Hyperbolic topological states with first-order and higher-order Chern numbers

The exploration of novel topological physics is one of the most fascinating frontiers in recent years. So far, most studies on topological states have focused on systems in Euclidean space. Here, we report on the construction of various topological phases in hyperbolic space, which is non-Euclidean space with constant negative curvature. We firstly reveal the boundary-dominated Chern topological states in hyperbolic spaces. Moreover, based on the unique property of non-Abelian Fuchsian translation group, the 2D hyperbolic topological states with non-zero second-order Chern numbers are constructed. These exotic hyperbolic topological states have been experimentally observed by electric circuit networks.