QUANTUM DISORDERED-SYSTEMS WITH A DIRECTION

Models of disorder with a direction (constant imaginary vector potenti al) are considered. These non-Hermitian models can appear as a result of computation for models of statistical physics using a transfer-matr ix technique, or they can describe nonequilibrium processes. Eigenener gies of non-Hermitian Hamiltonians are not necessarily real, and a joi nt probability density function of complex eigenvalues can characteriz e basic properties of the systems. This function is studied using the supersymmetry technique, and a supermatrix sigma model is derived. The sigma model differs from that already known by a new term. The zero-d imensional version of the sigma model turns out to be the same as the one obtained recently,for ensembles of random weakly non-Hermitian or asymmetric real matrices. Using a new parametrization for the supermat rix Q, the density of complex eigenvalues is calculated in zero dimens ion for both the unitary and orthogonal ensembles. The function is dra stically different in these two cases. It is everywhere smooth for the unitary ensemble but has a delta-functional contribution for the orth ogonal one. This anomalous part means that a finite portion of eigenva lues remains real at any degree of the non-Hermiticity. All details of the calculations are presented.