Exact solutions for even-number correlations of the square Ising model

Abstract

To obtain an unusually large set of exact solutions for even-number localized correlations of the standard square Ising model, a method is developed that combines traditional Pfaffian techniques with linear-algebraic systems of correlation identities having interaction-dependent coefficients. Two eight-site clusters (subclusters of the ``Greek cross'' cluster) are studied, and altogether sixty different even-number correlations are determined exactly at all temperatures. Besides demonstrating the existence of degeneracies within the set of exact solutions, the solution curve for an eight-site correlation is evidently the first of such large order to be displayed for an Ising model on any regular planar lattice. Significant time as well as labor efficiency of the method is clearly demonstrated since relatively few correlations need to be actually calculated by Pfaffian techniques in order to obtain large additional numbers of multisite correlation solutions using much simpler linear-algebraic procedures.