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The Quantum Monte-Carlo simulations of the two-dimensional Hubbard model arepresented for the half filling. The method based on the direct-space proposedby Suzuki and al., and Hirsch and al. was used. The states generated by thismethod are basis states in occupation number representation built with Wannierstates localised on each site of the square array. The configurations offermions can be observed on the real 2D array. An antiferromagnetic factor isdefined and calculated for each temperature. The curves of energy, specificheat, conducivity and antiferromagnetic factor are presented for differentvalues of the repulsive coulombian on site interaction U. There is ametal-insulator transition at low temperature for small values of U. Thistransition corresponds with a paramagnetic-ferromagnetic first ordertransition. Indeed, for these interaction values, the energy curves show a gapwhich is a characteristic of a first order transition. An hysteresis phenomenonappears 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 withoutobservable effect on the energy and the specific heat. The metal-insulatortransition does not exist any more, the conductivity stays very small.Isotherms of the physical quantities versus U/t show a transition which seemsto be the metal-insulator Mott transition. These results allow to draw a phasediagram with two first order transition lines.

Simulation Monte-Carlo du Mod\`ele de Hubbard \`a deux dimensions