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Canadian Journal of Physics

The general product average decomposition equation and the Ising ferromagnet

B. Frank, K. Nath

Published Mar 1, 1985• 2 citations
Publisher

Abstract

A method is found whereby the product average decomposition (PAD) approximation of Frank and Mitran may be directly applied to the thermal average of any well-behaved function of Oi = ΣjJijSj, for the spin-1/2 Ising model with nearest-neighbour interactions. A general formula is derived by considering a fictitious spin system wherein the PAD is exact. It is applied to the calculation of transition temperatures in d dimensions, for d = 3 to d = 20.

DOI
10.1139/p85-061

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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%.

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Approximate Accounting of Spin Correlations in the Ising Model

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Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.

Physics of the Solid State•Sep 1, 2021•2 citations
A method is found whereby the product average decomposition (PAD) approximation of Frank and Mitran may be directly applied to the thermal average of any well-behaved function of Oi = ΣjJijSj, for the spin-1/2 Ising model with nearest-neighbour interactions. A general formula is derived by considering a fictitious spin system wherein the PAD is exact. It is applied to the calculation of transition temperatures in d dimensions, for d = 3 to d = 20.
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%.
An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.
The difference equation method of Frank, Cheung, and Mouritsen (FCM) is made self-consistent by relaxing an assumption of the theory which pertains to the correlation of a spin with its nearest neighbour. The full set of A = 1 and I − δ equations and U equations is brought into play, with the U2 = T2 = (1/4) (1 − 1/F(1)) double equation replaced by a single relation between U2 and T2. The resulting ferromagnetic transition temperatures for the cubic lattices are consistent with the series results to within about 2%. The correlation functions are much closer to the series results than those of FCM.
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.
Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.