  Based on the Dirac equations in the two-dimensional $\pi-$ flux model, we
study the interaction effects both in nontrivial gapped and gapless Dirac
equations with numerical exact diagonalization method. In the presence of the
nearest and next nearest neighbor interactions: for nontrivial gapped Dirac
equation, the topological phase is robust and persists in a finite region of
the phase diagram; while for gapless Dirac equation, charge-density-wave and
stripe phases are identified and the phase diagram in $(V_1, V_2)$ plane is
obtained. When the next-next-nearest neighbor interaction is further included
to gapless Dirac equation, the topological phase expected in the mean-field
theory is absent. Our results are related to the possibility of dynamically
generating topological phase from the electronic correlations.
