# Logarithmic Corrections in the 2D XY Model

## TL;DR

This study investigates the finite-size scaling behavior of the 2D XY model near the Kosterlitz-Thouless transition, focusing on logarithmic corrections, and finds results consistent with recent theoretical and numerical analyses.

## Contribution

It provides high-precision Monte Carlo analysis of logarithmic corrections in the XY model, clarifying the value of the correction exponent r near criticality.

## Key findings

- Estimate of r = -0.0270(10) at criticality
- Estimate of r = 0.0560(17) in the high-temperature phase
- Agreement with recent Monte Carlo and series expansion results

## Abstract

Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square $L \times L$ lattices, the scaling behavior of the susceptibility $\chi$ and correlation length $\xi$ at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections $(ln L)^{-2r}$ in the finite-size scaling region and $(ln \xi)^{-2r}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to $512^2$ we obtain $r = -0.0270(10)$, in agreement with recent work of Kenna and Irving on the the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to $\xi \approx 140$ in the high-temperature phase, however, we arrive at quite a different estimate of $r = 0.0560(17)$, which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.