We investigate the two-dimensional Hubbard model with the long-range Coulomb interaction. We assume the form of the effective Coulomb interaction, which is treated as a perturbation. For the half-filling case, the self-energy is numerically evaluated up to the second-order terms of the perturbation expansion with the full momentum dependence included. The k -mass is calculated from this self-energy. It is found that the first-order term makes a dominant contribution to the reduction of the k -mass.
We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as
GW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of
GW in
GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.