Landau singularities and singularities of holonomic integrals of the Ising class

Abstract

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 38 (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model χ (3) at the points 1+3w +4w 2 = 0 and show that χ (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).

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

“…Let us consider the first unexpected singularities 1 + 3w + 4w 2 = 0 we found [126,129] for the Fuchsian linear differential equation of χ (3) , and also found in other n-fold integrals of the Ising class [29]. This polynomial condition reads in the s variable, 2s 2 + s + 1 s 2 + s + 2 = 0.…”

“…This polynomial condition reads in the s variable, 2s 2 + s + 1 s 2 + s + 2 = 0. We have shown [29] that χ (3) itself is not singular at the roots of the first polynomial whose roots are such that |s| < 1, but is actually singular at the roots of the second polynomial. In the variable k = s 2 , these singularities read:…”

“…13 Denoted Y (n) (w) in [29]. 14 Surprisingly the integrand with ( n j=1 yj) 2 yields second order linear differential equations [29], and consequently, we have been able to totally decipher the corresponding singularity structure.…”

“…The two families of integrals we have considered in [29] are very rough approximations of the integrals (7.2). For the first family 13 , we considered the n-fold integrals corresponding to the product of (the square 14 of the) y i 's, integrated over the whole domain of integration of the φ i (thus getting rid of the factors G (n) and R (n) ).…”

“…These results provide a large set of "candidate singularities" for the χ (n) . From the resolution of the Landau conditions [29,35] for (7.5), we have shown that the singularities of (the linear ODEs of) these multiple integrals actually reduce to the concatenation of the singularities of (the linear ODEs of) a set of one-dimensional integrals. We discuss the mathematical, as well as physical, interpretation of these new singularities.…”