Tunable Bands in 1D Fractional Quantum Media

Abstract

Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index $q$ governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning $q$ in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height $V_0$, barrier thickness $L$, and well width $W$ using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at $q=2$, separating into distinct regimes of dispersion for $q>2$ and $q<2$. For $q>2$, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from $k=0$ toward $k=\pm \pi /a$ with increasing $q$. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The $q$ at which inversion completes scales as $q\propto V_0^{-0.28\pm 0.05}$, $q\propto L^{-0.35\pm 0.08}$, and $q\propto W^{-0.49\pm 0.06}$ when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for $q<2$, the ground band hardens around $k=0$, with a dispersion following $C|k{|}^q+E_0$ near $k=0$. This suggests an effective mass of 0 for $1<q<2$ at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.

References in corpus (6)

• The scaling laws of human travel
• Fractional Quantum Mechanics and Levy Path Integrals
• Lévy walks
• Fractional Schrödinger equation in optics
• $\mathcal{P}\mathcal{T}$ symmetry in a fractional Schrödinger equation
• Existence of Mexican-hat dispersion and symmetry group of a layer