Hyperbolic non-Abelian semimetal

We extend the notions of topologically protected band crossings and semimetals to hyperbolic lattices in negatively curved space. Due to their distinct translation group structure, such lattices support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. By adapting the Hamiltonian of a four-dimensional quantum Hall insulator, we devise a hyperbolic tight-binding model whose non-Abelian Bloch states constitute a peculiar kind of two-dimensional topological semimetal. In particular, by combining diverse numerical and analytical approaches, we uncover a quartic scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five inside the reciprocal space. The band crossing is topologically protected by a non-zero value of the second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number. We investigate the anomalous nonlinear topological response of this two-dimensional system that emanates from the novel type of semimetallic band structure.