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We study fermions in a lattice, with on-site and nearest neighbor attractiveinteractions between two spin species. We consider two geometries: both spinsin a triangular lattice, and a mixed geometry with up-spins in honeycomb anddown-spins in triangular lattices. We focus on the interplay betweenspin-population imbalance, on-site and valence bond pairing, and orderparameter symmetry. The mixed geometry leads to a rich phase diagram oftopologically non-trivial phases. In both geometries, we predict orderparameters with simultaneous time-reversal and translational symmetry breaking.

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