattice.1.IntroductionTheHamiltonianoftheBlume-Capelmodel(1,2)isgivenby,HI=−JXijSiSj+DXiS2i(1)wheresi=−1,0,1andthefirstsumiscarriedoutoverallnearest-neighboring(nn)spinpairsonathree-dimensionalsimplecubiclattice.Theparameters1oftheJandDarethebilinearinteractionenergyandsingle-ionanisotropyconstant,respectively.Themodelisknowntohavearichcriticalbehaviorandconsideredbybothnumericalandanalyticalmethods.Recently,mostofthesestudiesmainlyfocusedondeterminingthetricriticalpointofthemodel(3−8).Inpre-viouspaper(9),weobtainedthetricriticalpointvalueofthe3-dBlume-CapelmodelasD/J=2.82.Thisresultsagreeswithaseriesexpansion(3−4),theclustervariationmethod(5),theeffectivefieldtheory(6)andBethe-Peierlsapproximationresults(7).Althoughthecriticalbehavioronthesecond-orderphasetransitionregion(D/J2.82)hasbeenstudiedcommonly,thefinite-sizeeffectsinthefirst-orderphasetransitionregion(D/J2.82)isnotinvestigatedexactly.OuraimofthispaperistoidentifythephasetransitioninthefirstorderphasetransitionregionoftheBlumeCapelmodelbythefinite-sizeeffectsandthehistogramsofenergydistributionP(E)andorderparameterdistributionP(M).Itiswellknownthatthefirstorderphasetransitionsinvolvethecoex-istenceoftwodistinctphasesandarecharacterizedbytheexistenceofadiscontinuityintheenergyandmagnetizationfortheinfinitesystems.Asaresultsofthesediscontinuities,thespecificheatandthesusceptibilityshowsingularitiesofthedeltafunction.However,thecharacteristicsingularitiesanddiscontinuitiesinthefirstorderphasetransitionsappearsasroundedandsmearedinfinitesystemsemployedincomputersimulations(10,11).Thesebehaviorsatthefirstorderphasetransitionsinfinite-sizesystemsarequalita-2tivelysimilartothefinite-sizeeffectsatsecond-orderphasetransitions.ThefinitesizeeffectsatthefirstorderphasetransitionhavebeeninvestigatedbythedifferentmethodssuchastheMonteCarlo(12−15),thephenomenologicalrenormalizationgroup(16),transfermatrixmethod(17),renormalization-groupanalysis(18).Accordingtotheresultsofthesestudies,thefinite-sizeeffectsatthefirstorderphasetransitiondependonthevolumeofthesystemLd.ThespecificheatandthesusceptibilityincreasewithLdandthetransitiontemperaturewhicharelocationsoftheirextrema,approachthetransitiontemperatureintheinfinitesystemasL−d.Inaddition,thehistogramsofenergydistributionP(E)andtheorderparameterdistributionP(M)arethereliablemethodtostudyfirst-orderphasetransitions(10,12,13,19).Whiletheprobabilitydistributionofenergyshowsasinglepeakinasecond-orderphasetransition,itshowsadoublepeakinafirst-orderphasetransition.InthispaperwesimulatethethreedimensionalBlume-CapelmodelwiththecoolingalgorithmwhichimprovefromtheCreutzCellularAutomatoninthefirstorderphasetransitionregion.Thefinite-sizescalingandtheprobabilitydistributionforenergyandorderparameterareusedtodeterminethenatureofthephasetransition.Theremainderofthepaperisorganizedasfollows.ThedataareanalyzedandtheresultsarediscussedinSection2andtheconclusionisgiveninSection3.3.ResultsanddiscussionThethreedimensionalBlume-Capelmodelissimulatedwiththecool-ingalgorithm(9)whichimprovedfromtheCreutzCellularAutomaton.ThesimulationshavebeenmadeattheD/J=2.9anisotropyparametervalue3inthefirstorderphasetransitionregiononsimplecubiclatticesLxLxLofthelineardimensionsL=8,10,12,14,16,18and20withperiodicboundaryconditions.Atthealgorithm,thecoolingrateisequalto0.01HkpersiteforD/J=29/10valueandthekineticenergyofthesystemreducedbythedifferentcoolingamountspersitebecausethekineticenergy,Hk,isanintegervariableintheinterval(0,24J).Thecomputedvaluesofthequantitiesareaveragesoverthelatticeandoverthenumberoftimesteps(1.000.000)withdiscardofthefirst100.000timestepsduringwhichthecellularautomatondevelops.Thesimulationsweredoneabout10timeswithdifferentinitialconfig-urationsateachlatticesize.MeasurementsshownthatthevariationofthermodynamicquantitiesandthehistogramofP(E)andP(M)foreachdifferentinitialconfigurationaredifferentfromeachotheraroundthetransi-tiontemperature.Atthesametime,someofthesimulationsdonotexhibitthedoublepeakinthehistogramoftheP(E)andthethreepeaksinthehistogramoftheP(M)whichcharacterizethefirstorderphasetransition.Therefore,wechoosethesimulationswhichexhibitthecharacteristicdoublepeakinthehistogramofP(E)andthethreepeaksinthehistogramoftheP