Square lattice Ising model $\tilde{\chi}^{(5)}$ ODE in exact arithmetic

TL;DR

This paper derives an exact order 24 linear differential operator for the fifth particle contribution to the susceptibility of the square lattice Ising model, analyzing its factorization, critical behavior, and singularities.

Contribution

It provides the first exact derivation of the differential operator for ^{(5)} and proves its indecomposability, advancing understanding of the model's mathematical structure.

Findings

Proves no further factorization of the order 12 operator is possible.

Analyzes the behavior of ^{(5)} at the critical point.

Identifies singularities at w=1/2 on multiple branches.

Abstract

We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(\Phi^{(5)}) = E^{(5)}$, where $\Phi^{(5)} =\tilde{\chi}^{(5)}-\tilde{\chi}^{(3)}/2+\tilde{\chi}^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {\bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(\rm left)} \cdot L_{12}^{(\rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(\rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $\tilde{\chi}^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $\tilde{\chi}^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {\bf 41} 455202 (2008)), we find that $\tilde{\chi}^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.