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Zbl 1113.82020
Zenine, N.; Boukraa, S.; Hassani, S.; Maillard, J.-M.
Square lattice Ising model susceptibility: series expansion method and differential equation for $\chi^{(3)}$.
(English)
[J] J. Phys. A, Math. Gen. 38, No. 9, 1875-1899 (2005). ISSN 0305-4470

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 $\chi^{(3)}$, 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 $(n-1)$-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 $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^2)$, where $s=\sinh(2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.
MSC 2000:
*82B27 Critical phenomena
34M55 Painlevé and other special equations
82B10 Quantum equilibrium statistical mechanics (general)

Citations: Zbl 1073.82014

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