1 October 2001

Physics Letters A 288 (2001) 271–276www.elsevier.com/locate/pla

The parallel updating in a fully-frustrated Ising model:a damage-spreading analysis

Fernando D. Nobre∗, Amadeu A. JúniorDepartamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário,

Caixa Postal 1641, 59072-970 Natal, RN, Brazil

Received 26 April 2001; received in revised form 14 August 2001; accepted 21 August 2001Communicated by C.R. Doering

Abstract

The fully-frustrated square-lattice Ising model is investigated through the damage-spreading technique, with all spins of thelattice being updated simultaneously (full parallel updating). Surprisingly, the frustration does not provide any serious difficultyfor the parallel updating, in the sense that the Hamming distance presents the same qualitative behaviour as the one obtained bymeans of the standard sequential updating. 2001 Elsevier Science B.V. All rights reserved.

PACS: 05.50.+q; 64.60.-i; 75.40.MgKeywords: Ising model; Frustration; Damage spreading; Parallel updating

1. Introduction

Monte Carlo simulations [1,2] represent nowadaysone of the most important tools in the study of many-body systems. Its major drawback lies in the limita-tion of computers, restricted to work with both finitetimes and number of particles, whereas one is typi-cally interested in thermodynamic-limit results (whichrequire long times and number of particles approach-ing infinity). The Monte Carlo method, as proposedoriginally by Metropolis et al. [3], is essentially a se-quential updating (SU) process: the execution of anew move depends on the previously accepted stateand cannot be performed independently in a straight-forward way. However, after the creation of parallelcomputers, new algorithms have been produced, al-

* Corresponding author.E-mail address: nobre@dfte.ufrn.br (F.D. Nobre).

lowing for the possibility of parallelization; as a con-sequence, one obtains a reasonable gain in computingperformance. Among the several types of parallel al-gorithms [4], one may single out two of the most com-monly employed, namely, the event and geometric par-allelisms. In the former case, one deals with tasks thatcan be carried out independently of each other; as anexample, one may refine the statistics by replicatingthe program among the available processors, allocat-ing to each processor a different initial configuration.In the latter case, one divides the volume of the sys-tem into equal-sized portions that are assigned to thevarious processors; as an example, one may mentionthe “checkerboard” procedure: for spin models withnearest-neighbor couplings, defined on hypercubic lat-tices, one divides the system into interpenetrating sub-lattices, with all the spins in a given sublattice beingupdated simultaneously through a local-updating al-gorithm. It is important that no interacting spins areupdated at the same time; the checkerboard algorithm,

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01)00568-0

272 F.D. Nobre, A.A. Júnior / Physics Letters A 288 (2001) 271–276

for which each sublattice is assigned to a differentprocessor, is defined in such a way to avoid this possi-bility.

The full parallelization, or parallel updating (PU),whereall spins of the system are updated simultane-ously, may be carried in a careful way, since it maylead to unpredictable results, e.g., for the antiferro-magnetic Ising model on a triangular lattice phasetransitions may appear, which are not observed withinthe conventional SU [5,6]. In fact, if one considers anIsing system defined through the Hamiltonian

(1)H = −∑〈ij〉

Jij SiSj (Si = ±1),

where the∑

〈ij〉 applies to all distinct pairs of nearest-neighbor spins of a given lattice, the detailed balancecondition, concerning the ratio of transition probabili-ties from statesI → J andJ → I , is satisfied for bothSU and PU, and it may be written as

(2)W(I → J )

W(J → I)= exp[−βH(J )]

exp[−βH(I)] .

In the SU,H(I) corresponds to the energy of the sys-tem at stateI , as defined in Eq. (1), whereas in the PUone has [7,8]

(3)H(I) = − 1

β

N∑i=1

ln

[cosh

(β

∑j (i)

Jij Sj (I)

)],

whereN denotes the total number of spins of the sys-tem and the sum

∑j (i) applies to all nearest-neigh-

bor sitesj of a given sitei. Obviously, the energyfunctions in Eqs. (1) and (3) are very different fromone another and there is no a priori reason why theyshould yield similar results. One of the most strikingdifferences between these two energy functions con-cerns the fact that the one in Eq. (3) is invariant under asign inversion of all coupling constants (Jij → −Jij ),and consequently, it is not able to distinguish a pureferromagnet from a pure antiferromagnet; such an ef-fect has already been observed through a damage-spreading technique for the Ising model on a triangularlattice [5,6].

Herein, we present a fully-frustrated Ising model forwhich the SU and PU yield essentially the same qual-itative behavior within a damage-spreading analysis,although at low temperatures, some small quantitative

discrepancies between the two updatings may be ob-served. In Section 2, we define the model and the nu-merical formalism; in Section 3, we present and dis-cuss our results.

2. The model and the numerical procedure

Let us consider an Ising model defined throughthe Hamiltonian of Eq. (1); the sum is restricted tonearest-neighbor pairs of spins on a square lattice oflinear sizeL (N = L2), with the coupling constantsfavoring antiferromagnetism (Jij = −J ) for alternatevertical bonds, and ferromagnetism (Jij = J ), other-wise. This is a fully-frustrated Ising model, introducedby Villain [9], who showed that it exhibits no sta-tic phase transition at finite temperatures. Regardlessof its large ground-state degeneracy, the free-energylandscape is smooth, and no diverging barrier heightsbetween states appear in the thermodynamic limit;therefore, no ergodicity breaking occurs.

We are interested in investigating how an initial per-turbation propagates in time throughout the system.This may be done through a technique usually denom-inated “damage spreading” [10–14], which consists infollowing the time evolution of the Hamming distance(or damage) between two configurations of the system,({SA

i } and{SBi }),

(4)D(t) = 1

2N

N∑i=1

∣∣SAi (t) − SB

i (t)∣∣.

First of all, one takes configuration{SAi } to equilib-

rium, by letting it evolve fort0 Monte Carlo (MC)steps. At this time, which we will choose ast = 0, acopy of such a configuration is made, which will cor-respond to the second configuration ({SB

i }); a fractionof spins of the copyB is flipped, introducing the initialdamageD(0). After submitting both copies to a newthermalization process (t0 MC steps), where the dam-aged copy is taken to equilibrium, one starts to mea-sureD(t) for an extra time interval oft0 MC steps.Both copies should always evolve under the same dy-namics and random numbers, so as to ensure that anypossible difference between them should be only dueto the initial damageD(0). It is obvious that once thetwo copies become identical at a given timet , they

F.D. Nobre, A.A. Júnior / Physics Letters A 288 (2001) 271–276 273

will remain identical at any later time. The simula-tion is then repeated forNsamp samples (i.e.,Nsampdifferent sequences of random numbers), such as toallow for the calculation of the average Hamming dis-tance,〈D(t)〉.

Two dynamical procedures have been used in thiswork, namely, the Glauber and heat-bath dynamics.Both are defined in terms of a probability associatedto each sitei of a copy{Sµ

i } (µ = A,B),

(5a)pµi (t) = {

1+ exp[−2h

µi (t)

]}−1,

where

(5b)hµi (t) = β

∑j

Jij Sµj (t)

is the local field acting on sitei, at timet , in the con-figuration{Sµ

i }. As usual, a uniform random number,0 � zi(t) � 1, is generated for each sitei, at timet , tobe compared with the flipping probabilitiespµ

i (t). Inthe heat-bath dynamics the new variableS

µi (t + 1) is

obtained through the rule

(6)Sµi (t + 1) =

{1, if zi(t) � p

µi (t),

−1, if zi(t) > pµi (t),

whereas in the Glauber dynamics, the value ofS

µi (t + 1) depends also onSµ

i (t) and is defined as

(7a)Sµi (t + 1) =

{1, if zi(t) � p

µi (t),

−1, if zi(t) > pµi (t),

whenSµi (t) = −1,

(7b)Sµi (t + 1) =

{−1, if zi(t) � 1− p

µi (t),

1, if zi(t) > 1− pµi (t),

whenSµi (t) = 1.

Although our numerical simulations were per-formed on a single-processor machine, we tried to em-ulate the action ofN processors working in parallel(each processor assigned to a spin variable). For that,the spins on both copies were visited in a “typewriter”sequence and submitted to one of rules (6) or (7); theinformation on the new variableSµ

i (t + 1) was stored,in such a way that only after all the spins were vis-ited, the updating process was executed for theN spinsof each copy simultaneously. Herein,all spins of thelattice are updated at once (full parallelization); sucha procedure should be distinguished from the usual

checkerboard algorithm where even and odd sublat-tices are updated in parallel independently. It is im-portant to remind that in the standard sequential algo-rithm, each spin is updated right after it is visited. Asusual, a Monte Carlo step corresponds to a completesweep of the lattice. Due to the simultaneous updatingof the spins, an efficient algorithm may be easily im-plemented for a multi-processor machine, resulting ina considerable reduction in computational time. How-ever, the object of the present work is not the presen-tation of an efficient parallel algorithm, but rather theinvestigation of the effects generated by the simulta-neous updating of interacting pairs of spins.

3. Results and discussion

We have simulated square lattices of linear sizeL = 60, with periodic boundary conditions. The ther-malization processes, as well as the measuring ofHamming distances, were carried fort0 = N/2 =1800 MC steps, and the sample averages were takenover Nsamp= 400 different sets of random numbers.As usual, for the Glauber dynamics we started withan infinitesimal damage (D(0) = 1/N ) [13]; in theheat-bath case, several initial conditions were used(D(0) = 0.05,0.5,1.0) [14]. As a reference, our tem-peratures are defined in units of the critical tempera-ture of the ferromagnetic Ising model on a square lat-tice (kBTCF ≈ 2.2692J ).

In Fig. 1 we exhibit the average damage〈D(t)〉,as a function of the temperature, obtained throughthe Glauber dynamics; the results of the PU (emptysymbols) are compared to those of the SU (full sym-bols). One notices that below a certain value of thetemperature,T ∗/TCF ≈ 0.30, the two updatings yieldslightly different results. However, forT > T ∗ both al-gorithms converge to the well-known saturation limitof the Glauber dynamics, i.e.,〈D(t)〉 = 1/2, which isusually associated with a paramagnetic state in the cor-responding static treatment. Despite the fact that somequantitative discrepancy is observed at low tempera-tures, the two updating algorithms yield the same qual-itative behavior, i.e., only the propagating regime (forwhich a small initial perturbation propagates through-out the system) was found for all temperatures. Sucha result is in agreement with the fact that the fully-frustrated Ising model on the square lattice presents

274 F.D. Nobre, A.A. Júnior / Physics Letters A 288 (2001) 271–276

Fig. 1. The average damage〈D(t)〉 as a function of the scaledtemperature, for the Glauber dynamics (the temperature is expressedin units of the critical temperature of the ferromagnetic Ising modelon the square lattice,kBTCF ≈ 2.2692J ). The results obtained fromthe parallel updating (empty symbols) are compared to those of thesequential updating (full symbols). AroundT /TCF = 0.30, 〈D(t)〉stops depending on the updating algorithm.

no static magnetic phase transition at finite temper-atures [9], and that damage propagation within theGlauber dynamics usually occurs throughout para-magnetic phases [10,11,13]. AsT → 0, one expectsto get〈D(t)〉 → 0 abruptly, which would be presum-ably associated with the zero-temperature static phasetransition. Although both updatings indicate a slightdecrease in〈D(t)〉 at low temperatures, as shown inFig. 1, we were not able to detect such jump due tolow-temperature numerical difficulties.

Such results for the Glauber dynamics should becontrasted with those obtained in a similar problem,i.e., the antiferromagnetic Ising model on a triangu-lar lattice [5]: whereas the SU provided a propagat-ing regime for all temperatures, the PU produced a dy-namic phase transition at a temperature slightly higherthan the Curie temperature associated with the cor-responding ferromagnetic model. Therefore, the sub-stantial disagreement between the two updating algo-rithms in the antiferromagnetic Ising model on a tri-angular lattice should be more a consequence of thelattice topology, rather than of the frustration itself.

In Fig. 2 we present the average damage〈D(t)〉within the heat-bath dynamics, for different initialconditions (D(0) = 0.05,0.5,1.0). The results of thePU (Fig. 2) are qualitatively identical to those achieved

Fig. 2. The average damage〈D(t)〉 as a function of the scaled tem-perature, for the heat-bath dynamics, within the parallel updating,for three distinct initial conditions (the temperature scale is the sameas the one in Fig. 1).

Fig. 3. The average damage〈D(t)〉 as a function of the scaledtemperature, for the heat-bath dynamics, with three distinct initialconditions (the temperature scale is the same as the one in Fig. 1).The results of the parallel updating (empty symbols) are comparedwith those of the sequential updating (full symbols). AroundT /TCF = 0.30, two effects are observed:〈D(t)〉 stops dependingon the initial conditions, as well as on the updating algorithm.

through the SU [15]; indeed, above a certain valueof the temperature (roughly the sameT ∗ as in theGlauber dynamics), the two updating algorithms agreewithin the error bars, as shown in Fig. 3, where the re-sults of both updatings are superposed. A rich struc-ture is verified, similar to those already obtainedfrom sequential heat-bath algorithms on the classicaltwo-dimensional XY [16,17] and three-dimensional

F.D. Nobre, A.A. Júnior / Physics Letters A 288 (2001) 271–276 275

Heisenberg [18] models, antiferromagnetic Ising mod-el on the triangular lattice [19], and Ising spin glass-es [14]. Three distinct regimes are present:

(i) a low-temperature one (T < T2), for which〈D(t)〉depends onD(0);

(ii) an intermediate regime (T2 < T < T1), for which〈D(t)〉 in nonzero and independent ofD(0);

(iii) a high-temperature one (T > T1), where〈D(t)〉vanishes independently ofD(0).

From Figs. 2 and 3 one obtains from both PU andSU algorithms,T2/TCF ≈ 0.30 (kBT2/J ≈ 0.68) andT1/TCF ≈ 0.75 (kBT1/J ≈ 1.70).

The low-temperature regime for which the finaldamage depends on the initial conditions, is also sen-sitive to the particular updating algorithm. In thisregime, the barriers between states are high enough insuch a way that different initial conditions yield dis-tinct values for〈D(t)〉.

Within the PU it is more difficult for the system tojump some barriers, and the exploration of the wholephase space becomes troublesome; as a consequence,the dependence on the initial conditions is more pro-nounced in the PU than in the SU, as shown in Fig. 3.In general, as the temperature increases, the barrierheights get reduced, in such a way that atT = T2,〈D(t)〉 becomes independent of the initial conditions;in the same way, at a given temperature(T ∗), 〈D(t)〉turns out to be the same whether one uses the PUor the SU algorithms. Our simulations show thatT2andT ∗, if not coincident, are indeed very close; thissuggests a correlation between these two temperatures,for the fully-frustrated square-lattice Ising model. ForT > T ∗, the particular choice between the PU and SUdoes not play an important role, at least in what con-cerns the average Hamming distance.

As far as we know, the temperatureT2 presents nointerrelated static analog. However, the higher temper-ature (kBT1 ≈ 1.70J ) is very close to the correspond-ing cluster-percolation transition temperature foundby Cataudella [20] (kBTp/J = 1.69± 0.05). Such acorrelation has also been verified through sequentialheat-bath algorithms in other frustrated systems, likethe antiferromagnetic Ising model on a triangular lat-tice [19], as well as two- [20], and three-dimensional[21] ±J Ising spin glasses. If the temperaturesT1 andTp do really coincide, parallel damage-spreading al-gorithms should come out as useful tools in the study

of the cluster-percolation transition, in the present sys-tem, and possibly in other frustrated systems as well.

It is also interesting to compare our results withthe Grassberger conjecture [22]. According to Grass-berger, all damage-spreading transitions should fol-low in the same universality class of directed perco-lation, unless they coincide with other static transi-tions. Within such a conjecture, directed percolationshould be interpreted as a dynamic process, for whichtime is considered as an additional direction; there-fore, damage-spreading transitions ind-dimensionalsystems should follow in the same universality class of(d + 1)-dimensional directed percolation. An analysisof the data shown in Figs. 2 and 3, close to the damage-spreading transition at temperatureT1, follows reason-ably the power-law behavior〈D(t)〉 ∼ (T1−T )β , withan exponentβ = 0.677± 0.035. Although close, ourestimate does not coincide with the corresponding di-rected percolation exponent,β = 0.584(4) [23]. It ispossible that such a disagreement may be due to thesmall size of the lattice investigated herein.

To conclude, we have investigated the fully-frus-trated Ising model on a square lattice by means ofa damage-spreading technique, using a full parallel-updating algorithm (all spins of the lattice are updatedsimultaneously). The resulting average Hamming dis-tance is qualitatively similar to the one obtainedthrough a standard sequential algorithm. We havefound a “threshold temperature” (kBT ∗/J ≈ 0.68),below which the average Hamming distance is quan-titatively different on the two algorithms; above thistemperature, both updating algorithms agree withinthe error bars. The temperatureT ∗ seems to be thesame whether one studies the spreading of damagethrough the Glauber or heat-bath dynamics; within theheat-bath dynamics,T ∗ seems to coincide with thetemperature at which the Hamming distance stops de-pending on the particular initial conditions. Such aneffect should be in some way related to a smooth-ing of the energy barriers, which is usually associatedwith some important type of physical phenomena. Toour knowledge, there are no evidences of any changeof behavior aroundT ∗, observed in studies of a sin-gle copy of such a system, up to the moment. How-ever, the possibility of a new type of phase transi-tion (or even a crossover) occurring in a single copyof the present model may not be discarded; furtherinvestigations are required to clarify this matter. Al-

276 F.D. Nobre, A.A. Júnior / Physics Letters A 288 (2001) 271–276

though the parallel updating has given adequate resultsfor the average Hamming distance in the present sys-tem, one cannot guarantee it will work out properlyfor the investigation of other properties; indeed, quan-tities which depend directly on nearest-neighbor cor-relations are not recommended to be studied througha full-parallel algorithm, at least for low temperatures.However, it is possible that other physical quantitiesmay present their own threshold temperatures, like theone found herein for the Hamming distance, in sucha way that above such a temperature the particularchoice between the sequential or parallel algorithmsbecomes irrelevant; in these cases, the parallel updat-ing may be safely implemented, resulting in a reason-able gain in computational time.

Acknowledgements

It is a pleasure to thank Prof. Marcelo L. Lyrafor fruitful discussions. The partial financial supportfrom CNPq and Pronex/MCT (Brazilian agencies) areacknowledged.

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