Finite-size scaling method for the Berezinskii-Kosterlitz-Thouless transition

TL;DR

This paper introduces an improved finite-size scaling method using logarithmic corrections to accurately determine the BKT transition temperature in the 2D XY model, highlighting the importance of sub-leading corrections.

Contribution

The authors develop a reliable finite-size scaling approach incorporating sub-leading logarithmic corrections for precise BKT transition temperature estimation.

Findings

The method yields a transition temperature of 0.8935(1), higher than previous estimates.

Sub-leading logarithmic corrections significantly influence the extrapolation accuracy.

GPU computing enables high-precision data for large system sizes up to L=512.

Abstract

We test an improved finite-size scaling method for reliably extracting the critical temperature $T_{\rm BKT}$ of a Berezinskii-Kosterlitz-Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness $\rho_s$ at $T_{\rm BKT}$ in combination with the Kosterlitz-Nelson relation between the transition temperature and the stiffness, $\rho_s(T_{\rm BKT})=2T_{\rm BKT}/\pi$, we define a size dependent transition temperature $T_{\rm BKT}(L_1,L_2)$ based on a pair of system sizes $L_1,L_2$, e.g., $L_2=2L_1$. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining $T_{\rm BKT}(L_1,L_2)$. For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for $L$ up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result $T_{\rm BKT}=0.8935(1)$ is several error bars above the previously best estimates of the transition temperature; $T_{\rm BKT} \approx 0.8929$. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated $T_{\rm BKT}$ because of neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.