Universality of Hofstadter Butterflies on Hyperbolic Lattices.

Motivated by recent realizations of hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. Utilizing large hyperbolic lattices with periodic boundary conditions, we obtain the true bulk spectrum unaffected by boundary states. The butterfly spectrum with large extended gapped regions prevails, and its shape is universally determined by the fundamental tile, while the fractal structure is lost. We explain how these features originate from Landau levels in hyperbolic space and can be verified experimentally.