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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster-variation–Padé-approximants method and the critical exponents of the fcc Ising model

Alessandro Pelizzola

Published Jun 1, 1996• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.53.5825

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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.
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.
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.
On the basis of the Pad\'e approximant method we deduce from the exact series expansions for the Ising model that the reduced magnetic susceptibility behaves at the critical point as ${\ensuremath{\chi}}_{\mathrm{fcc}}\ensuremath{\approx}{[\frac{0.09923}{(0.101767\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{bcc}}\ensuremath{\approx}{[\frac{0.152773}{(0.1561789\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sc}}\ensuremath{\approx}{[\frac{0.22138}{(0.218156\ensuremath{-}w}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{t}}\ensuremath{\approx}{[\frac{0.2432}{(2\ensuremath{-}\sqrt{3}\ensuremath{-}w)}]}^{\frac{7}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sq}}\ensuremath{\approx}{[\frac{0.35724}{(\sqrt{2}\ensuremath{-}1\ensuremath{-}w)}]}^{\frac{7}{4}}$, and ${\ensuremath{\chi}}_{\mathrm{h}}\ensuremath{\approx}{[\frac{0.4506}{(\frac{1}{\sqrt{3}}\ensuremath{-}w)}]}^{\frac{7}{4}}$, where w=tanh(JkT)w=tanh(\frac{J}{\mathrm{kT}})w=tanh(kTJ​) and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{fcc}}\ensuremath{\approx}{[12.5(0.664658\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{bcc}}\ensuremath{\approx}{[10.4(0.5326607\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{sc}}\ensuremath{\approx}{[10.9(0.411940\ensuremath{-}{z}^{2})]}^{0.3}$, where $z=\mathrm{exp}(\ensuremath{-}\frac{2J}{\mathrm{kT}})$ and again the last place quoted is somewhat uncertain. The numbers 54\frac{5}{4}45​ and 74\frac{7}{4}47​ have an error of at most ${10}^{\ensuremath{-}3}$, and 0.3 of at most ${10}^{\ensuremath{-}2}$. The lattices referred to are fcc, face-centered cubic; bcc, body-centered cubic; sc, simple cubic; t, triangular; sq, simple quadratic; and h, honeycomb.
Series expansions for certain thermodynamic variables have been computed for various standard model lattices. The high-temperature series for the magnetic susceptibility of the Ising model of three-dimensional SC, BCC and FCC lattices allows estimates of the critical index gamma of each series to be made. Dramatic progress in this direction was made by the Pade Dlog method, and here a similar technique is introduced. Estimates of gamma from row sequences of generalised inverse vector-valued Pade approximants are made. The results point to the conclusion that gamma =1.2406+or-0.0017 is the critical index common to all three lattices.
The susceptibility of the spin-sss nearest-neighbor fcc Ising model is studied at zero magnetic field by twelve-term high-temperature series. We find that the data near criticality are consistent with $\ensuremath{\chi}=A(s){t}^{\frac{\ensuremath{-}5}{4}}[1+B(s){t}^{{\ensuremath{\Delta}}_{1}}]$, $t=1\ensuremath{-}\frac{{T}_{c}(s)}{T}$. The correction-to-scaling exponent has the universal value ${\ensuremath{\Delta}}_{1}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ in good agreement with $\ensuremath{\epsilon}$-expansion estimates. The confluent amplitude B(s)B(s)B(s) vanishes at s=12s=\frac{1}{2}s=21​ to within uncertainties. Values of the critical parameters are tabulated for a variety of spins.