Landau singularities and singularities of holonomic integrals of the Ising class

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 ($|s|>1$).

arXiv comment34 pages, 1 figure

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• Square lattice Ising model susceptibility: connection matrices and singular behavior of ${\chi }^{(3)}$ and ${\chi }^{(4)}$
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• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Lattice Green Functions: the seven-dimensional face-centred cubic lattice
• On conjectured local generalizations of anisotropic scale invariance and their implications
• Canonical decomposition of linear differential operators with selected differential Galois groups
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?