Quantum phase diagram of disordered hyperbolic Dirac fermions

A family of two-dimensional hyperbolic lattices, realized on a curved space with a constant negative curvature, features gapless Dirac excitations near the half-filling displaying a vanishing density of states therein. This outcome can be established from a simple nearest-neighbor tight-binding model with spin-independent hopping amplitudes. Despite the corresponding Hamiltonian belonging to the orthogonal symmetry class, we find that in the presence of random pointlike on-site charge impurities, two-dimensional hyperbolic Dirac fermions undergo two distinct phase transitions. First they encounter a semimetal-to-metal transition at moderate disorder, followed by the Anderson metal-insulator transition at much stronger disorder. We establish these findings, that are in stark contradistinction to Dirac systems on flat Euclidean space such as graphene, by computing the average and typical density of states on hyperbolic Dirac lattices with open boundary conditions and analyzing the spatial variation of the latter one with increasing strength of disorder.