Topological Fermi liquids in the doped Honeycomb lattice

We study spontaneous symmetry broken phases of spinless electrons  in the Honeycomb lattice using a tight binding model with nearest neighbors Coulomb interactions $V$. When the unit cell is enlarged to allow non-zero, selfconsistent currents inside it we find within a variational mean field approach a metallic phase with broken time reversal symmetry T around 1/3 filling, very close in parameter space to a Pomeranchuk instability. Within the T broken region the predominant configuration is an anomalous Hall phase with non zero Hall conductivity, a realization of a topological Fermi liquid. A T broken phase with zero Hall conductivity is stable in a small region of the parameter space for lower values of $V$.