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Electron correlations tend to generate local magnetic moments that usuallyorder if the lattices are not too frustrated. The hexagonal compoundSrRu$_2$O$_6$ has a relatively high Neel temperature but small local moments,which seem to be at odds with the nominal valence of Ru$^{5+}$ in the$t_{2g}^3$ configuration. Here, we investigate the electronic property ofSrRu$_2$O$_6$ using density functional theory (DFT) combined withdynamical-mean-field theory (DMFT). We find that the strong hybridizationbetween Ru $d$ and O $p$ states results in a Ru valence that is closer to $+4$,leading to the small ordered moment $\sim1.2\mu_B$. While this is consistentwith a DFT prediction, correlation effects are found to play a significantrole. The local moment per Ru site remains finite $\sim2.3\mu_B$ in the wholetemperature range investigated. Due to the lower symmetry, the $t_{2g}$manifold is split and the quasiparticle weight is renormalized significantly inthe $a_{1g}$ state, while the renormalization in $e_g'$ states is about afactor of 2--3 weaker. Our theoretical Neel temperature $\sim700$~K is inreasonable agreement with experimental observations. SrRu$_2$O$_6$ is a uniquesystem in which localized and itinerant electrons coexist with the proximity toan orbitally-selective Mott transition within the $t_{2g}$ sector.

Localized-itinerant dichotomy and unconventional magnetism in SrRu$_2$O$_6$