Symmetry-enforced topological phononic quadratic Dirac points, type-III nodal rings, and nodal surfaces in BaNa₂

Topological gapless phonons have been extensively studied both in crystal symmetry and materials realization due to their spinless properties. Based on degeneracy and geometry, they can be categorized into the topological nodal point, line, and surface as well as other complex topological phonons. However, the coexistence of more than two types of topological phonons in a single material is rarely reported due to strict symmetry constraints. By means of symmetry analysis, k center dot p model, and first-principles calculations, we found that the quadratic Dirac point (QDP), type-III nodal ring (NR), and nodal surface (NS) can simultaneously emerge under the symmetry protection of space group P6(3)/mmc. We also predicted a series of candidate materials for experimental realizations. As a representative, we performed a thorough investigation of BaNa2 in the C14 Laves phase, analyzing its bulk phonon dispersion relationships, topological invariants, and topological surface states. We unveiled the underlying mechanism for the coexistence of QDPs, NRs, and NSs. This study not only highlights the significance of crystal symmetry in realizing novel topological phonons but also lays the foundation for the analysis of topological materials with hybrid topological characteristics.