Pseudo-Hourglass Nodal Surface Phonons in XF (X = Ba, Sr, Ca)

Condensed-matter systems have been a wonderland of topological quasiparticles, e.g., topological fermions and topological phonons. In this work, we focus on the topological phonons. Here, we find that hourglass nodal loops (HNLPs)/lines (HNLEs) emerge in the phonon spectra of materials XF (X = Ba, Sr, Ca). Specifically, the HNLPs are enforced by nonsymmorphic symmetry, while the HNLEs which are throughout the Brillouin zone are guaranteed by simultaneous operations of symmorphic and nonsymmorphic symmetries. Notably, the hourglass nodal loops are tangent to four hourglass nodal lines, thus forming an hourglass cage located between two double-degenerate terminals. The two terminals, i.e., nodal surface (NS) and quadratic nodal lines (QNLs), exhibit quite flat dispersion. In addition, when going from NS to QNLs along a generic path, states will undergo a partner switching, leading to a band crossing in the hourglass shape. As a consequence, the neck points of hourglass shape dispersion finally form an hourglass nodal surface in its 3D Brillouin zone, which has not been predicted before. Furthermore, this family of materials enjoys the spacetime inversion symmetry, so that the nodal loops/lines feature a quantized Berry phase. This leads to drumhead surface states on their projection surfaces. Our findings present a novel topological phonon in realistic materials.