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Bloch and Bethe ansatze for the Harper model: A butterfly with a boundary
- Source :
- Phys. Rev. B 104, 165140 (2021)
- Publication Year :
- 2021
-
Abstract
- Based on a recent generalization of Bloch's theorem, we present a Bloch ansatz for the Harper model with an arbitrary rational magnetic flux in various geometries, and solve the associated ansatz equations analytically. In the case of a cylinder and a particular boundary condition, we find that the energy spectrum of edge states has no dependence on the length of the cylinder, which allows us to construct a quasi-one-dimensional edge theory that is exact and describes two edges simultaneously. We prove that energies of bulk states, generating the so-called Hofstadter's butterfly, depend on a single geometry-dependent spectral parameter and have exactly the same functional form for the cylinder and the torus with general twisted boundary conditions, and argue that the (edge) bulk spectrum of a semi-infinite cylinder in an irrational magnetic field is (the complement of) a Cantor set. Finally, realizing that the bulk projection of the Harper Hamiltonian is a linear form over a deformed Weyl algebra, we introduce a Bethe ansatz valid for both cylinder and torus geometries.<br />Comment: 13+3 pages, 2+1 figures
- Subjects :
- Condensed Matter - Mesoscale and Nanoscale Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- Phys. Rev. B 104, 165140 (2021)
- Publication Type :
- Report
- Accession number :
- edsarx.2107.10393
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1103/PhysRevB.104.165140