# Global Delocalization Transition in the de Moura-Lyra Model

## TL;DR

This paper provides numerical and analytical evidence for a delocalization transition at alpha=1 in the 1D de Moura-Lyra model, showing the localization length diverges and clarifying the nature of localization in correlated disordered systems.

## Contribution

First numerical demonstration of a delocalization transition at alpha=1 in the de Moura-Lyra model, with a detailed analysis of finite-size effects and correlation impacts.

## Key findings

- Localization length diverges as (1-alpha)^{-1} at alpha=1
- Finite-size scaling affected by slow convergence of correlations
- Localization effectively caused by uncorrelated Anderson model at small scales

## Abstract

The possibility of having a delocalization transition in the 1D de Moura-Lyra class of models (having a power-spectrum $\propto q^{-\alpha})$ has been the object of a long standing discussion in the literature, filled with ambiguities. In this letter, we report the first numerical evidences that such a transition happens at $\alpha=1$, where the localization length (measured from the scaling of the conductance) is shown to diverge as $(1-\alpha)^{-1}$. 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. This last conclusion leads to the re-interpretation of the localization in these models to be caused by an effective Anderson uncorrelated model at small length-scales. Finally, the numerical results are confirmed by analytical perturbative calculations which are built on previous work.