Topological bands in hyperbolic geometry
 
 
In a pioneering work by Alicia J. Kollár et al. [Nature 571, 45 (2019)], networks of coplanar waveguide resonators were utilized to create a class of materials that constitute lattices in an effective hyperbolic space with constant negative curvature. This work opens a broad avenue for investigating the interplay between geometrical characteristics of curved spaces and exotic topological phases. In this talk, I will review some recent theoretical advances in exploring topological phases in such hyperbolic lattices. I will introduce the hyperbolic Haldane model and the hyperbolic Kane-Mele model, both obtained by replacing the hexagonal cells of their Euclidean counterparts with 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 the bulk and boundary density of states, modeling propagation of edge excitations, and their robustness against disorder.