Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection
Discrete fermionic and bosonic models on hyperbolic lattices have attracted broad interest following their experimental realization in metamaterial platforms and the emergence of hyperbolic crystallography. A central ingredient remains largely unexplored: fermions in curved space couple to geometry through the spin connection, as required by general covariance. Here we develop a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices that incorporates this spin-curvature coupling through a discrete spin connection. Starting from the continuous symmetries of the Poincaré disk, we identify the isometry algebra and principal-series spectral basis of the continuum Dirac theory. For massless Dirac fermions, the continuum theory yields a finite zero-energy density of states (DOS) at any finite curvature in $D-$dimensional hyperbolic space for $2\leq D\leq4$, implying enhanced susceptibility to interaction-driven instabilities at weak coupling. We then derive the discrete rotational and translational symmetries of $\{p,q\}$ hyperbolic lattices and construct the corresponding lattice spin-connection factor. For finite open $\{10,3\}$ lattices, we implement the geodesic Wilson-line phase as a spin-dependent nearest-neighbor hopping. Numerical calculations employing kernel-polynomial method reveal a robust low-energy DOS enhancement that persists across the studied system sizes at fixed numerical resolution. At fixed lattice geometry, this behavior provides qualitative lattice-level support for the continuum prediction of a nonvanishing low-energy DOS. Our results establish a symmetry-based framework for spin-curvature coupling on hyperbolic lattices and motivate further studies of correlated Dirac phases and experimental spin-curvature effects in metamaterial platforms.