  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.
