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.