  We use exact diagonalization and cluster perturbation theory to address the
role of strong interactions and quantum fluctuations for spinless fermions on
the honeycomb lattice. We find quantum fluctuations to be very pronounced both
at weak and strong interactions. A weak second-neighbor Coulomb repulsion $V_2$
induces a tendency toward an interaction-generated quantum anomalous Hall
phase, as borne out in mean-field theory. However, quantum fluctuations prevent
the formation of a stable quantum Hall phase before the onset of the
charge-modulated phase predicted at large $V_2$ by mean-field theory.
Consequently, the system undergoes a direct transition from the semimetal to
the charge-modulated phase. For the latter, charge fluctuations also play a key
role. While the phase, which is related to pinball liquids, is stabilized by
the repulsion $V_2$, the energy of its low-lying charge excitations scales with
the electronic hopping $t$, as in a band insulator.
