  We revisit the effect of local interactions on the quadratic band touching
(QBT) of Bernal stacked bilayer graphene models using renormalization group
(RG) arguments and quantum Monte Carlo simulations of the Hubbard model. We
present an RG argument which predicts, contrary to previous studies, that weak
interactions do not flow to strong coupling even if the free dispersion has a
QBT. Instead they generate a linear term in the dispersion, which causes the
interactions to flow back to weak coupling. Consistent with this RG scenario,
in unbiased quantum Monte Carlo simulations of the Hubbard model we find
compelling evidence that antiferromagnetism turns on at a finite $U/t$, despite
the $U=0$ hopping problem having a QBT. The onset of antiferromagnetism takes
place at a continuous transition which is consistent with a dynamical critical
exponent $z=1$ as expected for 2+1 d Gross-Neveu criticality. We conclude that
generically in models of bilayer graphene, even if the free dispersion has a
QBT, small local interactions generate a Dirac phase with no symmetry breaking
and that there is a finite-coupling transition out of this phase to a
symmetry-broken state.
