The diagonal Ising susceptibility

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 ${\chi }_d^{(5)}(t)$, 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 $|\sinh 2E_v/kT\sinh 2E_h/kT|=1$ as $n\to \infty$.

18 pages

References in corpus (2)

• Holonomy of the Ising model form factors
• Landau singularities and singularities of holonomic integrals of the Ising class

Cited by in corpus (3)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve