Energy bands and Wannier functions of the fractional Kronig-Penney model

• Main Author: Vellasco-Gomes, Arianne [UNESP]
• Publication Date: 2020
• Other Authors: de Figueiredo Camargo, Rubens [UNESP], Bruno-Alfonso, Alexys [UNESP]
• Format: Article
• Language: eng
• Source: Repositório Institucional da UNESP
• Download full: http://dx.doi.org/10.1016/j.amc.2020.125266; http://hdl.handle.net/11449/201692
• topic: asymptotic behavior; Fractional Schrödinger equation; Riesz fractional derivative; Symmetry; Wannier function

Energy bands and Wannier functions of the fractional Schrödinger equation with a periodic potential are calculated. The kinetic energy contains a Riesz derivative of order α, with 1 < α ≤ 2, and numerical results are obtained for the Kronig-Penney model. Bloch and Wannier functions show cusps in real space that become sharper as α decreases. Energy bands and Bloch functions are smooth in reciprocal space, except at the Γ point. Depending on symmetry, each Wannier function decays as a power-law with exponent −(α+1) or −(α+2). Closed forms of their asymptotic behaviors are given. Each higher band displays anomalous behavior as a function of potential strength. It first narrows, becoming almost flat, then widens, with its width tending to a constant. The position uncertainty of each Wannier function follows a similar trend.