Jahn-Teller coupling to moir\`e phonons in the continuum model formalism for small angle twisted bilayer graphene

We show how to include the Jahn-Teller coupling of \mo phonons to the electrons in the continuum model formalism which describes small-angle twisted bilayer graphene. These phonons, which strongly couple to the valley degree of freedom, are able to open gaps at most integer fillings of the four flat bands around the charge neutrality point. Moreover, we derive the full quantum mechanical expression of the electron-phonon Hamiltonian, which may allow accessing phenomena such as the phonon-mediated superconductivity and the dynamical Jahn-Teller effect.