Butterfly-shape hourglass type-II nodal birdcage and multiple quadratic nodal-line phonons in Ba X N 2 ( X = Ti, Zr, Hf)

The exploration of various topological nodal -line phonons with novel topological properties and unique geometrical configurations has long been considered one of the central topics in topological physics. In this work we find that Weyl nodal lines are composed of two basic categories, i.e., linear and quadratic. According to the dispersion relations of degeneracy points in them, both linear and quadratic nodal lines can be divided into three types, i.e., types I, II, and III. Based on these basic types, two unusual nodal lines, i.e., anisotropic and hybrid, may be defined here. Unfortunately, real materials which may simultaneously generate these basic and unusual nodal -line phonons with different types and categories have still rarely been reported, owing to the highly required symmetry conditions. By using symmetry analysis and first -principles calculations, we uncover that a material family Ba X N 2 ( X = Ti, Zr, Hf) in space group 129 may generate novel butterfly -shape nodal -birdcage phonons that are composed of three hourglasslike type -II nodal rings and four straight nodal lines. More importantly, multiple unusual nodal lines, including anisotropic and hybrid ones, also exist in these materials. The topologically nontrivial features of these various nodal -line phonons are confirmed by the nonzero Berry phases and the drumheadlike surface states. Our theoretical results provide an ideal material platform to study the topological properties of various nodal -line phonons, including three unusual types.