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The question of spatial locality of electronic correlations beyond GWapproximation is one of the central issues of the famous combination of GW anddynamical mean field theory, GW+DMFT. In this work, the above question isaddressed directly (for the first time) by performing calculations with andwithout assumption of locality of the corresponding diagrams. For this purposewe use sc(GW+G3W2) approach where the higher order part (G3W2) is evaluatedwith fully momentum dependent Green's function G and screened interaction W andwith "local" variant, where the single site approximation is assumed for both Gand W. For all three materials studied in this work (NiO, $\alpha$-Ce, LiFeAs),we have found the spatial non-locality effects to be strong. For NiO and LiFeAsthey, in fact, are decisive for the proper evaluation of vertex corrections.The results of this study have direct impact on our understanding ofapproximations made in practical implementations of GW+DMFT method, where alldiagrams beyond GW (DMFT part) are assumed to be local. Taking into account thefact that the first diagrams beyond GW represent the most importantcontribution also in GW+DMFT calculations, we conclude that the basicassumption of GW+DMFT, namely the locality of diagrams evaluated in the DMFTpart, is not as good as it is believed to be.

Spatial non-locality of electronic correlations beyond GW approximation