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arXiv:cond-mat/9910294 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster-variation–Padé-approximant method for the simple cubic Ising model

Alessandro Pelizzola

Published May 1, 2000• 7 citations
arXivHTMLPDF

Abstract

The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.

DOI
10.1103/physreve.61.4915

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The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.
In the present paper we show how nonclassical, quite accurate, critical exponents can be extracted in a very simple way from the Pad\'e analysis of the results obtained by mean-field-like approximation schemes, and in particular by the cluster variation method. We study the critical behavior of the Ising model on several lattices (quadratic, triangular, simple cubic and face centered cubic) and two problems of surface critical behavior. Both unbiased and biased approximants are used, and results are in very good agreement with the exact or numerical ones.
We report results of the application of a recently proposed technique, which combines the cluster variation method (CVM) and Pad\'e approximants, to the analysis of the critical behavior of the face-centered-cubic Ising model. The highest-order CVM approximation ever considered for this lattice is used to estimate magnetization and susceptibility, which are then analyzed by means of Dlog Pad\'e approximants in order to extract information about the critical behavior, taking also into account the effects of confluent singularities. Our results are in good agreement with the most recent simulation and series expansion estimates and require a remarkably small numerical effort. \textcopyright{} 1996 The American Physical Society.
The coherent-anomaly scaling is investigated for the approximations derived on the basis of the cluster-variation method (CVM). All of the mean-field critical coefficients are calculated in the Weiss, the Bethe and the Kikuchi approximations for the three-dimensional Ising model on the simple cubic lattice. The dynamical coefficients of the relaxation time are also obtained in the corresponding approximations for the kinetic Ising model. The non-classical values of critical exponents as well as the true value of critical temperature are estimated by extrapolating these approximants by the coherent-anomaly method. The results suggest that the approximations by the CVM possess the property of coherent-anomaly scaling.
Pad\'e approximants have been used to estimate both the critical temperature and the critical-point exponents of the Ising model. In the standard Pad\'e analysis the approximants near the diagonal are given more credence. In the present study of two-dimensional models more credence is given to those entries for which the estimated critical temperature is close to the exact value. For three-dimensional models an analogous method is employed using the "exact" critical temperature estimated from the high-temperature compressibility series. The rationale for selecting these particular approximants is based on the fact that one is estimating critical-point behavior by extrapolating from a series representation. Thus, if the estimated critical point is shifted from its true value, one should expect a corresponding shift in the estimated exponent. Indeed, this is what is found.
The critical exponent ν of correlation length is calculated on the basis of the coherent anomaly method (CAM) by using the pair, the cactus-square and the square approximations of the cluster variation method (CVM) in the D (≥2)-dimensional cubic lattice of the Ising model. These simple analytical approximations give good values of critical exponent ν and critical temperature in the two- and three-dimensional lattices. However, further consideration is needed to determine a critical dimension in the D -dimensional Ising model by the CAM.
Pad\'e-approximant techniques are applied to the high-temperature series for the temperature derivative of the specific heat of the Ising model. This method indicates a critical exponent $\ensuremath{\alpha}$ in the neighborhood of 18\frac{1}{8}81​ for those Pad\'e approximants which yield an estimate of the critical temperature in substantial agreement with that determined from the susceptibility series.