THE CROSSOVER BETWEEN ORTHOGONAL AND UNITARY SYMMETRY IN SMALL DISORDERED-SYSTEMS - A SUPERSYMMETRY APPROACH

We consider the spectral statistics of independent electrons moving at zero temperature in a weakly disordered metallic ring threaded by a m agnetic flux. The analysis is based on the supersymmetry method involv ing both commuting and anticommuting variables. Besides, we consider a n ensemble of Gaussian distributed symmetric random matrices (Gaussian orthogonal ensemble) which are perturbed by a small time reversal sym metry breaking contribution. For energies smaller than the inverse dif fusion time around the ring E(c), the spectral correlation functions o f 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. Thes e matrix integrals are to a large extent universal, i.e. they depend o nly on two physical parameters: the mean level spacing and a symmetry breaking parameter which is identified as the typical sensitivity of l evels to the time reversal symmetry breaking perturbation. We parametr ize 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, we present a detailed calculation of the level-level correlation function. The ba sic formalism, however, applies quite generally and can be used for th e investigation of different types of correlation functions and system geometries as well.