Electronic Instabilities of the AA-Honeycomb Bilayer

We use a functional renormalization group approach to study the instabilities due to electron-electron interactions in a bilayer honeycomb lattice model with AA stacking, as it might be relevant for layered graphene with this structure. Starting with a tight- binding description for the four $\pi$-bands, we integrate out the modes of the dispersion by successively lowering an infrared cutoff and determine the leading tendencies in the effective interactions. The antiferromagnetic spin-density wave is an expected instability for dominant local repulsion among the electrons, but for nonlocal interaction terms also other instabilities occur. We discuss the phase diagrams depending on the model parameters. We compare our results to single-layer graphene and the more common AB-stacked bilayer, both qualitatively and quantitatively.