  Twisted bilayer graphene exhibits a panoply of many-body phenomena that are
intimately tied to the appearance of narrow and well isolated electronic bands
near magic-angle. The microscopic ingredients that are responsible for the
complex experimental phenomenology include electron-electron (phonon)
interactions and non-trivial Bloch wavefunctions associated with the narrow
bands. Inspired by recent experiments, we focus here on an interplay of two
independent interaction-induced phenomena on superconductivity. We analyze the
combined effects of Coulomb interaction driven band-flattening and
phonon-mediated attraction due to the exchange of multiple electron-phonon
umklapp processes, as a function of filling and twist angle. The former leads
to a filling-dependent enhancement of the renormalized density of states, which
contributes to a robust increase in the tendency towards pairing in a range of
angles near magic-angle. In addition, the minimal spatial extent associated
with the Wannier functions develops a non-trivial enhancement as a result of
these many-body renormalizations, which can further contribute towards
stabilizing the superconducting state over a wider range of fillings and
twist-angles.
