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Journal of Physics C Solid State Physics

Transition temperature in a new integral representation for the Ising ferromagnet

B. Frank, O Mitran

Published Jul 28, 1977• 37 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/10/14/016

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Journal of Physics C Solid State Physics•Nov 28, 1979•3 citations
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 set of coupled integral equations for the correlation functions of the 3-D Ising ferromagnet is generalised to the case of a ferromagnet with the additional feature that beside the conventional two-spin interaction terms (J 2 S i z S j z ) it contains four-spin interaction terms (J 4 S i z S j z S k z S l z ) in its Hamiltonian. The second equation is linearised yielding the two-spin correlation function which is found to be the same as the two-spin correlation function for the pure two-spin interaction ferromagnet. The Ward identity is shown to be satisfied. The approximations which reproduce the Weiss and spherical results in a generalised form are indicated. Using a generalised non-uniform magnetic field method, a relation between the transition temperature for the combined two- and four-spin interaction ferromagnet and the transition temperature for the pure two-spin interaction is derived.
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.
High-temperature series expansions for the Ising model's spin-spin correlation function are found to order (JkT)12{(\frac{J}{\mathrm{kT}})}^{12}(kTJ​)12 on the simple cubic, bcc, and fcc latices. Analysis of moment series indicates that the scaling form of the correlation function is correct, at least in the limit $\ensuremath{\kappa}r$ fixed, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$. Values are obtained for the correlation indices $\ensuremath{\nu}$ and $\ensuremath{\eta}$-the latter by two methods, thereby providing a direct check on the scaling law.
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.
Two improvements to a previous calculation of the transition temperature for the three-dimensional Ising ferromagnet with combined two- and four-spin interaction are presented. The first improvement follows from the complete implementation of a refined product average decomposition (PAD) method which distinguishes between nearest-neighbour and non-nearest-neighbour pairs. The second improvement comes from a comparison of the value for the ordinary two-spin interaction average of order four calculated using the PAD method, with its more accurate value calculated using a critical correlation-function approximation method. The comparison with the available values of the transition temperature for the FCC lattice obtained from low-temperature expansions shows that the errors are below 0.1% at alpha = infinity ( alpha =J 4 /J 2 ), below 5.8% for alpha <or=2, and below 2.4% for alpha <or=1. At alpha =0 the errors are less than 0.3%. For intermediate values of alpha (4<or= alpha <or=8, for example) the errors are about 11%.