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 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 $V_0$, barrier thickness $L$, and well width $W$ using an imaginary-time evolution algorithm, and supplement the discrete energy dispersion through Gaussian process regression. 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 $V_0^{-0.28\pm0.05}L^{-0.34\pm0.08}W^{-0.49\pm0.06}$, indicating tunable sensitivity to geometry. For $q < 2$, the effective mass near $k = 0$ decreases exponentially with $q$, yielding a universal effective mass of $0.15\pm0.01$ as $q \to 1$, 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.

Figures (7)

• Figure 1: Ground-band evolution within the first Brillouin zone as the fractional order $q$ varies from 1.0 to 3.0. For $q < 2$, the band sharpens around $k = 0$, corresponding to a lower effective mass of the ground state. For $q > 2$, the band undergoes inversion: new minima emerge symmetrically about $k = 0$ and migrate toward the zone edge, merging at $k = \pi/a$ as inversion completes.
• Figure 2: Ground band tuning from fractional order 2.0 to 3.0. Two different cases are shown: (a) reference potential and (b) increased well height. Amplifying $V_0$ increases the ground band's sensitivity to inversion monotonically across fractional orders. (a) Referential case, the ground band finishes inversion between orders 2.8 and 2.9. (b) Increased potential height ($V_0$), the ground band finishes inversion around order 2.3.
• Figure 3: Gaussian process regression (GPR) interpolation of the ground band to approximate a continuous $k$-space and capture the shifting of band minima. Points denote the positions of the ground band minima obtained from the GPR, and dashed curves are power-law fits showing how the minima shift with fractional order $q$ for different potential parameters ($V_0$, $L$, $W$). The intersection of each fit with $k=\pm \pi/a$ marks the fractional order at which band inversion is complete.
• Figure 4: Tuning of the ground band minima at fixed fractional order $q > 2$. Varying the potential parameters $V_0$, $L$, and $W$ shifts the location of the minima in $k$-space (only $0 \leq k \leq \pi/a$ is shown). The fitted curves yield the scaling exponents that characterize inversion sensitivity, with the fractional order at which inversion completes scaling as $q \propto V_0^{-0.28\pm0.05}L^{-0.35\pm0.08}W^{-0.49\pm0.06}$.
• Figure 5: Band gap $\Delta (q)$ for three well widths $W$. A kink appears when only the ground band inverts, and its location shifts with $W$. Post-inversion, $\Delta (q)$ shows reduced $q$-sensitivity.