Can vertex-corrected perturbative approaches be improved through symmetry breaking?

Many-body perturbation theory (MBPT) approaches are ubiquitous in the study of materials. The most extended such approach is arguably the GW approximation, which includes classical, electrostatic screening on top of a mean-field reference, but neglects multi-quasiparticle interactions. In principle, such terms can be recovered by introducing non-trivial components of the interaction vertex Γ, set to zero in GW. However, it is only recently that systematic (in the Hedin sense) implementations of vertex-corrected GWΓ have started to be applied in ab initio calculations. Hence, important questions regarding the computational and physical nature of these approaches remain open. In this talk, we address one such issue: the impact of the mean-field reference on the MBPT result, particularly regarding symmetry broken (SB) solutions, in the context of self-consistent GWΓ (scGWΓ). The introduction of symmetry breaking formally absent in the true many-body ground state can improve mean-field approximations. However, scGW cannot typically recover the symmetry, resulting in ultimately worse results than the SB mean-field. We test whether it is possible to formulate vertex corrections that recover symmetries at self-consistency, thus allowing the use of SB mean-field reference states.