Topological continuum charges of acoustic phonons in two dimensions and the Nambu-Goldstone theorem

We analyze the band topology of acoustic phonons in 2D materials by considering the interplay between spatial/internal symmetries and additional constraints that arise from the physical context. These supplemental constraints trace back to the Nambu-Goldstone theorem and the requirements of structural stability. We show that this interplay can give rise to previously unaddressed nontrivial nodal charges that are associated with the crossing of the acoustic phonon branches at the center ($\mathrm{\ensuremath{\Gamma}}$ point) of the phononic Brillouin zone. We moreover apply our perspective to the concrete context of graphene, where we demonstrate that the phonon spectrum harbors these kinds of nontrivial nodal charges. Apart from its fundamental appeal, this analysis is physically consequential and dictates how the phonon dispersion is affected when graphene is grown on a substrate. Given the generality of our framework, we anticipate that our strategy, which thrives on combining physical context with insights from topology, should be widely applicable in characterizing systems beyond electronic band theory.