Topological flat bands in hyperbolic lattices

Topological flat bands (TFBs) provide a promising platform to investigate intriguing fractionalization phenomena, such as fractional Chern insulators (FCIs). Most TFB models are established in two-dimensional Euclidean lattices with zero curvature. In this paper, we systematically explore TFBs in a class of two-dimensional non-Euclidean lattices with constant negative curvature, i.e., the hyperbolic analogs of the kagome lattice. Based on Abelian hyperbolic band theory, TFBs have been respectively found in the heptagon-kagome, the octagon-kagome, the nonagon-kagome, and the decagon-kagome lattices by introducing staggered magnetic fluxes and next-nearest-neighbor hoppings. The flatness ratios of all hyperbolic TFB models are more than 15, which suggests that hyperbolic FCIs can be realized in these TFB models. We further demonstrate the existence of a $\ensuremath{\nu}=1/2$ FCI state with open boundary conditions when hard-core bosons fill into these hyperbolic TFB models.