Antiferromagnetic symmetry breaking in the half filled Hubbard model in infinite dimensions

We study the half filled Hubbard model on a hypercubic lattice in infinite dimensions in the presence of a staggered magnetic field. An exact Ward-identity between vertex functions and self-energies is derived, that holds in any phase without broken symmetry for all values of $U$. Making the reasonable assumptions that for small enough on-site repulsion $U$ the high-temperature phase is a Fermi liquid, and that in the weak coupling regime the effective Anderson impurity model can be studied perturbatively, we proof that Hartree-Fock theory and the random-phase approximation are very accurate for small $U$, and that the system develops long-range antiferromagnetic order at a finite temperature.