Topological edge modes and Weyl hypersurfaces in hyperbolic Maxwell lattices

We generalize topological mechanics to lattices with negative intrinsic curvature. We find that the mechanical stability of the lattice, quantified by Maxwell's counting, now depends on the spatial symmetries of the lattice while the eigenspace of deformations splits into sectors which can be under-, over- or critically constrained. Distortions of hyperbolic Maxwell lattices can lead to edge-localization of the zero-energy deformations and to partial or complete lattice polarization. We impose periodic boundary conditions by compactifying the lattice's unit cell into surfaces of high genus g>1, and compute topological indices using curved-space Bloch states. The lattice's curvature translates into a constraint between the Bloch wave functions, leading to precise correlations in the directions of localization of edge modes. Finally, we find that in certain hyperbolic lattices, the Brillouin zone contains not only Weyl points and lines but also higher dimensional Weyl hypersurfaces and define their topological charge.