Bending rigidity and sound propagation in graphene.

Despite the interest raised by graphene and 2D materials, their mechanical and acoustic properties are still highly debated. Harmonic theory predicts a quadratic dispersion for the flexural acoustic mode. Such a quadratic dispersion leads to diverging atomic fluctuations and a constant linewidth of in-plane acoustic phonon modes at small momentum, which implies that graphene cannot propagate sound waves. Many works based on membrane theory questioned the robustness of the quadratic dispersion, arguing that the anharmonic phonon-phonon interaction linearizes it, which implies a divergent bending rigidity (stiffness) in the long wavelength limit. However, these works are based on effective low-energy models that explicitly break the rotational invariance. Here we show that rotational symmetry protects the quadratic flexural dispersion against phonon-phonon interaction, and that the bending stiffness of graphene is unaffected by temperature and quantum fluctuations. Nevertheless, our non-perturbative anharmonic calculations predict that sound propagation coexists with such a quadratic dispersion. Since our conclusions are universal properties of membranes, they apply not just to graphene, but to all 2D materials.