We introduce and discuss the one-dimensional Lévy crystal as a probable candidate for an experimentally accessible realization of space-fractional quantum mechanics (SFQM) in a condensed-matter environment. The discretization of the space-fractional Schrödinger equation with the help of shifted Grünwald-Letnikov derivatives delivers a straightforward route to define the Lévy crystal of order $\alpha \in (1,2]$. As key ingredients for its experimental identification we study the dispersion relation as well as the density of states for arbitrary $\alpha \in (1,2]$. It is demonstrated that in the limit of small wave numbers all interesting properties of continuous-space SFQM are recovered, while for $\alpha \to 2$ the well-established nearest-neighbor one-dimensional tight-binding chain arises.

• Received 8 May 2013

DOI:https://doi.org/10.1103/PhysRevE.88.012120