Staggered magnetic flux induced higher-order topological insulators and topological flat bands on the ruby lattice

Quantum anomalous Hall states, or Chern insulators (CIs), have been investigated in various lattice models by introducing staggered magnetic fluxes. Considering long-distance hopping terms, some topological flat band (TFB) models have been proposed based on CI models. In this work, we construct a ruby lattice with staggered fluxes threading hexagons and find that staggered fluxes give rise to higher-order topological insulators (HOTIs) and TFBs. At $1/3$ filling, this model supports the phase transition from a CI with Chern number 2 to a HOTI phase by tuning the staggered fluxes. This HOTI can be identified based on the gapped edge state, robust corner state, and quantized topological invariants. Simultaneously, we find the best TFB with a flatness ratio of about 18 at $1/6$ filling. Furthermore, we explore fractional Chern insulator states when hard-core bosons and interacting spinless fermions fill into these TFB models.