Degradation of phonons in disordered moir\'e superlattices

The elastic collective modes of a moir\'e superlattice 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. Here, we show that disorder in the relative orientation between the layers and thermal fluctuations associated with their sliding motion degrade the propagation of sound in the moir\'e superlattice. Specifically, the phason modes become overdamped at low energies and acquire a finite gap, which displays a universal dependence on the twist-angle variance. Thus, twist-angle inhomogeneity is manifested not only in the non-interacting electronic structure of moir\'e systems, but also in their phonon-like modes. More broadly, our results have important implications for the electronic properties of twisted moir\'e systems that are sensitive to the electron-phonon coupling.