Tunable Bands in 1D 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 fractional calculus to explore a particle in a periodic potential, where the Schrödinger equation is generalized to its fractional form. This framework allows 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 V₀, barrier thickness L, and well width W using an imaginary-time evolution algorithm, and supplement the discrete energy dispersion through Gaussian process regression. Our analysis reveals a qualitative shift in the band structure at q=2, separating into regimes for q>2 and q<2. For q > 2, energy bands undergo an inverting transformation as symmetric minima emerge within the first Brillouin zone, shifting from k=0 toward k=pm 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 response of the ground band scales with fractional order as V₀^-0.28pm0.05L^-0.34pm0.08W^-0.49pm0.06, indicating tunable sensitivity to geometry. For q < 2, the effective mass near k = 0 decreases exponentially with q, yielding a universal effective mass of 0.15pm0.01 as q → 1, demonstrating that the Lévy index serves as a tunable degree of freedom capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics.