Tunable Bands in 1D Fractional Quantum Media

Fractional calculus [1, 2] has proven useful in geophysics [3], optics [4, 5], and biological systems [6, 7] to more accurately describe system phenomena. In this talk, we extend its foundation in Lévy paths to a particle subject to a periodic potential, where the Schrödinger equation (SE) is generalized to the fractional Schrödinger equation (FSE) [1, 2]. This allows us to characterize how the Lévy index, q, of a system influences energy bands and infer potential engineering and technological applications of varying the additional degree of freedom q in periodic single-particle-well systems. We solve the FSE for a range of periodic rectangular potentials, varying the potential height V₀, potential thickness L, and well width W, with an imaginary time evolution algorithm [8]. We then used a Gaussian process regression (GPR) [9] to fit the energy dispersion. Two distinct regions emerge for q > 2 and q < 2. For q > 2, we observe energy band inversion characterized by symmetric minima emerging in the first Brillouin zone, shifting from k = 0 toward k = ± πa as q increases. The emergent symmetric minima are degenerate in energy, giving rise to a Bloch-momentum qubit. Valleytronics is the concept of using valleys in band structure as a degree of freedom for computation [10]. We apply this to the Bloch-momentum qubit to assert that the valleys, where the Bloch-momentum qubit resides, can be modified to control the qubit. The ground band’s sensitivity to inversion, for increasing q, scales approximately as V^{−0.28±0.05}L^{−0.34±0.08}W^{−0.49±0.06}. For q < 2, the effective mass at k = 0 decreases exponentially with fractional order, with the exponent depending on q, V₀, L, and W. This dependence becomes more dominated by q as q → 1, leading to the effective mass being ≈ 0.15 ± 0.01 with a maximum variance of 0.06 at q = 1 for all tested potentials. Thus, using the L´evy index as a degree of freedom for a quantum particle in a periodic potential opens up opportunities for tuning the energy band to induce a Bloch-momentum qubit, varying the nature of the band gap, and adjusting the effective mass around k = 0.