Subsystem localization in a two-leg ladder system

We consider a ladder system where one leg, referred to as the ``bath,'' is governed by an Aubry-Andr\'e (AA)-type Hamiltonian, while the other leg, termed the ``subsystem,'' follows a standard tight-binding Hamiltonian. We investigate the localization properties in the subsystem induced by its coupling to the bath. For the coupling strength larger than a critical value (${t}^{\ensuremath{'}}>{t}_c^{\ensuremath{'}}$), the analysis of the static properties shows that there are three distinct phases as the AA potential strength $V$ is varied: a fully delocalized phase at low $V$, a localized phase at intermediate $V$, and a weakly delocalized (fractal) phase at large $V$. The fractal phase also appears in a narrow region along the boundary between the delocalized and localized phases. An analysis of the projected wave-packet dynamics in the subsystem shows that the delocalized phase exhibits a ballistic behavior, whereas the weakly delocalized phase is subdiffusive. Interestingly, the narrow fractal phase shows a super- to subdiffusive behavior as we go from the delocalized to localized phase. When ${t}^{\ensuremath{'}}<{t}_c^{\ensuremath{'}}$, the intermediate localized phase disappears and we find a delocalized (ballistic) phase at low $V$ and a weakly delocalized (subdiffusive) phase at large $V$. Between those two phases, there is also an anomalous crossover regime where the system can be super- or subdiffusive. Beyond the ballistic phase observed at low $V$, we also identify a superdiffusive regime emerging in the limit ${t}^{\ensuremath{'}}/V\ensuremath{\ll}1$, which continuously approaches the ballistic behavior as ${t}^{\ensuremath{'}}\ensuremath{\rightarrow}0$. Finally, in some limiting scenario, we also establish a mapping between our ladder system and a well-studied one-dimensional generalized Aubry-Andr\'e (GAA) model.