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\nconsist of triptycene molecules and phenyls, exotic electronic structures such\nas Dirac cones and flat bands arise from the kagome-type network. In this\npaper, we theoretically investigate the tight-binding models for polymerized\ntriptycene, focusing on the origin of flat bands and the topological\nproperties. The mechanism of the existence of the flat bands is elucidated by\nusing the "molecular-orbital" representation, which we have developed in the\nprior works. Further, we propose that the present material is a promising\ncandidate to realize the two-dimensional second-order topological insulator,\nwhich is characterized by the boundary states localized at the corners of the\nsample. To be concrete, we propose two methods to realize the second-order\ntopological insulator, and elucidate the topological properties of the\ncorresponding models by calculating the corner states as well as the bulk\ntopological invariant, namely the $\\mathbb{Z}_3$ Berry phase.\n