Edge states of topological acoustic phonons in graphene zigzag nanoribbons

The tight-binding method is an effective way to study topological insulators. However, an intrinsic element, i.e., the acoustic sum rule (ASR) in phonon systems, has often been overlooked when constructing the Wannier tight-binding phonon Hamiltonian, which results in edge states of acoustic modes rarely observed in previous works. In the present work, we show that the topological acoustic edge states in graphene need to be assessed by ASR. Interestingly, when the inversion-time-reversal symmetry is broken, the distributions of some edge states will mutate from two edges of the nanoribbon to one edge. Moreover, some other topological phonon modes, which are different from previous reports, have also been observed. Our results would shed light on the method for searching topological phonon states, and provide effective strategies for designing phononic devices.