One-dimensional Lévy quasicrystal

Abstract

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.