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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 H ₀ 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 -phonon-favored orbital orders, K-phonon-favored inter-valley coherent orders, and the kinetic and Coulomb-favored orders. A Hund's interaction H ₇ with strengths 1 to 3 meV, which originates from the carbon atom Hubbard, and partly counteracts the effect of H ₀, 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 H ₀, ₇. Experimental features that distinguish such a state are discussed.
Wang et al. (Mon,) studied this question.