Coexistence of zero-, one-, and two-dimensional degeneracy in tetragonal phonons

Based on the dimension of degeneracy, topological electronic systems can roughly be divided into three parts: nodal point, line, and surface materials corresponding to zero-, one-, and two-dimensional degeneracy, respectively. In parallel to electronic systems, the concept of topology was extended to phonons, promoting the birth of topological phonons. Till date, few nodal point, line, and surface phonon candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, we prove that zero-, one-, and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material with structure. In contrast to the previously reported electronic systems, the topological phonons observed in are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, with multiple dimensions of degeneracy phonons is a good platform for studying the entanglement among nodal point, line, and surface phonons. Moreover, obvious phonon surface states are visible, which is beneficial for experimental detection.