Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals

The Majorana edge states of the Kitaev chain model have attracted extensive attention on their stability and experimental realization. One of the interesting aspects is finding the exotic proximity effect, which guarantees the presence of the Majorana modes, further enables efficient braidings between them. In this paper, we explore the superconducting proximity effect for quasi-periodic quantum wires and discuss how quasi-periodic patterns affect the stability of the Majorana modes. Considering the Kitaev chain model of the one-dimensional quasi-periodic system, we discuss the pattern-dependent proximity effects. First, we argue that the presence of quasi-periodic hoppings energetically induces the $p$-wave pairing also to be quasi-periodic rather than uniform pairing. More interestingly, when the normal metallic wire is adjacent to the quasiperiodic superconducting wire, we have found that the Majorana edge modes are being transferred to the edge of the normal metallic side with enhanced stability. Finally, we discover the proximity effect on the strengths of the quasi-periodicities with a general power-law relationship, whose power depends on the tiling pattern. Our results show how quasi-periodic patterns play an important role in the Kitaev chain and the stabilization of the Majorana mode.