Critical properties of the half-filled Hubbard model in three dimensions

By means of the dynamical vertex approximation (D$\Gamma$A) we include spatial correlations on all length scales beyond the dynamical mean field theory (DMFT) for the half-filled Hubbard model in three dimensions. The most relevant changes due to non-local fluctuations are: (i) a deviation from the mean-field critical behavior with the same critical exponents as for the three dimensional Heisenberg (anti)-ferromagnet and (ii) a sizable reduction of the N\'eel temperature ($T_N$) by $\sim 30$ for the onset of antiferromagnetic order. Finally, we give a quantitative estimate of the deviation of the spectra between D$\Gamma$A and DMFT in different regions of the phase-diagram.