One-dimensional L{\'e}vy Quasicrystal

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by L{\'e}vy flights. We introduce L{\'e}vy quasicrystal by discretizing the space-fractional Schr$\ddot{\text{o}}$dinger equation using the Gr$\ddot{\text{u}}$nwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schr$\ddot{\text{o}}$dinger equation maps to the Aubry-Andr{\'e} Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between L{\'e}vy quasicrystal and the Aubry-Andr{\'e} (AA) model with power-law hopping and show that the L{\'e}vy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.