Flat bands with high Chern numbers and multiple flat bands in multifold staggered-flux models

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Topological flat bands (TFBs) have been proposed in Chern-insulator lattice models. Some two-dimensional monolayer systems can host TFBs with the Chern number $|\mathcal{C}|=1$, and some TFBs with higher Chern numbers are achieved in multilayer or multicomponent systems. In this work, we construct a series of TFBs with higher Chern numbers in the monolayer star lattice by adding several distant-neighbor hopping terms with staggered magnetic fluxes. By considering the third and the fourth nearest-neighbor hopping terms, we find the $|\mathcal{C}|=2$ and $|\mathcal{C}|=3$ TFBs with large flatness ratios, respectively. We also obtain the $|\mathcal{C}|=4$ and $|\mathcal{C}|=5$ TFBs when the fifth and the sixth nearest-neighbor hopping terms are added. Intriguingly, we also observe multiple flat bands with various Chern numbers by tuning the distant-neighbor hopping amplitudes and the staggered-flux parameters. Furthermore, a series of fractional Chern insulator (FCI) states have been numerically observed when hard-core bosons are filled into these TFB models, with fractional fillings $\ensuremath{\nu}=1/3, 2/5, 3/7$ and $\ensuremath{\nu}=1/4, 2/7$ in the $|\mathcal{C}|=2$ and $|\mathcal{C}|=3$ TFBs, respectively. Our findings provide an approach to search for more TFBs with higher Chern numbers and further investigate more interesting FCIs.