Coexistence of zero-, one- and two-dimensional degeneracy in tetragonal SnO$_2$ phonons

Abstract

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 twodimensional 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 phonons candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, for the first time, we prove that zero-, oneand two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material SnO2 with P 42/mnm structure. In contrast to the previously reported electronic systems, the topological phonons observed in SnO2 are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, SnO2 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.