Nonperturbative aspects of supersymmetry in statistical mechanics

Abstract

It was shown by Parisi and Sourlas that a scalar field theory coupled to a random source corresponds to a certain kind of supersymmetric theory, and that, furthermore, the correlation functions, evaluated in perturbation theory, are those of the original field theory, without the random source, in two fewer dimensions. We give a new derivation of the first result, based on the so-called replica method, which more clearly exposes the assumptions involved. We also show that the second result holds nonperturbatively. Two applications of this nonperturbative result are discussed. The first shows how topological instanton effects restore the symmetry in the three-dimensional continuous spin Ising model in a random magnetic field. These are the analog of the kinds in the one dimensional field theory. The second application is to the problem of the singularities in a randomly diluted ferromagnet. Fifteen years ago, Griffiths showed the free energy of a magnet diluted with nonmagnetic impurities is a nonanalytic function of the external magnetic field at temperatures below the critical temperature of the pure system. We show that this system can be modelled by a supersymmetric cubic scalar field theory with a random imaginary field. The equivalent ( d - 2)-dimensional theory has instantons, which are the nonperturbative effects responsible for the Griffiths' singularities.