The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model

Abstract

We show that the intensity of magnetization I (z,x) where z=e−2βH and x=e−2βJ, for the ferromagnetic Ising model in arbitrary dimension, reduces, for rational values of x, to a Diophantine moment problem I (z) =∑∞0nkzk, where nk=∫Λ0σ (λ) λkdλ, σ (λ) is a positive measure, n0=1/2, and nk is integer for k≠0. The fact that the nk are positive integers puts very stringent constraints on the measure σ (λ). One of the simplest results we obtain is that for Λ<4, σ (λ) is necessarily a finite sum of Dirac δ functions whose support is of the form 4 cos2(pπ/m), p=0,1,2,...,m−1, with m a finite integer. For Λ=4, which correspond to the one-dimensional Ising model, we have the result that either I (z) is a rational fraction belonging to the previous class Λ<4, or I (z) = (1/2)(1−4z)−1/2 which corresponds precisely to the exact answer for dimension 1. For Λ≳4, which is associated with Ising models in dimension d⩾2 we show that all cases are reducible to Λ=6, by a quadratic transformation which transforms integers into integers and positive measures into positive measures. The fixed point of this type of transformation is analyzed in great detail and is shown to provide a devil’s staircase measure. Various other results are also discussed as well as conjectures.