QUANTUM CHAOS 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 describe nonequilibrium processes. Eigenenergies of no n-Hermitian Hamiltonians are not necessarily real and a joint probabil ity density function of complex eigenvalues can characterize basic pro perties of the systems. This function is studied using the supersymmet ry technique and a supermatrix sigma model is derived. Explicit calcul ation shows that the density function is drastically different in the cases of orthogonal and unitary ensembles. It is everywhere smooth for the unitary ensemble but has a delta-functional contribution for the orthogonal ensemble. The anomalous part means that a finite portion of eigenvalues remains real at any imaginary vector potential.