Degradation of sound in moiré superlattices

The collective modes of twisted bilayer graphene and other moiré superlattices arise not from vibrations of a rigid crystal but from the relative displacement between the constituent layers. Despite their similarity to acoustic phonons, these modes, called phasons, are not protected by any conservation law. We show that phason modes become overdamped at low frequencies, reflecting that the moiré pattern relaxes via internal diffusive processes rather than by collective oscillations at long wavelengths. Disorder in the relative orientation between layers opens a gap in the phason dispersion, which displays a universal dependence on the twist-angle variance. Our results have important implications for the electronic properties of twisted moiré systems that are sensitive to the electron-phonon coupling, as well as for the low-temperature thermodynamic properties of these quasiperiodic lattices.