Directed Quantum Chaos

Quantum disordered problems with a direction (imaginary vector potential) are discussed and mapped onto a supermatrix σ model. It is argued that the 0D version of the σ 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.