Flexural phonons in free-standing graphene

Rotation and reflection symmetries impose that out-of-plane (flexural) phonons of free-standing graphene membranes have a quadratic dispersion at long wavelength and can be excited by charge carriers in pairs only. As a result, we find that flexural phonons dominate the phonon contribution to the resistivity $\rho$ below a crossover temperature T_x where we obtain an anomalous temperature dependence $\rho\propto T^{5/2}_{}\ln T$. The logarithmic factor arises from renormalizations of the flexural phonon dispersion due to coupling between bending and stretching degrees of freedom of the membrane.