Putting modern many-body methods to the test: the two-dimensional Hubbard model at weak coupling

We provide a detailed synopsis and comparison of a comprehensive set of state-of-the art many body techniques for the weak-coupling regime of the two dimensional half-filled Hubbard model on a square lattice. We put each of these methods to the test, for both one- and two-particle observables, in relation to the salient physical crossovers of this model: upon cooling from the high-temperature incoherent regime, coherent quasiparticles are formed below T_QP. At lower T, magnetic correlations stemming from the antiferromagnetically ordered phase at T=0 are gradually enhanced, resulting in the opening of an electronic (pseudo-)gap at T_*. By covering the range of modern many-body techniques available today, from numerically exact benchmark methods (determinantal and diagrammatic QMC [CDet]), over (dynamical) mean field theory (RPA, DMFT) and its cluster (DCA) and vertex based extensions (DΓA, TRILEX, DF, DB) to well-known approximations like parquet approximation (PA), the two-particle-self-consistent approach (TPSC) and the functional renormalization group (fRG), the realm of their applicability is put into perspective and a reference and agenda for future improvements is provided.