Analytical computation of the magnetization probability density function for the harmonic 2D XY model

TL;DR

This paper analytically derives the probability density function of magnetization in the 2D harmonic XY model, confirming previous numerical and perturbative findings and revealing a transition from Gumbel to Gaussian distribution with temperature.

Contribution

It provides the first explicit analytical computation of the magnetization PDF in the harmonic 2D XY model, validating previous numerical and perturbative results.

Findings

First moments of the PDF match analytical predictions.

Distribution approaches Gaussian at high temperature and large volume.

Numerical evaluation at low temperature aligns with Gumbel distribution.

Abstract

The probability density function (PDF) of some global average quantity plays a fundamental role in critical and highly correlated systems. We explicitly compute this quantity as a function of the magnetization for the two dimensional XY model in its harmonic approximation. Numerical simulations and perturbative results have shown a Gumbel-like shape of the PDF, in spite of the fact that the average magnetization is not an extreme variable. Our analytical result allows to test both perturbative analytical expansions and also numerical computations performed previously. Perfect agreement is found for the first moments of the PDF. Also for large volume and in the high temperature limit the distribution becomes Gaussian, as it should be. In the low temperature regime its numerical evaluation is compatible with a Gumbel distribution.