The crossover between orthogonal and unitary symmetry in small disordered systems: a supersymmetry approach

Abstract

The authors consider the spectral statistics of independent electrons moving at zero temperature in a weakly disordered metallic ring threaded by a magnetic flux. The analysis is based on the supersymmetry method involving both commuting and anticommuting variables. Besides, they consider an ensemble of Gaussian distributed symmetric random matrices (Gaussian orthogonal ensemble) which are perturbed by a small time reversal symmetry breaking contribution. For energies smaller than the inverse diffusion time around the ring Ec, the spectral correlation functions of both models can be represented in terms of supermatrix integrals of identical structure. In conformity with recent numerical results, this implies that the spectral properties of the two models coincide. These matrix integrals are to a large extent universal, i.e. they depend only on two physical parameters: the mean level spacing and a symmetry breaking parameter which is identified as the typical sensitivity of levels to the time reversal symmetry breaking perturbation. The authors parametrize the relevant matrix coset space of the nonlinear sigma -model in a novel way which is particularly convenient for treating models in the crossover between the two symmetry classes. As an example, they present a detailed calculation of the level-level correlation function. The basic formalism, however, applies quite generally and can be used for the investigation of different types of correlation functions and system geometries as well.