Global delocalization transition in the de Moura–Lyra model

Abstract

The possibility of having a delocalization transition in the 1D de Moura-Lyra class of models (having a power-spectrum ∝ q −α ) has been the object of a long standing discussion in the literature. In this paper, we report the first numerical evidences that such a transition happens at α = 1, where the localization length (measured from the scaling of the conductance) is shown to diverge as (1 − α) −1 . The persistent finite-size scaling of the data is shown to be caused by a very slow convergence of the nearest-neighbor correlator to its infinite-size limit, and controlled by the choice of a proper scaling parameter. Our results for these models are consistent with a localization of eigenstates that is driven by a persistent small-scale noise, which vanishes as α → 1 − . This interpretation in confirmed by analytical perturbative calculations which are built on previous work. Finally, the nature of the delocalization transition is discussed and the conclusions are illustrated by numerical work done in the α > 1 regime.

Probing the Global Delocalization Transition in the de Moura-Lyra Model with the Kernel Polynomial Method

“…In this paper, we use the KPM to determine the localization length, based on an old-result by Thouless, which is valid for 1D systems in the thermodynamic limit [6]. This work is a sequel of previous work by Santos Pires et al [12], which presents an independent verification of its results, and also demonstrates the usefulness of the KPM for studying localization properties on finite disordered lattices, even in the presence of strong finite-size effects. An excellent agreement is found between the two methods.…”

“…As shown in Ref. [12], the localization of eigenstates in the de Moura-Lyra model mostly depends on the magnitude of the short-scale noise, rather than the power-law tails of the disorder correlator. The normalized single-bond discontinuity, defined as…”

“…for the de Moura-Lyra model [12], at an energy E. The expression of Eq. 7, can be used to calculate the localization length of the de Moura-Lyra model for any values of the energy and of the correlation controlling parameter α, in limit N → ∞.…”

“…This model was revisited recently [12], having been shown to suffer from rather strong and tricky to handle finite-size effects over the entire parameter region α ∈ [0, ∞[. For α > 1, any finite section of the system becomes ordered in the thermodynamic limit, because the disorder's correlator for any two sites at distance r becomes a function of r/N, where N is the system size [11].…”

“…For α > 1, any finite section of the system becomes ordered in the thermodynamic limit, because the disorder's correlator for any two sites at distance r becomes a function of r/N, where N is the system size [11]. By devising a way to control finite size effects, and suitably take the thermodynamic limit, a delocalization transition of all eigenstates was proved to happen at α = 1 in the thermodynamic limit [12]. This work measured the localization length using the scaling of the typical mesoscopic conductance with system size, as calculated from the Landauer formalism.…”