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

Abstract

We study the band-centre anomaly in the one-dimensional Anderson model with weak correlated disorder. Our analysis is based on the Hamiltonian map approach; the correspondence between the discrete model and its continuous counterpart is discussed in detail. We obtain analytical expressions of the localisation length and of the invariant measure of the phase variable, valid for energies in a neighbourhood of the band centre. By applying these general results to specific forms of correlated disorder, we show how correlations can enhance or suppress the anomaly at the band centre.

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

“…As shown in [18], a similar effect occurs for disorder with exponentially decaying, positive correlations. One should also add that exponentially decaying correlations with alternating sign can have the opposite effect and enhance the band-center anomaly [18].…”

“…This is due to the fact that 0 E  corresponds to /2 µ π  and for this value of the µ parameter the angle map (10) has almost periodic orbits of period 4, which ultimately lead to the modulation of ( ) ρ θ . The invariant distribution ( ) ρ θ close to the band center can be obtained with the method introduced in [12] for the case of uncorrelated disorder and extended in [18] to the case of correlated disorder. In this approach one first considers the fourth iterate of the map (10) with / 2 µ π  ; the continuum limit is then taken and the map is replaced with a stochastic differential equation for ( ) t θ .…”

“…More recently, the analysis of the band center anomaly has been extended to the case of the 1D Anderson model with correlated disorder [17,18]. In particular, in Ref.…”

“…In particular, in Ref. 18 it was shown how correlations of the disorder can either enhance or suppress the anomaly at the band center.…”

“…20 (see also [4] and references therein). Incidentally, this suppression of the band-center anomaly was the reason that stimulated some of us to conduct the research work published in [18].…”