Gaussian fluctuations for the classical XY model

TL;DR

This paper proves that in the classical XY model with Dirichlet boundary conditions, the fluctuation field converges to Gaussian white noise as temperature approaches zero faster than a specific rate, extending to similar gradient models.

Contribution

It establishes the Gaussian fluctuation limit for the XY model at low temperatures, providing new insights into the model's behavior in the zero-temperature limit.

Findings

Fluctuation field converges to Gaussian white noise at low temperatures.

Results apply to a broad class of gradient field models.

Convergence occurs when temperature decreases faster than a certain rate.

Abstract

We study the classical XY model in bounded domains of $\mathbb{Z}^{d}$ with Dirichlet boundary conditions. We prove that when the temperature goes to zero faster than a certain rate as the lattice spacing goes to zero, the fluctuation field converges to a standard Gaussian white noise. This and related results also apply to a large class of gradient field models.