  Many-body instabilities of the half-filled honeycomb bilayer are studied
using weak coupling renormalization group as well as strong coupling expansion.
For spinless fermions and assuming parabolic degeneracy, there are
4-independent four-fermion contact couplings. While the dominant instability
depends on the microscopic values of the couplings, the broken symmetry state
is typically a gapped insulator with either broken inversion symmetry or broken
time reversal symmetry, with a quantized anomalous Hall effect. Under certain
conditions, the dominant instability may appear in the particle-particle
(pairing) channel. For some non-generic fine-tuned initial conditions, weak
coupling RG trajectories flow into the non-interacting fixed point, although
generally we find runaway flows which we associate with ordering tendencies.
Additionally, a tight binding model with nearest neighbor hopping and nearest
neighbor repulsion is studied in weak and strong couplings and in each regime a
gapped phase with inversion symmetry breaking is found. In the strong coupling
limit, the ground state wavefunction is constructed for vanishing in-plane
hopping but finite inter-plane hopping, which explicitly displays the broken
inversion symmetry and a finite difference between the number of particles on
the two layers. Finally, we discuss the spin-1/2 case and use Fierz identities
to show that the number of independent 4-fermion contact couplings is 9. The
corresponding RG equations in the spin-1/2 case are also presented, and used to
show that, just as in strong coupling, the most dominant weak coupling
instability of the repulsive Hubbard model (at half-filling) is an
anti-ferromagnet.
