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Square lattice Ising model susceptibility: series expansion method and differential equation for ${\chi }^{\left(3\right)}$. (English) Zbl 1113.82020

Summary: In a previous paper [J. Phys. A 37, No. 41, 9651–9668 (2004; Zbl 1073.82014)] we gave the Fuchsian linear differential equation satisfied by ${\chi }^{\left(3\right)}$, 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) which allow one to obtain a long series in polynomial time. The method is based on series expansion in the variables that appear in the $\left(n-1\right)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi$. The integration rules are straightforward due to remarkable formulae we derive for these variables.

We obtain without any numerical approximation ${\chi }^{\left(3\right)}$ as a fully integrated series in the variable $w=s/2/\left(1+{s}^{2}\right)$, where $s=sinh\left(2K\right)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

##### MSC:
 82B27 Critical phenomena (equilibrium statistical mechanics) 34M55 Painlevé and other special equations; classification, hierarchies 82B10 Quantum equilibrium statistical mechanics (general)