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We propose an experimental setup using ultracold atoms to implement a bilayerhoneycomb lattice with Bernal stacking. In presence of a potential bias betweenthe layers and at low densities, Fermions placed in this lattice form anannular Fermi sea. The presence of two Fermi surfaces leads to interestingpatterns in Friedel oscillations and RKKY interactions in presence ofimpurities. Furthermore, a repulsive fermion-fermion interaction leads to aStoner instability towards an incommensurate spin-density-wave order with awave vector equal to the thickness of the Fermi sea. The instability occurs ata critical interaction strength which goes down with the density of thefermions. We find that the instability survives interaction renormalization dueto vertex corrections and discuss how this can be seen in experiments. We alsotrack the renormalization group flows of the different couplings between thefermionic degrees of freedom, and find that there are no perturbativeinstabilities, and that Stoner instability is the strongest instability whichoccurs at a critical threshold value of the interaction. The criticalinteraction goes to zero as the chemical potential is tuned towards the bandbottom.

Bilayer honeycomb lattice with ultracold atoms: Multiple Fermi surfaces and incommensurate spin density wave instability