Ising model susceptibility: Fuchsian differential equation for $\chi^{(4)}$ and its factorization properties

TL;DR

This paper derives the Fuchsian differential equation for the four-particle susceptibility contribution of the 2D Ising model, revealing factorization properties and connections to lower-particle contributions, suggesting a general structure.

Contribution

It introduces the Fuchsian differential equation for , uncovers its factorization properties, and conjectures a universal structure for all n-particle contributions in the Ising susceptibility.

Findings

The order ten differential operator for  has notable factorization properties.

The two-particle contribution  is a solution of this differential operator.

Similar structures are conjectured for all n-particle contributions.

Abstract

We give the Fuchsian linear differential equation satisfied by $\chi^{(4)}$, the ``four-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $\chi^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $\chi^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $\chi^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ \chi^{(n)}$.