Temperature dependence of antiferromagnetic order in the Hubbard model

We suggest a method for an approximative solution of the two dimensional Hubbard model close to half filling. It is based on partial bosonisation, supplemented by an investigation of the functional renormalisation group flow. The inclusion of both the fermionic and bosonic fluctuations leads in lowest order to agreement with the Hartree-Fock result or Schwinger-Dyson equation and cures the ambiguity of mean field theory . We compute the temperature dependence of the antiferromagnetic order parameter and the gap below the critical temperature. We argue that the Mermin-Wagner theorem is not practically applicable for the spontaneous breaking of the continuous spin symmetry in the antiferromagnetic state of the Hubbard model. The long distance behavior close to and below the critical temperature is governed by the renormalisation flow for the effective interactions of composite Goldstone bosons and deviates strongly from the Hartree-Fock result.