Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
arXiv:2104.13314 · doi:10.1088/1361-648X/ac24c4
Abstract
We explore the hyperbolic band theory under a magnetic field for the first time. Our theory is a general extension of the conventional band theory defined on a Euclidean lattice into the band theory on a general hyperbolic lattice/Riemann surface. Our methods and results can be confirmed experimentally by circuit quantum electrodynamics (cQED), which enables us to create novel materials in a hyperbolic space. To investigate the band structures, we construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood, by which they can be regarded as a global quantum gravity solution detectable in a laboratory. Besides this is the first explicit example of a massless Dirac state on a higher genus surface. Moreover we show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.
7 pages, 3 figures
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- Chern insulator in a hyperbolic lattice
- Higher-order topological insulators in hyperbolic lattices
- Higher-order Topological Hyperbolic Lattices
- Selberg trace formula in hyperbolic band theory
- Hyperbolic band theory through Higgs bundles
- On the hyperbolic Bloch transform
- The Random-Bond Ising Model and its dual in Hyperbolic Spaces
- Algebra of Hyperbolic Band Theory under Magnetic Field
- Bose-Einstein condensation on hyperbolic spaces
- Flat bands and band touching from real-space topology in hyperbolic lattices