Topological States with Broken Translational and Time-Reversal Symmetries in a Honeycomb-Triangular Lattice

We study fermions in a lattice, with on-site and nearest neighbor attractive interactions between two spin species. We consider two geometries: both spins in a triangular lattice, and a mixed geometry with up-spins in honeycomb and down-spins in triangular lattices. We focus on the interplay between spin-population imbalance, on-site and valence bond pairing, and order parameter symmetry. The mixed geometry leads to a rich phase diagram of topologically non-trivial phases. In both geometries, we predict order parameters with simultaneous time-reversal and translational symmetry breaking.