Enhanced Superconductivity in Quasi-periodic Crystals*

It is ubiquitous that superconductivity emerges in materials with incommensurate structure. Here we study superconductivity in a family of one-dimensional incommensurate system with s-wave pairing interaction. The incommensurate potential changes the characteristics of the electronic wave function either to extended, critical or localized states. Through standard Bogoliubov-de Gennes calculations and analytical analysis utilizing Anderson's idea of pairing the time-reversed exact eigenstates, we find that superconductivity is enhanced when the electronic wave functions are critical. The superconducting transition temperature obeys an unconventional power law relation with respect to the pairing interaction. Consequently, there exists a superconducting dome around the localization transition when the amplitude of the incommensurate potential is tuned, or near the mobility edge when the chemical potential is varied. Our results suggest a way to enhance superconducting transition temperature by introducing an incommensurate potential.