HAL Id: jpa-00211111
https://hal.archives-ouvertes.fr/jpa-00211111
Submitted on 1 Jan 1989
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The critical behaviour of the three-dimensional dilute Ising model : universality and the Harris criterion
Jian-Sheng Wang, Debashish Chowdhury
To cite this version:
Jian-Sheng Wang, Debashish Chowdhury. The critical behaviour of the three-dimensional dilute Ising model : universality and the Harris criterion. Journal de Physique, 1989, 50 (19), pp.2905-2910.
�10.1051/jphys:0198900500190290500�. �jpa-00211111�
2905
LE JOURNAL DE
PHYSIQUE
Short Communication
The critical behaviour of the three-dimensional dilute Ising model : universality and the Harris criterion
Jian-Sheng Wang (1) and Debashish Chowdhury (1,2)
(1) HLRZ, c/o Kernforschungsanlage Jülich, Postfach 1913, D-5170 Jülich 1, F.R.G.
(2) School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India (*) (Reçu le 10 juillet 1989, accepté le 26 juillet 1989)
Résumé. 2014 Des simulations de Monte Carlo du modèle d’Ising tridimensionnel dilué (le désordre de site étant gelé) sont effectuées à l’aide de l’algorithme de Swendsen-Wang. Les exposants critiques de
la susceptibilité et de la longueur de corrélation sont estimés respectivement à 03B3 = 1, 52 ± 0, 07, v = 0, 77 ± 0, 04 pour tout pc p 1, où p est la concentration des spins. A la différence de travaux
précédents, nos résultats corroborent fortement le concept d’universalité.
Abstract. 2014 Monte Carlo simulation of quenched site-diluted three-dimensional Ising model has
been carried out using the Swendsen-Wang algorithm. The susceptibility exponent and the correlation
length exponent are estimated to be 03B3 = 1.52 ± 0.07, v = 0.77 ± 0.04, respectively, for all
pc p 1, where p is the concentration of the spins. In contrast to some earlier work, our results strongly support the concept of universality.
1 Phys. France 50 (1989) 2905-2910 ler OCTOBRE 1989, 1
Classification
Physics Abstracts
05.50 - 64.60F - 75.10H
The effects of quenched random impurities on the critical behaviour of various spin mod-
els have attracted attention over the last two decades. It is now well established [1] that random
dilution does not destroy the long-range ferromagnetic order at low temperatures provided the
concentration of the spins, p, is larger than the percolation threshold, po So far as the effects of disorder on the critical exponents are concerned there are three distinct possibilities : (i) disorder
is an irrelevant variable, so that the critical behaviour of the disordered system would be the same as that of the pure system for all p > Pc; (ü) disorder is a relevant variable, so that the random model belongs to a universality class different from that of the pure model. A crossover from the
universality class of the pure system to that of the random model takes place, but on further dilu- tion the exponents do not vary continuously with the varying dilution until p becomes equal to po ;
(* ) Present and permanent address.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500190290500
2906
(iii) there exists a line of fixed points, one for each value of p, so that the critical exponents vary
continuously with p.
Using a set of heuristic arguments Harris [2] derived a criterion to decide whether or not a
given disordered model belongs to the same universality class as the corresponding model without
disorder. This criterion states that the disordered model would belong to a universality class dif-
ferent from that of the corresponding pure model provided the specific heat exponent apUre of the pure system is positive. Since for the Ising model apure m 0.1 in three dimensions a crossover is
expected with dilution. The Harris criterion is inclonclusive for the two-dimensional Ising model
for which apure = 0. So far all the analytic calculations [1] support the Harris conjecture. How-
ever, no consensus has emerged as to the precise values of the exponents for the dilute Ising model (DIM) in two and three dimensions.
The two-dimensional DIM, which has aroused lot of interest [3] and controversies [4] in the
past, is being currently investigated [5]. The three-dimensional DIM has frustrated a11 previous
numerical investigations one way or another. The earliest Monte Carlo (MC) simulation by Lan-
dau [6] using system sizes up to 303 could not detect any difference between the effective critical
exponents of the DIM and the corresponding exponents for the pure Ising model. In subsequent work, Marro, Labarta, and lejada [7], using system sizes up to 403, and Chowdhury and Stauffer [8], using system sizes 903, observed continuous variation of the affective exponent (3 with varying dilution, apparently violating the concept of universality. Braun and Fâhnle [11] carried out ex-
tensive calculation at p = 0.8, and confirmed that the exponent ,Q is larger than the pure Ising
value.
Large corrections to scaling may be one of the factors that can give rise to large differences between the effective and the true exponents. In order to observe the true critical exponents one
should carry out simulations at temperatures as close to the critical temperature as possible. On
the other hand, the closer to the critical temperature, the larger is the correlation length, and the stronger are the finite-size effects. In order to minimize the finite-size effects the system size should be as large as possible. But, the larger is the system size the longer is the CPU time required to update the spins during a computer simulation of the system. Moreover, as the critical temperature
is approached, the dynamics becomes sluggish because of the critical slowing down.
Swendsen and Wang [9] have developed an efhcient algorithm, based on cluster flipping tech- niques, that circumvents the critical slowing down. The algorithm goes as follows : (1) in an Ising spin configuration at temperature T, two nearest-neighbor spins are connected by a bond with probability 1 - e- 2J / kBT if the spins are parallel ; otherwise sites are not connected by a bond ; (2) clusters are then identified as collections of sites connected by bonds ; (3) each cluster receives
a new up or down spin with equal probability. This constitutes one MC step and the process is
repeated.
We use the clusters generated in the algorithm described above to calculate the susceptibility.
The susceptibility is related to the cluster sizes by
where sa is the number of sites belonging to cluster a ; Smax is the maximum cluster size in the
system. The angular brackets denote thermal average and square brackets denote average over random impurity configurations.
From the finite-size scaling theory [10], we know that the slope of the log-log plot of the sus- ceptibility maxima versus L yields the ratio of the susceptibility exponent y and the correlation
length exponent v. Therefore, if this ratio y/v for the DIM, computed from a finite-size analysis,
can be shown to be different from that for the pure Ising model the result would be consistent with the Harris conjecture. However, the converse is not necessarily true, because the disorder may
change the exponents in such a way that the ratio ,/v remains unchanged. Computation of the susceptibility peak requires the use of equation (2) at all temperatures. Figure 1 is a plot of the peak value of the susceptibility as a function of system size for three different spin concentrations.
We found y/v = 1.98::i: 0.02, 2.0 ± 0.1, 2.0 ± 0.1 for p = 0.8, 0.6, and 0.4, respectively.
Within the accuracy of computation, our data do not reveal any concentration dependence of -ilv.
Note that the ratio, / v is approximately 2 for both the p = 1 thermal phase transition and for the p = pc m 0.31 percolation transition. Thus for all Pc p 1, this ratio turns out to be identical with that of the corresponding pure model within the attainable accuracy. Therefore, this ratio is inadequate to check the validity of the Harris conjecture in the context of the three-dimensional DIM.
Fig. 1.- Susceptibility peak of the three-dimensional site-diluted Ising model as a function of system sizes.
lhe different symbols correspond to different spin concentrations ; circles : p = 0.8 ; stars : p = 0.6 ; pulses : p = 0.4.
2908
We have put a major effort to calculate the exponent y from susceptibility data. Accurate estimation of the exponents requires highly accurate location of the critical temperature Tc. We
have estimated Te by the location of the susceptibility maxima extrapolated to infinite system size limit and from the fourth-order cumulant intersection method [10] and checked their consistency.
Fig. 2.- Susceptibility above the transition temperatures plotted against t = [T - Tc(p)] /Tc (p) in
double logarithmic scale. The straight lines are guides to the eyes. Pulses : p = 1 ; stars : p =
0.8, Tc(0.8)/Tc(1) = 0.775 ; circles : p = 0.6, Tc (0.6) f Tc (1) = 0.536 ; crosses : p =
0.4, Tc (0.4) /Tc (1) = 0.2675. The system size for all the data is 903. Typical MC steps/spin are 400 to 1000, averaged over 10 to 30 different impurity configurations.
Figure 2 is a plot of susceptibility vs. reduced temperature t = [T - Te ( p)] / Tc (p) For
p = 0.8 (stars) we see that the slope gradually increases as t decreases. The same kind of behavior is observed for p = 0.6 (circles) but the (asymptotic) linear behavior is approached relatively quickly. For p = 0.4 (crosses) the asymptotic behavior is observed over a wider range of t. Our data clearly show the crossover from the pure Ising behavior to that of the DIM. Next we check whether or not the exponents vary with concentration p. The effective exponents are calculated by fitting four successive data points to a straight line in figure 2. These effective exponents are plotted against an average value of t in figure 3. For all three impurity concentrations the asymptotic values of y for small t appear to be the same. Our best estimate for the expunem is -j = 1.52 ± 0.07.
The pulses are data for the pure Ising case ; these are consistent with ypure m 1.24.
The data for concentration p = 0.4 do not suffer from crossover effect. On the finite-size
scaling plot, shown in figure 4, the data from size L = 4 to 90 scale well. Our estimates of the
exponents are thus confirmed by this independent scaling analysis.
Fig. 3.- Thé effective exponent T, obtained from a straight line fit to four successive data points in figure 2, plotted against average location (in log t) ofthe four points. Thé symbols have the same meaning as in figure
2.
Fig. 4.- A finite-size scaling plot of the susceptibility, XL T/Y vs. tLl/lI, for p = 0.4, with y/v = 1.98,
v = 0.77.
2910
We conclude that the critical exponents for the three-dimensional DIM are different from the corresponding exponents for the pure system ; this supports the Harris conjecture. Moreover,
the truè exponents for the DIM do not vary continuously with the dilution ; this is consistent with the concept of universality. We have demonstrated that the differences between the exponents for
p = 1 and for Pc p 1 are large enough to be detected by numerical simulation. This has been
possible because we simulated large and strongly disordered systems using an efficient algorithm.
Acknowledgements.
We thank D. Stauffer for stimulating discussions, useful suggestions, and a critical reading of
the manuscrit We have also benefited from discussions with D.R Landau and W Selke.
References
[1] STINCHCOMBE R.B., in Phase Transitions and Critical Phenomena, Vol. 7, Eds. C. Domb and J.L.
Lebowitz (Academic Press) 1983.
[2] HARRIS A.B., J. Phys C7 (1974) 1671.
[3] DOTSENKO VS. and DOTSENKO V.S., Adv. Phys. 32 (1983) 129.
[4] SHANKAR R., Phys. Rev. Lett. 58 (1987) 2466.
[5] SELKE W., private communication (1989).
[6] LANDAU D.P., Phys. Rev. B22 (1980) 2450.
[7] MARRO J., LABARTA A. and TEJADA J., Phys. Rev. B34 (1986) 347.
[8] CHOWDHURY D. and STAUFFER D., J. Stat. Phys. 44 (1986) 203.
[9] SWENDSEN R.H. and WANG J.-S., Phys. Rev. Lett. 58 (1987) 86.
[10] BINDER K., in Application of the Monte Carlo Methods in Statistical Physics, Ed. K. Binder, second
edition (Springer-Verlag) 1987, p.17.
[11] BRAUN P. and FAHNLE M., J. Stat. Phys. 52 (1988) 775 ;
see also BRAUN P., STAADEN U., HOLEY T, and FAHNLE M., preprint (1989).
