Replica Symmetry Breaking in the Random Field Ising Model

We discuss the critical behavior of the random field Ising model, using techniques of replica symmetry breaking familiar from the theory of spin glasses and random manifolds. Using an approximation that is valid in the limit of a large number of spin components, m, we find that the replica symmetric solution is unstable and obtain equations for solutions which break replica symmetry according to a "natural" scaling ansatz. Although we are unable to solve these equations exactly, we show that they lead to exponents which are different to order 1/m from the replica symmetric solution.