Second-order topological states in the artificial systems have been extensively studied, but the study in the phonons of atomic vibrations has been limited. In this paper, we propose that a phononic two-order topological phase can be realized in two-dimensional C3N, a previously synthesized and intensively investigated two-dimensional material, and its nontrivial phase arises from the mismatch between the Wannier center of the out-of-plane phonon modes and the atomic positions. Using a simplified force constant model, we find that gapped edge states and in-gap corner states can only exist on the structures with broken-pure-carbon-ring terminations, and this unexpected phenomenon can be explained by the electron-like filling anomaly for phonons. Further calculations reveal that these corner modes are robust to external disturbances. The nontrivial phononic phase in C3N provides an avenue in crystal for exploring higher-order topological phases in Bose systems.
