  Numerical results for ground state and excited state properties (energies,
double occupancies, and Matsubara-axis self energies) of the single-orbital
Hubbard model on a two-dimensional square lattice are presented, in order to
provide an assessment of our ability to compute accurate results in the
thermodynamic limit. Many methods are employed, including auxiliary field
quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of
dual fermions, density matrix embedding theory, density matrix renormalization
group, dynamical cluster approximation, diffusion Monte Carlo within a fixed
node approximation, unrestricted coupled cluster theory, and multi-reference
projected Hartree-Fock. Comparison of results obtained by different methods
allows for the identification of uncertainties and systematic errors. The
importance of extrapolation to converged thermodynamic limit values is
emphasized. Cases where agreement between different methods is obtained
establish benchmark results that may be useful in the validation of new
approaches and the improvement of existing methods.
