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Physical review. B, Condensed matter

The fee Ising model in the cluster variation approximation

J. M. Sánchez, Didier De Fontaine

Published Apr 1, 1978• 357 citations
Publisher

Abstract

The cluster-variation method was used to calculate the critical temperature for the fcc Ising ferromagnet in three different cluster approximations. A scheme for determing a set of independent cluster variables is presented, which considerably simplifies the minimization of the free energy. The system of equations arising from such minimization is solved, in the disordered state, by means of a simple iteration technique. The highest level of approximation treated in this work yields a critical temperature which is only 1.5% above the one estimated by the exact high-temperature expansion of the zero-field susceptibility. Furthermore, the high-temperature expansion for the specific heat gives four exact coefficients and the fifth is determined to within 0.4%.

DOI
10.1103/physrevb.17.2926

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The molecular-field-type approximations when applied to the Ising ferromagnet yield inaccurate and sometimes incorrect behaviors at the transition temperature. We have advanced the mathematical approximation of the cluster variation method, which is the generalization of the molecular-field-type approximation, a few more steps so that the four- and six-body correlations are taken into account. Some remarkable features became apparent from preliminary calculations for fcc Ising ferromagnets: (i) The transition temperature is predicted within a few percent of the exact value. ${T}_{c}(4\ensuremath{-}\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y})=0.83545$, ${T}_{c}(6\ensuremath{-}\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y})=0.83394$, ${T}_{c}(\mathrm{exact})=0.81627$, with ${T}_{c}(\mathrm{Weiss})=1.00000$; (ii) When closed-form equations for the low-temperature specific heat are expanded into infinite series, all of the known exact coefficients are reproduced; (iii) The extent of mathematical analysis and the amount of numerical computation are considerably less than those involved in diagram enumeration methods.

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The antiferromagnetic Ising model in a magnetic field: Linked-cluster expansion analysis

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Journal of Applied Physics•May 15, 1993•5 citations
The cluster-variation method was used to calculate the critical temperature for the fcc Ising ferromagnet in three different cluster approximations. A scheme for determing a set of independent cluster variables is presented, which considerably simplifies the minimization of the free energy. The system of equations arising from such minimization is solved, in the disordered state, by means of a simple iteration technique. The highest level of approximation treated in this work yields a critical temperature which is only 1.5% above the one estimated by the exact high-temperature expansion of the zero-field susceptibility. Furthermore, the high-temperature expansion for the specific heat gives four exact coefficients and the fifth is determined to within 0.4%.
The molecular-field-type approximations when applied to the Ising ferromagnet yield inaccurate and sometimes incorrect behaviors at the transition temperature. We have advanced the mathematical approximation of the cluster variation method, which is the generalization of the molecular-field-type approximation, a few more steps so that the four- and six-body correlations are taken into account. Some remarkable features became apparent from preliminary calculations for fcc Ising ferromagnets: (i) The transition temperature is predicted within a few percent of the exact value. ${T}_{c}(4\ensuremath{-}\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y})=0.83545$, ${T}_{c}(6\ensuremath{-}\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y})=0.83394$, Tc(exact)=0.81627{T}_{c}(\mathrm{exact})=0.81627Tc​(exact)=0.81627, with Tc(Weiss)=1.00000{T}_{c}(\mathrm{Weiss})=1.00000Tc​(Weiss)=1.00000; (ii) When closed-form equations for the low-temperature specific heat are expanded into infinite series, all of the known exact coefficients are reproduced; (iii) The extent of mathematical analysis and the amount of numerical computation are considerably less than those involved in diagram enumeration methods.
A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
Abstract : The Cluster Variation Method proposed by Morita is shown to yield a development of the entropy into an alternating series when applied to the Lenz- Ising Model. Truncation of this series to its first N terms gives an N-th order variationally based approximation to the exact statistical behavior of this system. The first four orders of this sequence of approximations have been carried out in this work. curious behaviors such as complex Curie temperatures have been observed by others and this work shows that such behavior is due to the alternating sign of the terms in the entropy expansion. The effect is large enough to cause a non-physical global minimum to appear on the boundary of the variation space, which is certainly unphysical. A modification of the Cluster Variation Method is effective in suppressing these undesired effects. The (2n+1) -cluster results are then an improvement over those of the (2n+1)-cluster approximation. The (2n)-cluster results are less affected by spurious minima. Keywords: Statistical mechanics; Ising model; Critical point; Ferromagnetism; Order disorder; Thermodynamics; Probability theory; Minimization; Free energy; Cooperative behavior; Curie temperature; Chaos; Variation methods; Entropy; Phase transitions; Clusters.
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.
Critical temperatures T c and mean-field critical coefficients [Formula: see text] for susceptibilities are calculated by a cluster-variation method (CVM) for a three-dimensional Heisenberg model on a simple cubic lattice. Both the full CVM based on a 4-spin cluster and the SCVM are applied and the same values for T c and [Formula: see text] are obtained by three different approximation in the SCVM. The results are compared with those obtained by the same approach for the three-dimensional Ising model on the simple cubic lattice.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.