Global partial density of states: Statistics and localization length in quasi-one-dimensional disordered wires

We study numerically the behavior of the distributions functions for diagonal and off-diagonal elements of the global partial density of states DOS in quasi-one-dimensional Q1D disordered wires as a function of disorder parameter from metal to insulator. We consider two different models for disordered Q1D wire: a set of two-dimensional N scatterers of potentials with arbitrary signs and strengths placed randomly and a tightbinding Hamiltonian with several modes M and on-site disorder. We show that the variances of global partial DOS in the metal to insulator crossover regime are crossing. The critical value of disorder wc, where we have crossover for given numbers of N scatterers and for modes M, can be used for calculating a localization length in Q1D systems. The matrix elements of Green’s function of Dyson’s equation in Q1D wires for the two models are calculated analytically. It is shown that the Q1D problem can be mapped to the 1D problem and that the poles of the Green’s function matrix elements, as well as the scattering matrix elements, are a determinant of rank N N, where N is the number of scatterers. It is shown that the determinant can be used to calculate the spectrum of an electron in the disordered Q1D wire, the DOS, the scattering matrix elements, etc., without determining the exact electron wave function.