Temperature dependent fluctuations in the two-dimensional XY model

TL;DR

This paper investigates how the probability density function of order parameter fluctuations in the 2D XY model depends on temperature, revealing a weak temperature dependence and approaching Gaussian behavior at higher temperatures.

Contribution

It analytically and numerically demonstrates the temperature dependence of the PDF in the 2D XY model's critical phase, resolving conflicting previous results.

Findings

PDF approaches a universal non-Gaussian limit at low T

Weak temperature dependence due to multiple loop graphs

PDF becomes Gaussian above T ~ 4π

Abstract

We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2dXY) model. In the low temperature critical phase of this model, the PDF approaches a universal non-Gaussian limit distribution in the limit T-->0. Our analysis resolves the question of temperature dependence of the PDF in this regime, for which conflicting results have been reported. We show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously-derived graphical expansion. This is confirmed by numerical simulations on two controlled approximations to the 2dXY model: the Harmonic and ``Harmonic XY'' models. The Harmonic model has no Kosterlitz-Thouless-Berezinskii (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ~ 4\pi. Near to that temperature we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.