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お茶の水女子大學自然科學報告

An Improvement in the Spherical Approximation of an Ising Spin System

Nagai Noriko, Yoko Ohkawa, Mayumi Shingu, Sachiko Yagi, Giiti Iwata

Published Dec 1, 1977• 0 citations
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Abstract

Rewriting the partition function of an Ising system as a multiple integral, dividing the integration space by a set of concentric spherical surfaces and sets of parallel planes into small domains, averaging separately the two factors constituting the integrand of the multiple integral over each of these domains, and replacing the two factors by their respective averages, logarithm of the partition function per spin is given by the extremum of a function with respect to a complex variable and a number of real variables. The specific heat curve of an Ising square lattice obtained by the present method is compared with those by the exact method, the spherical approximation and the mean field approximation.

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Rewriting the partition function of an Ising system as a multiple integral, dividing the integration space by a set of concentric spherical surfaces and sets of parallel planes into small domains, averaging separately the two factors constituting the integrand of the multiple integral over each of these domains, and replacing the two factors by their respective averages, logarithm of the partition function per spin is given by the extremum of a function with respect to a complex variable and a number of real variables. The specific heat curve of an Ising square lattice obtained by the present method is compared with those by the exact method, the spherical approximation and the mean field approximation.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
The partition functions of the finite Ising models of spin S =1, 3/2, 2, 5/2 and 3 have been calculated by a high speed computer. The periodicity condition at the boundaries has been imposed, and only the nearest neighbor interaction has been assumed. The specific heats of these finite systems have been calculated as a function of temperature. The maxima of the specific heat increase proportionally to long N in the two-dimensional Ising model with spin S =1, N being the number of lattice points, which indicates that the specific heat for the above lattice has a singularity such as C v / k ∼ A log | T - T c |+ B ± : A ∼0.10 and B - - B + =0.24. The difference ( B + - B - ) has been estimated from the asymmetric distribution of the zeros of the partition function in the complex temperature plane.
From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.
The partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.