Phononic quadratic nodal lines of different types in Li2NaN

The search for new topological states has been shifted from electronic fermionic states to phononic bosonic ones, and topological elements with conventional linear band crossing have also been extended into the high order dispersion field. In this Letter, based on theoretical calculations and effective model analysis, we predicted the quadratic phonon states in the hexagonal compound Li2NaN. Particularly, two quadratic nodal lines are found, which are distributed along the same straight Γ–A path, protected by the C6v point group symmetry. The detailed band surface distribution and effective Hamiltonian construction are elucidated, demonstrating the quadratic order dispersion. These two quadratic nodal lines were further categorized into type-I and type-III, and their surface spectral properties were calculated, in which clear drumhead surface states are observed for both quadratic nodal lines, and they stretched throughout the entire Brillouin zone. Furthermore, the surface states were well separated from the bulk band projection, which is extremely beneficial for experimental detection. Notably, phononic quadratic nodal line states have been rarely studied and, more importantly, a type-III line has even not been reported thus far. Therefore, our work can pave the way for related studies for both quadratic nodal lines and type-III states, and the presented material can serve as an ideal candidate to promote further experimental investigation.