  The Quantum Monte-Carlo simulations of the two-dimensional Hubbard model are
presented for the half filling. The method based on the direct-space proposed
by Suzuki and al., and Hirsch and al. was used. The states generated by this
method are basis states in occupation number representation built with Wannier
states localised on each site of the square array. The configurations of
fermions can be observed on the real 2D array. An antiferromagnetic factor is
defined and calculated for each temperature. The curves of energy, specific
heat, conducivity and antiferromagnetic factor are presented for different
values of the repulsive coulombian on site interaction U. There is a
metal-insulator transition at low temperature for small values of U. This
transition corresponds with a paramagnetic-ferromagnetic first order
transition. Indeed, for these interaction values, the energy curves show a gap
which is a characteristic of a first order transition. An hysteresis phenomenon
appears on the conductivity curves. There is a behaviour change for U/t=3.5.
For the values U>3.5 there is ferromagnetic-paramagnetic change without
observable effect on the energy and the specific heat. The metal-insulator
transition does not exist any more, the conductivity stays very small.
Isotherms of the physical quantities versus U/t show a transition which seems
to be the metal-insulator Mott transition. These results allow to draw a phase
diagram with two first order transition lines.
