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