Non-analyticity of the correlation length in systems with exponentially decaying interactions

Abstract

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on ${\mathbb{Z}}^d$ with long-range interactions of the form $J_x=\psi (x)e^{-|x|}$, where $\psi (x)=e^{\mathsf{o}(|x|)}$ and $|\cdot |$ is an arbitrary norm. We characterize explicitly the prefactors $\psi$ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $\beta$, magnetic field $h$, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $\psi$ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein--Zernike theory of correlations.

Minor typos corrected. Final version, to appear in Communications in Mathematical Physics