Tunable Bands in 1D Fractional Quantum Media

TL;DR

This paper extends fractional calculus to quantum systems, revealing how the Lévý index q controls energy band formation and inversion in a periodic potential, enabling tunable quantum behaviors and device functionalities.

Contribution

It introduces a fractional Schrödinger equation framework for periodic potentials, demonstrating how tuning q influences band structure, effective mass, and potential valleytronic applications.

Findings

Band inversion occurs at q=2, with symmetric minima emerging in the first Brillouin zone.

Effective mass near k=0 decreases exponentially with q, approaching a universal value.

Ground band response scales with potential height and geometry, indicating tunable sensitivity.

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 Schr\"odinger equation is generalized to its fractional form. This framework allows us to study how the L\'evy 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 Schr\"odinger 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 $k=\pm \pi/a$ 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 L\'evy index serves as a tunable degree of freedom capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics.