Moiré optical phonons coupled to heavy electrons in magic-angle twisted bilayer graphene

The authors propose here a projection scheme to simplify electron-phonon coupling in magic-angle twisted bilayer graphene, highlighting the coupling with low-energy heavy electrons. Several active moir\'e phonons that strongly couple to the flat band are identified. These modes exhibit unique envelope functions at the moir\'e scale and preserve monolayer oscillations at the atomic scale. Further incorporating them into an extended Holstein model will facilitate studies of correlated insulating and superconducting phases.