  The Quantum Monte Carlo simulations of the ionic Hubbard model on a
two-dimensional square lattice at half filling were performed. The method based
on the direct-space, proposed by Suzuki and al., Hirsch and al., was used.
Cycles of increasing and decreasing values of the Coulomb interaction $ U $
were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at
low temperature, the two insulator phases are separated by a metallic phase for
weak to intermediate values of the staggered potential $ \Delta $. For large
Coulomb repulsion the system is in a Mott insulator with an antiferromagnetism
order. On increasing and decreasing the Coulomb interaction $ U $ the
metal-Mott insulator transition shows an hysteresis phenomenon while the
metal-band insulator transition is continue. For large $ \Delta $ it seems that
the metallic region shrinks to a single metallic point. However, the band
insulator to the Mott insulator transition is not direct for the studied model.
A phase diagram is drawn for the temperature $ kT=0.01 $. For $ \Delta =0.5 $
cycles of increasing and decreasing temperature were programmed for different
values of the Coulomb interaction $ U $ . A behaviour change appears for $
U\simeq 1.75 $. This suggests that a crossover line divides the metallic region
of the phase diagram.
