Probability-Changing Cluster Algorithm for Two-Dimensional XY and Clock Models

We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff’s idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) XY and $q$-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\xi \propto \exp (c\mathrm{/}\sqrt{T\mathrm{/}{T}_{\mathrm{K}\mathrm{T}}-1})$, we determine the KT transition temperature and the decay exponent $\eta$ as $T_{\mathrm{K}\mathrm{T}}=0.8933(6)$ and $\eta =0.243(5)$ for the 2D XY model. We investigate two transitions of the KT type for the 2D $q$-state clock models with $q=6,8,12$, and {\it for the first time} confirm the prediction of $\eta =4\mathrm{/}{q}^2$ at $T_1$, the low-temperature critical point between the ordered and XY-like phases, systematically.