Fragile phonon topology on the honeycomb lattice with time-reversal symmetry

We use the methods of topological quantum chemistry to explore the topology of phonons on time-reversal symmetric crystals with the structure of the planar honeycomb (layer group p6/mmm). This approach is not tied to a particular model of atomic vibrations, but is applied to the most general dynamical matrix constrained only by the symmetries of the system. We show that four distinct fragile topological phonon phases are generically possible. Truncating the dynamical matrix to third-nearest neighbors yields a model that realizes the different phonon topologies, characterized by the existence of phononic edge and corner modes and by Wilson loops with winding numbers one and two. Fitting the dynamical matrix to the ab initio phonon bands shows that graphene is not very far from a topologically nontrivial phonon phase.