Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

TL;DR

This paper investigates the mathematical structures of differential equations related to the susceptibility of the square lattice Ising model, revealing new singularities and properties of the associated Fuchsian equations.

Contribution

It introduces a novel analysis of the Fuchsian differential equations for $ ext{χ}^{(3)}$ and $ ext{χ}^{(4)}$, including factorization, singularities, and differential Galois groups, advancing understanding of the model's mathematical framework.

Findings

Identification of new Landau-like singularities.

Development of an efficient method to compute connection matrices.

Insights into the differential Galois groups of the equations.

Abstract

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $ \chi^{(3)}$ and $ \chi^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $ \chi^{(3)}$ and $ \chi^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $ \chi^{(3)}$ and $ \chi^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $ n$-particle contributions $ \chi^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlev\'e oriented approach.