Finite-size scaling of correlation ratio and generalized scheme for the probability-changing cluster algorithm

We study the finite-size scaling (FSS) property of the correlation ratio, the ratio of the correlation functions with different distances. It is shown that the correlation ratio is a good estimator to determine the critical point of the second-order transition using the FSS analysis. The correlation ratio is especially useful for the analysis of the Kosterlitz-Thouless (KT) transition. We also present a generalized scheme of the probability-changing cluster algorithm, which has been recently developed by the present authors, based on the FSS property of the correlation ratio. We investigate the two-dimensional spin-$\frac{1}{2}$ quantum $\mathrm{XY}$ model of with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.