The first-order phase transition in the one-dimensional q-state Potts model with long-range interactions decaying with distance as 1/r 1+σ , has been studied by Monte Carlo numerical simulations for 0  2. On the basis of thefinite-size scaling analysis of interface free energy ∆FL, specific heat and Binder’s fourth order cumulant, we obtain the first-order transition which occurs for σ below a threshold value σc (q ).U jednodimenzijskom Pottsovom modelu q–stanja s dugodosežnim međudjelovanjima koja opadaju s udaljenošću kao 1/r 1+σ , Monte Carlo simulacijama je promatran fazni prijelaz prvog reda za 0  2. Na temelju scaling analize slobodne energije međuplohe, specifične topline i Binderovog kumulanta četvrtog reda, dobivamo prijelaz prvoga reda za σ manji od granične vrijednosti σc(q)

Glumac, Zvonko

Uzelac, Katarina

English

HRČAK - Portal of Croatian Scientific and Professional Journals

ISSN1330–0016CODEN FIZBE7LETTERTO THE EDITORFIRST-ORDERPHASETRANSITION IN 1d POTTSMODEL WITH LONG-RANGEINTERACTIONSKATARINA UZELAC andZVONKO GLUMACInstituteof Physics,Bijenička46,P. O. Box304,HR-10000Zagreb,CroatiaReceived28August1997UDC 536.763PACSnumbers05.50.+q,64.60.CnThefirst-orderphasetransitionin theone-dimensionalq-statePottsmodelwith long-rangeinteractionsdecayingwith distanceas1 r1 σ, hasbeenstudiedby MonteCarlonumericalsimulationsfor 0  σ  1 andintegervaluesof q  2.Onthebasisof thefinite-sizescalinganalysisof interfacefreeenergy ∆FL, specificheatandBinder’s fourthordercumulant,weobtainthefirst-ordertransitionwhichoccursfor σ below athresholdvalueσc  q .Thesubjectof our studyis theone-dimensional(1d) Pottsmodelwith ferromagneticlong-range(LR) interactionsdecayingwith distanceas1 r1 σ, definedby theHamiltonianH 	 ∑i  jJi  j  1 σ δ  si  sj   (1)whereJ  0, si denotestheq-statePottsvariableat site i, δ is theKroneckersymbol,andsummationis takenover all pairsin thesystem.Thephasetransitionat nonzerotemper-ature,shown rigorously[1] for the Ising  q  2 casewith σ  1 andby renormalizationgroupfor thecontinuousn-componentmodelswith σ  1 [2], existsalsoin model(1) forσ  1 andq  0 [3,4] andgoesthrougha varietyof universalityclassesby variationof qandσ. Model(1) hasbeenusedasarelevantmodelfor describinganumberof phenomenaFIZIKA B 6 (1997)3, 133–139 133UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .involvingLR interactions,from spinglassesto neuralnetworks,but mayalsobeof interestfor possibleanalogies[5], in somecasesvery direct [6], with short-range(SR)modelsinhigherdimensions.An importantfeatureof Pottsmodelswith SRinteractionsis theonsetof thefirst-orderphasetransitionfor q above somethresholdvalueqc  d  , which dependson dimension-ality [7]. For example,in d  2 [8] andd  4 [9] analyticalresultsyield qc equalto 4and2, respectively, while for d  3, theapproximatemethodsgivea non-integervalueforqc, slightly lower than3 [10], so that the 3d 3-statePottsmodelhasan extremelyweakfirst-ordertransition,very difficult to detect[11]. Suchdistinctsituationscreatedby vari-ationof d andq make this modela canonicalexamplefor thestudyof variousaspectsoftemperature-drivenfirst-orderphasetransitions[12].Onecanexpectsimilar behaviour to occurfor the Pottsmodelwith LR interactions,althoughcertainrelatedquantities(e.g. the interfaceenergy) may requiredifferentinter-pretation.Generally, this modelhasbeenmuchlessexploredthanits SRversion,duetonon-localityof interactions,which makesdifficult the applicationof standardrenormal-izationgrouptechniquesin directspace,otherwiseappropriatefor discretemodels.A fewstudiesthathave beendoneconcernmostly the mean-fieldregion andits vicinity by us-ing anε  expansionwithin Ginzburg-Landaucontinuousformalism[13], andthespecialcaseof σ  1 [3]. We haverecentlyproposedthefinite-rangescaling(FRS)approach[14]suitablefor studyof theHamiltonian(1), with bothσ andq arbitraryandcontinuous[4].This approachis, however, inherentlyinsensitive to discernthefirst-ordertransitionsandremainedinconclusivein thisrespect.Theproblemwasnot resolvedin otherrecentworks[15,16]on theLR Pottsmodeleither.For this reasonwepresentheretheresultsbasedonsimplenumericalsimulationsper-formedwith intentionto examinetheexistenceof a first-ordertransitionin thismodelandgeta qualitativeestimateof its dependenceonq andσ.It hasbeenrecentlypointedout [17], thatthetemperature-drivenfirst-ordertransitionscanbeidentifiedfrom finite-sizescaling(FSS)analysisof maximaof theenergy probabil-ity distributionPL  E  1ZL  K L  E  e KE  (2)whereK  J  kBT, ZL  K  is thepartitionfunction,  L  E  is thenumberof configurationswith energy E, andindex L denotesthesystemsize.Dueto thecoexistenceof phases,atthetemperatureof a first-orderphasetransitionPL  E  hastwo maxima,correspondingtothetwo wellsof thefreeenergy. Thebarrierseparatingthem,whichrepresentstheinterfacefreeenergy, is definedby∆FL  ln PL  EminPL  EmaxKL (3)whereEmin andEmax denotethe energiescorrespondingto the minimum andoneof thetwo maxima,respectively, thefinite-sizetemperatureKL beingadjustedsoasto make thetwo maximaequal.For a first-orderphasetransition,∆FL shoulddiverge for L  ∞. Insystemswith SRinteractionsit scaleslike a surface,i.e.  Ld  1, while in thepresentcaseit is expectedto scaleratherlike avolume,i.e.  L.134 FIZIKA B 6 (1997)3, 133–139UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .The calculationswere performedon chainsof size 100  L  400 with periodicboundaryconditions. We used the simple Metropolis single-spin-flipalgorithm with1  106  3  106 Monte Carlo (MC) sweepsper spin. The numberof necessaryrunsfor thepreciselocalizationof eachof thesize-dependentcritical temperaturesKL hasbeenreducedby applyingtheFerrenberg andSwendsen[18] histogrammethod.Weconsideredintegervaluesof q  2 in theinterval 0  σ  1. While for q  2 (Isingcase),thesimulationsat Tc show only a singlemaximumin PL  E  in theentirerangeofσ, for highervaluesof q, thetwo peaksemergefor σ sufficiently low. They becomemorepronouncedwith increasingq or decreasingσ. For illustration, in Fig. 1 areshown themaximacorrespondingto differentvaluesof σ, takenfrom threetypicalsetsof simulationswith thefixedvaluesq  3 andL  400.−0.65−0.60−0.55−0.50−0.45−0.40E’!−4.2−4.1−4.0−3.9−3.8ln PL" (E’)σ = 0.3σ = 0.4σ = 0.5Fig. 1. Maximaof lnPL # E $ % for q & 3 ' L & 400andσ & 0 ( 3 ' 0 ( 4 and0 ( 5, takenat respectivevaluesof KL. E $ & E ) *E0 # σ % * , whereE0 # σ % standfor thezero-temperaturenergies.We reportherethe systematicresultsfor two chosenvalues,q & 3 andq & 5, in thewholeinterval 0 + σ , 1 takenwith increment0 ( 1.Figures2aandb summarizetheresultsfor thefreeenergy barrierplottedasa functionof chainsizeL for q & 3 andq & 5, respectively. Thecorrespondingcritical temperaturesextrapolatedto L - ∞, givenin Table1,arefoundto bein goodagreementwith ourearlierFRSresults[4], aswell aswith otherknown approximateresults[16].FIZIKA B 6 (1997)3, 133–139 135UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .100.200.300.400.500. L/0.00.10.20.30.400.50.6 ∆ F1L100.200.300.400.500. L/0123456 ∆ F1Lq = 3q = 5 σ = 0.1σ = 0.2σ = 0.3σ = 0.4σ = 0.1σ = 0.2σ = 0.3σ = 0.4σ = 0.5 σ = 0.6 σ = 0.7(a) (b)  Fig.2.MC resultsof thefreeenergybarrier∆FL for sizesL 2 100to 400for: (a)q 2 3 3 σ 20 4 1 to 0 4 4, (b) q 2 5 3 σ 2 0 4 1 to 0 4 7. Noticethattheslopeis by anorderof magnitudelargerin thecase(b). Thesizeof thenumericalerrorbarsis comparableto or smallerthanthesizeof thepoints.TABLE 1. Inversecritical temperatures(KMCe ) obtainedby extrapolationof KL comparedto FRSextrapolatedvalues(KFRSe ) [4].σ KMCe KFRSe KMCe KFRSeq 2 3 q 2 50.1 0.190 0.136 0.262 0.280.2 0.279 0.270 0.333 0.450.3 0.380 0.386 0.492 0.5760.4 0.489 0.494 0.637 0.6900.5 0.771 0.8030.6 0.901 0.9200.7 1.019 1.046For bothconsideredvaluesof q, thereis a wide rangeof σ, where∆FL increaseswithsize,indicatingthefirst-ordertransition.As expectedfor theLR interactions,∆FL is pro-portionalto thevolumeratherthansurfaceanddependslinearly on L. Theslopeis largerby an orderof magnitudefor q 2 5 in comparisonto q 2 3, showing that the first-ordercharacterbecomesstrongerwith increasingq, like for theSR interactions.In both casestheslopedecreaseswith increasingσ to thepoint where∆FL becomesof theorderof nu-mericalerror. Beyondthisvalue,at leastfor thesizesconsideredhere,thePL 5 E 6 exhibitsasinglemaximumindicatingtheonsetof thesecond-orderphasetransition.In presentcal-culation,takenwith a roughincrementof 0.1in σ, thischangeis observedaroundσ 2 0 4 5andσ 2 0 4 8 for q 2 3 andq 2 5, respectively. Thesevaluesshouldbetakenwith cautionandonly asa lower limit for thethresholdvalueσc betweenthefirst- andthesecond-ordertransition. Namely, the presentmodel allows the continuousapproachto the thresholdvalueσc, wherebythefirst-ordertransitionbecomesarbitrarily weakandvery difficult todetect,comparableto thesituationwith the3d 3-statePottsmodelwith SR interactions.Theabove results,however, stronglysuggesthatσc is considerablylargerfor q 2 5 than136 FIZIKA B 6 (1997)3, 133–139UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .for q 7 3, andthatdependenceσc 8 q9 shouldbeexpected,analogousto the thresholdde-pendencedc 8 q9 in theSRmodel.Two otherenergy-relatedquantitiesaremoreconventionally[19] usedfor determina-tion of thefirst-ordertransitionin context of FSSanalysisof MC simulationresults:spe-cific heatandBinder’s fourthordercumulant[20], whichbothcanbederivedfrom PL 8 E 9 ,andexpressedin termsof higherenergy momenta: En ; L 7 ∑E EnPL 8 E 9 . Thespecificheatis givenbyCL 7 K2Ld < : E2 ; L = : E ; 2L >@? (4)Accordingto theFSStheory, for second-ordertransitionsits maximumscalesasCmaxL ALα B ν, whereα and ν are the critical exponentsof the specificheatand the correlationlength, respectively. Whenthe transitionis of the first order, it scalesasa volume, i.e.CmaxL A Ld. Insteadof theBinderfourthcumulantV C 4DL 7 1 = U C 4DL E 3, weconsiderheretheratioU C 4DL 7GF E4 H LF E2 H 2L ?(5)For the first-order transitions, limL I ∞ U C 4DL 7 1 when T E7 Tc, while at T 7 TclimL I ∞ U C 4DL 7 const H 1. For thesecond-ordertransitionsit alwaystendsto one.Wepresenthebehaviour of thosetwo quantitiesontwo examples:q 7 5 J σ 7 0 ? 2 andq 7 3 J σ 7 0 ? 8, representativeof thefirst- andsecond-orderregimes,respectively.In Figs. 3a andb, onecanobserve two kinds of behaviour of CmaxL in the two cases:linear and power law. The fit to the form CmaxL A Lx in the latter casegives the valuex 7 0 ? 24 for q 7 3 J σ 7 0 ? 8. Thebareextrapolationerrorbarsfor x areestimatedto beoforderof 10%.Thehyper-scalingrelationwith thesubstitutionof FRSresult[4] ν 7 1 ? 74givesα E ν 7 0 ? 15. The differencecanbeattributedto thegeneraldifficulty in extractingthecritical exponentsfrom thespecificheat,andto thefactthatthecalculatedexponentissmall,andadditionalcorrectiontermsdueto finite-sizegainimportance.Theconvergenceof themaximaof U C 4DL with sizeis presentedin Figs.3c andd. Thepoints for the Ising model with σ 7 0 ? 5 are addedas a referencefor the second-ordertransitionbehaviour. Within the limits of accuracy, thecaseq 7 3 J σ 7 0 ? 8 shows a clearconvergencetowards1. The points for q 7 5 J σ 7 0 ? 2 converge towardsa much largervalue,which is approximately2 ? 4 whenwe take into accountlargervaluesof L andusethelinearextrapolation.Thus,thetwo quantitiesconfirmearlierconclusionsbaseduponthebehaviour of ∆FL.However, at thepresentstage,we couldnot extractfrom thesequantitiesany betterpreci-sionin thedeterminationof σc 8 q9 , sowedonot reproduceany systematicstudyof them.FIZIKA B 6 (1997)3, 133–139 137UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .100K200K300K400K LL2.53.03.54.04.5cL max102030405060cL max0.000 K0.005 K0.010 K 1/L1.001.021.041.06Max [ UML (4) ]q = 2   σ = 0.5q = 3   σ = 0.81.001.502.002.503.00Max [ UNL (4) ]q = 5 σ = 0.2 q = 3 σ = 0.8q = 5 σ = 0.2  (a) (b) (c) (d) Fig.3.MC datafor sizesL O 100to400: specificheatmaximaareplottedfor (a)q O 3 P σ O0 Q 8, (b) q O 5 P σ O 0 Q 2; ratio U R 4SL maximaareplottedin functionof inversesizefor: (c)q O 3 P σ O 0 Q 8, (d) q O 5 P σ O 0 Q 2 (full circles).Thediamondscorrespondto q O 2 P σ O 0 Q 5taken asa reference.Lines show the linear extrapolations.Notice the commonscaleonx-axis,but differentscaleson y-axis.Thesizeof thenumericalerrorbarsis smallerthanthesizeof thepoints.In summary, by simplenumericalcalculations,in combinationwith FSSarguments,we have shown thatthe1d LR Pottsmodelfor integerq T 2 exhibits thefirst-orderphasetransitionfor σ below somethresholdvalueσc generallydependingon q. Thefirst-ordercharacterbecomesweaker with the increaseof σ, which representsa continuousparam-eter leadingfrom the first- to the second-orderphasetransitionregime. More intensivenumericalapproach[21] is neededin thefuturein orderto determinethethresholdvalueσc U qV .References1) F. J.Dyson,Commun.Math.Phys.12 (1969)91;2) J.M. Kosterlitz,Phys.Rev. Lett. 37 (1976)1577;3) J.L. Cardy, J.Phys.A 14 (1981)1407;4) Z. GlumacandK. Uzelac,J.Phys.A 26 (1993)5267;5) C. Tsallis,Fractals5 (1995)541;6) E. Luijten andH. W. J.Blöte,Phys.Rev. Lett. 76 (1996)1557;7) F. Y. Wu, Rev. Mod. Phys.54 (1982)235;8) R. J.Baxter, J.Phys.C: SolidStatePhys.6 (1973)L445;9) A. Aharony andE. Pytte,Phys.Rev. B 23 (1981)362;10) seee.g.H. W. J.BlöteandR.H. Swendsen,Phys.Rev. Lett. 43 (1979)799,andreferencesin [7];11) W. Janke andR. Villanova,Nucl. Phys.B 489 [FS] (1997)679;138 FIZIKA B 6 (1997)3, 133–139UZELAC AND GLUMAC: FIRST-ORDER PHASE TRANSITION IN . . .12) seee.g.A. Billoire, Nucl. Phys.Proc.Suppl.42 (1995)21,andreferencestherein;13) R. G. PriestandT. C. Lubensky, Phys.Rev. B 13 (1976)4159; W. K. TheumannandM. A.Gusm̃ao,Phys.Rev. B 31 (1985)379;14) K. UzelacandZ. Glumac,J.Phys.A 21 (1988)L421;15) E. Luijten andH. W. J.Blöte,Int. J.Mod. Phys.C 6 (1995)359;16) S.A. CannasandA. C. N. deMagalh̃aes,J.Phys.A 30 (1997)3345;17) J.LeeJ andJ.M. Kosterlitz,Phys.Rev. Lett. 65 (1990)137;18) A. M. Ferrenberg andR. H. Swendsen,Phys.Rev. Lett. 61 (1988)2635;19) K. BinderandH. J.Herrmann,MonteCarlo Simulationin StatisticalPhysics, eds.M. Cardona,P. Fulde,K. vonKlitzing andH.-J.Queisser, Springer-Verlag,Berlin (1992);20) K. Binder, Phys.Rev. Lett. 47 (1981)693;21) Z. GlumacandK. Uzelac,in preparation.FAZNI PRIJELAZPRVOGA REDA U 1d POTTSOVOM MODELU SDUGODOSĚZNIM MED– UDJELOVANJEMU jednodimenzijskom Pottsovom modelu q–stanjas dugodosěznim med–udjelovanjimakoja opadajus udaljenǒsću kao 1W r1X σ, Monte Carlo simulacijamaje promatranfazniprijelazprvogredaza0 Y σ Z 1 i cjelobrojnevrijednostiq [ 2. Natemeljuscalinganalizeslobodneenergije med–uplohe,specifǐcnetopline i Binderovog kumulantačetvrtogreda,dobivamoprijelazprvogaredazaσ manjiodgranǐcnevrijednostiσc \ q] .FIZIKA B 6 (1997)3, 133–139 139

Fazni prijelaz prvoga reda u 1d Pottsovom modelu s dugodosežnim međudjelovanjem

Abstract

Similar works

Full text

Available Versions

HRČAK - Portal of Croatian Scientific and Professional Journals