  Topological flat bands have gained extensive interest as a platform for
exploring the interplay between nontrivial band topology and correlation
effects. In recent studies, strongly correlated phenomena originating from a
topological flat band were discussed on a periodically 1/6-depleted honeycomb
lattice, but the fundamental topological nature associated with this lattice
structure remains unexplored. Here we study the band structure and topological
phase diagram for the Haldane model on this lattice, which we call the
Bishamon-kikko lattice. We also extend our study to the model connecting the
Bishamon-kikko and honeycomb lattices. We show that these models exhibit richer
topological characteristics compared to the original Haldane model on the
honeycomb lattice, such as topological insulating states with higher Chern
numbers, metallic states with nontrivial band topology even at commensurate
electron fillings, and metal-insulator transitions between them.7 Our findings
offer a playground of correlated topological phenomena and stimulate their
realization in a variety of two-dimensional systems, such as van der Waals
materials, graphene nanostructures, and photonic crystals.
