On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

TL;DR

This paper benchmarks various approximate diagrammatic methods against diagrammatic Monte Carlo for the 2D Hubbard model in the weak-coupling regime, revealing limitations of common schemes and proposing improved hybrid approaches.

Contribution

It systematically compares diagrammatic approximation schemes to Monte Carlo results, highlighting their inaccuracies and suggesting more reliable hybrid methods like GW+DMFT with restricted nonlocal diagrams.

Findings

Dynamical mean-field theory accurately captures local self-energy.

Partial summation schemes like GW and FLEX often perform worse than second order perturbation.

Hybrid GW+DMFT with restricted nonlocal diagrams improves results.

Abstract

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.