Two-size probability-changing cluster algorithm

We propose a self-adapted Monte Carlo approach to automatically determine the critical temperature by simulating two systems with different sizes at the same temperature. The temperature is increased or decreased by checking the short-time average of the correlation ratios of the two system sizes. The critical temperature is achieved using the negative feedback mechanism, which can be regarded as an Ehrenfest model for diffusion with a central force. Moreover, the thermal average near the critical temperature can be calculated precisely. The proposed approach is a general method to treat second-order phase transition, first-order phase transition, and Berezinskii–Kosterlitz–Thouless transition on the equal footing. Keywords: first-order phase transition, second-order phase transition, Berezin skii–Kosterlitz–Thouless transition, cluster Monte Carlo algorithm