Empirical relations between static and dynamic exponents for Ising model cluster algorithms

Abstract

We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is Zim^^cr/v. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that Z §0!E =/J/v for the Ising model. PACS numbers: 64.60.Ht, 05.50.+q, 11.15.Ha The Monte Carlo cluster update algorithms of Swendsen and Wang (SW) [1] and Wolff [2] can dramatically reduce critical slowing down in computer simulations of spin models, and thus greatly increase the computational efficiency of the simulations (for reviews of cluster algorithms, see Refs. [3,4]). There is little theoretical understanding of the dynamics of these algorithms. In particular, little is known as to why they seem to eliminate critical slowing down completely in some cases and not others. There is no known theory which can predict the value of the dynamic critical exponent z for any spin model, although a rigorous bound on z for the SW algorithm for Potts models has been derived [5]. Another problem which is not well understood is why the SW and Wolff algorithms give similar values of z for the 2D Potts model [6], but have very different behaviors for other models, such as the Ising model in more than two dimensions [7,8].The measurement of dynamic critical exponents is notoriously difficult, and both very good statistics and very large lattices are required in order to obtain accurate results. This is certainly the case for the Ising model, where a number of different measurements have given conflicting results. For the two-dimensional Ising model, initial results suggested z ~ y for both the SW and Wolff algorithms [1,8]. Further work [7] gave z~ y, and it was later shown that the data were consistent with a logarithmic divergence, suggesting that z =0 [9]. Recent results show that it is very difficult to distinguish between a logarithm and a small power [6], Measurements on the three-dimensional model have proven to be just as difficult, with values of z for the SW algorithm ranging from 0.339(4) to 0.75(1) [1,7,10]. For the Wolff algorithm, Tamayo, Brower, and Klein [8] obtained 0.44(10), while Wolff found a value of 0.28(2) for the energy autocorrelations [7]. We have examined Wolff's data and found that they are also fitted well by a logarithm, so that z =0 is also a possibility.In four dimensions only one result is known, which is z = -0.05(15) for the Wolff algorithm [8]. Simulations have also been done on the mean-field Ising model, which 10

Dynamic critical behavior of the Swendsen?Wang algorithm for the three-dimensional Ising model

“…β Potts = 0.44330918), which is Blöte et al's [39] best estimate of the critical temperature and is very near to the estimates by other workers [40,41] (see also the review [42]). We studied lattice sizes L = 4, 6,8,12,16,24,32,48,64,96,128,192,256 and performed between 2.9 × 10 7 and 5 × 10 8 SW iterations for each lattice size (see Table 1). The total data set at each L corresponds to ≈ 10 6 τ on the largest lattices (L = 192, 256), at least 10 7 τ on all L ≤ 64, and nearly 10 8 τ at L = 16 (see again Table 1).…”

“…• the nearest-neighbor connectivity (which is an energy-like observable [11]) 16) where γ xy equals 1 if both ends of the bond xy belong to the same cluster, and 0 otherwise. More generally, the connectivity γ ij can be defined for an arbitrary pair i, j of sites:…”

“…For the dynamic critical exponents associated to the integrated autocorrelation times of the "energy-like" observables, we find z int,N = z int,E = z int,E ′ = 0.459 ± 0.005 ± 0.025 , (1.4) where the first error bar represents statistical error (68% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68% subjective 5 The only numerical study of which we are aware is [16], which yielded z int,E = 0.86 ± 0.02, based on lattices of size 4 ≤ L ≤ 16 (which by present-day standards are much too small). In addition, there have been numerical [22,23] and analytic [22][23][24] studies of the Swendsen-Wang algorithm for the Ising ferromagnet on the complete graph (also known as the Curie-Weiss or "mean-field" model), which indicate z = 1.…”

Static and dynamic critical behaviour of 3d random-site Ising model: Different Monte Carlo algorithms

“…However, whereas in the pure model they can be related to the combination of static exponents [2], the very existence of such a relation in the diluted case remains unsettled.…”

“…In this paper we treat cluster algorithms not only as an efficient simulational tools but also study their dynamics for RIM, particularly concentrating on the dynamical critical index z. This is also interesting from the point of view of the general physical considerations relating z to certain combinations of static critical indices [2].…”