One-dimensional L{é}vy Quasicrystal
arXiv:2210.10772 · doi:10.1088/1361-648X/acf9d4
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{é}vy flights. We introduce L{é}vy quasicrystal by discretizing the space-fractional Schrdinger equation using the Grnwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrdinger equation maps to the Aubry-Andr{é} Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between L{é}vy quasicrystal and the Aubry-Andr{é} (AA) model with power-law hopping and show that the L{é}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.
12 pages, 10 figures
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