Symmetry-enforced type-II Weyl phonons and hybrid Weyl nodal-line phonons in P4¯m2 -carbon

The exploration of unusual Weyl phonons has long been regarded as one of the central topics in topological physics. Our symmetry analysis shows that symmetry-enforced type-II Weyl phonons may exist under the protections from two mirror symmetries (${\mathcal{M}}_{x}$ and ${\mathcal{M}}_{y}$), a twofold rotation symmetry (${C}_{2z}$) and time-reversal symmetry ($\mathcal{T}$), and meanwhile, another kind of unusual Weyl phonon, i.e., hybrid Weyl nodal-line phonons, may be constructed on a square and four straight Weyl nodal lines. Our first-principles calculations demonstrate that these two unusual Weyl phonons may coexist in a real material $P\overline{4}m2$-carbon. More importantly, another kind of nontrivial surface states, i.e., which we refer to as hybrid surface states, are produced both by type-II Weyl phonons and by the hybrid Weyl nodal-line phonons. Our work provides an ideal material platform to study type-II Weyl phonons and hybrid Weyl nodal-line phonons in phononic systems.