The susceptibility of the square lattice Ising model: New developments

Abstract

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N^6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T-T_c|^{9/4}, though high-low temperature symmetry is still preserved. At terms of order |T-T_c|^{17/4} and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T-T_c)^p(log|T-T_c|)^q with p \ge q^2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.