Parallel spin wave for the Villain model

TL;DR

This paper establishes optimal bounds for the decay of parallel spin correlations in the Villain model on high-dimensional lattices at low temperature, advancing understanding of phase behavior in such spin systems.

Contribution

It provides the first sharp bounds for correlation decay in the Villain model in three dimensions, improving previous results and employing novel analytical techniques.

Findings

Proved optimal upper and lower bounds for correlation decay in 3D Villain model.

Improved bounds on correlation decay in general dimensions.

Connected Villain model analysis to vector-valued interface models using duality and renormalisation.

Abstract

In this paper, we study the Villain model in $\mathbb{Z}^d$ in dimension $d\geq 3$. It is conjectured, that the parallel correlation function in the infinite volume Gibbs state, i.e., the map $$ x \mapsto \langle \cos\theta(0) \cos\theta(x) \rangle_{\mu_{\mathrm{Vil}, \beta}} -\left( \langle \cos\theta(0) \rangle_{\mu_{\mathrm{Vil}, \beta}} \right)^2, $$ decays like $|x|^{-2(d-2)}$ as $|x| \to \infty$ at low temperature. The results of Bricmont, Fontaine, Lebowitz, Lieb, and Spencer (1981) show that for the related XY model, this correlation decays at least as fast as $|x|^{2-d}$. We prove the optimal upper and lower bounds for the Villain model in $d=3$, up to a logarithmic correction, and also improve the upper bound in general dimensions. Our proof builds upon the approach developed in our previous article, which in turn is inspired by a key observation of Fr\"{o}hlich and Spencer (1982): in the low temperature regime, a combination of duality transformation and renormalisation allows certain properties of the Villain model to be analysed in terms of a (vector-valued) $\nabla \varphi$ interface model. This latter model can be investigated using the Helffer-Sj\"{o}strand representation formula combined with tools of elliptic and parabolic regularity.