Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface

• URL: http://arxiv.org/abs/2104.13314v4
• Date: Fri, 23 Jul 2021 16:34:02 GMT
• Title: Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
• Authors: Kazuki Ikeda, Shoto Aoki, Yoshiyuki Matsuki
• Abstract summary: We explore the hyperbolic band theory under a magnetic field for the first time. We construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood. We also show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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