Tunable bands in one-dimensional fractional quantum media.

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 the success of fractional calculus in physical models to explore a particle in a periodic potential, where the Schrödinger equation is extended to its fractional form. This framework enables us to study how the Lévy index q governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors and device functionalities by tuning q in periodic quantum systems. We solve the fractional Schrödinger equation for periodic rectangular potentials of varying height V_{0}, barrier thickness L, and well width W using an imaginary-time evolution algorithm, and supplement the discrete energy dispersion through Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at q=2, separating into distinct regimes of dispersion that define behavior for q>2 and q<2. For q>2, the energy bands undergo an inverting transformation as symmetric minima emerge within the first Brillouin zone and shift from k=0 toward k=±π/a with increasing q. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom used in valleytronics. The q at which the inversion completes scales as q∝V_{0}^{-0.28±0.05}, q∝L^{-0.35±0.08}, and q∝W^{-0.49±0.06} when varying potential parameters individually, indicating a tunable transformation sensitivity to potential geometry. In contrast, for q<2, the ground band hardens around k=0, with a dispersion following the functional form of 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 Lévy 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.