1D Anderson model revisited: Band center anomaly for correlated disorder

Abstract

We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for the localization length and the invariant measure of the phase variable. The analytical expressions are complemented by numerical data.

The band-centre anomaly in the 1D Anderson model with correlated disorder

“…In this work we derive analytical results that show how the band-centre anomaly is modified when disorder possesses arbitrary correlations. The general expressions that we obtained allow us to treat the correlations considered in [17] as a particular case and to show how, in general, different kinds of disorder correlations can either enhance or suppress the anomaly at the centre of the energy band. To derive an analytical expression for the localization length in a neighbourhood of the band centre, we relied on the Hamiltonian map formalism [10,19] and we replaced a map for the angle variable with its continuum limit.…”

“…In the literature on the band-centre anomaly, almost no attention has been given to the case of correlated disorder, the main exception being [16] which, however, was focused on specific cases of very short-ranged correlations. Only very recently the effects of correlated disorder on the localization of the band-centre states have begun to be investigated, in the wake of the discovery that the band-centre anomaly can be strongly reduced if the site energies exhibit exponentially decaying, positive correlations [17].…”

Numerical study of the transverse localization of waves in one-dimensional lattices with randomly distributed gain and loss: Effect of disorder correlations

“…In [59], it was shown numerically that the two-point correlation function β i β j exhibits an exponential decay. This exponential decay is not completely smooth due to the presence of the prefactor preceding the exponential term in the two-point correlation function [60]. However, this does not affect the short-range correlated nature of the disorder distribution.…”

“…Short-range correlated random sequences can be generated using several different methods [9,[57][58][59][60]. In this study, we follow the procedure described in [59] and [60].…”

“…Short-range correlated random sequences can be generated using several different methods [9,[57][58][59][60]. In this study, we follow the procedure described in [59] and [60]. We first generate a large number of independent random numbers {η m } uniformly distributed in the interval [−1/2, 1/2].…”