Classification and materials realization of topologically robust nodal ring phonons

Nodal ring phonons with topologically nontrivial properties have triggered much interest in recent years. Unlike fermions inevitably affected by the spin-orbit interaction, nodal ring phonons are topologically robust. In this paper, we investigate the classification of nodal rings in phonon systems based on symmetry arguments and $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ models. We identify that all the nodal ring phonons can be categorized into three types, i.e., in-plane nodal rings protected by mirror symmetry, twisted nodal rings protected by inversion and time-reversal symmetries, and the hybrid type protected by a combination of these symmetries. On the basis of first-principles calculations, we propose that these three types of nodal ring phonons can emerge in two different phases of silver oxide. The calculation results also show clear drumheadlike surface states along high-symmetry paths and nontrivial arc states in the isofrequency surfaces. This work exposes various appearances of nodal rings and also promotes the development of topological phonons.