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

We test an improved finite-size scaling method for reliably extracting the critical temperature T_BKT of a Berezinskii-Kosterlitz-Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ_s at T_BKT in combination with the Kosterlitz-Nelson relation between the transition temperature and the stiffness, ρ_s(T_BKT) = 2T_BKT/π, we define a size-dependent transition temperature T_BKT(L₁,L₂) based on a pair of system sizes L₁,L₂, e.g., L₂ = 2L₁. 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_BKT(L₁,L₂). 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_BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T_BKT ≈ 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_BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.