  Using a controlled weak-coupling renormalization group approach, we establish
the mechanism of unconventional superconductivity in the vicinity of spin or
charge ordered excitonic states for the case of electrons on the Bernal stacked
bilayer honeycomb lattice. With one electron per site this system exhibits
nearly parabolically touching conduction and valence bands. Such a state is
unstable towards a spontaneous symmetry breaking, and repulsive interactions
favor excitonic order, such as a charge nematic and/or a layer antiferromagnet.
We find that upon adding charge carriers to the system, the excitonic order is
suppressed, and unconventional superconductivity appears in its place, before
it is replaced by a Fermi liquid. We focus on firmly establishing this
phenomenon using the RG formalism within an idealized model with parabolic
touching.
