Hybrid nodal-ring phonons with hourglass dispersion in AgAlO2

Topological nodal lines are open or closed one-dimensional manifolds formed by symmetry-protected band crossings in momentum space. Here we propose one hybrid type of closed nodal lines, i.e., the intersecting nodal rings, which exhibit hourglass dispersions guaranteed by two glide mirror symmetries. By performing first-principles calculations, we identify that type-I and type-II topological phonons coexist in a realistic material AgAlO2, forming hybrid-type nodal rings. Moreover, the hybrid nodal-ring phonons display unique saddlelike instead of drumheadlike surface states due to their exotic frequency dispersion. Our findings trigger the study of topological materials with hybrid topological features.