Flat bands and higher-order topology in polymerized triptycene: Tight-binding analysis on decorated star lattices

In a class of carbon-based materials called polymerized triptycene, which consist of triptycene molecules and phenyls, exotic electronic structures such as Dirac cones and flat bands arise from the kagome-type network. In this paper, we theoretically investigate the tight-binding models for polymerized triptycene, focusing on the origin of flat bands and the topological properties. The mechanism of the existence of the flat bands is elucidated by using the "molecular-orbital" representation, which we have developed in the prior works. Further, we propose that the present material is a promising candidate to realize the two-dimensional second-order topological insulator, which is characterized by the boundary states localized at the corners of the sample. To be concrete, we propose two methods to realize the second-order topological insulator, and elucidate the topological properties of the corresponding models by calculating the corner states as well as the bulk topological invariant, namely the $\mathbb{Z}_3$ Berry phase.