Symmorphic and nonsymmorphic symmetries jointly-protected hourglass phonons.

Hourglass dispersion is generally believed to be solely protected by nonsymmorphic symmetries, because these symmetries can introduce high-dimensional projective representations. Here, based on symmetry arguments, we propose that the hourglass dispersion can be jointly protected by the symmorphic and nonsymmorphic symmetries, instead of the only conventional nonsymmorphic symmetry. Moreover, using first-principles calculations, we realize our proposal in phonon spectra of realistic materials that share an antiperovskite structure with space group \emph{P}4/\emph{nmm}. Importantly, the neck points of these hourglass dispersions trace out two nodal rings tangent to four nodal lines, forming a unique hourglass nodal cage in the bulk Brillouin zone. The Berry phase analysis reveal the nontrivial topology of these nodal rings and nodal lines. In addition, the nontrivial surface states and isofrequency surface arcs are visible, facilitating their experimental confirmation of such exotic quasiparticles. Our work not only offers a new insight into the hourglass dispersion, but also expands the prospect for studying promising topological quasiparticles in condensed-matter systems.