Improvement of the Cluster-Variation Method

Abstract

It is shown that efficient convergence of the cluster-variation method (of cooperative phenomena in statistical mechanics) for the two-dimensional problem can be achieved by increasing the size of the basic cluster one dimensionally, rather than two dimensionally, and by formulating the degeneracy factor anisotropically. The method is illustrated with the Ising model in a square lattice, and the following are presented: (a) an angle-shaped (or a V-shaped) basic cluster of three points can give the same result as a square basic cluster; (b) a general case is formulated in which a zigzag shape of n V's is used as the basic cluster; and (c) the case of the W cluster (two V's) is calculated using the general formulation mentioned above. It is shown that, when they are plotted against the reciprocal of the number of points in a cluster, the Curie points calculated by different methods lie very close to a straight line.