Phonon-mediated nematic nodal superconductivity in twisted bilayer graphene

Recently, a topological heavy fermion model for the magic-angle twisted bilayer graphene has been proposed, which enlightens that in the heavy Fermi liquid (hFL) regime, the strong Coulomb repulsion is screened and gives way to, e.g., phonon induced, attractive interactions to mediate Cooper pairs. In the present work, we investigate superconducting instabilities of the fully symmetric hFL and various valley-ordered hFL's, where valley-polarized (VP), inter-valley coherent (IVC), and Kramer's inter-valley coherent (KIVC) orders are assumed respectively. As the kinetic energy of heavy electrons is small compared to interaction, a strong coupling picture applies - local Cooper pairs form at a higher energy scale and phase coherence developes at a relatively lower energy scale. Phase diagrams are numerically obtained for various parameters and match well with the strong coupling expansion. We find that the fully symmetric hFL favors the s-wave spin-singlet pairing that opens a full gap; while the valley-ordered hFL's - in a wide range of realistic parameters - can favor nematic nodal pairings, where the C_2zT symmetry is preserved but C_6z is spontaneously broken. In particular, in the VP and KIVC phases, the nematic pairing is a p-wave-like spin-singlet, and we mathematically prove that 2+4n (n∈Z) pairing nodes are enforced on the inner Fermi surface by the C_2zT symmetry and a π Berry phase. In the IVC phase, the nematic pairing is generally (s+d)-wave-like spin-singlet, and the existence of nodes is not topologically protected, but depends on interaction details.