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

Flat bands in magic-angle twisted bilayer graphene serve as the arena where exotic correlation physics unfolds. Previous studies have shown that phonons, through mediating an effective electron-electron interaction, can play a crucial role in determining various electronic phases. In this study, we project the full electron-phonon coupling vertex, derived from microscopic tight-binding lattice calculations, onto the basis of the topological heavy fermion model [Song and Bernevig, PRL 129, 047601 (2022)], and identify the significance of each phonon mode. With phonon fields integrated out, an on-site anti-Hund's interaction ĤA on the moir\'e-scale local f-orbitals is obtained, with strengths 1 to 4 meV. We solve the phonon-induced multiplet splittings, and then elaborate on phonon-favored symmetry-breaking orders at even-integer fillings. Through systematic self-consistent Hartree-Fock calculations, we uncover a tight competition between Γ-phonon-favored orbital orders, K-phonon-favored inter-valley coherent orders, and the kinetic and Coulomb-favored orders. A Hund's interaction ĤH with strengths 1 to 3 meV, which originates from the carbon atom Hubbard, and partly counteracts the effect of ĤA, is also discussed. In the end, we 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 ĤA,H. Experimental features that distinguish such a state are discussed.