Topological Invariant of Acoustic Phonons in 2D materials

2D materials that are allowed to flex out-of-plane display an unusual acoustic phonon mode – the flexural mode. This mode disperses quadratically away from the center of the Brillouin zone, as opposed to the linear dispersion of the in-plane modes. This leads to an unusual triple degeneracy at the zone center, with two linear and one quadratic band touching. This band touching is unusual because it is enforced by Goldstone’s theorem rather than symmetry, as will be discussed. I will also discuss the topological invariant associated with this crossing and how this invariant affects the physics. The invariant is a priori of quaternionic type, but it reduces to a Z2 invariant under rather general assumptions. This turns out to have important implications for the band splitting that occurs when a 2D material is grown on a substrate. This splitting will be discussed for the specific case of graphene, where the Z2 invariant associated with the acoustic bands turns out to be non-trivial.