Interplay of dimerization and quasiperiodicity in the superconducting proximity effect of a one-dimensional hybrid ring

Recent studies of the superconducting proximity effect in quasicrystalline and topological systems have opened up a new research direction for exploring how quasiperiodicity and topology influence proximity induced superconductivity. In this work, we investigate spatial variation of the proximity induced pairing amplitude in a hybrid ring composed of a spin singlet superconductor and a normal region described by three different lattice models using the self-consistent Bogoliubov-de Gennes formalism. We first consider the normal region described by the diagonal Aubry-André-Harper (AAH) model. Increasing the quasiperiodic potential enhances spatial fluctuations in the induced order parameter, while progressively suppressing its magnitude in the normal region. Beyond the localization transition, proximity-induced pairing is strongly diminished due to the localized nature of the underlying electronic states. The normal region is then modeled by the Su-Schrieffer-Heeger (SSH) chain to investigate the effect of hopping dimerization. Weak dimerization introduces oscillatory modulations in the induced pairing that extend deep into the normal region, whereas strong dimerization confines these oscillations and significantly reduces the penetration of superconducting correlations. Finally, we study the combined SSH-AAH model to explore the interplay between the dimerized hopping and quasiperiodicity. The results show that the SSH dimerization determines the oscillatory behavior and penetration of the induced pairing, while the AAH potential enhances spatial inhomogeneity and further reduces its magnitude. Together, these two effects provide a versatile means of controlling proximity-induced superconductivity in quasiperiodic hybrid systems.