  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.
