  The Hubbard model represents the fundamental model for interacting quantum
systems and electronic correlations. Using the two-dimensional half-filled
Hubbard model at weak coupling as a testing ground, we perform a comparative
study of a comprehensive set of state of the art quantum many-body methods.
Upon cooling into its insulating antiferromagnetic ground-state, the model
hosts a rich sequence of distinct physical regimes with crossovers between a
high-temperature incoherent regime, an intermediate temperature metallic regime
and a low-temperature insulating regime with a pseudogap created by
antiferromagnetic fluctuations. We assess the ability of each method to
properly address these physical regimes and crossovers through the computation
of several observables probing both quasiparticle properties and magnetic
correlations, with two numerically exact methods (diagrammatic and
determinantal quantum Monte Carlo) serving as a benchmark. By combining
computational results and analytical insights, we elucidate the nature and role
of spin fluctuations in each of these regimes. Based on this analysis, we
explain how quasiparticles can coexist with increasingly long-range
antiferromagnetic correlations, and why dynamical mean-field theory is found to
provide a remarkably accurate approximation of local quantities in the metallic
regime. We also critically discuss whether imaginary time methods are able to
capture the non-Fermi liquid singularities of this fully nested system.
