Square lattice variational approximations applied to the Ising model

Kazuyuki Tanaka, Tsuyoshi Horiguchi, Tohru Morita

The spin-pair correlation functions in the Ising model on the square lattice are obtained in three approximations of the simplified cluster variation method. We find good estimates of the critical exponents ην, γ and ν, by applying the coherent-anomaly method to the obtained spin-pair correlation functions.

R. J. Baxter

Alessandro Pelizzola

Tohru Morita

A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.

B. V. Thompson

The partition function of the Ising model with $N$ spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site $l$ 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.

Ivan Franko