Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.18574 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915634724470784 |
|---|---|
| author | Guyette, Brenden R. Lewis, Joshua M. Carr, Lincoln D. |
| author_facet | Guyette, Brenden R. Lewis, Joshua M. Carr, Lincoln D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tunable Bands in 1D Fractional Quantum Media Guyette, Brenden R. Lewis, Joshua M. Carr, Lincoln D. Quantum Physics 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. |
| title | Tunable Bands in 1D Fractional Quantum Media |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2511.18574 |