Combinatorics

Physics

Topology (electrical circuits)

Mathematics

Hyperbolic triangle

Hyperbolic space

Topological insulator

Mathematical analysis

Position and momentum space

Hyperbolic geometry

Differential geometry

Quantum mechanics

Hyperbolic manifold

Hyperbolic function

Physical Review Letters

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their nontrivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modeling propagation of edge excitations, and by their robustness against disorder.

Hyperbolic Topological Band Insulators