The diagonal Ising susceptibility

TL;DR

This paper analyzes the susceptibility of the square Ising model with a diagonal magnetic field using form factor expansions, deriving differential equations for multi-particle contributions and revealing their singularity structure.

Contribution

It provides exact evaluations and differential equations for multi-particle susceptibility contributions, highlighting their structural and singularity properties.

Findings

Exact evaluation of one and two particle contributions.

Differential equations for three, four, and five particle contributions.

Singularities at roots of unity becoming dense on the unit circle.

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 $\chi_{d}^{(1)}$ and $\chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${\chi}^{(5)}_d(t)$, 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 $\chi_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $\chi_{d}^{(n)}$ 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$.