Quantum chaos with a direction

Abstract

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 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 σ model is derived. Explicit calculation 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 δ-functional contribution for the orthogonal ensemble. The anomalous part means that a finite portion of eigenvalues remains real at any imaginary vector potential.