Grassmann variables and supersymmetry in the theory of disordered systems

Abstract

In the theory of disordered systems anticommuting variables are often more convenient for averaging over the randomness than replica. This is illustrated here on the example of a Gaussian ensemble of random matrices ; the averaged resolvent, for instance, may be written this way as a double integral for any size of the matrices. In some other exceptional circumstances it is advantageous to introduce Grassmannian “dimensions”. Thus in certain problems, such as the random field Ising model in the description of Parisi and Sourlas or in the problem of an electron gas in the presence of a strong magnetic field, a symmetry of the interaction between space and superspace allows one to reduce the problem from d to (d-2) ordinary dimensions.