tempering simulations in the four dimensional ±j ising spin glass in a magnetic field

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Physica A 250 (1998) 46–57 Tempering simulations in the four dimensional ± J Ising spin glass in a magnetic eld Marco Picco a , Felix Ritort b; * a LPTHE, Universit e Pierre et Marie Curie, PARIS VI, Universit e Denis Diderot, PARIS VII, Boite 126, Tour 16, 1 er etage, 4 place Jussieu, F-75252 Paris Cedex 05, France b Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands Received 30 June 1997 Abstract We study the four dimensional (4D) ± J Ising spin glass in a magnetic eld with the simulated tempering algorithm recently introduced by Marinari and Parisi. We compute numerically the order parameter function P(q) and analyze the temperature dependence of the rst four cumulants of the distribution. We discuss the evidence in favor of the existence of a phase transition in a eld. Assuming a well dened transition we are able to bound its critical temperature. c 1998 Elsevier Science B.V. All rights reserved. PACS: 75.50.Lk; 64.60.Cn; 02.50.Ng; 05.50.+q Keywords: Monte Carlo studies; Spin glasses; Tempering simulations 1. Introduction Spin glasses are systems which deserve considerable theoretical interest due to the interplay between randomness and frustration [ 1 – 3]. The role of the frustration in the statics and dynamics is essential to understand the nature of the low temperature phase. Despite great progress during the last decade in the understanding of the mean-eld theory of spin glasses, a large number of topics are still poorly understood. In partic- ular, it is completely unclear which features of the mean-eld theory survive in nite dimensions. This problem has recently received considerable attention [4 –7] and has become the cornerstone to validate the correct description of the spin glass state. The reason why this topic still remains open relies on the absence of a convinc- ing nal theory for the spin glass state. Eorts to construct a eld theory for spin glasses, based on the Parisi solution to the mean-eld theory, have been done mainly by * Corresponding author. E-mail: [email protected]. 0378-4371/98/$19.00 Copyright c 1998 Elsevier Science B.V. All rights reserved PII S0378-4371(97)00545-1

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Page 1: Tempering simulations in the four dimensional ±J Ising spin glass in a magnetic field

Physica A 250 (1998) 46–57

Tempering simulations in the four dimensional±J Ising spin glass in a magnetic �eld

Marco Picco a, Felix Ritort b; ∗a LPTHE, Universit�e Pierre et Marie Curie, PARIS VI, Universit�e Denis Diderot, PARIS VII,

Boite 126, Tour 16, 1er �etage, 4 place Jussieu, F-75252 Paris Cedex 05, Franceb Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65,

1018 XE Amsterdam, The Netherlands

Received 30 June 1997

Abstract

We study the four dimensional (4D) ±J Ising spin glass in a magnetic �eld with the simulatedtempering algorithm recently introduced by Marinari and Parisi. We compute numerically theorder parameter function P(q) and analyze the temperature dependence of the �rst four cumulantsof the distribution. We discuss the evidence in favor of the existence of a phase transition in a�eld. Assuming a well de�ned transition we are able to bound its critical temperature. c© 1998Elsevier Science B.V. All rights reserved.

PACS: 75.50.Lk; 64.60.Cn; 02.50.Ng; 05.50.+qKeywords: Monte Carlo studies; Spin glasses; Tempering simulations

1. Introduction

Spin glasses are systems which deserve considerable theoretical interest due to theinterplay between randomness and frustration [ 1–3]. The role of the frustration in thestatics and dynamics is essential to understand the nature of the low temperature phase.Despite great progress during the last decade in the understanding of the mean-�eldtheory of spin glasses, a large number of topics are still poorly understood. In partic-ular, it is completely unclear which features of the mean-�eld theory survive in �nitedimensions. This problem has recently received considerable attention [4–7] and hasbecome the cornerstone to validate the correct description of the spin glass state.The reason why this topic still remains open relies on the absence of a convinc-

ing �nal theory for the spin glass state. E�orts to construct a �eld theory for spinglasses, based on the Parisi solution to the mean-�eld theory, have been done mainly by

∗Corresponding author. E-mail: [email protected].

0378-4371/98/$19.00 Copyright c© 1998 Elsevier Science B.V. All rights reservedPII S0378-4371(97)00545 -1

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 47

De Dominicis, Kondor and Temesvari [8]. Despite a large number of new results, aclear answer to the �nite dimensional issue is still missing.After the Parisi solution to the mean-�eld theory, a new phenomenological approach

to the spin-glass state (based on the Migdal–Kadano� approximation in the renormal-ization group theory) was proposed by McMillan [10], Bray and Moore [11] and lateron analyzed in detail by Koper and Hilhorst [12] and Fisher and Huse [13,14]. In thisapproach (now called the droplet model), the zero-temperature �xed point completelydetermines the properties of the low temperature phase. In the droplet model the ther-modynamics is determined by two Gibbs states (related by spin inversion symmetry)plus a spectrum of excitations which correspond to the inversion of compact domainsof �nite size (droplets). This picture of the spin-glass state lacks the most peculiarfeature of the mean-�eld theory, i.e. the coexistence of a large number of phases orstates in the spin-glass phase.Quite recently, Newman and Stein [4,5] and also Guerra [6] have analyzed the ques-

tion about which features of the spin glass state, present in the Sherrington–Kirkpatrick(SK) model [9], survive in �nite dimensions. Numerical simulations are one of the fewtools we can use to investigate this problem and clarify the controversy [15,16]. Withthe aid of numerical simulations, two main questions in spin glasses have been ad-dressed. The �rst one concerns the low temperature behavior of the model in zeromagnetic �eld. The second one concerns the existence of a paramagnet–spin-glasstransition line in a magnetic �eld like has been found in mean-�eld theory (the ATline [17]). A clearcut answer to these questions would be very useful as a guide forconstructing a �nal theory of the spin glass state in �nite dimensions. While the �rstproblem has received considerable attention, very few results have been obtained forthe second one.The purpose of this work is the study of the existence of spin-glass phase in a

magnetic �eld. This work is the natural continuation of previous numerical simulationsdone in the SK model in a magnetic �eld, where the existence of a replica symmetrybroken phase, as predicted by Parisi [19,20], was veri�ed through the study of theoverlap probability distribution P(q) [21]. In that work we also studied the four di-mensional (4D) ±J Ising spin glass in a magnetic �eld in the low T phase but didnot �nd evidence for a P(q) of the mean-�eld type, even though we were not sure thatequilibrium was achieved for the largest sizes. 1 In order to investigate the existenceof a transition line in a magnetic �eld we have done a detailed investigation runningextensive tempering Monte Carlo simulations in the ±J Ising spin glass in four dimen-sions. Our interest in this work is two fold. First, we will try to ascertain the generalfeatures of the low temperature equilibrium behavior of spin glasses in a magnetic �eld.To our knowledge this is the �rst time the simulated tempering algorithm is used forthe study of spin glasses in a magnetic �eld. This algorithm is certainly more e�cientto get equilibrated data than usual simulated annealing algorithms. Second, this study is

1 We chose to study 4D instead of 3D because the evidence in favor of a phase transition at zero �eld isless obvious in this last case [ 22–25]. Moreover the 4D model is easier to thermalize than in 3D.

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48 M. Picco, F. Ritort / Physica A 250 (1998) 46–57

a direct check of the performance of the simulated tempering algorithm introduced byMarinari and Parisi [26] to simulate random systems. We will see that the e�ciency ofthe simulated tempering method is compromised for large sizes and low temperaturesdue to the existence of low lying energy con�gurations where the system can remaintrapped for very large times.

2. The model and the numerical algorithm

We have considered the model described by the Hamiltonian

H = −∑(i; j)

Jij�i�j − h∑i

�i ; (1)

where the spin variables �i take the values ±1, the Jij are random discrete ±1 quenchedvariables and h denotes the magnetic �eld. The spins are located in the sites of a 4Dcubic lattice of size L and N = L4 sites with periodic boundary conditions.In order to reach the maximum e�ciency in the Monte Carlo simulations we have

used the tempering method introduced by Marinari and Parisi [26] (for a general reviewon multicanonical algorithms see Ref. [27]). This is a Monte Carlo method in which thetemperature is a dynamical variable and the system can change the temperature beingalways in thermal equilibrium. The system does a random walk in temperature insuch a way that low temperature equiprobable con�gurations separated by high energybarriers can be e�ciently sampled. The main drawback of the algorithm (compared toother multicanonical algorithms like the parallel tempering method recently proposedby Hukushima et al. [25]) is that the energy at each temperature needs to be estimatedat the beginning. If the system falls in a state with energy far from the estimated one e�it can remain trapped in that state for all the time of the simulation. For a descriptionand details about this algorithm, the reader is referred to [28].In what follows we brie y describe the numerical procedure. Samples are cooled

down, at constant magnetic �eld h, starting from the high temperature phase (abovethe critical temperature at zero �eld Tc ' 2:0 [18]) down to T =1:0 and the internalenergy e�= 〈H〉 is estimated as a function of � for a selected set of N� di�erentvalues of � (N�=50 for the largest sizes). The separation �� between the di�erentvalues of � is taken such that the tails of the probability distributions of the energy fordi�erent neighboring temperatures do superimpose. For sake of simplicity the di�erentvalues of � were taken equidistant with �� = 0:03 for the largest sizes. It is importantto note that the simulated tempering (and, in general, all multicanonical methods) areexpected to work if the thermodynamic chaos (to be discussed below) is small. Theweakness of chaotic e�ects in temperature for �nite sizes was numerically checked forthe SK model [29] as well as for 4D ±J Ising spin glasses [30].Starting from a random initial condition and an initial temperature �r , all the spins

are sequentially updated at each Monte Carlo step (MCS) and single spin ips are ac-cepted with a probability given by the heat bath algorithm. After each MCS a change in

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 49

temperature is proposed �r → �r+1 or �r → �r−1, each with a probability 1=2.The change in temperature �� is accepted with probability exp(−��(E(�) − (e� +e�+��)=2:)). The spins are again updated and the change of temperature is again pro-posed. In this way one is able to compute the equilibrium values of di�erent observablesfor all values of �. In order to increase the statistics we have simulated 8 di�erentreplicas in parallel in a multispin coding program.Before presenting the numerical results, we will comment about chaoticity e�ects in

spin glasses since these determine the value of the magnetic �eld we chose for thesimulations.

3. The value of the magnetic �eld and chaos in spin glasses

One of the main properties of spin glasses is the existence of chaotic e�ects whensome external parameter like the temperature or the magnetic �eld is changed [11,15; 16]. This feature is present in the mean-�eld approach as well as in the dropletmodel at zero magnetic �eld. In the framework of mean-�eld theory of spin glasses,the physical meaning of thermodynamic chaos is rather intuitive. It is related to the factthat small energy perturbations can redistribute the (small) free energy di�erences ofthe many equilibrium states modifying completely their equilibrium statistical weights.In the framework of droplet models, energy perturbations can strongly modify spincorrelations due to the fractal nature of the droplet domain walls. According to thedroplet model, the e�ect of a uniform magnetic �eld is to suppress the spin glass phase,hence chaotic e�ects in temperature disappear if the system becomes magnetized. Tomeasure chaoticity in spin glasses we de�ne the chaos correlation length associated tothe q− q correlation function at large spatial distances x [13,14,31],

Cchaos(x) = 〈qi qi+x〉 ∼ x−�exp(− x�c

); (2)

where qi= �i�i and �i; �i denote the spins of the unperturbed (H0(�)) and perturbedsystem (Hp(�)), respectively, and the expectation value 〈··〉 is taken over the equi-librium Boltzmann distribution associated to the full Hamiltonian H(�; �) =H0(�) +Hp(�). � is a positive exponent. The chaos correlation length �c gives an estimate ofthe typical size of spatial regions which are similar in the unperturbed and perturbedsystem. When the intensity of the perturbation goes to zero, the chaos correlation length�c diverges and the exponent associated to the divergence is related to the particulartype of perturbation.For magnetic �eld perturbations, we know that chaotic e�ects are quite strong. 2 This

e�ect sets a lower limit for the value of the magnetic �eld we can use in the simulations.This is the most relevant parameter in the simulations because it determines how closewe are to the h = 0 spin-glass phase. The value of h cannot be too large otherwise, if

2 In contrast with chaotic e�ects in the presence of temperature perturbations which are small [29,31,32].

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50 M. Picco, F. Ritort / Physica A 250 (1998) 46–57

a spin glass transition exists, it will be pushed down to very low temperatures. Also itcannot be too small otherwise the results are strongly a�ected by the h = 0 spin-glassphase for the �nite sizes one can study. The crossover between the h = 0 behaviorand the �nite h behavior depends on the chaos correlation length �c(h) de�ned inEq. (2). The value of the magnetic �eld h is chosen in such a way that �c(h)¡ L forthe explored lattice sizes but not too large as explained previously. We have found agood compromise for the value h = 0:4 which yields a macroscopic magnetization atlow temperatures of order 0:15. Then, we can estimate from independent Monte Carlosimulations (see [29]) that �c ' 5 for the lowest temperature T = 1. We expect thatsimulations for sizes above L = 5 can yield convincing results on the existence orabsence of phase transition at this value of the �eld.

4. Numerical results

Simulations were performed for the sizes L = 3; 5; 7; 9 with 1000; 325; 120; 130 sam-ples and N� = 20; 40; 50; 50, respectively, ranging from Tmin = 1:0 up to Tmax = 2:5for L = 7; 9 and Tmax = 3:0 for the smallest sizes L = 3; 5. For the largest size L = 9 itwas not possible to achieve equilibrium at low temperatures for the reason mentionedin the previous section. That is, at low temperatures the system often got trapped inlow energy states. Hence we will show the data only for temperatures above T ' 1:5for that size. In Fig. 1 we present results for the magnetization M = 1

N

∑i �i at di�er-

ent temperatures and sizes. Instead of plotting directly the magnetization we plot theratio r(T; L) = MT=h. This quantity (due to a local gauge symmetry of the disorder[22]) should be equal to 1 above Tc(h = 0) ' 2:0 in the limit of very small h. For�nite h, because of the divergence of the spin-glass susceptibility at zero �eld, r issmaller than 1 (at T = 2:5 it is of order 0.7) but converges to 1 quite fast at hightemperatures where the spin-glass susceptibility vanishes like �3. The important resultwhich emerges in Fig. 1 is that, below Tc(h = 0), r is linear with T . Consequently,the magnetization is nearly constant in the low temperature phase. This is a typicalresult found in spin glasses. This feature is also present in the mean-�eld theory andhas been observed in the 3D case [33] as well as in �eld cooled experiments in spinglasses [ 1–3].More information about a possible phase transition can be obtained by directly mea-

suring the spin-glass order parameter Q between two di�erent replicas {�; �} with thesame set of Jij, Q = (1=N )

∑i �i�i and its associated probability distribution,

PJ (q) = 〈�(q− Q)〉 ; (3)

where 〈(·)〉 denotes the thermal Gibbs average. The PJ (q) shows big uctuations fromsample to sample. Hence it is convenient to de�ne P(q) = PJ (q) where (·) means aver-age over the disorder. P(q) is the natural order parameter for spin glasses and measureshow equilibrium states are distributed in phase space. To each value of the overlapq there is associated a Hamming distance in phase space d2 = (qEA − q)=2 [ 1–3]

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 51

Fig. 1. Parameter r = MT=h as a function of temperature. From top to bottom L = 3; 5; 7; 9. Data for L = 9is hardly distinguishable from those for L = 7.

where qEA is the maximum overlap or Edwards–Anderson parameter. This parameteris a measure of the spin-glass ordering in that state. A phase space dominated by asingle equilibrium state with a large statistical weight corresponds to P(q)= �(q−qEA).A phase space with several equilibrium states randomly distributed in phase space withsimilar statistical weights implies P(q) = �(q). This does not necessary mean that theredoes not exist any kind of structure between the states in phase space. For intermedi-ate situations the P(q) always re ects the existence of an underlying structure in thedistribution of equilibrium states.In Figs. 2 and 3 we show the P(q) for a high and a low temperature T = 2:5, T =

1:5, respectively. At T = 2:5 P(q) is strongly peaked around qEA = 0:125 showing thatthere is a single state (magnetized) which has a very large statistical weight. Instead,at T = 1:5 the P(q) shows a long tail extending down to negative values of q. Still theP(q) is peaked around qEA = 0:48 but a tail extending down to small values of q is stillpresent. Obviously, the question is how this tail behaves in the thermodynamic limitL → ∞. If we are in a phase dominated by a single state the tail should dissappear(even if the contrary assessment is not true). It is di�cult to extrapolate the data ofFigs. 2 and 3 to the thermodynamic limit (such an extrapolation has been consideredfor zero magnetic �eld [34]). While T = 1:5 is probably slightly above the criticaltemperature (see below) the same behavior is still found at lower temperatures wherethe left tail of P(q) persists. In order to gather more information from P(q) it is moreconvenient to analyze the cumulants of the distribution. They give information abouthow P(q) deviates from a perfect Gaussian.We calculated for each sample the �rst four moments of the distribution of Eq. (3).

We have computed the mean value, the variance X , the skewness Y and the kurtosis Z

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52 M. Picco, F. Ritort / Physica A 250 (1998) 46–57

Fig. 2. P(q) at T = 2:5 for L = 3; 5; 7; 9. The variance of the P(q) decreases with the size.

Fig. 3. P(q) at T = 1:5 for L = 3; 5; 7; 9. The value of P(q = qEA) increases with the size.

of the distribution P(q). The skewness and the kurtosis are a measure of the asymmetryand Gaussianity, respectively, of the overlap distribution. More precisely, if we de�nethe following averages [f(q)] =

∫dqf(q)P(q) we have

X = [(q− [q])2] ; (4)

Y =[(q− [q])3][(q− [q])2]3=2 ; (5)

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 53

Fig. 4. Mean value [q] as a function of temperature. From bottom to top L = 3; 5; 7; 9. Data for L = 9 arehardly distinguishable from those for L = 7.

Z =12

(3− [(q− [q])4]

[(q− [q])2]2): (6)

In Fig. 4, we plot the mean value [q] as a function of the temperature for di�erentsizes. Data for L = 9 above T ' 1:5 is nearly indistinguishable from data for L = 7. Asshown in Fig. 4, we expect that [q] converges to a value close to 1 at zero temperature(but smaller than 1 if there is ground state degeneracy). The cumulants X; Y; Z givemore information about a possible phase transition. They are expected to vanish in theL → ∞ limit in the paramagnetic phase. Within an ordered phase of the mean-�eldtype, where several pure states contribute to the Gibbs average, we expect X; Y; Z tobe �nite. In Figs. 5–7 we show NX; Y; Z (where N = L4) as a function of temperaturefor four di�erent lattice sizes L = 3; 5; 7; 9.Fig. 5 is quite appealing. We observe the presence of a maximum in the spin-glass

susceptibility for sizes L = 5; 7. This maximum moves to higher temperatures as thesize increases (for L = 3 such a maximum is not observed in the range of temperaturesexplored). Note that the error bars in Fig. 5 are large. Also the maximum of thespin-glass susceptibility moves to higher temperatures as the size increases. We do notexclude the possibility that this is a subtle non-equilibrium e�ect at low temperatures.Unfortunately, it is di�cult to disprove this result since tempering simulations aredi�cult to control at very low temperatures. On the other hand, the maximum observedin Fig. 5 is quite di�erent to what is observed in a normal paramagnet where (contrarilyto our case) �nite-size corrections are not so big.Figs. 6 and 7 show the parameters Y; Z for the di�erent sizes. How a second order

phase transition should manifest? For large enough sizes, it is expected the adimen-sional quantities Y; Z to scale like Y ≡ Y (L(T − Tc)�) (the same for Z). Consequently

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54 M. Picco, F. Ritort / Physica A 250 (1998) 46–57

Fig. 5. Spin glass susceptibility NX as a function of temperature. From bottom to top L = 3; 5; 7; 9.

Fig. 6. Skewness Y as a function of temperature. Dotted line corresponds to L = 3, �lled circles to L = 5,�lled triangles to L = 7 and �lled squares to L = 9.

they should display a crossing point for di�erent sizes at T = Tc like happens at zeromagnetic �eld [18]. The lines in Figs. 6 and 7 corresponding to L = 5; 7; 9 sizes havebeen indicated by full symbols in order to distinguish the general trend of the data fromthe results for L = 3. Fig. 6 shows that the skewness is negative for �nite sizes andabove T ' 1:5 it goes to zero when the size increases as expected in the paramagneticphase. The same tendency is also observed in Fig. 7 for the kurtosis. It is interesting

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 55

Fig. 7. Kurtosis Z as a function of temperature. Dotted line corresponds to L = 3, �lled circles to L = 5,�lled triangles to L = 7 and �lled squares to L = 9.

to note that the curves for L = 5; 7 for both the skewness and the kurtosis cross atthe same temperature T ' 1:5 which is an upper bound for of an hypothetical criticaltemperature. Unfortunately, we have not covered a large enough range of sizes (L = 3is too small) in order to have clear evidence of such a crossing point. The main resultswhich emerge from Figs. 6 and 7 (which are di�cult to infer from Figs. 2 and 3)is that deviations of the P(q) from a perfect Gaussian are quite large and increase atlow temperatures. Our results suggest the existence of paramagnetic ordering at leastabove T ' 1:5 and we cannot exclude the existence of crossing point and hence aphase transition below that temperature.

5. Conclusions

From our data we reach the following conclusions: (1) From Figs. 6 and 7 we canconclude that a phase transition, if exists, appears below Tc(h = 0) ' 2:0. Hence thetransition temperature is pushed down by the magnetic �eld. (2) We clearly observea change of behavior above L = 5 where the trend of the skewness and kurtosis as afunction of the size changes. This crossover length is in agreement with the estimatedchaos correlation length �c ' 5 for the value of the magnetic �eld h = 0:4 and isan estimate of the minimum size L above which the tail of the P(q) extending downto negative values of the overlap is suppressed. (3) Fig. 5 shows the existence ofa maximum in the spin-glass susceptibility. To our knowledge, this e�ect has neverbeen observed at zero magnetic �eld. We cannot exclude that this maximum is athermalization e�ect due to the di�culty of the numerical algorithm to work e�ciently

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56 M. Picco, F. Ritort / Physica A 250 (1998) 46–57

at low temperatures. On the other hand, we believe that this e�ect is connected tothe nature of the phase transition in a �eld. This transition should be located aboveT ' 1:2 where the maximum of the spin-glass susceptibility for L = 9 is observed.Such a transition should be of a di�erent type to that found in zero magnetic �eld.(4) Support to the previous assertion is obtained from Figs. 6 and 7. There, the lowT large deviations of the third and fourth cumulants of P(q) from a perfect Gaussianindicate a behavior far from being paramagnetic. Indeed they suggest a phase transitiondominated by a �xed point with strong non-Gaussian properties, i.e large values of thecritical amplitudes Yc = Y (0) and Zc = Z(0).We note that the results we are presenting here are probably seen in a very narrow

range of �elds. For �elds larger than h = 0:4, the cusps in X will move to lowertemperatures and should be di�cult to see them numerically since tempering does notwork e�ciently for very low temperatures. On the other hand, for smaller �elds, thechaos correlation length would be larger and this would require to simulate much largerlattices in order to start to see the trend of the data. A similar analysis on the Gaussiancase (to avoid the ground state degeneracy) but using the (more e�cient) paralleltempering method [25] would be welcome in order to check the main conclusions ofthis work.A word of caution is essential at this point. Spin glasses are extremely di�cult to

thermalize and this is probably the reason why small progress has been done in theunderstanding of their equilibrium properties. The simulated tempering method is muchmore e�cient than more standard simulation techniques used in the past (like simulatedannealing). But still, it is not e�cient enough to reach a de�nite conclusion about thetransition in a �eld. Mainly because at low temperatures the system remains trapped incon�gurations with energy below the average equilibrium energy e�. In our simulationsthis e�ect started to be dramatic even for L = 9.Our data show indications of a phase transition in a �eld in a di�erent universality

class than the zero �eld one. This result is supported mainly by Figs. 5–7. The transitionshould be above T ' 1:25 (where the maximum of X in Fig. 5 for the largest size islocated) and below T ' 1:5 (where the crossing point for the skewness and the kurtosisis observed). The �xed point which describes the phase transition in a �eld is stronglydi�erent from the zero-�eld �xed point, in particular a large negative value for thecritical amplitude of the third cumulant Yc = Y (0) is found. It is di�cult to go beyondsuch a conclusion and extremely careful work with more e�cient numerical algorithmsare needed in order to clarify this scenario.

Acknowledgements

We acknowledge helpful discussions with S. Franz, E. Marinari and G. Parisi. We aregrateful to G. Parisi for a careful reading of the manuscript. One of us (F.R.) is gratefulto E.N.S in Paris for its kind hospitality where part of this work was done. The workby F.R has been supported by FOM under contract FOM-67596 (The Netherlands).

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M. Picco, F. Ritort / Physica A 250 (1998) 46–57 57

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