Fate of localization in coupled free chain and disordered chain

TL;DR

This paper investigates how coupling a disordered chain with a free chain affects localization, revealing regimes where localization is suppressed or maintained, challenging traditional beliefs about 1D disordered systems.

Contribution

It demonstrates that coupling between disordered and free chains leads to distinct localization behaviors without a phase transition, extending understanding of localization phenomena.

Findings

Localization length can be comparable to system size despite strong disorder.

Resonant coupling localizes states in the overlapped regime.

Suppression of localization in the un-overlapped regime depends on coupling strength and energy shift.

Abstract

It has been widely believed that almost all states in one-dimensional (1d) disordered systems with short-range hopping and uncorrelated random potential are localized. Here, we consider the fate of these localized states by coupling between a disordered chain (with localized states) and a free chain (with extended states), showing that states in the overlapped and un-overlapped regimes exhibit totally different localization behaviors, which is not a phase transition process. In particular, while states in the overlapped regime are localized by resonant coupling, in the un-overlapped regime of the free chain, significant suppression of the localization with a prefactor of $\xi^{-1} \propto t_v^4/\Delta^4$ appeared, where $t_v$ is the inter-chain coupling strength and $\Delta$ is the energy shift between them. This system may exhibit localization lengths that are comparable with the system size even when the potential in the disordered chain is strong. We confirm these results using the transfer matrix method and sparse matrix method for systems $L \sim 10^6 - 10^9$. These findings extend our understanding of localization in low-dimensional disordered systems and provide a concrete example, which may call for much more advanced numerical methods in high-dimensional models.