Attenuation of flexural phonons in free-standing crystalline two-dimensional materials

We develop the theory for dynamics of the out-of-plane deformations in flexible two-dimensional materials. We focus on study of attenuation of flexural phonons in free-standing crystalline membranes. We demonstrate that the dynamical renormalization does not involve the ultraviolet divergent logarithmic contributions contrary to the static ones. This fact allows us to find the scaling form of the attenuation, determine its small- and large-frequency asymptotes, and to derive the exact expression for the dynamical exponent of flexural phonons in the long-wave limit: $\mathsf{\text{z}}=2\ensuremath{-}\ensuremath{\eta}/2$. Here $\ensuremath{\eta}$ is the universal exponent controlling the static renormalization of the bending rigidity. Also we determine the dynamical exponent for the long-wave in-plane phonons: ${\mathsf{\text{z}}}^{\ensuremath{'}}=(2\ensuremath{-}\ensuremath{\eta}/2)/(1+\ensuremath{\eta}/2)$. We discuss implication of our results to experiments on phonon spectra in graphene and dynamics of graphene-based nanomechanical resonators.