Kekule Lattice in Graphdiyne: Coexistence of Phononic and Electronic Higher-Order Band Topology.

The topological physics has been extensively studied in different kinds of bosonic and fermionic systems, ranging from artificial structures to natural materials. However, the coexistence of topological phonon and electron in one single material is seldom reported. Recently, graphdiyne is proposed to be a two-dimensional (2D) electronic second-order topological insulator (SOTI). In this work, based on density-functional tight-binding calculations, we found that graphdiyne is equivalent to the Kekule lattice, also realizing a 2D phononic SOTI in both out-of-plane and in-plane modes. Depending on edge terminations, the characterized topological corner states can be either inside or outside the bulk gap, which are tunable by local corner potential. Most remarkably, a unique selectivity of space and symmetry is revealed in electron-phonon coupling between the localized phononic and electronic topological corner states. Our results not only demonstrate the phononic higher-order band topology in a real carbon material, but also provide an opportunity to investigate the interplay between phononic and electronic higher-order topological states.