Charged moiré phonons in twisted bilayer graphene.

Moiré superlattices are generically incommensurate. The collective vibrations of the moiré pattern (moiré phonons) include two acoustic branches (phasons) protected by the invariance of the free-energy against relative displacements of the layers. Electrons scatter off moiré phonons differently than in-phase monolayer modes. Besides opposite pseudogauge fields and deformation potentials within each layer, electron-moiré phonon coupling (EPC) includes events involving interlayer tunneling of electrons. In non-polar materials such as twisted bilayer graphene, moiré phonons do not carry intrinsic dipole moments, yet infrared-active modes can give rise to optical absorption due to EPC when the system is charged (charged phonons). This is a consequence of the inhomogeneous distribution of charge in the moiré unit cell. If the chemical potential lies within a gap of the electronic spectrum, the optical response of phasons is quantized as a manifestation of the adiabatic drag of charge by the sliding of the superlattice. The result is a Drude-like peak whose magnitude is fixed by the total sliding Chern number, an invariant related to the amount of charge added to or substracted from the system. Our findings bring out the hybridized nature of charge (electrons) and stacking (moiré phonons) collective modes, of relevance for any consistent theory of superconductivity in twisted bilayer graphene.