  The bulk-boundary correspondence is a generic feature of topological states
of matter, reflecting the intrinsic relation between topological bulk and
boundary states. For example, robust edge states propagate along the edges and
corner states gather at corners in the two-dimensional first-order and
second-order topological insulators, respectively. Here, we report two kinds of
topological states hosting anomalous bulk-boundary correspondence in the
extended two-dimensional dimerized lattice with staggered flux threading. At
$\frac{1}{2}$-filling, we observe isolated corner states with no fractional
charge as well as metallic near-edge states in the $\mathcal{C}=2$ Chern
insulator states. At $\frac{1}{4}$-filling, we find a $\mathcal{C}=0$
topologically nontrivial state, where the robust edge states are well localized
along edges but bypass corners. These robust topological insulating states
significantly differ from both conventional Chern insulators and usual
high-order topological insulators.
