Topological states in a dimerized system with staggered magnetic fluxes

Robust edge states propagate along the edges and corner states gather at the corners in two-dimensional (2D) first-order and second-order topological insulators, respectively. Here, we report two kinds of topological states in an extended 2D dimerized lattice with staggered flux threading. At $\frac{1}{2}$ filling, we observe isolated corner states as well as metallic near-edge states in the $\mathcal{C}=2$ Chern insulator states. At $\frac{1}{4}$ filling, we find a $\mathcal{C}=0$ topological state, where the robust edge states are well localized along the edges but bypass corners. These topological insulator states differ from both conventional Chern insulators and the usual high-order topological insulators.