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We study the one-band Hubbard model on the honeycomb lattice using acombination of quantum Monte Carlo (QMC) simulations and static as well asdynamical mean-field theory (DMFT). This model is known to show a quantum phasetransition between a Dirac semi-metal and the antiferromagnetic insulator. Theaim of this article is to provide a detailed comparison between theseapproaches by computing static properties, notably ground-state energy,single-particle gap, double occupancy, and staggered magnetization, as well asdynamical quantities such as the single-particle spectral function. At thestatic mean-field level local moments cannot be generated without breaking theSU(2) spin symmetry. The DMFT approximation accounts for temporal fluctuations,thus captures both the evolution of the double occupancy and the resultinglocal moment formation in the paramagnetic phase. As a consequence, the DMFTapproximation is found to be very accurate in the Dirac semi-metallic phasewhere local moment formation is present and the spin correlation length small.However, in the vicinity of the fermion quantum critical point the spincorrelation length diverges and the spontaneous SU(2) symmetry breaking leadsto low-lying Goldstone modes in the magnetically ordered phase. The impact ofthese spin fluctuations on the single-particle spectral function --\textit{waterfall} features and narrow spin-polaron bands -- is only visible inthe lattice QMC approach.

The Hubbard model on the honeycomb lattice: from static and dynamical mean-field theories to lattice quantum Monte Carlo simulations