Electron phonon coupling in the twisted bilayer graphene

We discuss several aspects of the electron phonon coupling (EPC) in the twisted bilayer graphene. We first project the full EPC vertex, derived from microscopic tight-binding lattice calculations, onto the basis of the topological heavy fermion model [Song and Bernevig, Phys. Rev. Lett. 129, 047601 (2022)]. We identify the significance of each phonon mode, and reveal that EPC contributes an on-site anti-Hund's interaction $\hat{H}_{\rm A}$ on the moir\'e-scale local $f$-orbitals, with strengths 1 to 4 meV. It combines with a Hund's interaction $\hat{H}_{\rm H}$ of strengths 1 to 3 meV that originates from the carbon atom Hubbard, and serves to split the $f$-impurity many-electron configurations. We elaborate on phonon-favored symmetry-breaking orders at even-integer fillings, and also explore the possibility of finding an exotic Dirac semi-metal at the charge-neutrality point which is formed solely by $c$-electrons, while $f$-impurities exhibit a symmetric Mott gap by forming non-degenerate singlets under $\hat{H}_{\rm A,H}$. Finally, we also discuss the charge pumping induced by the relative motion of the two layers.