  Despite many of the applications of graphene rely on its uneven stiffness and
high thermal conductivity, the mechanical properties of graphene, and in
general of all 2D materials, are still elusive. The harmonic theory predicts a
quadratic dispersion for the flexural acoustic vibrational mode, which leads
the unphysical result that long wavelength in-plane acoustic modes decay before
vibrating one period, preventing the propagation of sound. The robustness of
the quadratic dispersion has been questioned by arguing that the anharmonic
phonon-phonon interaction linearizes it. However, this implies a divergent
bending rigidity in the long wavelength limit not reproduced experimentally.
Here we show that rotational symmetry protects the quadratic flexural
dispersion against phonon-phonon interactions and that, consequently, the
bending stiffness is non-divergent irrespective of the temperature. Our
non-perturbative anharmonic calculations also determine that sound propagation
coexists with a quadratic dispersion. We also show that the temperature
dependence of the height fluctuations of the membrane, known as ripples, is
fully determined by thermal or quantum fluctuations, but without the anharmonic
suppression of their amplitude previously assumed. The universality of our
conclusions reconcile experimental evidence and theory not just in graphene,
but all 2D materials.
