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We explore the hyperbolic band theory under a magnetic field for the firsttime. Our theory is a general extension of the conventional band theory definedon a Euclidean lattice into the band theory on a general hyperboliclattice/Riemann surface. Our methods and results can be confirmedexperimentally by circuit quantum electrodynamics (cQED), which enables us tocreate novel materials in a hyperbolic space. To investigate the bandstructures, we construct directly the hyperbolic magnetic Bloch states and findthat they form Dirac cones on a coordinate neighborhood, by which they can beregarded as a global quantum gravity solution detectable in a laboratory.Besides this is the first explicit example of a massless Dirac state on ahigher genus surface. Moreover we show that the energy spectrum exhibits anunusual fractal structure refracting the negative curvature, when plotted as afunction of a magnetic flux.

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