Hourglass phonons jointly protected by symmorphic and nonsymmorphic symmetries

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 symmorphic and nonsymmorphic symmetries, instead of the conventional nonsymmorphic symmetry. Moreover, using first-principles calculations, we realize our proposal in the phonon spectra of realistic materials that share an antiperovskite structure with space group $P4$/nmm. Importantly, the neck points of these hourglass dispersions trace out two nodal rings tangential to four nodal lines, forming a unique hourglass nodal cage in the bulk Brillouin zone. The Berry phase analysis reveals the nontrivial topology of these nodal rings and nodal lines. Furthermore, nontrivial surface states and isofrequency surface arcs are visible, facilitating their experimental confirmation of such exotic quasiparticles. work not only offers different insights into the hourglass dispersion, but also expands aspects for studying promising topological quasiparticles in condensed-matter systems.