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 $\mathit{k}·\mathit{p}$ 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.

Figure 1

(a) Nodal rings protected by mirror symmetries. (b) A nodal ring protected by inversion and time-reversal symmetries. (c) Hybrid nodal rings protected by mirror, inversion, and time-reversal symmetries.

Figure 2

(a) Crystal structure of AgO with space group $P{2}_1/c$. (b) Bulk Brillouin zone and projected Brillouin zone of the (100) and (010) surfaces. (c) Calculated phonon spectrum of AgO.

Figure 3

(a) Phonon spectrum along the $H-\mathrm{\Gamma }-Y$ direction. (b) Nodal ring protected by $M_x$ mirror symmetry. (c) Phonon surface states along the $\overline{Y}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{A}$ direction of the (100) surface. (d) Phonon surface arcs on the (100) surface Brillouin zone. The units of color bars in (c) and (d) are in arbitrary units.

Figure 4

(a) Phonon spectrum along the $Z-\mathrm{\Gamma }-H$ direction. (b) Nodal rings protected by inversion and time-reversal symmetries. (c) Phonon surface states along the $\tilde{Y}\text{−}\tilde{\mathrm{\Gamma }}\text{−}\tilde{A}$ direction of the (010) surface. (d) Phonon surface arcs on the (010) surface Brillouin zone. The units of color bars in (c) and (d) are in arbitrary units.

Figure 5

(a) Crystal structure of AgO with space group $I{4}_1/a$. (b) Bulk Brillouin zone and projected Brillouin zone of the (001) surface. (c) Calculated phonon spectrum of this phase of AgO. (d) Nodal rings in the Brillouin zone. (e), (f) Projections of nodal rings on the A and B surfaces.

Figure 6

(a) The phonon surface states along the $\overline{N}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ direction of the (100) surface. (b) The phonon surface Fermi arcs on the (001) surface Brillouin zone.