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This year, Liu \textit{et al}. [Phys. Rev. B \textbf{104}, L041405 (2021)]proposed a new class of topological phonons (TPs; i.e., one-nodal surface (NS)phonons), which provides an effective route for realizing one-NSs in phononsystems. In this work, based on first-principles calculations and symmetryanalysis, we extended the types of NS phonons from one- to three-NS phonons.The existence of three-NS phonons (with NS states on the $k_{i}$ = $\pi$ ($i$ =$x$, $y$, $z$) planes in the three-dimensional Brillouin zone (BZ)) is enforcedby the combination of two-fold screw symmetry and time reversal symmetry. Wescreened all 230 space groups (SGs) and found nine candidate groups (with theSG numbers (Nos.) 19, 61, 62, 92, 96, 198, 205, 212, and 213) hosting three-NSphonons. Interestingly, with the help of first-principles calculations, weidentified $P2_{1}$2$_{1}$2$_{1}$-type YCuS$_{2}$ (SG No. 19), $Pbca$-typeNiAs$_{2}$ (SG No. 61), $Pnma$-type SrZrO$_{2}$ (SG No. 62),$P4_{1}$2$_{1}$2-type LiAlO$_{2}$ (SG No. 92), $P4_{3}$2$_{1}$2-type ZnP$_{2}$(SG No. 96), $P2_{1}$3-type NiSbSe (SG No. 198), $Pa\bar{3}$-type As$_{2}$Pt(SG No. 205), $P4_{3}$32-type BaSi$_{2}$ (SG No. 212), and $P4_{1}$32-typeCsBe$_{2}$F$_{5}$ (SG No. 213) as realistic materials hosting three-NS phonons.The results of our presented study enrich the class of NS states in phononsystems and provide concrete guidance for searching for three-NS phonons andsingular Weyl point phonons in realistic materials.

Three-nodal surface phonons in solid-state materials: Theory and material realization