Physica D 168–169 (2002) 318–324

Critical exponents for extended dynamical systems withsimultaneous updating: the case of the Ising model

Gabriel Pérez∗, Francisco Sastre, Rubén MedinaDepartamento de F´ısica Aplicada, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional,

Unidad Mérida, Apartado Postal 73 “Cordemex”, 97310 Mérida, Yucatán, Mexico

Abstract

The behavior of a simultaneously updated Metropolis simulation of the two-dimensional (2D) Ising model is explored. Theconfigurations generated by such a simulation are not constructed to be equilibrium configurations, and may therefore displaya different critical behavior, assuming a phase transition is present. This is relevant to the recently posed question of how theupdating scheme affects the critical exponents for chaotic extended systems that show continuous phase transitions. It is foundthat the simultaneously updated algorithm drives the lattice, for all non-zero temperatures, into blinking maximum energyconfigurations, making it of little use. A small modification, given by a reduction in the acceptance probabilities for spin flips,restores the ability of the algorithm to generate ordered and disordered states, with a well-defined phase transition betweenthem. The critical temperature becomes a function of the acceptance probability for energy lowering spin flips. The criticalexponents still fall into the 2D Ising universality class. This happens even though the simultaneously updated Metropolisalgorithm does not really simulate the Ising model in equilibrium.© 2002 Elsevier Science B.V. All rights reserved.

PACS:05.10.Ln; 05.45.Ra; 05.50.+q

Keywords:Metropolis algorithm; Ising model; Phase transition

1. Introduction

Continuous order–disorder phase transitions havebeen found recently in extended two-dimensional(2D) systems, with both chaotic[1] and stochastic[2] local dynamics, and diffusive coupling. In all themodels looked up until now, the local dynamics isscalar and odd-symmetric, and so they were expectedto fall in the 2D Ising universality class[1,3]. How-ever, in most cases it has been found that the criticalexponents are close, but not equal, to those of the 2DIsing model[2,4,5]. In practice, one finds thatν ≈ 0.9,

∗ Corresponding author.E-mail address:gperez@kin.cieamer.conacyt.mx (G. Perez).

instead of the Ising valueν = 1, while the ratiosβ/ν

andγ /ν remain at their Ising values of 1/8 and 7/4,respectively. It has been conjectured that this may berelated to the fact that updating in all these systemsis done simultaneously over the whole lattice, sincethey do go back to the 2D Ising universality classwhen the updating is changed to asynchronous[4].This would mean then that the updating scheme is arelevant parameter in the vicinity of the critical point.

It is therefore important to ask what happens to stan-dard, well known statistical models if one tries to forcethem to do simultaneous updating. This proposition inthe abstract may seem to make no sense, since equilib-rium statistical mechanics is usually studied in termsof ensembles, with no need for a time flow. But for

0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0167-2789(02)00519-5

G. Perez et al. / Physica D 168–169 (2002) 318–324 319

the purposes of this work, what we are thinking aboutis the particular simulation algorithms that introducea “time” in the models, in the sense of an ordered se-quence of iteration steps. Of these many algorithms wewill concentrate in the Metropolis[6] single-spin-flipMonte Carlo algorithm for the 2D nearest-neighborIsing model, which is simple and well known[7,8].

In order to give a faithful sampling of the equi-librium distribution, this algorithm requires that theupdating be done asynchronously and one can findin the literature many examples where this is accom-plished following the order of the lattice (“typewriterupdating”), or choosing sites at random. One is also al-lowed to update any non-interacting part of the latticeat once; for square lattices, half of the lattice can thenbe updated simultaneously (“checkerboard updating”).By mistake, full simultaneous updating have been tried[9], but it was found soon enough that in this case theMetropolis algorithm drives square lattices into blink-ing maximum energy patterns, which we will discusslater on. It has been assumed then that simultaneousupdating is of no use for the Metropolis algorithm[10].

The rest of this paper is arranged as follows. InSection 2we will review the Metropolis algorithm,and will discuss why its simultaneous updated ver-sion always gets trapped into blinking configurations.We will then introduce a slight modification in thealgorithm, so that it will once more show a continu-ous phase transition as the temperature is changed. InSection 3we will show the results of the simulationsdone with this modified algorithm, and inSection 4we will present our conclusions.

2. The Metropolis and simultaneous Metropolisalgorithms for the Ising model

The Metropolis algorithm for the simulation ofphysical systems in the canonical ensemble[6] hasbeen the workhorse of numerical statistical mechan-ics for almost half a century. For completeness, wewill give a short description of the algorithm here,following the very detailed presentation given in[7].It consists of the construction of a sequence (moreexactly, a Markov chain) of configurations, which

we will denote with greek indicesµ, ν, etc., whichappear with frequencies proportional to their Boltz-mann weight,Wµ = exp(−βEµ), whereEµ is theenergy of the configurationµ and β ≡ (kBT )−1.Equilibrium average values of observables are thengiven by simple averages over this sequence. TheMarkov chain of configurations one generates has toobey the following two restrictions: (1) it should beergodic, i.e., for a finite system, one should be ableto reach any valid configurationν starting from anyother valid configurationµ and (2) it should exhibitdetailed balance, which means that the ratio betweenthe probability of going from one configurationµ toanotherν, and the probability for the opposite transi-tion, ν to µ, should be proportional to the ratio of thecorresponding Boltzmann weights.

Since we will be dealing here only with its applica-tion to the Ising model, let us describe the Metropolisalgorithm in terms of spin configurations. The stepsfor one iteration of the algorithm are the following:starting form some arbitrarily generated spin config-uration (1), choose one spin. As mentioned in the in-troduction, this may be done in some order throughthe lattice, or at random, and one may even choose agroup of spins, as long as they do not interact with eachother. (2) One then flips the chosen spin and calcu-lates the change in energy E this leads to. (3) If this E is negative, the spin flip lowers the energy of thelattice, and is always accepted. If not (4), the flip in-creases the energy of the lattice, and is given a chanceof happening proportional to the Boltzmann weightinvolved in the transition,W = exp(−β E). This isimplemented with the generation of a random numberη, uniformly distributed between 0 and 1, and accept-ing the spin flip ifη < W . (5) Go back to step (1).

It is easy to see the ergodicity of this algorithm forT �= 0, since going from any configurationµ to anyother configurationν is always possible, requiring onlythe generation of a finite string of random numbersabove certain thresholds. For the condition of detailedbalance, one notices the following: letµ and ν beconfigurations that differ only in the orientation of onespin. The probability of going fromµ to ν, denotedP(µ → ν), is composed of two parts: the probabilityof generatingν starting fromµ, which is just 1/N ,

320 G. Perez et al. / Physica D 168–169 (2002) 318–324

where N is the number of sites in the lattice, andthe probability ofacceptingthe new configuration,A(µ → ν). In terms of these quantities, the conditionof detailed balance is

P(µ → ν)

P (ν → µ)= A(µ → ν)/N

A(ν → µ)/N= exp(β(Eµ − Eν)).

The Metropolis algorithm is then assigning the maxi-mum possible acceptance probabilityA(µ → ν) = 1for energy lowering transitions, and setsA(ν → µ) =exp(−β(Eµ − Eν)) otherwise, giving the correctacceptance ratio for the previous equation.

As mentioned, this needs to be done one spin ata time. When the previously described steps are ap-plied on every site of a square lattice simultaneously,one finds that in a few iterations the lattice reaches astate of maximum energy (for ferromagnetic coupling,neighboring spins pointing opposite to each other), andthat all spins flip at every iteration. This effect has beencalled “blinking checkerboard catastrophe”[9], but isnot limited to square lattices. In fact, when simulta-neous updating is attempted in a triangular lattice, theconfiguration soon evolves into one of the many localmaxima of energy that the frustration of the systemoffers, and then continues flipping the whole lattice atevery iteration.

It is not difficult to understand what is actuallyhappening in these cases. Suppose a checkerboardcluster develops by chance. Then, for every site inthis cluster, one finds that energy can be loweredflipping the local spin. If one then precedes to do justthat, the whole cluster is flipped and therefore goesback in two iterations to the original situation. Thesame applies to sites in touch with the cluster, whichthen grows until covering the whole lattice.1 The endreason for this behavior is the fact that spin flips thatlower the energy of the system arealwaysaccepted.

The solution for this anomaly becomes then clear:allow the algorithm to reject a given fraction of theenergy lowering spin flips. At the same time, in or-der to maintain detailed balance in its asynchronousversion, a similar extra fraction of the energy-raising

1 The possibility for the formation of large checkerboard-likedomains with static walls between them cannot be discarded. Asituation like this has not appeared in our simulations.

spin flips should be rejected. The spin-flip acceptanceis fixed asA(µ → ν) = A0, for Eµ > Eν , andA(µ → ν) = A0 exp(β(Eµ − Eν)) otherwise, whereA0 is some constant with 0< A0 < 1. For the asyn-chronously updated algorithm this change means thata fraction 1− A0 of all attempted moves are alwaysdiscarded, and the algorithm is simply a less efficientversion of the original Metropolis.

Before getting into the results, it is worth to restatethat this algorithm is no longer simulating the Isingmodel in equilibrium. For what we want to studyhere, it could be considered simply as an stochasticextended dynamical system, which borrows most ofits structure from the Metropolis simulation of Isingmodel, but is not really intended to be a simulation ofany physical model in equilibrium. In particular, no-tice that there is no clear way of establishing detailedbalance here. On the other hand, it should be noticedthat our evaluation of critical quantities is based infinite size scaling[11], which strictly speaking isjustified only for equilibrium models. The motivationfor its use here is mostly heuristic.

3. Numerical results for the Ising modelsimultaneous Metropolis algorithm

The simultaneous Metropolis algorithm was imple-mented in a simulation of the Ising model in a 2D

Fig. 1. Critical temperature as a function of the acceptance factorA0. The line is provided only as a guide to the eye.

G. Perez et al. / Physica D 168–169 (2002) 318–324 321

Fig. 2. Crossing of fourth-order cumulantsUL(T ), used to locatethe critical point. The actual data contains 20 points for each valueof L, covering a larger range. The values ofL are, starting fromthe smallest slope,L = 42, 48, 56, 64, 74, 84, 98, 112 and 128.The lines are second-order polynomial fits to the full data, andgive the best polynomial fit in the sense of minimizingχ2 perdegree of freedom.

triangular lattice, and order–disorder continuous phasetransitions were found in a large range ofA0. All thedeterminations of critical quantities were done usingarguments from finite size scaling. First, we did apreliminary exploration of the phase diagram, and theresults are shown inFig. 1. The critical temperaturebecomes a function ofA0, and for the values we havechecked stays above the correct value in triangularlattices,Tc = 4/ ln (3) ≈ 3.641. This determinationwas done via crossing of fourth-order cumulants[12]

UL(T ) = 1 − m4L(T )

3(m2L(T ))2

,

wheremkL(T ) is the average value found for thekth

power of the magnetization density.2 The two endpoints are not included, since, as already mentioned,A0 = 1 gives the blinking maximum energy patterns,and A0 = 0 would leave any configuration unmod-ified. The behavior of the model withA0 close tothese two extremes is left for future study.

2 As usual when dealing with finite size lattices, it is necessaryto consider the absolute value of the instantaneous magnetization,since otherwise the average values of all its odd moments wouldgo to zero for large runs. Since this systems is not in the thermo-dynamic limit, it cannot display ergodicity breaking.

A detailed simulation of the model forA0 = 1/2was performed, using sides ranging fromL = 42 to128. Periodic boundary conditions were implemented.For each size, the total number of iterations was around1400 times the correlation time. A dynamical exponentz ≈ 2 was found.

First we located the critical point, using againfourth-order cumulants. The results are given inFig. 2, and out estimation for this parameter is

Tc(A0 = 12) = 3.7475(10).

Here, the digits between parenthesis indicate theuncertainty in the corresponding last digits of thequantity. Using this value, we evaluated the critical

Fig. 3. (a) Scaling of derivatives of cumulantsU and other quanti-ties used to calculateν. Starting from the lower line, the quantityAcorresponds to∂T UL(T ), ∂T ln (〈m〉), ∂T ln (〈m2〉), ∂T ln (〈m3〉)and ∂T ln (〈m4〉). The lines have slope 1. (b) Same quantities,after removing the leading termL1/ν = L from their scaling.

322 G. Perez et al. / Physica D 168–169 (2002) 318–324

exponentν using the scaling behavior at the criticalpoint of the following quantities:

∂T UL(Tc) ∝ L1/ν, ∂T ln mkL(Tc) ∝ L1/ν .

For the uncertainty levels we have, we may assumethat we are already in the scaling regime. Therefore,these equations do not include finite size corrections.The results are shown inFig. 3(a) and (b), which showthe scaling of the derivatives forU and for ln(mk),with k = 1, 2, 3 and 4. The combined result obtainedfrom these five lines is

ν = 1.0070(81)

perfectly compatible with the exact Ising model resultν = 1. The magnetization exponentβ was calculated

Fig. 4. (a) Scaling of the first four moments of the magnetizationdensity. Starting from the lower line, we havem, m2, m3 andm4.The lines have slope−1/8. (b) First four moments of the magne-tization density, after removing the leading termL−kβ/ν = L−k/8

from their scaling.

using the scaling relations

mkL(Tc) ∝ L−kβ/ν,

where again we runk from 1 to 4, and no finite sizecorrections are used. The results are given inFig. 4(a)and (b), and the combined value we find is

β

ν= 0.1290(30).

The Ising valueβ/ν = 1/8 differs slightly from thisresult, but is still within its 2σ range. Next, we checkedthe magnetic susceptibility exponent, usingχL(T ) =N(m2

L(T ) − (mL(T ))2), and its scaling behavior

χL(T ) ∝ Lγ/ν.

Fig. 5. (a) Scaling of the specific heat, assuming a power-lawdependence. The line corresponds to the best linear fit with slope0.2232. (b) Scaling of the specific heat, after removing the leadingpower law term. It is clear that a simple power law does not fitwell with the specific heat.

G. Perez et al. / Physica D 168–169 (2002) 318–324 323

From the results obtained (not shown) the value foundis

γ

ν= 1.740(21)

again, compatible with the Ising valueγ /ν = 7/4.Finally, one can explore the behavior of the spe-

cific heat, given byc = N(e2L(T ) − (eL(T ))2), where

ekL(T ) is the average value found for thekth power

of the energy density. For the Ising model, this quan-tity shows a logarithmic divergence as the lattice sizegrows, which is interpreted as a zero value for theexponentα in a power-law divergence. For the simul-taneous Metropolis simulation, and assuming that thescaling relationα + 2β + γ = 2 remains valid, thevalues already found forν, β/ν andγ /ν imply that

Fig. 6. (a) Scaling of the specific heat, assuming a logarithmicdependence. The line corresponds to the best linear fit, with slopec0 = 9.88. (b) Scaling of the specific heat after removing thelogarithmic term.

α ≈ −0.012(26), compatible with the Ising result.The actual scaling found shows that in factc divergesas ln(L) (seeFig. 5(a) and (b)), and cannot be fittedto a power law (Fig. 6(a) and (b)).

4. Conclusions and final comments

As mentioned inSection 1, the phase transitionsthat appear in diffusively coupled chaotic 2D latticeswith scalar order parameter do not fall into the 2DIsing universality class, as was originally expected.Since they do become Ising-like when the updatingis changed to asynchronous, the only explanationoffered up to now has been to give to the updatingscheme the role of a relevant parameter. In order totest this possibility, we have explored a simultaneousversion of the well-known Metropolis algorithm forthe simulation of the 2D Ising model.

It is found that the change from asynchronous tosimultaneous updating in this algorithm always drivesthe lattice into a blinking maximum energy configu-ration, which are really not relevant to the propertiesone expects the model to show, even with such achanged simulation. This anomalous result can beavoided via the reduction of the acceptance probabil-ity A0 for energy lowering spin flips. This reductionshould then be also applied to energy raising flips, inorder to preserve detailed balance in the asynchronousversion of the algorithm. This modification generatedone-parameter family of algorithms, indexed byA0.

For A0 = 1/2, the simultaneous Metropolis simu-lation of the 2D Ising model gives the same criticalbehavior of the original model,even if the simulationitself is not designed to explore its equilibrium con-figurations. This difference with a correct simulationdoes show in a different critical temperature, whichactually becomes a function ofA0. One then has thefollowing main conclusion: for the universality classof the Ising model in the Metropolis simulation, theissue of synchronicity in its updating is irrelevant. Itshould be mentioned, however, that it could be pos-sible that the reduction in acceptance in energy low-ering spin flips may be acting as an effective sourceof asynchrony in the updating[13]. We believe that

324 G. Perez et al. / Physica D 168–169 (2002) 318–324

the possible asynchrony thus generated is not enoughas to explain the present results, but the possibilitydoes merit some extra checking. The simplest way ofdoing so would be to raiseA0 to some value closeto 1. Another option, which is now being studied,is to consider the heat-bath algorithm, which shouldnot need the introduction of an acceptance loweringfactor.

One is then left with the same puzzle: What makesthe Miller–Huse[1] and some other chaotic models[4], and a similar stochastic model[2], show a crit-ical behavior different from that of the Ising model?At the moment, two other options may need to beconsidered: (1) presence of still not well understoodcorrections, or (2) existence in parameter space ofsome close but not yet identified critical point, whichwould generate cross-over behavior in this dynamicalmodels[13]. These are questions that certainly needto be addressed.

Acknowledgements

This work was supported by CONACyT (Mexico)through Grant No. 28383E.

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