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

TL;DR

This paper investigates the conditions under which the correlation length in lattice spin systems with exponentially decaying interactions becomes non-analytic, revealing new behaviors especially in one-dimensional cases and linking it to the Ornstein--Zernike theory.

Contribution

It explicitly characterizes the functions a that lead to non-analytic correlation lengths in long-range interacting spin systems across any dimension.

Findings

Correlation length non-analyticity occurs for summable a in 1D systems.

Non-analyticity correlates with qualitative changes in the 2-point function.

Links non-analyticity to the failure of the mass gap condition in Ornstein--Zernike theory.

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.