Square lattice Ising model susceptibility: Series expansion method and differential equation for $\chi^{(3)}$

TL;DR

This paper details a method to derive long series expansions for the three-particle susceptibility contribution in the square lattice Ising model, using series expansion and differential equations without numerical approximation.

Contribution

It introduces a series expansion method based on integral variables and formulas that efficiently computes the susceptibility series for the Ising model's three-particle contribution.

Findings

Derived the Fuchsian differential equation for $ ext{chi}^{(3)}$

Obtained a fully integrated series in variable $w$ without numerical approximation

Provided insights and tools for calculating susceptibility series in lattice models

Abstract

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi $. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $ s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.