Abstract

Non-analyticity of the correlation length in systems with exponentially decaying interactions Y. Aoun, D. Ioffe, S. Ott, Y. Velenik Commun. Math. Phys. 386, 433-467 (2021). 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. Key words: Ising model, Potts model, XY model, random cluster model, Ornstein-Zernike asymptotics, correlation length, analyticity Files: PDF file, Published version, bibtex, slides, talk