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 transfer matrix technique or describe non-equilibrium 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 $\sigma$-model is derived. The $\sigma$-model differs from already known by a new term. The zero-dimensional version of the $\sigma$-model turns out to be the same as that 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 $0D$ 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 $\delta$-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.