Landau singularities and singularities of holonomic integrals of the Ising class

TL;DR

This paper analyzes the singularities of Ising class integrals and their associated differential equations, showing how Landau conditions predict these singularities and revealing their impact on duality and susceptibility behavior.

Contribution

It demonstrates that Landau conditions can accurately identify singularities of Ising class integrals and their differential equations, clarifying their relation to duality and physical properties.

Findings

Landau conditions predict integral singularities accurately.

Singularities break Kramers-Wannier duality in certain variables.

Analytic continuation reveals singularities outside the unit circle.

Abstract

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= \sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {\bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $\chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $\chi^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $ 2+s+s^2=0$ oustside the unit circle ($| s | > 1$).