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Canadian Journal of Physics

Lower bound approximation to the free energy of an Ising model on a simple cubic lattice

Chin‐Kun Hu, Wen‐Den Chen, Yu-Ming Shih, Dong-Chung Jou, C. K. Pan, H. M. Huang, Kuo-Gen Chen, W. S. Lee, D. S. Wan, Hsien-Che Tseng

Published Oct 1, 1981• 1 citation
Publisher

Abstract

We apply a modified Kadanoff's variational method to calculate the lower bound zero-field free energies and their derivatives for an Ising model on the simple cubic lattice. We find a critical point at Kc = 0.2393769 with precision ±10−7.
DOI
10.1139/p81-169

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The modified Kadanoff's variational method recently considered by us is applied to calculate the free energies and their derivatives for a three-dimensional Ising model with bcc structure and without external magnetic field. We find a critical point at kc=0.162134{k}_{c}=0.162134kc​=0.162134 and the associated critical indices $\ensuremath{\alpha}=0.12$, ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.095$.
We present a systematic procedure for the direct calculation of the free energy and its first and second derivatives for the Ising model in one to three dimensions with a wide class of symmetry properties. This renormalization group method is based on the modified Kadanoff's variational method (MKUM). This method is not only general but also very accurate numerically both near and far from the critical point. Further it includes correction to scaling effects not present in the standard linearized renormalization group treatment. This work describes the techniques and presents some illustrative results for the square latttice and body-centered cubic lattice with nearest neighbor interactions. A full treatment of our results will be given elsewhere. (Author)
The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.
A variational approximation to the Ising model in a magnetic field (T< TcT<~{T}_{c}T< Tc​), based on the zero-field solution, yields an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. In two dimensions, these are numerically evaluated, and compared with the results of series expansions.