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The diagonal Ising susceptibility

S. Boukraa·S. Hassani·J. -M. Maillard·B. M. McCoy·N. Zenine
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Abstract

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions $χ_{d}^{(1)}$ and $χ_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution $χ^{(5)}_d(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $ n$-particle contributions $χ_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $χ_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $ n\to \infty$.

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S. Boukraa, S. Hassani, J. -M. Maillard, B. M. McCoy, N. Zenine. 2007-03-01. The diagonal Ising susceptibility. https://doi.org/10.1088/1751-8113/40/29/002

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