Phasons in twisted trilayer graphene

Twisted trilayer graphene (TTG) is the latest addition to the growing family of novel moiré systems. Motivated by recent experimental advances in this system, we study its long-wavelength lattice fluctuations on the moiré scale and, in particular, its phason modes. Unlike twisted bilayer graphene, TTG hosts two sets of phasons, which can be classified as either odd or even under the mirror symmetry of the system. Taking into account the important effects of lattice relaxation, we calculate the phason spectra by solving the continuum equations of motion for infinitesimal vibrations and illustrate the differences between the mirror-even and odd sectors. An out-of-plane displacement field provides an experimental knob to tune the physics of TTG, and we analyze the effects of such a field, which breaks the mirror symmetry and couples the even and odd modes. Finally, we discuss the interactions between the phasons and the electronic degrees of freedom and consider the implications thereof for superconductivity.