Electron phonon coupling in the topological heavy fermion model of twisted bilayer graphene

On flat bands of the magic-angle twisted bilayer graphene, exotic correlation physics unfolds. Phonons, through mediating an effective electron-electron interaction, can play a crucial role in selecting various electronic phases. In this study, we derive the full electron-phonon coupling (EPC) vertex from the microscopic tight-binding lattice, and identify the significance of each phonon mode. We then project the EPC vertices onto the topological heavy fermion (THF) basis [Song and Bernevig, Phys. Rev. Lett. 129, 047601 (2022)], and show that an anti-Hund's interaction $\hat{H}_{\rm A}$ is induced on each moir\'e-scale local $f$-orbital, with strengths 1 to 4 meV. We analyze the phonon-induced multiplet splittings, which can significantly affect the local correlation. As an example, we elaborate on the phonon-favored symmetry-breaking orders at even-integer fillings. Through systematic self-consistent Hartree-Fock calculations, we uncover a tight competition between $\Gamma$-phonon-favored orbital orders, $K$-phonon-favored inter-valley coherent orders, and the kinetic and Coulomb-favored orders. Contrary to EPC, the carbon atom Hubbard repulsion induces an on-$f$-site Hund's interaction $\hat{H}_{\rm H}$ with strengths 1 to 3 meV that partly counteracts the effect of $\hat{H}_{\rm A}$. The combined influence of $\hat{H}_{\rm A,H}$ on the multiplet splitting and symmetry-breaking states is discussed. In the end, we explore the possibility of finding an exotic Dirac semi-metal formed solely by $c$-electrons at the charge-neutrality point, while $f$-impurities exhibit a symmetric Mott gap by forming non-degenerate singlets under $\hat{H}_{\rm A,H}$. Experimental features that distinguish such a state are discussed.