Glassy Behavior in the Ferromagnetic Ising Model on a Cayley Tree

Abstract

We present a detailed study of the nearest-neighbor ferromagnetic Ising model on a Cayley tree. In the limit of zero field, the system displays glassy behavior below a crossover temperature, $T_g$, that scales inversely with the logarithm of the number of generations; thus $T_g$ is inversely proportional to the logarithm of the logarithm of the number of sites. Non-Gaussian magnetization distributions are observed for $T < T_g$, reminiscent of that associated with the central spin of the Edwards-Anderson model on the same tree; furthermore a dynamical study indicates metastability, long relaxation times and ageing consistent with the development of glassy behavior for a finite but macroscopic number of sites.