Skeleton diagrammatic expansions with screened Hubbard interaction versus multivaluedness of the Luttinger-Ward functional

We systematically study properties of high-order bold-diagrammatic expansions, which may converge to unphysical answers for the Hubbard interaction due to the multivaluedness of the Luttinger-Ward functional (LWF), by the prototypical example of the Hubbard atom. The diagrammatic Monte Carlo method with fully dressed propagator G is adopted to generate the high-order series. By varying the level of screening in the interaction line; bare U, random-phase approximation W_rpa, and fully dressed W_exact, we present the convergence properties of the different bold series in connection with multiple branches of the LWF. In particular, we find that the bold-GW_exact and bold-GW_rpa series diverge well below the branching point of the LWF, but admit the analytic continuation beyond their convergence radius by standard techniques. We further explore the possibility of using the bold diagrammatic series in the strongly correlated regime to obtain precise results with controlled accuracy.