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

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.