Replica symmetry and massless modes in the Ising spin glass

Abstract

The n to 0 replica method is used to study the Ising spin glass below its transition temperature. The order parameter correlation functions are shown to contain massless modes ('replicon' modes) within mean-field theory. These become unstable at first order in perturbation theory about the mean-field solution. An alternative formulation of perturbation theory, however, yields a family of new solutions which break, in general, the symmetry between the replicas. Stability requirements single out a unique member of this family which exhibits (i) massless moded, and (ii) a restoration of replica symmetry. Within the framework of the new solution, perturbation theory for the order parameter (Qalpha beta ) breaks down for dimensionalities d<or=4 due to an infrared divergence. Four dimensions is interpreted as the 'lower critical dimensionality' of the theory, below which there can be no order of the Edwards-Anderson kind (1975).