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Physica Scripta

Transition temperature for the Ising model with arbitrary finite and infinite spinsin a new correlation reduction theory

Ladislav Šamaj

Published Oct 1, 1988• 2 citations
Publisher

Abstract

An analytic method for finding the highly accurate values of the critical point for cubic Ising models with arbitrary spin is presented. The thermodynamic quantities of interest are expressed through some higher-order correlations with a reference spin excluded. A reduction scheme for these higher-order correlations is constructed. It leads to two types of resulting equations for correlation functions. The coincidence of two different representations of critical correlations is ensured by the introduced parameter of the reduction scheme. The estimation of critical points for all cubic lattices is compatible with series works to within 1.69%, 0.70% for the limiting values of spin, ½ and ∞, respectively.

DOI
10.1088/0031-8949/38/4/003

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An analytic method for finding the highly accurate values of the critical point for cubic Ising models with arbitrary spin is presented. The thermodynamic quantities of interest are expressed through some higher-order correlations with a reference spin excluded. A reduction scheme for these higher-order correlations is constructed. It leads to two types of resulting equations for correlation functions. The coincidence of two different representations of critical correlations is ensured by the introduced parameter of the reduction scheme. The estimation of critical points for all cubic lattices is compatible with series works to within 1.69%, 0.70% for the limiting values of spin, ½ and ∞, respectively.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
An infinite set of coupled integral equations is found, with the use of a spectral theorem for the correlation functions of the spin-1/2 three-dimensional Ising model. The second equation is linearised to yield the two spin correlation function. The Ward identity is then satisfied. Those approximations which give the Weiss and spherical model results are indicated. The transition temperature is calculated for SC, BCC and FCC lattices using a product average decomposition of the higher-order correlation functions. The scheme is seen to converge toward the respective values 0.738, 0.789 and 0.823 of 4/ beta c J(0) which are within 2% of the exact values 0.752, 0.794 and 0.816.
A new method of correlation reduction near the critical temperature is introduced for a classical spin system. This method is based on an assumption much less restrictive than the usual mean-field approximation. It permits an important parameter to be determined realistically. The transition temperature of the Ising ferromagnet is expressed very simply in a closed form. The values of the transition temperature evaluated with various approximations in this method are presented for three-dimensional Ising lattices.
A theory is presented for the determination of the transition temperature T c for spin- 1 / 2 Ising models on cubic lattices with pair (J 2 ) and quartet (J 4 ) interactions. A linear relationship between T c and J 4 /J 2 is found to have a base wider in scope than that of previous theories. The calculation of the linear coefficient is performed through the derivation of a new equation for special four-spin correlation functions which is solved in a novel way by the use of a ratio approximation. The results are compared with those derived from previous theories, series analysis and Monte Carlo calculations which are also presented here. The linear coefficient is shown to depend on a parameter which turns out to have approximately the same value for all cubic lattices, thus suggesting a set of scaled variables in terms of which the phase boundary for all cubic lattices may be fitted by a common curve. From the authors' work a coherent picture emerges for the phase behaviour of cubic Ising lattices with pair and quartet interactions, involving regions of continuous and first-order transitions separated by a tricritical point which occurs at the same value of the scaled variables for all cubic lattices.
In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M 2n (n=3,4,5,...) in terms of the two- and four-spin correlation functions M 2 and M 4 , at the critical point. As an example, the expression for M 6 is M 4 2 /M 2 -9M 2 M 4 -6M 2 3 . These expressions (for up to M 60 ) are applied to the calculation of T c for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2 / 3 % of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).
The rational values beta = 5 / 16 , delta =5, alpha = 1 / 8 , nu = 5 / 8 , and eta =0 of the critical exponents for the three-dimensional spin 1 / 2 Ising ferromagnet are shown to be consistent with a recently-derived linearised equation for the Ising two-spin correlation function. This equation arises from an exact integral representation which previously, on application of a straightforward approximation for higher-order cumulant averages, was shown to yield transition temperatures T c to within 2% of their correct values for the cubic lattices. In the search for beta , the requirement is imposed that the correction terms for the magnetisation and for M 2 , a quantity closely related to the system internal energy, be analytic in some power of T c -T (which turns out to be 1 / 8 ) when the external magnetic field h is zero.