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We investigate the 5d transition metal oxide BaOsO$_3$ within a combinationof density functional theory (DFT) and dynamical mean-field theory (DMFT),using a matrix-product-state impurity solver. BaOsO$_3$ has 4 electrons in thet$_{2g}$ shell akin to ruthenates but stronger spin-orbit coupling (SOC) and isthus expected to reveal an interplay of Hund's metal behavior with SOC. Weexplore the paramagnetic phase diagram as a function of SOC and Hubbardinteraction strengths, identifying metallic, band (van-Vleck) insulating andMott insulating regions. At the physical values of the two couplings we findthat BaOsO$_3$ is still situated inside the metallic region and has a moderatequasiparticle renormalization $m^*/m \approx 2$; consistent with specific heatmeasurements. SOC plays an important role in suppressing electroniccorrelations (found in the vanishing SOC case) through the splitting of avan-Hove singularity (vHs) close to the Fermi energy, but is insufficient topush the material into an insulating van-Vleck regime. In spite of the strongeffect of SOC, BaOsO$_3$ can be best pictured as a moderately correlated Hund'smetal.

BaOsO$_3$: A Hund's metal in the presence of strong spin-orbit coupling