Temperature-dependent fluctuations in the two-dimensional XY model

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-Berezinskiĭ (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ap 4π. Near to that temperature, we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.