  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 non-trivial 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 modelling propagation of edge excitations, and by their
robustness against disorder.
