Directed quantum chaos

Quantum disordered problems with a direction (imaginary vector potential) are discussed and mapped onto a supermatrix sigma model. It is argued that the OD version of the sigma model may describe a broad class of phenomena that can be called directed quantum chaos. It is demonstrated by explicit calculations that these problems are equivalent to those of random asymmetric or non-Hermitian matrices. A joint probability of complex eigenvalues is obtained. The fraction of states with real eigenvalues proves to be always finite for time reversal invariant systems.