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

27 pages no figures