Fate of localization in a coupled free chain and a disordered chain

It has been widely accepted that almost all states in one-dimensional 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 localization-delocalization transition. In particular, while states in the overlapped regime are localized by resonant coupling, in the un-overlapped regime of the free chain the localization length is extremely large, which can be beyond the capability of state-of-art numerical methods, due to the prefactor ${t}_{\text{v}}^{4}/{\mathrm{\ensuremath{\Delta}}}^{4}$, where ${t}_{\text{v}}$ is the interchain coupling strength and $\mathrm{\ensuremath{\Delta}}$ is the band center offset between them. We confirm these results using the transfer-matrix method and sparse-matrix method for systems $L\ensuremath{\approx}{10}^{6}\text{--}{10}^{9}$. 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.