Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices

Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potential for a new class of topological metamaterials. One important feature of hyperbolic lattices is that their translation group is nonabelian, permitting high-dimensional irreducible representations (irreps), in contrast to abelian translation groups in Euclidean lattices. Here we introduce a general framework to construct wave eigenstates of high-dimensional irreps of infinite hyperbolic lattices, thereby generalizing Bloch's theorem, and discuss its implications on unusual mode counting and degeneracy, as well as bulk-edge correspondence in hyperbolic lattices. We apply this method to a mechanical hyperbolic lattice, and characterize its band structure and zero modes of high-dimensional irreps.

Automated summary

Periodic lattices in hyperbolic space exhibit non-Euclidean crystallographic symmetries and non-abelian translation groups, allowing for high-dimensional irreducible representations. This paper presents a general framework for constructing wave eigenstates of high-dimensional irreps in infinite hyperbolic lattices, generalizing Bloch's theorem, and discusses its implications on mode counting, degeneracy, and bulk-edge correspondence.