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We study spontaneous symmetry breaking in a system of spinless fermions inthe Honeycomb lattice paying special emphasis to the role of an enlarged unitcell on time reversal symmetry broken phases. We use a tight binding model withnearest neighbor hopping t and Hubbard interaction V1 and V2 and extract thephase diagram as a function of electron density and interaction within a meanfield variational approach. The analysis completes the previous work done inPhys. Rev. Lett. 107, 106402 (2011) where phases with non--trivial topologicalproperties were found with only a nearest neighbor interaction V1 in theabsence of charge decouplings. We see that the topological phases aresuppressed by the presence of metallic charge density fluctuations. Theaddition of next to nearest neighbor interaction V2 restores the topologicalnon-trivial phases.

Charge instabilities and topological phases in the extended Hubbard model on the honeycomb lattice with enlarged unit cell