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

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

2005-02-07

arxiv.org/abs/cond-mat/0502155v1

J.Phys.A38 (2005) 4149-4173

Physics

Condensed Matter

Statistical Mechanics

27 pages no figures

Scientific paper

10.1088/0305-4470/38/19/007

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)}$.

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