  We investigate the fate of interaction driven phases in the half-filled
honeycomb lattice for finite systems via exact diagonalization with nearest and
next nearest neighbour interactions. We find evidence for a charge density wave
phase, a Kekul\'e bond order and a sublattice charge modulated phase in
agreement with previously reported mean-field phase diagrams. No clear sign of
an interaction driven Chern insulator phase (Haldane phase) is found despite
being predicted by the same mean-field analysis. We characterize these phases
by their ground state degeneracy and by calculating charge order and bond order
correlation functions.
