A Study of Dynamic Finite Size Scaling Behavior of the Scaling Functions-Calculation of Dyn

CriticalIndexofWolffAlgorithm

SEMRAGUND¨UC¨¸,MEHMETDILAVER˙,MERALAYDIN†andY˙IG˘IT˙GUND¨UC

¨¸HacettepeUniversity,PhysicsDepartment,

06532Beytepe,Ankara,Turkey

Abstract

Inthisworkwehavestudiedthedynamicscalingbehavioroftwoscalingfunctionsandwehaveshownthatscalingfunctionsobeythedynamicfinitesizescalingrules.Dynamicfinitesizescalingofscalingfunctionsopenspossibilitiesforawiderangeofapplications.Asanapplicationwehavecalculatedthedynamiccriticalexponent(z)ofWolff’sclusteralgorithmfor2-,3-and4-dimensionalIsingmodels.Configurationswithvanishinginitialmagnetizationarechoseninordertoavoidcomplicationsduetoinitialmagnetization.TheobserveddynamicfinitesizescalingbehaviorduringearlystagesoftheMonteCarlosimulationyieldszforWolff’sclusteralgorithmfor2-,3-and4-dimensionalIsingmodelswithvanishingvalueswhichareconsistentwiththevaluesobtainedfromtheautocorrela-tions.Especially,thevanishingdynamiccriticalexponentweobtainedford=3impliesthattheWolffalgorithmismoreefficientineliminatingcriticalslowingdowninMonteCarlosimulationsthanpreviouslyreported.

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Keywords:Isingmodel,scalingfunctions,dynamicscaling,timeevolutionofthemagne-tizationandthescalingfunctions,dynamiccriticalexponent.

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1Introduction

Finitesizescalinganduniversalityargumentshavebeenusedtostudythecriticalparame-tersofspinsystemsovertwodecades[1].Jansen,SchaubandSchmittmann[2]showedthatforadynamicrelaxationprocess,inwhichasystemisevolvingaccordingtoadynamicsofModelA[3]andisquenchedfromaveryhightemperaturetothecriticaltemperature,auniversaldynamicscalingbehaviorwithintheshort-timeregimeexists[4,5,6].TheexistenceoffinitesizescalingevenintheearlystagesoftheMonteCarlosimulationhasbeentestedforvariousspinsystems[5,6,7,8,9,10,11,12],thedynamiccriticalbehavioriswell-studiedandithasbeenshownthatthedynamicfinitesizescalingrelationholdsforthemagnetizationandforthemomentsofthemagnetization.Forthekthmomentofthemagnetizationofaspinsystem,dynamicfinitesizescalingrelationcanbewrittenas[2]

M(k)(t,ǫ,m0,L)=L(−kβ/ν)M(k)(t/τ,ǫL1/ν,m0Lx0)

(1)

whereListhespatialsizeofthesystem,βandνarethewell-knowncriticalexponents,tisthesimulationtime,ǫ=(T−Tc)/Tcisthereducedtemperatureandx0isanindependentexponentwhichistheanomalousdimensionoftheinitialmagnetization(m0).InEq.(1)τistheautocorrelationtime,τ∼Lzandzisthedynamiccriticalexponent.

TherelationgiveninEq.(1)canbeusedtostudytheknowncriticalexponentsaswellasexponentszandx0.Momentsofthemagnetizationhavetheirownanomalousdimensions(kβ/ν)Andusingthesequantities(inordertoobtaindynamicexponentszandx0)one

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mayexpectsomeambiguitiesduetocorrectiontoscalinganderrorsondeterminingtheanomalousdimensionofthegiventhermodynamicquantity.Theambiguitiesduetotheanomalousdimensionofthethermodynamicquantitycanbeavoidedifoneconsidersquan-titieswhicharethemselvesscalingfunctions.Moreover,scalingfunctionsareextremelypowerfultoidentifytheorderofthephasetransition,aswellaslocatingthetransitionpointofstatisticalmechanicalsystemsonfinitelattices.

Inthisworkweproposethatthedynamicfinitesizescalingrelationalsoholdsforthescalingfunctionsandthescalingrelationcanbewrittensimilarlytothemomentsofthemagnetization,

O(t,ǫ,m0,L)=O(k)(t/τ,ǫL1/ν,m0Lx0).

(2)

OuraimistostudydynamicfinitesizescalingbehaviorofthescalingfunctionsbyusingEq.(2).

Inourcalculationstwodifferentscalingfunctionsareused.ThefirstsuchquantityisBinder’scumulant[13,14,15].Binder’scumulantiswidelyusedinordertoobtainthecriticalparametersaswellastodeterminethetypeofthephasetransition.Thisquantityinvolvestheratioofthemomentsofthemagnetizationorenergy.InthisworkwehaveusedthedefinitionofBinder’scumulantwhichinvolvestheratioofthemomentsofthemagnetization.Simplestsuchquantitycanbegivenas

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B=

n2(t)

.(4)

Heretheaveragesarecalculatedovertheconfigurationsobtainedateachiteration.InthisworkweuseBinder’scumulantforn=2byusingthetherelation

B2(t)=

(t)

F=

(6)

whereSiisthesumofthespinsintheithsurface.SimilartothecalculationsofBinder’scumulant,iteration-dependentcalculationofFrequirestheconfigurationaverageswhichareobtainedforeachiterationyieldingaMonteCarlotimedependentexpression,

F(t)=(t).

(7)

F(t)canbeusedincalculatingthedynamicfinitesizescalingrelationgiveninEq.(2).

InWolff‘salgorithm[22],onlyspinsbelongingtoacertainclusteraroundtheseedspinareconsideredandupdatedateachMonteCarlostep.Inequilibrium,thedynamiccriticalexponentoftheWolff’salgorithmcannotbeobtaineddirectlyfromtheobservedauto-correlationtimes(τW),insteadtheautocorrelationtime(τW)isgovernedbytheaveragesize(oftheclusters.τWcanbeobtainedbytherelation,

′

τW=τW

′

smallpower[27].Forthe3-dimensionalcase,Tamayoetal[28]calculatedthedynamiccriticalexponentasz∼0.44(10).Wolffcalculatedasmallervalueofz=0.28(2)[25]usingenergyautocorrelations.In4-dimensions,Tamayoetal[28]obtainedzwithavanishingvalue.Thisresultisalsoconsistentwiththemean-fieldsolutionfortheIsingmodelinfourandhigherdimensions.Inarecentpublicationithasbeenshownthatvariousalternativeclusteralgorithmspossessimilardynamicbehavior[29]

TheefficiencyoftheWolff‘salgorithmisdirectlyrelatedtothesizeoftheupdatedclusters,hencetheefficiencyincreasesduringthequenchingprocess,asthenumberofiterationsincreases.Boththeaverageclustersizeandsusceptibilityhavethesameanomalousdi-mension,henceinobtainingτfromtheobservedbehaviorofthedynamicvariable,onecanreplaceby.InourcalculationsbothquantitieshavebeenusedinordertoscaletimevariableforquantitiesB2(t)andF(t)considered.

2SimulationsandResults

WehavestudieddynamicscalingforscalingfunctionsB2(t)andF(t)for2-,3-and4-dimensionalIsingmodelsevolvingintimebyusingWolff’salgorithm.Wehavepreparedlatticeswithvanishinginitialmagnetizationandtotalrandominitialconfigurationsarequenchedatthecorrespondinginfinitelatticecriticaltemperature.WehaveusedthelatticesL=256,384,512,640,L=32,48,64,80andL=16,20,24for2-,3-and4-dimensionalIsingmodels,respectively.Foreachlatticesize,independentinitialconfigu-rationsarecreated.Thenumberofinitialconfigurationsvariesdependingonthelattice

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size.Onaverage,tenbinsofonethousandruns,twentybinsoftwentythousandrunsandtenbinsoftenthousandrunshavebeenperformedfor2-3-,and4-dimensionalIsingmodels,respectively.Errorsarecalculatedfromtheaveragevaluesforeachiterationobtainedindifferentbins.

Inthedynamicfinitesizescaling,forthealgorithmsinwhichallspinsarecheckedforupdatingtheMonteCarlotime,tscalesast/Lz.InWolff’salgorithm,oneclusterisupdatedateachiteration,hencethereisaneedtousetheaveragenumberofupdatedspinsateachiteration.Ifthetimeisnotscaledbytheaverageclustersize,usingonlyLzasafactorshiftsthecurvestowardseachotherandcurvescrossatsomepoint,butscalingcannotbeobserved.Inordertoseeagoodscaling,thereisaneedtouseafactorwhichistheaverageclustersize((t))oralternatively(t).Dynamicscalingusing(t)and(t)asthefactorintimescalingresultsinthesamevalueofthedynamiccriticalexponentz.Thedynamiccriticalexponentziscalculatedusingtherelation

z=z′−(2YH−d)

whichisobtainedfromtherelation

(9)

τ=τ′(10)

wherez′andτ′arethemeasuredvaluesofthedynamiccriticalexponentandtheau-tocorrelationtime.Inthesecalculations,YHistakenasYH=

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YH=2.4808[30,31],YH=3(mean-fieldsolution)forthe2-,3-and4-dimensionalmod-els,respectively.Sinceandscaleinthesameform,inourpresentationwehavescaledtimeaxiswitht/Lz.

′

InFigure1wehavepresentedBinder’scumulant(B2(t))beforeandafterthedynamicfinitesizescalingfor2-dimensionalIsingmodelforthelatticesizesconsidered.Figure1a)showsthetimeevolutionofB2(t)duringtherelaxationofthesystem.Duringtherelaxationprocessthecorrelationlengthtendstogrowuntilitreachesthelatticesize.Whenthecorrelationlengthapproachesthelatticesize,Binder’scumulantexhibitsanabruptchangeandfinallyitsettlestoanew,longcorrelationlengthvalue.Thepositionofthisabruptchangealongtimeaxisdependsonthelinearsize(L)ofthelattice.Buttheinitialandthefinalvaluesareexactlythesameforalllatticesizes.InFigure1b)thescalingofBinder’scumulant(B2(t))canbeseen.Asitisseenfromthisfigure,B2(t)scaleswithtimeast(t)/Lz.Asaresultofscaling,thevaluez′=1.725±0.03isobtainedfromminimizingdistancesbetweenB2(t)datafordifferentlatticesizes.

′

Figure2showsthesurfacerenormalizationfunctionF(t)forthe2−dimensionalIsingmodelforthesamelatticesizesgiveninFigure1.Figure2a)showsthesimulationdataandFigure2b)showsthescalingbyuseof(t)asthefactorintimescaling.Thisfunctionalsoexhibitsanabruptchangefrominitialvanishingvaluetoacertainconstantvalueasthecorrelationlengthreachesthesizeofthelattice.Asitisseenfromthisfigure,timetoreachtheplateauisproportionaltothelinearsize(L)ofthesystem.The

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positionsoftheabruptchangesforbothF(t)andB2(t)arethesameforeachlatticesize.AsinthecaseofB2(t)giveninFigure1,agoodscalingisobservedforthesamevalueofz′=1.725±0.03.

Figures3and4showthesimulationdataandthedynamicscalingforB2(t)andF(t)for3-dimensionalIsingmodel,respectively.Inbothfiguresa)showsthetimeevolutionofthescalingfunctionsandb)showsthefunctionsafterdynamicscaling.Scalinggivesz′=1.95±0.05forbothB2(t)andF(t)forthe3-dimensionalIsingmodel.Similarly,figures5and6showthesimulationdataandthedynamicscalingforB2(t)andF(t)for4-dimensionalIsingmodel,respectively.ForthismodelscalingofdataforB2(t)andF(t)resultsinz′=2.0±0.2andz′=2.1±0.2,respectively.InallthesefiguressimulationdataforfunctionsB2(t)andF(t)showthesamebehaviorandscalingisverygood.Theerrorsinthevaluesofz′areobtainedfromthelargestfluctuationsinthesimulationdataforB2(t)andF(t).ThevaluesofthedynamiccriticalexponentzarecalculatedusingEq.(9)for2-,3-and4-dimensionalIsingmodelsandthesevaluesaregiveninTable1.Theliteraturevaluesarealsogivenforcomparison.

3Conclusion

Wolff’salgorithmisoneofthemostdifficultalgorithmstocalculatethedynamiccriti-calexponent.Simplythedifficultyarisesfromthecomparisonbetweenthenumberofupdatedspinsandthetotalnumberofspins.Ateachiterationonlyasingleclusterisupdated.Intheliterature,for2-,3-and4-dimensions,smalldynamiccriticalexponents

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areobtained[22,23,24,25,26,27,28],butfurtherstudiesofthedatasuggestthatforallthreedimensionsthedynamiccriticalexponentoftheIsingmodelcanbeconsideredaszero.Themeasurementofthedynamiccriticalexponentinthermalequilibriumisex-tremelydifficult,sincethecorrelationlengtharoundthephasetransitionpointisaslargeasthesizeofthelattice.Indynamicfinitesizescaling,sincethecorrelationlengthremainssmallerthanthelatticesize,itisexpectedthatstatisticallyindependentconfigurationsleadtobetterstatisticssincetherearenofinitesizeeffects.

InthisworkwehaveconsideredthedynamicscalingbehaviorofBinder’scumulant(B2(t))andtherenormalizationfunction(F(t))for2-,3-and4-dimensionalIsingmodels.Wehaveobservedthatthesescalingfunctionscanbeusedtoidentifythecriticalpointandthecriticalexponentsduringtheinitialstagesofthethermalization.Inourcalculations,wehaveobservedthatourresultsareconsistentwithvanishingdynamiccriticalexponent.Despitethefactthatobtaininggoodstatisticsisextremelytimeconsumingforlargelattices,finitesizeeffectsdonotplayanyroleinobtainingtheresults.Onecanseefromtheresultsofdynamicscalingthatscalingisverygoodandtheerrorsareverysmall,hencethismethodisagoodcandidatetocalculatethedynamiccriticalexponentforanyspinmodelandforanyalgorithm.Themoststrikingresultofourcalculationsisthatthedynamiccriticalexponentfor3-dimensionalIsingmodelisobtainedasz=0.02±0.09,insteadofpreviouslyreportedrangeofvaluesz=0.28−0.44[28,25].ThisisaclearindicationthattheefficiencyoftheWolff’salgorithmisbetterthanpreviouslythought,especiallyineliminatingcriticalslowingdownofMonteCarlosimulations.Thismeans

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thatusingthisalgorithm,verylargelatticesatcriticalitycanbeconsidered,withoutunusuallylargestatisticalerrorsbuildingup.

Acknowledgements

We

greatfully

acknowledge

Hacettepe

University

Research

(Projectno:0101602019)andHewlett-Packard’sPhilanthropyProgramme.

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Fund

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TableCaptions

Table1.

Thevaluesofcalculateddynamiccriticalexponents(z)(usingEq.(9))for2-,3-and4-dimensionalIsingmodels.FirsttwocoulumnsarethevaluesobtainedfromscalingfunctionsB2(t)andF(t),respectivelyandthethirdcolumnincludestheliteraturevalues.

FigureCaptions

Figure1a)Bindercumulantdata(B2(t))for2-dimensionalIsingModelforlinearlatticesizesL=256,384,512,640asafunctionofsimulationtimet,b)scalingofB2(t)datagivenina)using(t)asthefactorintimescaling,

Figure2a)Simulationdatafortherenormalizationfunction(F(t))asafunctionofsimu-lationtimetfor2-dimensionalIsingmodelforlinearlatticesizesL=256,384,512,640,b)scalingofF(t)datagivenina)using(t)asafactorintimescaling.

Figure3.SimulationdataforB2(t)asafunctionofsimulationtimetfor3-dimensionalIsingmodelforlinearlatticesizesL=32,48,64,80,b)scalingofB2(t)datagivenina)using(t)asthefactorintimescaling.

Figure4.SimulationdataforF(t)asafunctionofsimulationtimetfor3-dimensionalIsingmodelforlinearlatticesizesL=32,48,64,80,b)scalingofF(t)datagivenina)

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using(t)asthefactorintimescaling.

Figure5.SimulationdataforB2(t)asafunctionofsimulationtimetfor4-dimensionalIsingmodelforlinearlatticesizesL=16,20,24,b)scalingofB2(t)datagivenina)using(t)asthefactorintimescaling.

Figure6.SimulationdataforF(t)asafunctionofsimulationtimetfor4-dimensionalIsingmodelforlinearlatticesizesL=16,20,24,b)scalingofF(t)datagivenina)using(t)asthefactorintimescaling.

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dz(F)20.02±0.0530.02±0.094

−0.13±0.19

2d=21.8L=256L=384L=512L=640B2(t)1.61.41.21010000t2000030000(a)2d=21.8B2(t)1.61.41.2100.05 t < S > / L2z0.1(b)

Figure1:

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d=20.8L=256L=384L=512L=6400.6 F(t)0.40.20010000 t2000030000(a)d=20.80.6F(t)0.40.2000.020.040.06 t < S > / L2z0.080.10.12(b)

Figure2:

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2d=3L=32L=48L=64L=801.8B2(t)1.61.41.2010000t2000030000(a)

d=31.8 B2(t)1.61.41.200.05t < S > / L 2z0.1(b)

Figure3:

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0.8L=320.6L=48d=3L=64L=80F(t)0.40.2001000020000t3000040000(a)0.8d=30.6 F(t)0.40.2000.05 t < S > / L2z0.1(b)

Figure4:

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d=41.8L=20L=16L=24B2(t)1.61.405000t1000015000(a)

d=4B2(t)1.61.400.05 t < S > / L2z0.1(b)

Figure5:

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0.8d=4L=160.6L=20L=24F(t)0.40.200500010000t1500020000(a)0.8d=40.6 F(t)0.40.2000.05 t < S > / L2z0.1(b)

Figure6:

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