Non-analyticity of the Correlation Length in Systems with Exponentially Decaying Interactions

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