Beyond the Mean Field Approximation for Spin Glasses

We study the d-dimensional random Ising model using a Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model is mapped into a single Ising model with a random gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions.