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Based on first-principles calculations, we predict that the superconductingMgB$_2$ with a AlB$_2$-type centrosymmetric lattice host the so-called phononictopological Weyl nodal lines (PTWNLs) on its bulk phonon spectrum. These PTWNLscan be viewed as countless Weyl points (WPs) closely aligned along the straightlines in the $-$H-K-H direction within the three-dimensional Brillouin zone(BZ). Their topological non-trivial natures are confirmed by the calculatedBerry curvature distributions on the planes perpendicular to these lines. Theselines are highly unique, because they exactly locate at the high-symmetryboundary of the BZ protected by the mirror symmetry and, simultaneously,straightly transverse the whole BZ, in different from known classificationsincluding nodal rings, nodal chains or nets, and nodal loops. On the(10$\bar{1}$0) crystal surface, the PTWNLs-induced drumhead-like non-trivialsurface states appear within the rectangular area confined by the projectedlines of the PTWNLs with opposite chirality. Moreover, when the mirror symmetryis broken, the double-degenerate PTWNLs are further lifted to form a pair WPswith opposite chirality. Our results pave the ways for future experimentalstudy on topological phonons on MgB$_2$ and highlights similar results in aseries of isostructural AlB$_2$-type metallic diborides.

Phononic Weyl Nodal Straight Lines in High-Temperature Superconductor MgB$_2$