Fig. 5. Acoustic phonon of tellurium.

a The lattice structure of tellurium with space group P3₁21. b By using the values of the stiffness tensor in Materials Project⁶⁸, we find that away from the triple point, the red nodal lines occur only between the two lowest energy bands so that $\mathfrak{q}$ can be defined. c The Wilson loop spectrum shows trivial winding structure, and therefore, $\mathfrak{e}=0$. d The wavefunction texture of the highest energy band shows trivial skyrmion texture, which is consistent with the constraint that $2{\mathfrak{n}}_{\mathrm{sk}}=\mathfrak{e}$. e, We perturb the dynamical matrix such that the degeneracy of TATP in b is lifted (note, however, that these perturbations break the continuous translation symmetry, since the triply degenerate Goldstone mode disappears.). The black nodal lines formed by the two highest energy bands are created. As opposed to the case with nontrivial $\mathfrak{q}$ in Fig. 3d, the nodal structure in e does not have link between the red and black nodal lines because $\mathfrak{q}$ is zero.