  We propose an experimental setup using ultracold atoms to implement a bilayer
honeycomb lattice with Bernal stacking. In presence of a potential bias between
the layers and at low densities, Fermions placed in this lattice form an
annular Fermi sea. The presence of two Fermi surfaces leads to interesting
patterns in Friedel oscillations and RKKY interactions in presence of
impurities. Furthermore, a repulsive fermion-fermion interaction leads to a
Stoner instability towards an incommensurate spin-density-wave order with a
wave vector equal to the thickness of the Fermi sea. The instability occurs at
a critical interaction strength which goes down with the density of the
fermions. We find that the instability survives interaction renormalization due
to vertex corrections and discuss how this can be seen in experiments. We also
track the renormalization group flows of the different couplings between the
fermionic degrees of freedom, and find that there are no perturbative
instabilities, and that Stoner instability is the strongest instability which
occurs at a critical threshold value of the interaction. The critical
interaction goes to zero as the chemical potential is tuned towards the band
bottom.
