17 June 2002

Physics Letters A 298 (2002) 325–329

www.elsevier.com/locate/pla

Mixed initial conditions to estimate the dynamic critical exponentin short-time Monte Carlo simulation

Roberto da Silva, Nelson A. Alves∗, J.R. Drugowich de Felício

Departamento de Física e Matemática, FFCLRP Universidade de São Paulo, av. Bandeirantes 3900,CEP 014040-901 Ribeirão Preto, SP, Brazil

Received 4 October 2001; received in revised form 19 April 2002; accepted 26 April 2002

Communicated by A.R. Bishop

Abstract

We explore the initial conditions in short-time critical dynamics to propose an alternative way to evaluate the dynamicexponentz. Estimates are obtained with high precision for the 2D Ising model and the 2D Potts model with three and four statesby performing heat-bath Monte Carlo simulations. 2002 Elsevier Science B.V. All rights reserved.

PACS: 64.60.Fr; 64.60.Ht; 02.70.Lq; 75.10.Hk

Keywords: Short-time dynamics; Critical phenomena; Dynamic exponent; Ising model; Potts model; Monte Carlo simulations

The description of static critical phenomena interms of finite size scaling (FSS) relations, developedby Fisher et al. [1,2] has been extended by Halperin,Hohenberg, Ma and Suzuki [3,4] to include dynamicalproperties of the system. Later, Janssen et al. [5],and independently Huse [6] found evidence for anuniversal behavior far from equilibrium.

As discussed by Janssen et al. one finds an universalbehavior already in the early stages of the relaxationprocess for systems prepared at an initial state char-acterized by non-equilibrium values of the order para-meter. As a consequence, they could advance the exis-tence of a new critical exponentθ , independent of the

* Corresponding author.E-mail addresses: rsilva@dfm.ffclrp.usp.br (R. da Silva),

alves@quark.ffclrp.usp.br (N.A. Alves), drugo@usp.br(J.R. Drugowich de Felício).

known set of static exponents and of the dynamic crit-ical exponentz. This new exponent characterizes theso-called “critical initial slip”, the anomalous increas-ing of the magnetization when the system is quenchedto the critical temperatureTc.

That new universal stage has been exhaustively in-vestigated to confirm theoretical predictions and to en-large our knowledge on phase transitions and criticalphenomena. In this sense, several models and algo-rithms [7–9] have been used, as toy models, in orderto check the ability of the new approach in obtainingdynamic and static critical exponents. Results are ingood agreement with pertinent results for static expo-nents and seems to be confident even for the new crit-ical exponentθ . However, a reliable technique to ob-tain the dynamic exponentz is lacking. A first proposalby Li et al. [10] using a time-dependent Binder cu-mulant yields estimates with low precision when com-

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00571-6

326 R. da Silva et al. / Physics Letters A 298 (2002) 325–329

pared with other techniques [11,12]. Another kind ofcumulant, proposed by Zheng [13], gives the right an-swer for the 2D Ising model but does not produce theexpected estimates forz at least for two well-knownmodels: the 3-state Potts model [13] and the Isingmodel with multispin interactions [14].

In this Letter, we introduce and check a differentapproach in the context of short-time simulations toobtain the exponentz, combining the behavior of theorder parameter and its second moment when thesystem is submitted to different initial conditions.

Before presenting our approach, we shall reviewthe main results in short time dynamics.

Although Halperin et al. [3] have studied systemswith different dynamics, we consider here only sys-tems without conservation laws, the so-called model A[15], because our discussion aims dynamics generatedby heat-bath (HB) dynamics. Following Janssen et al.we start from a magnetic system prepared at high tem-perature (T � Tc) with a small non-zero magnetiza-tion m0 and quenched to the critical temperatureTc

without any external magnetic field. If the system isallowed to relax towards equilibrium with the dynam-ics of model A, the magnetization obeys the followingscaling relation (generalized to thekth moment),

M(k)(t, τ,L,m0)

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

).

Here b is an arbitrary spatial scaling factor,t is thetime evolution andτ is the reduced temperature,τ =(T − Tc)/Tc. M(k) = 〈Mk〉 are thekth moments ofmagnetization. The exponentsβ and ν are the staticcritical exponents, whilez is the dynamic one. Thisscaling relation depends on the initial magnetizationm0 and gives origin to a new, independent criticalexponentx0, the scaling dimension of the initialmagnetization, which is related toθ .

From Eq. (1) we can derive the power law increas-ing of the magnetization observed in the initial stageof the dynamic relaxation. For this purpose, we con-sider large lattice sizesL at τ = 0 with b = t1/z. Thisleads to the scaling relation

(2)M(t,m0) = t−β/νzM(1, tx0/zm0

),

for the first moment of the magnetizationM(1) ≡ M.By expanding this equation for smallm0, we obtain

the following power law,

(3)M(t) ∼ m0tθ ,

where, as anticipated, we identifyθ = (x0 − β/ν)/z.Here we also have the condition thattx0/zm0 is small,which sets a time scalet0 ∼ m

−z/x00 [5,16] where that

phenomenon can be observed.On the other hand, it has been realized the exis-

tence of another important dynamic process from aninitial ordered state [18–20], which represents anotherfixed point in the context of renormalization group ap-proach. This leads to a different universal behaviorof the dynamic relaxation process also described byEq. (1) withm0 = 1. In particular, dealing with largeenough lattice sizes at the critical temperature (τ = 0),one obtains a power law decay of the magnetization

(4)M(t) ∼ t−β/νz,

which follows from Eq. (1) when we chooseb−zt = 1and shows the average magnetization is not zero aswould be expected from a disordered initial state.

Eq. (1) and their particular forms in (3) and (4) canbe used to determine relations involving static criticalexponents and the dynamic exponentz [13,16].

The observables in short-time analysis are de-scribed by different scaling relations according to theinitial magnetization. In particular, with initial condi-tion m0 = 0 the second momentM(2)(t,L),

M(2) =⟨(

1

Ld

N∑i=1

σi

)2⟩

(5)= 1

L2d

⟨N∑

i=1

σ 2i

⟩+ 1

L2d

N∑i =j

〈σiσj 〉,

behaves asL−d since in the short-time evolution thespatial correlation length is very small compared withthe lattice sizeL. Thus we have [8,13]

(6)

M(2)(t,L) = t−2β/νz M(2)(1, t−1/zL

)∼ t(d−2β/ν)/z.

Moreover, under this initial condition one can also de-fine the time-dependent fourth-order Binder cumulantat the critical temperature,

(7)U(t,L) = 1− M(4)(t,L)

3(M(2)(t,L))2,

R. da Silva et al. / Physics Letters A 298 (2002) 325–329 327

which leads atT = Tc to the FSS relation

(8)U(t,L) = U(b−zt, b−1L

),

and the exponentz can be independently evaluatedthrough scaling collapses for the cumulant from dif-ferent lattice sizes [10,17].

In order to obtain more precise estimates for thedynamic exponentz, another cumulant has been pro-posed [13]. It is given by

(9)U2(t,L) = M(2)(t,L)

(M(t,L))2− 1,

which is supposed to behave as

(10)U2(t) ∼ td/z,

when one starts from ordered states. In this case,curves for all the lattices would lay on the samestraight line without any re-scaling in time and resultsin more precise estimates forz. However, as men-tioned before, the application of this procedure hasnot confirmed previous estimates for the exponentz atleast in two well-known models: the two-dimensionalq = 3 Potts model [13] and the Ising model with threespin interactions in just one direction [14].

Concerned about that disturbing disagreement [13]we decided to study the evolution of the ratioF2 =M(2)/M2 but using different initial conditions to cal-culate each one of the mean values. The reason iswe know the behavior of the second moment of themagnetization when samples are initially disordered(m0 = 0) and also the dependence on time of the mag-netization of samples initially ordered(m0 = 1). Un-der the above mentioned conditions the ratio behavesas

(11)

F2(t,L) = M(2)(t,L)|m0=0

(M(t,L))2|m0=1∼ t(d−2β/ν)/z

t−2β/νz= td/z,

which has the same power law that the cumulant men-tioned before but requires two independent simula-tions instead of one used for calculatingU2.

In short-time Monte Carlo (MC) simulations, thetime scalet is settled in units of whole lattice updates,and does not depend on the initial conditions. How-ever, the same dynamics (Metropolis, Glauber or heat-bath) should be used in both simulations [7,8].

Now we report our estimates obtained for the 2DIsing model,q = 3 andq = 4 Potts models.

We have performed independent heat-bath MCsimulations for a large lattice(L = 144) accordingto the required initial conditions and the evolutionshave been done until the maximum timet = 200 MCsweeps withN = 10000 samples. This gives averagesfor the magnetization and its second moment. We haveperformed this kind of simulation 20 times to obtainour final estimates for each moment in function oft .Since all runs are independent due to the choice ofrandom numbers, we have a total of 400 time seriesfor Eq. (11).

Our results are shown on a log–log scale for thetime interval[10,100] in Figs. 1 and 2, respectively,for three- and four-state Potts model (a similar figureis obtained for 2D Ising model) and they appear tobe clearly consistent with a power law in that timeinterval. The error bars are also presented in thosefigures but they are hardly seen in that scale.

Fig. 1. Time evolution ofF2(t) for the 2D three-state Potts model.

Fig. 2. Time evolution ofF2(t) for the 2D four-state Potts model.

328 R. da Silva et al. / Physics Letters A 298 (2002) 325–329

Table 1Dynamic exponentz for 2D Ising model, three- and four-state Pottsmodel

Reference (year) Ising q = 3 q = 4

This work 2.156(2) 2.197(3) 2.290(3)

[12] (2000) 2.1667(5)

[9]1 (1999) 2.153(2) 2.191(6)

[13]2 (1998) 2.153(4) 2.203(11)[13]3 (1998) 2.16(2) 2.14(3)

[22] (1998) 2.137(8)

[23] (1997) 2.166(7)

[8]4 (1997) 2.155(3) 2.196(8)

[11] (1995) 2.172(6)

[24] (1995) 2.16(4)

[25] (1993) 2.165(10)[26] (1992) 2.16(2)

[27] (1987) 2.16(5) 2.16(4) 2.18(3)

[28] (1986) 2.43(15) 2.36(20)

1 2D TP lattice;2 from scaling collapse (Table 3);3 applying Eq. (9);4 from HB algorithm, whilez = 2.137(11) andz = 2.198(13),

respectively, for Ising andq = 3 Potts model from Metropolisalgorithm.

Our estimates forz comes from a least squarefitting in the time interval[ti , tf ]. Due to our smallstatistical errors, we can make a systematic study forthe range int where we find acceptable goodness-of-fit Q [21]. As examples, we obtain for 2D Ising model,z = 2.1435(2) in the time interval[10,200] with Q =10−250, z = 2.1359(3) in [50,200], with Q = 10−41,and the most acceptableQ = 0.99 in [30,90], whichyieldsz = 2.1565(7). Here we observe that our resultis in agreement with most of the recent estimatesshown in Table 1, within two standard-deviations.This indicates our statistical errors are presumablyunderestimated possibly due to corrections to scalingand correlations between successive measurementsin the time series. This occurs mainly because ofthe initial ordered condition in our samples whenwe calculate the evolution of the magnetization. Toeliminate in part this problem we have reanalized ourdata taking measurements at every 5th sweep. In thiscase we obtainz = 2.156(2), keeping the averagevalue but enlarging our previous estimate for the errorbar.

To complete the overview in Table 1 we nextrecall other estimates also presented in Ref. [23].Their estimates have been obtained from the long time

behavior of the magnetization for the square lattice(sq),z = 2.168(5), for the triangle (TP),z = 2.180(9),and for the honeycomb (hc) lattice,z = 2.167(8),while from damage spreading in short-time the authorsquote 2.166(7), 2.164(7) and 2.170(10) for sq, TP andhc lattices.

For q = 3 Potts model, our study monitoringQ gives z = 2.198(2) in the interval [50,90] withQ = 0.82. This result agrees withz = 2.203(11)(Ref. [13]) obtained from the Binder cumulant inEq. (7), and with the estimatez = 2.196(8) obtainedin [8] from short-time analysis of the autocorrelationfunction. The value in [9] refers to the triangularlattice, presenting further numerical evidence to thedynamic universality. Ref. [13] also presents the value2.14(3), obtained from Eq. (9), in clear disagreementwith 2.203(11) as commented in [13]. On the otherhand, our estimate from Eq. (11) gives a value in fullagreement with the fourth-order Binder cumulant andautocorrelation function analysis in short-time. Now,to take into account the correlations within time seriesdata we consider once more every 5th measurement,which leads to our final estimatez = 2.197(3).

The caseq = 4 has been less studied. Our analysisgivesz = 2.290(3) in the interval[60,90] with Q =0.72. We stress again the importance of monitoringQ,since we may find values forz as large asz =2.3483(2) in [10,200] but with unacceptable valuefor Q. Finally, we have extended our analysis ofq = 4 Potts model to a longer interval of time. Theresult for z is 2.293(3) in the interval [60,1000],which is in complete agreement with the precedingresult although, following our procedure where wechoose the time interval according to the values ofQ, we should discard such a large range in viewof its unacceptableQ. In this last analysis we haveconsidered measurements taken at every 10th sweepto evaluate the exponentz.

As a final comment, the suspectedz as beingweakly [28] or even independent onq [27] is notsupported by the most recent results presented inTable 1.

Acknowledgements

The authors acknowledge support by Brazilianagencies (CAPES, CNPq and FAPESP). Thanks are

R. da Silva et al. / Physics Letters A 298 (2002) 325–329 329

also due to DFMA for computer facilities at Institutode Física (USP).

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