Global phase diagram of two-dimensional dirty hyperbolic Dirac liquids

Within the framework of the canonical nearest-neighbor tight-binding model for spinless fermions, a family of two-dimensional bipartite hyperbolic lattices hosts massless Diraclike excitations near half-filling with the iconic vanishing density of states (DOS) near zero energy. We show that a collection of such ballistic quasiparticles remains stable against sufficiently weak pointlike charge impurities, a feature captured by the vanishing average [$\rho_{a}(0)$] and typical [$\rho_{t}(0)$] DOS at zero energy, computed by employing the kernel polynomial method in sufficiently large $\{ 10, 3\}$ hyperbolic lattices (Schl\"afli symbol) with more than $10^8$ and $10^5$ sites, respectively, with open boundary conditions. However, at moderate disorder the system enters a metallic state via a continuous quantum phase transition where both $\rho_{a}(0)$ and $\rho_{t}(0)$ become finite. With increasing strength of disorder, ultimately an Anderson insulator sets in, where only $\rho_{t}(0) \to 0$. The resulting phase diagram for dirty Dirac fermions living on a hyperbolic space solely stems from the background negative spatial curvature, as confirmed from the vanishing $\rho_{t}(0)$ for arbitrarily weak disorder on honeycomb lattices, fostering relativistic fermions on a flatland, as the thermodynamic limit is approached.