  We consider two-dimensional Fermi systems with quadratic band touching and
$C_3$ symmetry, as realizable in Bernal-stacked honeycomb bilayers. Within a
renormalization-group analysis, we demonstrate the existence of a quantum
critical point at a finite value of the density-density interactions,
separating a semimetallic disordered phase at weak coupling from a gapped
ordered phase at strong coupling. The latter may be characterized by, for
instance, antiferromagnetic, quantum anomalous Hall, or charge density wave
order. In the semimetallic phase, each point of quadratic band touching splits
into four Dirac cones as a consequence of the nontrivial interaction-induced
self-energy correction, which we compute to the two-loop order. We show that
the quantum critical point is in the $(2+1)$-dimensional Gross-Neveu
universality class characterized by emergent Lorentz invariance and a dynamic
critical exponent $z=1$. At finite temperatures, $T > 0$, we hence conjecture a
crossover between $z=2$ at intermediate $T$ and $z=1$ at low $T$, and we
construct the resulting nontrivial phase diagram as function of coupling
strength and temperature.
