Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection

Abstract

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 Poincar\'e 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.