Hyperbolic topological flat bands

Topological flat bands, which are regarded as the cornerstone of various topological states induced by the many-body interaction, have aroused great interest in the fields of physics and material science. To date, most of the established topological flat bands have been employed in Euclidean space. Here, we report on an observation of hyperbolic topological flat bands in non-Euclidean space. By introducing magnetic flux into the hyperbolic octagon-kagome lattice, energy bands in four-dimensional $k$ space with the nontrivial Chern vector can be created via the formalism of hyperbolic band theory. The bandwidth of hyperbolic topological bands can be significantly reduced by tuning the next-nearest-neighbor coupling strength. Numerical results of finite hyperbolic models with fully and partially open boundary conditions clearly demonstrate the existence of topological boundary states induced by hyperbolic topological flat bands. Moreover, we design and fabricate electric circuits to observe hyperbolic topological flat bands in experiments. Site-resolved impedance responses and robust voltage dynamics demonstrate the coexistence of flat-band dispersion and topological boundary states. This letter may act as a foundation for exploring exotic hyperbolic fractional Chern insulators.