1997/06/06 by K. B. Efetov · 4 citations
Mathematics · Physics and Astronomy · #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectroscopy and Quantum Chemical Studies #cond-mat.dis-nn #cond-mat.mes-hall
paper · pdf · doi:10.1103/physrevb.56.9630
37 pages, Latex, the manuscript has been submitted to Phys.Rev.B on April 10
arxiv created 1997/06/06 · openalex publication_date 1997/10/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 σ-model is derived. The σ-model differs from already known by a new term. The zero-dimensional version of the σ-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 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.