Unpaired Weyl phonon systems in NaHPO₃NH₂

Phonon-based Weyl points (WPs) are usually considered to appear in pairs with opposite chirality. Here, we report the phonon dispersion in a realistic material monosodium phosphoramidate, NaHPO3NH2, which has two types of unpaired WP phonon systems: (i) two hourglass charge-3 (C-3) WPs and six C-1 conventional WPs forming a topological Weyl complex; (ii) one C-2 WP, with a topologically charged Weyl nodal surface, forming a topologically charged Weyl point-surface system. Remarkably, the simultaneous presence of the C-1, C-2, and hourglass C-3 WPs in NaHPO3NH2 results in the formation of clean and long phononic double and sextuple-helicoid surface states on the (10 1 over bar 0) and (001) surfaces, respectively. The phononic double and sextuple-helicoid surface states span the entire first surface Brillouin zones of the (10 1 over bar 0) and (001) surfaces, which will facilitate their detection in future experimental studies.