A STUDY OF THE FREE ENERGY OF THE LENZ-ISING MODEL USING THE CLUSTER VARIATION METHOD OF MORITA (CRITICAL POINT, COOPERATIVE PHENOMENA, CURIE TEMPERATURE, CHAOS)
The Cluster Variation Method proposed by Morita is shown to yield a development of the entropy into an alternating series when applied to the Lenz-Ising model. Truncation of this series to its first N terms gives an N-th order variationally based approximation to the exact statistical behavior of this system. The first four orders of this sequence of approximations have been carried out in this work. The first level of approximation reproduces the familiar mean field results; among these results is a second order phase transition for all lattices of all dimensions. The second level reproduces the familiar Bethe-Peierls approximation; there is no phase transition in one dimension, while a second order phase transitions is found in every lattice of higher dimensions. Many researchers have studied higher order approximations, usually finding curious behaviors such as complex Curie temperatures. This work shows that such behavior is due to the alternating sign of the terms in the entropy expansion. The effect is large enough to cause a non-physical global minimum to appear on the boundary of the variation space, starting with the third order approximation. Even in the second approximation, the entropy becomes negative for certain regions of the variation space, which is certainly unphysical. A modification of the Cluster Variation Method suggested by common practices in numerical methods is effective in suppressing these undesired effects. The (2n + 1)-cluster results are then an improvement over those of the (2n)-cluster approximation. The (2n)-cluster results are less affected by spurious minima.