Hofstadter's butterfly in the hyperbolic plane

Applying a magnetic field to an electron in a two-dimensional crystal lattice produces a spectrum that is widely known as Hofstadter's butterfly. Hofstadter's results however only apply to lattices that are embedded in flat (Euclidean) space. In light of the recent interest in hyperbolic lattices, we re-consider Hofstadter's problem on hyperbolic {p,q} lattices beyond the Euclidean {4,4} square lattice of the original calculation. For this, we implement periodic boundaries in hyperbolic space, eliminating the extensive edge mode contributions that would otherwise cloud the spectrum. We present the resulting magnetic spectra for a variety of {p,q} lattices and observe distinct features of these hyperbolic Hofstadter butterflies such as the loss of fractality and a systematic dependence between the type of lattice and spectral features.