Relaxation to equilibrium of single cluster dynamics of the Ising model

The approach to equilibrium at the critical temperature of the two-dimensional Ising model from an initial ordered spin configuration is investigated using the single cluster Monte Carlo dynamics introduced by Wolff. An initial configuration of all up spins and its dual, a checkerboard state, are considered. We give simple mean-field and scaling arguments for the existence of power-law decay. Because the usual arguments for time rescaling might not be applicable when the system is approaching equilibrium, we directly determine the relation of the number of cluster flips to the number of Monte Carlo steps per spin. For both initial configurations we find that the effective dynamical critical exponent determined from the power-law behavior and the proper time rescaling differs from the estimates of the equilibrium dynamical exponent and depends on the initial conditions for the system sizes and times we have investigated.