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

TL;DR

This paper develops a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices, explicitly incorporating spin-curvature coupling, and predicts a finite density of states at zero energy due to curvature effects.

Contribution

It introduces a discrete spin connection into hyperbolic lattice models, linking continuum Dirac theory with lattice symmetries, and provides explicit constructions for various Schl"afli symbols.

Findings

Finite density of states at zero energy for curved hyperbolic spaces.

Explicit construction of discrete spin connection as hopping phases.

Framework enables experimental and numerical exploration of spin-curvature effects.

Abstract

Discrete fermionic and bosonic models for hyperbolic lattices have attracted significant attention across a range of fields since the experimental realization of hyperbolic lattices in metamaterial platforms, sparking the development of hyperbolic crystallography. However, a fundamental and experimentally consequential aspect remains unaddressed: fermions propagating in curved space inherently couple to the underlying geometry via the spin connection, as required by general covariance - a feature not yet incorporated in studies of hyperbolic crystals. Here, we introduce a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices, explicitly incorporating spin-curvature coupling via a discrete spin connection. Starting from the continuous symmetries of the Poincar\'e disk, we classify the irreducible representations and construct a symmetry-adapted basis, establishing a direct correspondence to the continuum Dirac theory. We show that this continuum theory predicts a finite density of states at zero energy for any finite curvature in $D-$dimensional hyperbolic space with $2\leq D \leq 4$, suggesting enhanced susceptibility of Dirac fermions to interaction-driven instabilities at weak coupling. We then derive explicit forms of discrete translational and rotational symmetries for lattices characterized by Schl\"afli symbols $\{p,q\}$, and explicitly construct the discrete spin connection, represented as hopping phases, via parallel transport. Our results pave the way for experimental realization of spin-curvature effects in metamaterial platforms and systematic numerical studies of correlated Dirac phases in hyperbolic geometries.