f the727]. Inodel withodel. Thetialent

Physics Letters A 333 (2004) 277–283

www.elsevier.com/locate/pla

Critical dynamics of the Potts model:short-time Monte Carlo simulations

Roberto da Silvaa,∗, J.R. Drugowich de Felíciob

a Departamento de Informática Teórica, Instituto de Informática, Universidade Federal do Rio Grande do Sul,Av. Bento Gonçalves, 9500, CEP 91501-970 Porto Alegre, RS, Brazil

b Departamento de Física e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo,Av. Bandeirantes, 3900, CEP 014040-901 Ribeirão Preto, SP, Brazil

Received 25 March 2004; accepted 15 October 2004

Available online 27 October 2004

Communicated by A.R. Bishop

Abstract

We calculate the new dynamic exponentθ of the 4-state Potts model, using short-time simulations. Our estimatesθ1 =−0.0471(33) andθ2 = −0.0429(11) obtained by following the behavior of the magnetization or measuring the evolution otime correlation function of the magnetization corroborate the conjecture by Okano et al. [Nucl. Phys. B 485 (1997)addition, these values agree with previous estimate of the same dynamic exponent for the two-dimensional Ising mthree-spin interactions in one direction, that is known to belong to the same universality class as the 4-state Potts manomalous dimension of initial magnetizationx0 = zθ + β/ν is calculated by an alternative way that mixes two different iniconditions. We have also estimated the values of the static exponentsβ andν. They are in complete agreement with the pertinresults of the literature. 2004 Elsevier B.V. All rights reserved.

PACS:05.50.+q; 05.10.Ln; 05.70.Fh

Keywords:Monte Carlo simulations; Short time dynamics; Potts model

eenin

n-min

e-

at-ta-

1. Introduction

Several results about critical phenomena have brecently obtained using Monte Carlo simulations

* Corresponding author.E-mail addresses:rdasilva@inf.ufrgs.br(R. da Silva),

drugo@servermail.ffclrp.usp.br(J.R. Drugowich de Felício).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.10.042

short-time regime[1]. Such simulations are more covenient than traditional ones done in the equilibriuregime because they circumvent a known problemcomputational physics: the critical slowing down phnomena.

Investigating Monte Carlo simulations beforetaining equilibrium permits to us obtaining the stic critical exponents (β and ν) but also leads to

.

278 R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283

tionne-en-gy

g-ts

derthe

w-nse-ad-

and

c

ter

po-ti-rgehe-

iza-iti-vee

needart

as

lht ofoalst

sal-

ical

g.av-y-ent

ng(atni-

po-ni-aled

singon

n.del

iornd

e-

of

the less known dynamical ones. The scaling equafor non-equilibrium regimewas obtained by Janseet al. [2], on the basis of renormalization group thory. For systems without conserved quantities likeergy and magnetization (model A in the terminoloof Halperin and Hohenberg[3]), it is written as

M(k)(t, τ,L,m0)

(1)= b−kβ/νM(k)(b−zt, b1/ντ, b−1L,bx0m0

),

wherem0 is the initial magnetization,x0 = x0(m0)

[4] is the anomalous dimension of the initial manetization,β and ν are the known static exponenand z is the dynamic one (τ ∼ ξz). The values ofM(k) are thekth moments of magnetization, defineby M(k) = 〈Mk〉, where〈·〉 indicates an average ovseveral samples randomly initialized but satisfyingcondition of having the same magnetization(m0) atthe beginning. The relevance of this result is alloing to include the dependence on the initial conditioof the dynamic systems in their non-equilibrium rlaxation. As an important consequence, they couldvance the existence of a new critical exponentθ , whichis independent of the known set of static exponentseven of the dynamic exponentz.

By considering large systems and choosingτ = 0(T = Tc); b−zt = 1 in Eq.(1), we obtain the dynamiscaling law for the magnetization:

(2)M(t,m0) = t−β/νzM(1, tx0/zm0

).

ExpandingM around the zero value of the parameu = tx0/zm0, one obtains:

(3)M(t,m0) = m0tθ + O

(u2)

which leads to the power law:

(4)M(t) ∼ m0tθ

whereast < t0 ∼ m−z/x00 since in this regimeu � 1.

This new universal stage, characterized by the exnent θ = (x0 − β

ν)/z, has been exhaustively inves

gated to confirm theoretical predictions and to enlaour knowledge on phase transitions and critical pnomena.

Indeed, the anomalous behavior of the magnettion at the beginning of relaxation, also called crcal initial slip, was originally associated to a positivalue ofθ . This kind of behavior was confirmed in thkinetic q = 2- and 3-state Potts models[1], as well

as in irreversible models like the majority voter o[5] and the probabilistic cellular automaton proposby Tomé and Drugowich de Felício to describe pof the immunological system[6]. However, the ex-ponentθ can also be negative. This possibility wobserved in the Blume–Capel model (analytical[7]and numerically[8]) and in the Baxter–Wu mode(numerically [9]), an exactly solvable model whicshares with the 4-state Potts model the same secritical exponents. In addition, a negative but close tzero value forθ was obtained for the two-dimension(2D) Ising model with three-spin interactions in juone direction (IMTSI)[10], a model which is alsoknown to behave to the 4-state Potts model univerity class.

However, at the best of our knowledge a numerdetermination of the exponentθ for this special case(q = 4) of the Potts model continues to be lackinThus, we decided to investigate the short-time behior of that model in order to better understand its dnamics and also to learn about the role of the exponx0 of the non-equilibrium magnetization concernithe well established classification of the modelsleast in what concerns the static behavior) in the uversality classes.

As we mentioned above the estimates for the exnent θ for two models that belong to the same uversality class as the 4-state Potts model have revediscrepant results. Whereas the two-dimensional Imodel with three spin interactions in one directi(IMTSI) obeys the conjecture by Okano et al.[11] (θshould be negative and close to zero)[10] the Baxter–Wu model[9] strongly disagrees from that predictioPutting in numerical values, whereas the IMTSI moexhibits a small and negative value(−0.03± 0.01)for θ , when studying the BW model we foundθ =−0.186± 0.002.

In this Letter we study the short-time behavof the two-dimensional four-state Potts model apresent numerical estimates forθ using two differentmethods. First, we fix the initial value of the magntization and follow its time evolution to obtainθ(m0)

which in turn can be extrapolated to lead toθ whenm0 → 0. Second, we deal with the time correlationthe magnetization defined by[5]:

(5)C(t) = 1

N2

⟨M(t)M(0)

⟩,

R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283 279

ini-bey-

edllym--there-

ingthe

taperthes.e-

ngy

nee-onin

redus

de

g-f

sde-rhen-

hf

an

,phe-

s)ult

ec-the

d innher-s not

where the average is done over samples whosetial magnetizations are randomly chosen but o〈M(0)〉 = 0. In Ref. [5] it was shown that this quantity exhibits atT = Tc the power-law behavior:

(6)C(t) ∼ tθ .

This approach has severaladvantages when comparto the other technique but its application was initiarestricted to the models which exhibit up–down symetry. Recently, Tomé[12] has shown that this result is more general and can be applied whenevermodel presents any group of symmetry operationslated to the Markovian dynamics. Eq.(6)was shown tobe valid, for instance, in the antiferromagnet ordermodel, models with one absorbent state and even inBaxter–Wu model[9] which exhibits aZ(2) ⊗ Z(2)

symmetry.It is well known that theq �= 2 Potts model does no

have up–down symmetry. However, based on the pof Tomé[12] we can use this approach to calculateexponentθ of the three- and four-state Potts model

We checked this possibility working with the threstate Potts model. Our estimate isθ = 0.072(1), ingood agreement with the result obtained by Zhe[1] (θ = 0.070(2)) and with the estimate obtained bBrunstein and Tomé[13] using a cellular automatowhich exhibits C3v symmetry, the same as the thrstate Potts model. A plot of the correlation functiof the magnetizationC(t) in this case is presentedFig. 1.

In the sequence we estimated the exponentθ of the4-state Potts model which agrees with the conjectuby Okano et al. In addition, they agree with previo

Fig. 1. Log–log plot ofC(t) × t to 3-states Potts model.

estimates forθ obtained by Simões and DrugowichFelício[10].

To confirm our first estimate we simulated the manetization evolution for four different initial values om0 which after extrapolated (m0 → 0) led to a similarresult.

It is clear that having the exponentθ the anomalousdimensionx0 follows. But inspired in previous studie[14] we decided to investigate a direct manner interminingx0 using mixed initial conditions. In ordeto build the adequate function we remember that win contact with a heat bath atT = Tc the magnetization of completely ordered samples (m0 = 1) decayslike a power law

(7)M(t) ∼ t−β/νz.

Thus, according to the scaling relations(4) and(7)it is enough to work with the ratio

(8)F3(t,L) = 〈M〉m0→0

〈M〉m0=1

to obtain a power law which decays astθ+β/νz = tx0/z.Usingz as input[14] we can achieve the value ofx0.On the other hand, by following the relation(7) weestimate the ratioβ/νz which can be compared witthe exact resultβ/ν = 1/12 after using the value othe dynamical exponentz.

Finally, using derivatives of the magnetization atearly time and once more the value ofz as input weobtain the critical exponentν of the correlation lengthwhose numerical estimates by usual techniques (nomenological renormalization group, Hamiltonianstudies and equilibrium Monte Carlo simulationare always very different from the pertinent res(ν = 2/3).

The Letter is organized as follows. In the next stion we present a brief review and some details ofsimulation. The results are presented in Section3 andour conclusions are in Section4.

2. The kinetic 4-states Potts model

In time-dependent simulations we are interestefinding power laws for physical quantities even whethe system is far from equilibrium. In this regime tmagnetizationM(t) must be calculated as an aveage over several samples because the system doe

280 R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283

r-all

arplessne-

theil-

rests-

x-

asri-ag-

rage

edl-theyngenull

y

g toba-

igh-

s,orr-

les

rnt.

e

forbe-e

far-

he

obey any a priori probability distribution. The aveage can be done in different ways: we can preparethe samples with the same initial magnetization (shpreparation) or generate samples which satisfy arestrict criteria like to have mean value of the magtization equal to zero.

Theq states Potts model ferromagnetic withoutpresence of an external field is defined by the Hamtonian[15]:

(9)H = −J∑〈i,j〉

δσi,σj ,

whereJ denotes the interaction between the neaneighbors〈i, j 〉 and σi can assume different valueσi = 0,1, . . . , q −1. If two spins are parallel they contribute with energy−J , else the energy is null.

The critical temperature of this model, is known eactly [15],

(10)J

kBT= log

[1+ √

q].

The magnetization, different of other modelsIsing and Blume–Capel is not only the sum of vaables of spin. A general expression used for the mnetization in the Potts model that considers an aveover the sites and over the samples is written as

(11)⟨M(t)

⟩ = 1

Ns(q − 1)N

Ns∑j=1

Ld∑i=1

(qδσi,j (t),1 − 1),

whereσi,j denotes the spini of thej th sample at thet th MC sweep. HereNs denotes the number of thsamples andLd is the volume of the system. This kinof simulation is performedNB times to obtain our finaestimates as a function oft . In order to prepare a lattice with a given magnetization we need to choosestatesσi = 0,1,2,3 in each site with equal probabilit(1/4). Next we measure the magnetization and chastates in sites randomly chosen in order to obtain avalue for the magnetization. Finally, we changeδ sitesoccupied byσi = 0,2 or 3 and substitute byσi = 1.The initial magnetization in this case will be given b

(12)m0 = 4δ

3N,

whereN = Ld .We have chosen to update the spins accordin

the heat bath algorithm, which means that the probility that the spinσx localized at the sitex can assume

another valueσf

x is given by:

(13)

P(σx → σ

f

x) =

exp(− J

kBT

∑k δ

σx+k(t),σf

x

)∑3

σf

x =0exp

(− JkBT

∑k δ

σx+k(t),σf

x

) ,

where∑

k denotes the sum about the nearest nebors sites of the spin at the sitex.

3. Results

We performed Monte Carlo simulations forNB = 5different bins and four different initial magnetizationm0 = 4δ/3N ≈ 0.033, 0.049, 0.066 and 0.082, fa lattice of sizeL = 90. These magnetizations corespond respectively toδ = 200, δ = 300, δ = 400,δ = 500 according to the expression(12). To estimatethe errors we ran 5 different bins, with 35 000 sampeach one.

In Figs. 2 and 3we present the log–log plot fotwo different initial magnetizations and two differevalues ofm0 = 0.033 andm0 = 0.082, respectivelyThe fit had excellent goodness(Q = 0.99) in the en-tire range[ti , tf ] ⊂ [0,200]. We have chosen the rang[10,100] to estimate the value toθ for all initial mag-netizations. We must note that the slope is positivem0 = 0.082 but changes the signal at some valuetween 0.066 and 0.049. This trend to a negative valucan be also certified inFig. 4 that shows the plot oθ = θ(m0). It is worth to mention that there is a clelinearity in the plot ofθ versus the initial magnetization (seeFig. 4) but big fluctuations are observed in t

Fig. 2. Time evolution of the magnetization form0 = 0.033.

R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283 281

).

luet

Wu

b-

teate-les

ate.

tsns.

f ai-thatfar

lyne-e

is

-ing

Fig. 3. Time evolution of the magnetization form0 = 0.082.

Fig. 4.θ as function ofm0 in the 4-state Potts model (extrapolation

evolution of the magnetization in each case.The vafound for θext is −0.0471(33) is in fair agreemenwith the result[10] for the IMTSI model and stronglydisagrees with the result obtained for the Baxter–model.

We also have ran Monte Carlo simulations to otain the time-evolution correlationC(t), and the dy-namic exponentθ according to the relation(6). Forthe same interval[10,100] we have gotten the estimaθ = −0.0429(11) that corroborates the above estim−0.0471(33) obtained from the evolution of the nonequilibrium magnetization. We used 35 000 sampand once more 5 different bins to make this estimIn Fig. 5 it is shown the log–log plot ofC(t) × t forthe four-state Potts model.

Equilibrium simulations for the four-state Potmodel are ever accompanied by strong fluctuatio

Fig. 5. Log–log plot ofC(t) × t to 4-states Potts model.

Fig. 6. Plot ofF3 function for casem0 = 0.066.

The reason is the presence in the Hamiltonian omarginal operator characterized by an anomalous dmension equal to zero. If we hope some signal ofanomalous behavior in short-time simulations donefrom equilibrium we should look for that in the onnew anomalous dimension: that of the initial magtization x0. In order to confirm the null value for thanomalous dimension we estimate directlyx0 usingthe reported functionF3. Here we usedNb = 5 binsfor each initial condition and again theNs = 35 000realizations. A plot when the initial magnetizationm0 = 0.082 can be seen inFig. 6. Extrapolating tom0 → 0 we found(x0/z)ext = 0.0077(33) with thesame set initial magnetizationsm0 that was used toextrapolatingm0. Using as inputz = 2.290(3), we ob-tainx0 = 0.0176(75).

The value ofβ/ν can be obtained directly of magnetization decay of a initial ordered state, consider

282 R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283

cay

-

r

e:

e

them-e ofnza-te

therthee-

tedg-e

, al-eatorion

urre-

es-l-

andtoni-za-

ial

the relation:

(14)β

ν=

(β

νz

)c

· z.

The value of the exponent obtained from the de(M ∼ t−β/νz), was (β/νz)c = 0.0547(1) with Q =0.84. Using as the input the same value forz value weachievedβ/ν = 0.12526(28) that is in a good agreement with the exact resultβ/ν = 0.125[15].

Finally to find the exponentν, we calculated theexpected decay of derivative

D(t) = ∂

∂τlnM(t, τ )

∣∣∣∣τ=0

(15)N= lnM(t, τ + δ) − lnM(t, τ − δ)

2δ.

According to Ref.[1], for m0 = 1, we expect ofD(t) the power law behavior

(16)D(t) ∼ t1/νz

and so

ν =(

1

νz

)−1

c

1

z.

In Fig. 7 we illustrate the power law(16), perform-ing simulations withNs = 8000 samples withNb = 5to estimate errors andtmax = 1000 MC steps. Oubest estimative is 1/νz = 0.65453(46), in the inter-val [10,1000] with Q = 0.99. The value found toν atthis interval isν = 0.667(1). The value(1/νz)c also is

Fig. 7. Time dependence of derivativeD(t) of the 4-state Pottsmodel. The slope gives 1/νz.

used to estimateβ . This minimizes the errors becaus

(17)β =(

β

νz

)c

(1

νz

)−1

c

.

The value found toβ is 0.0836(2). These values arin complete agreement with conjectured resultsν =2/3= 0.6̄ andβ = 1/12= 0.083̄.

4. Conclusions

We calculated the dynamic exponentθ of the four-state Potts model using two different techniques:time-evolution of the magnetization when the saples are prepared with a small and nonzero valum0 for four different values of initial magnetizatioand through of the time correlation of the magnetition. The values found forθ in both cases corroborathe conjecture by Okano et al.[11] and are in com-plete agreement with the estimate obtained for anomodel which is in the same universality class as4-state Potts model: the 2D Ising model with threspin interactions in one direction. We also estimadirectly the anomalous dimension of the initial manetizationx0 which results very closed to zero. Thsame does not occur in the Baxter–Wu model thatthough exhibiting the sameleading exponents as th4-state Potts model, does not own a marginal operwhich prevents us from determine with good precisthe critical exponents.

We also estimated the static exponentsβ andν us-ing the short-time Monte Carlo simulations and oresults are in surprisingly agreement with pertinentsults.

As an additional contribution we present a newtimate for the exponentθ of the 3-state Potts modeworking with the time evolution of the time correlation of the magnetizationC(t). This kind of ap-proach has been previously used by BrunsteinTomé when studying a four-state cellular automawith C3v symmetry[13]. The result agrees with estmates obtained from the evolution of the magnetition [1] after the extrapolation (m0 → 0).

Acknowledgements

J.R. Drugowich thanks CNPq for partial financsupport.

R. da Silva, J.R. Drugowich de Felício / Physics Letters A 333 (2004) 277–283 283

89)

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