I think you need to be a bit more specific. Your question doesn't depend only on the lattice geometry and the fact that you are considering the "occupation/filling" of bands by fermions, but on the form of the interactions you are considering in the model for your system. In other words, you need to also supply the Hamiltonian for the system before anyone can help you address this question. Are you using a tight binding model? If so, are the nearest neighbour hopping overlaps (t coefficients) associated with the two discretely translationally invariant directions of space equal? Are there next nearest neighbour interactions etc?