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The dual boson approach [Ann. Phys. 327, 1320 (2012)] provides a means toconstruct a diagrammatic expansion around the extended dynamical mean-fieldtheory (EDMFT). In this paper, we present the numerical implementation of theapproach and apply it to the extended Hubbard model with nearest-neighborinteraction $V$. We calculate the EDMFT phase diagram and study the effect ofdiagrams beyond EDMFT on the transition to the charge-ordered phase. Includingdiagrammatic corrections to the EDMFT polarization shifts the EDMFT phaseboundary to lower values of $V$. The approach interpolates between the randomphase approximation in the weak coupling limit and EDMFT for strong coupling.Neglecting vertex corrections leads to results reminiscent of the EDMFT+$GW$approximation. We however find significant deviations from the dual bosonresults already for small values of the interaction, emphasizing the crucialimportance of fermion-boson vertex corrections.

Beyond extended dynamical mean-field theory: Dual boson approach to the two-dimensional extended Hubbard model