Phonon topology and winding of spectral weight in graphite

The topology of electronic and phonon band structures of graphene is well studied and known to exhibit a Dirac cone at the K point of the Brillouin zone. Here, we applied inelastic x-ray scattering (IXS) along with $\textit{ab initio}$ calculations to investigate phonon topology in graphite, the 3D analogue of graphene. We identified a pair of modes that form a very weakly gapped linear anticrossing at the K point that can be essentially viewed as a Dirac cone approximant. The IXS intensity in the vicinity of the quasi-Dirac point reveals a harmonic modulation of the phonon spectral weight above and below the Dirac energy, which was previously proposed as an experimental fingerprint of the nontrivial topology. We illustrate how the topological winding of IXS intensity can be understood in terms of atomic displacements, and highlight that the intensity winding is not in fact sensitive in telling quasi- and true Dirac points apart.