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We investigate the interplay of spin-orbit coupling, electronic correlations,and lattice distortions in the $5d^1$ double perovskite Ba$_2$MgReO$_6$.Combining density-functional theory (DFT) and dynamical mean-field theory(DMFT), we establish the Mott-insulating character of Ba$_2$MgReO$_6$ in bothits cubic and tetragonal paramagnetic phases. Despite substantial spin-orbitcoupling, its impact on the formation of the insulating state is minimal,consistent with theoretical expectations for $d^1$ systems. We furthercharacterize the electronic properties of the cubic and tetragonal phases byanalyzing spectral functions and local occupations in terms of multipolemoments centered on the Re sites. Our results confirm the presence offerroically ordered $z^2$ quadrupoles in addition to the antiferroic$x^2-y^2$-type order. We compare two equivalent but complementary descriptionsin terms of either effective Re-\ttg frontier orbitals or more localizedatomic-like Re-d and O-p orbitals. The former maps directly on a physicallyintuitive picture in terms of nominal $d^1$ Re cations, while the latterexplicitly demonstrates the role of hybridization with the ligands in thespin-orbit splitting and the formation of the charge quadrupoles around the Resites. Finally, we compare our DFT+DMFT results with a previous DFT+$U$ studyof the tetragonal paramagnetic state. We find good qualitative agreement forthe dominant charge quadrupoles, but also notable differences in thecorresponding spectral functions, underscoring the need for more comparativestudies between these two methods.

Probing the Mott-insulating behavior of Ba$_2$MgReO$_6$ with DFT+DMFT