By means of the dynamical vertex approximation ($\mathrm{D}\Gamma \mathrm{A}$) 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 nonlocal 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 Néel 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 $\mathrm{D}\Gamma \mathrm{A}$ and DMFT in different regions of the phase diagram.

• Received 5 April 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.256402