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The band connectivity as imposed by the compatibility relations between theirreducible representations of little groups can give rise to the exotichourglass-like shape composed of four branches of bands and five band crossings(BCs). Such an hourglass band connectivity could enforce the emergence ofnontrivial excitations like Weyl fermion, Dirac fermion or even beyond them. Onthe other hand, the bosons, like phonons, magnons, and photons, were also shownto possess nontrivial topology and a comprehensive symmetry classification ofthe hourglass bosonic excitations would be of great significance to bothmaterials design and device applications. Here we firstly list all concretepositions and representations of little groups in the Brillouin zone (BZ)related with the hourglass bosonic excitations in all the 1651 magnetic spacegroups and 528 magnetic layer groups, applicable to three dimensional (3D) andtwo dimensional (2D) systems, respectively. 255 (42) MSGs (MLGs) are found toessentially host such hourglass BCs: Here ``essentially'' means that thebosonic hourglass BC exists definitely as long as the studied system iscrystallized in the corresponding MSG/MLG. We also perform first-principlescalculations on hundreds of 3D nonmagnetic materials essentially hostinghourglass phonons and propose that the 2D material AlI can host hourglassphonons. We choose AuX (X=Br and I) as illustrative examples to demonstratethat two essential hourglass band structures can coexist in the phonon spectrafor both materials while for AuBr, an accidental band crossing sticking twohourglasses is found interestingly. Our results of symmetry conditions forhourglass bosonic excitations can provide a useful guide of designingartificial structures with hourglass bosonic excitations.

All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups