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

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 \( \chi^{(n)}\).