Coexistence of hourglass Weyl and Dirac nodal line phonons in orthorhombic-type KCuS.

In parallel to electronic systems, the concept of topology has been extended to phonons, which has led to the birth of topological phonons. Very recently, nodal point phonons, nodal line phonons, and nodal surface phonons have been proposed, and a handful of candidate materials have been predicted in solid-state materials. In this work, based on symmetry analysis and first-principles calculations, we propose that hourglass Weyl nodal line (HWNL) phonons and Dirac nodal line (DNL) phonons coexist in the phonon dispersion of a single material, KCuS, with a $Pnma$-type structure. The HWNLs and DNLs are relatively flat in frequency and well separated from other phonon bands. The corresponding topological phonon surface states appear only in the [100] and [001] surfaces, unlike those of typical phononic nodal-line materials. The reason for this phenomenon is explained based on Zak phase calculations. Our work, for the first time, proves that phononic nodal lines with different types of degeneracies can be achieved in one single material. Thus, KCuS can be viewed as a good platform to investigate the entanglement between HWNL phonons and DNL phonons.