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Acta Physica Sinica

CLOSED-FORM APPROXIMATION FOR THE 3-DIMENSIONAL ISING MODEL (Ⅳ)——THE APPROXIMATE INTERPOLATION FORMULA FOR THE PARTITION FUNCTION

SHI HE, Bailin Hao, 中国科学院

Published Jan 1, 1981• 0 citations
Publisher

Abstract

A class of approximate interpolation formula for the partition function of the 3-dimensional Ising model is proposed on the basis of a previous analysis of the 2-dim rigorous solution and 3-dim Q-approximation. In principle, the present knowledge about the location of the singular point and the low- and high-temperature expansion coefficients can be incorporated provided a sufficient amount of computation is carried out. One and the same analytical expression leads to good low- and high-temperature behaviour except for the specific heat singularity.

DOI
10.7498/aps.30.1234

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A class of approximate interpolation formula for the partition function of the 3-dimensional Ising model is proposed on the basis of a previous analysis of the 2-dim rigorous solution and 3-dim Q-approximation. In principle, the present knowledge about the location of the singular point and the low- and high-temperature expansion coefficients can be incorporated provided a sufficient amount of computation is carried out. One and the same analytical expression leads to good low- and high-temperature behaviour except for the specific heat singularity.
An analytical expression for the partition function of the 3-dimensional Ising Model is obtained (see formula (32)). It is approximate, but has several rational features: (1) It is valid on the whole temperature range from zero to infinity and has only one singular point at uc = 0.2612 in comparision to 0.22 from the series expan-sion; (2) It leads to a nonlogarithmic singularity for the specific heat; (3) It can be reduced into the exact Onsager 2-dimensional solution for planar rectangle lattice in the anisotropic case (see formula (34)) ; (4) It has a high temperature expansion, the first two coefficients are correct and the higher ones diminish systematically but remain positive.
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.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
A simple Monte Carlo high-temperature expansion method to evaluate the partition function of quantum spin systems which was proposed in the previous paper l) is improved so that all of the needed expansion-coefficients are obtained from interpolating points of less than ten with an interpolation formula. By this improvement, this method is applicable to the finite spin-systems of rather large spin size regardless of dimensionality of the system. As an example, the method is applied to the one-dimensional spin-l/2 isotropic Heisenberg ferromagnetic system.
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.
The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.