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

Yacine Aoun,Dmitry Ioffe,Sébastien Ott,Yvan Velenik

doi:10.1007/s00220-021-04038-6

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.

Highlights

The correlation length plays a fundamental role in our understanding of the properties of a statistical mechanical system
The usual way of defining it precisely is as the inverse of the rate of exponential decay of the 2-point function
Is λsat always smaller than λexp? While this work provides precise criteria to decide whether λsat(s) > 0, we were only able to obtain an upper bound in a limited number of cases. It would in particular be very interesting to determine whether it is possible that λsat coincides with λexp, that is, that the correlation length remains constant in the whole high-temperature regime

Summary

Introduction and Results

While this work provides precise criteria to decide whether λsat(s) > 0, we were only able to obtain an upper bound in a limited number of cases It would in particular be very interesting to determine whether it is possible that λsat coincides with λexp, that is, that the correlation length remains constant in the whole high-temperature regime. The inverse correlation length νs is a continuous function of λ at λsat(s) Once this is settled, one should ask more refined questions, including a description of the qualitative behavior of νs(λ) close to λsat(s), to what was done in [19] in a case where a similar saturation phenomenon was analyzed in the context of a Potts model/FK percolation with a defect line. Provide a necessary and sufficient condition ensuring that λsat(s) > 0 in a direction s ∈ Sd−1 in which ∂U fails to be quasi-isotropic

Some Basic Properties

Ising with positive field

Ising with a positive field

XY model