Kekulé Lattice in Graphdiyne: Coexistence of Phononic and Electronic Second-Order Topological Insulator.

Topological physics has been extensively studied in different kinds of bosonic and Fermionic systems, but the coexistence of topological phonons and electrons in one single material has seldom been reported. Recently, graphdiyne has been proposed as a two-dimensional (2D) electronic second-order topological insulator (SOTI). In this work, we found that graphdiyne is equivalent to Kekulé 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 and are tunable by the local corner potential. Most remarkably, a unique selectivity of space and symmetry is revealed in the 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.