Hyperbolic lattice waves, band theory and boundary modes of high dimensional representations of infinite hyperbolic lattice

Regular lattices in non-Euclidean space offer a new platform of exotic wave states due to both the non-abelian nature of their translation groups and the curvature of the embedding space. Waves in Euclidean lattices are governed by Bloch's theorem, which originates from one-dimensional representations of their abelian translation groups. In contrast, non-Euclidean lattices are beyond Bloch's theorem. In this talk, we propose a fundamental framework for characterizing wave states in non-Euclidean lattices, where we introduce a new formulation for compatibility and equilibrium matrices, derive a band theory for infinite hyperbolic lattices where waves carry high-dimensional irreducible representations of their non-abelian translation groups, and a representation-based method of finding boundary modes in hyperbolic lattices. These results shed light on the dispersion, response, and boundary modes of hyperbolic lattices, which are fundamentally distinct from those of conventional crystals.