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arXiv · 2210.10772

One-dimensional L{\'e}vy Quasicrystal

Pallabi Chatterjee·Ranjan Modak
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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.

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BibTeXRIS

Pallabi Chatterjee, Ranjan Modak. 2022-10-19. One-dimensional L{\'e}vy Quasicrystal. https://doi.org/10.1088/1361-648x%2Facf9d4

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