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Many-body instabilities of the half-filled honeycomb bilayer are studiedusing weak coupling renormalization group as well as strong coupling expansion.For spinless fermions and assuming parabolic degeneracy, there are4-independent four-fermion contact couplings. While the dominant instabilitydepends on the microscopic values of the couplings, the broken symmetry stateis typically a gapped insulator with either broken inversion symmetry or brokentime reversal symmetry, with a quantized anomalous Hall effect. Under certainconditions, the dominant instability may appear in the particle-particle(pairing) channel. For some non-generic fine-tuned initial conditions, weakcoupling RG trajectories flow into the non-interacting fixed point, althoughgenerally we find runaway flows which we associate with ordering tendencies.Additionally, a tight binding model with nearest neighbor hopping and nearestneighbor repulsion is studied in weak and strong couplings and in each regime agapped phase with inversion symmetry breaking is found. In the strong couplinglimit, the ground state wavefunction is constructed for vanishing in-planehopping but finite inter-plane hopping, which explicitly displays the brokeninversion symmetry and a finite difference between the number of particles onthe two layers. Finally, we discuss the spin-1/2 case and use Fierz identitiesto show that the number of independent 4-fermion contact couplings is 9. Thecorresponding RG equations in the spin-1/2 case are also presented, and used toshow that, just as in strong coupling, the most dominant weak couplinginstability of the repulsive Hubbard model (at half-filling) is ananti-ferromagnet.

Interacting fermions on the honeycomb bilayer: from weak to strong coupling