Some properties of the one-dimensional L\'{e}vy crystal

TL;DR

This paper introduces the one-dimensional Lévy crystal as a potential physical realization of space fractional quantum mechanics, analyzing its dispersion relation and density of states for various parameters.

Contribution

It provides a discretization approach using Grünwald-Letnikov derivatives to define the Lévy crystal and explores its properties relevant for experimental detection.

Findings

Dispersion relation and density of states are derived for the Lévy crystal.

In the limit of small wavenumbers, properties of continuous space SFQM are recovered.

For α approaching 2, the model reduces to the standard tight-binding chain.

Abstract

We introduce and discuss the one-dimensional L\'{e}vy 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 Schr\"{o}dinger equation with the help of shifted Gr\"{u}nwald-Letnikov derivatives delivers a straight-forward route to define the L\'{e}vy 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 wavenumbers 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.