  Motivated by the physics of spin-orbital liquids, we study a model of
interacting Dirac fermions on a bilayer honeycomb lattice at half filling,
featuring an explicit global SO(3)$\times$U(1) symmetry. Using large-scale
auxiliary-field quantum Monte Carlo (QMC) simulations, we locate two
zero-temperature phase transitions as function of increasing interaction
strength. First, we observe a continuous transition from the weakly-interacting
semimetal to a different semimetallic phase in which the SO(3) symmetry is
spontaneously broken and where two out of three Dirac cones acquire a mass gap.
The associated quantum critical point can be understood in terms of a
Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition towards
an insulating phase in which the SO(3) symmetry is restored and the U(1)
symmetry is spontaneously broken. While strongly first order at the mean-field
level, the QMC data is consistent with a direct and continuous transition. It
is thus a candidate for a new type of deconfined quantum critical point that
features gapless fermionic degrees of freedom.
