Open Momentum Space Method for Hofstadter Butterfly and the Quantized Lorentz Susceptibility
| Autor: | Lian, Biao, Xie, Fang, Bernevig, B. Andrei |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Phys. Rev. B 103, 161405 (2021) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1103/PhysRevB.103.L161405 |
| Popis: | We develop a generic $\mathbf{k}\cdot \mathbf{p}$ open momentum space method for calculating the Hofstadter butterfly of both continuum (Moir\'e) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the Moir\'e model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hofstadter gaps can be understood as edge states on a momentum space boundary, from which one can determine the two integers ($t_\nu,s_\nu$) of a gap $\nu$ satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While $t_\nu$ is known as the Chern number, our theory identifies $s_\nu$ as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility $\gamma_{xy}=eBs_\nu$. Comment: 5+30 pages, 3+4 figures |
| Databáze: | arXiv |
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