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The electronic structure of a LaNiO$_3$ bilayer grown along the [111]direction and confined between insulating layers of LaAlO$_3$ is theoreticallyinvestigated using a combination of first principle calculations and effectivemulti-orbital lattice models. The LDA band structure is well reproduced by atight-binding model for the Ni-$e_g$ orbitals defined on the buckled honeycomblattice. We highlight peculiar properties of this model which include almostflat bands as well as linear and quadratic band crossing points. The effect oflocal correlations is discussed within the LDA$+U$ scheme and within theHartree-Fock approximation for interacting multi-orbital lattice models. Over awide range of interaction parameters we find that a ferromagnetic phase isenergetically favored. We discuss the possibility of additional orbital orderwhich could stabilize a spontaneous Chern insulator with chiral edge modes or astaggered orbital phase with a $\sqrt{3}\times\sqrt{3}$ reconstruction of theunit cell. By studying an interacting nickel-oxygen lattice model we find thatthe stability of these orbitally ordered phases also depends on the value ofthe charge-transfer energy. Controlling the charge-transfer energy mighttherefore be an important step towards engineering exotic electronic phases incertain classes of oxide heterostructures.

Electronic structure of (LaNiO$_3$)$_2$/(LaAlO$_3$)$_N$ heterostructures grown along [111]