Topological flat bands in hyperbolic lattices

TL;DR

This paper explores topological flat bands in hyperbolic lattices with negative curvature, revealing potential for fractional Chern insulators and demonstrating a fractionalized state in these non-Euclidean systems.

Contribution

It introduces hyperbolic analogs of kagome lattices, finds topological flat bands in them, and shows the possibility of fractional Chern insulators in these non-Euclidean lattices.

Findings

Topological flat bands found in hyperbolic kagome lattices.

Flatness ratios exceed 15, indicating strong flat bands.

Demonstration of a $ u=1/2$ fractional Chern insulator state.

Abstract

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