Exactness of the Replica Method and Critical Renormalization on a Generalized Gaussian Model with Randomness

Abstract

A two-dimensional Ising model with quenched bond disorder is mapped to a generalized Gaussian model (GGM) with randomness. In a perturbation expansion theory on the GGM with respect to a suitably chosen virtual regular system, it is shown that the replica method yields the exact result, and that critical divergences appearing spuriously in each term in the expansion are renormalized with certain self-consistent equations to give a finite value of the internal energy when the corresponding regular model exhibits a phase transition. The method used here is a generalization of the coherent potential approximation (CPA), and is applicable to a variety of random models to which corresponding regular models are soluble.