Competing instabilities in a two band Hubbard model on a square lattice

We study a two band Hubbard model on a two dimensional square lattice. In particular, we focus on the cases wherein one band is doped to have a small electron pocket while the other band is doped to have a hole pocket and the Fermi lines of these two pockets are nearly nested. Similar models have been studied extensively in the context related to the Iron-based material where the interactions between electrons are always repulsive. Here we investigate the generalized cases that the interactions between the fermions within the same band $U_1$ and $U_2$ and the interactions between electrons in different bands $U_{12}$ can be tuned independently. Such models can potentially be realized in a cold atom system where the manipulation of the interaction is possible by taking advantage of the Feshbach resonance. The freedom of tuning the strength and the sign (repulsive or attractive) of the interactions, combined with the nearly nested Fermi lines, allows both the density wave phases and the pairing phases to be potential candidates for the ground state. We employ the functional renormalization group approach so that we can investigate the competition between these possible instabilities on an equal footing.