Critical behaviour of the two-dimensional Ising susceptibility

Abstract

We report computations of the short-distance and the long-distance (scaling) contributions to the square-lattice Ising susceptibility in zero field close to T_c. Both computations rely on the use of nonlinear partial difference equations for the correlation functions. By summing the correlation functions, we give an algorithm of complexity O(N^6) for the determination of the first N series coefficients. Consequently, we have generated and analysed series of length several hundred terms, generated in about 100 hours on an obsolete workstation. In terms of a temperature variable, τ, linear in T/T_c-1, the short-distance terms are shown to have the form τ^p(ln|τ|)^q with p>=q^2. To O(τ^14) the long-distance part divided by the leading τ^{-7/4} singularity contains only integer powers of τ. The presence of irrelevant variables in the scaling function is clearly evident, with contributions of distinct character at leading orders |τ|^{9/4} and |τ|^{17/4} being identified.