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A generalization of the Aubry-Andr\'e model, the non-interacting GPD modelintroduced in S. Ganeshan et al.,[ Phys. Rev. Lett. 114, 146601 (2015)], isknown analytically to possess a mobility edge, allowing both extended andlocalized eigenstates to coexist. This mobility edge has been hypothesized tosurvive in closed many-body interacting systems, giving rise to a newnon-ergodic metallic phase. In this work, coupling the interacting GPD model toa thermal bath, we provide direct numerical evidence for multiple qualitativebehaviors in the parameter space of disorder strength and energy level. Inparticular, we look at the bath-induced saturation of entanglement entropy toclassify three behaviors: thermalized, non-ergodic extended, and localized. Wealso extract the localization length in the localized phase using the long-timedynamics of the entanglement entropy and the spin imbalance. Our workdemonstrates the rich localization landscape of generalized Aubry-Andr\'emodels containing mobility edges in contrast to the simple Aubry-Andr\'e modelwith no mobility edge.

Localization spectrum of a bath-coupled generalized Aubry-Andr\'e model in the presence of interactions