Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices

TL;DR

This paper extends Bloch's theorem to hyperbolic lattices with nonabelian translation groups, enabling the analysis of high-dimensional irreducible representations and boundary modes, with applications to mechanical hyperbolic lattices.

Contribution

It introduces a framework for constructing wave eigenstates of high-dimensional irreps in hyperbolic lattices, generalizing Bloch's theorem to non-Euclidean geometries.

Findings

Characterized band structure and zero modes of a mechanical hyperbolic lattice.

Demonstrated implications for mode-counting and degeneracy in hyperbolic systems.

Established bulk-edge correspondence in hyperbolic lattice models.

Abstract

Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potentials 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.