Quantum disordered systems with a direction

Models of disorder with a direction (constant imaginary vector potential) are considered. These non-Hermitian models can appear as a result of computation for models of statistical physics using a transfer-matrix technique, or they can describe nonequilibrium processes. Eigenenergies of non-Hermitian Hamiltonians are not necessarily real, and a joint probability density function of complex eigenvalues can characterize basic properties of the systems. This function is studied using the supersymmetry technique, and a supermatrix σ model is derived. The σ model differs from that already known by a new term. The zero-dimensional version of the σ 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 supermatrix $Q,$ the density of complex eigenvalues is calculated in zero dimension for both the unitary and orthogonal ensembles. The function is drastically different in these two cases. It is everywhere smooth for the unitary ensemble but has a δ-functional contribution for the orthogonal one. This anomalous part means that a finite portion of eigenvalues remains real at any degree of the non-Hermiticity. All details of the calculations are presented.