Statics and dynamics of the two-spin–facilitated kinetic Ising model

Abstract

The two-spin–facilitated kinetic Ising model, or Fredrickson–Andersen model, in two dimensions is investigated, where down-spins are energetically favored over up-spins due to a suitably oriented magnetic field and the only interaction between spins is a dynamic constraint which allows any spin to flip only if it has at least two nearest up-spin neighbors. From Monte Carlo simulations it is known that the model exhibits glasslike behavior where the autocorrelation time of a spin as a function of temperature is well described by the Adam–Gibbs expression. Due to the dynamic constraint, the state space is divided into nonconnected partitions the most important of which is the high temperature partition which includes the all-spins-up state. A proof given earlier by Fredrickson and Andersen is completed, which states that for infinitly large lattices and for a temperature larger than zero, the probability that a randomly picked state is part of the high temperature partition approaches unity. It is also shown that the system is ergodic, i.e., that no glass transition may occur at a temperature larger than zero. These results hold also for the model in higher dimensions than two and for the three-spin–facilitated kinetic Ising model in three dimensions. It is suggested that the model equilibrates by cooperative diffusion of a critical droplet of up-spins.