Abstract
Electron correlations tend to generate local magnetic moments that usually order if the lattices are not too frustrated.
The hexagonal compound SrRu$_2$O$_6$ has a
relatively high N{\'e}el 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 and magnetic properties of SrRu$_2$O$_6$ using density functional theory (DFT) combined with dynamical mean field theory (DMFT).
We find that the strong hybridization between Ru $d$ and O $p$ states results in a Ru valence that is closer to $+4$, leading to the small ordered moment,
consistent with a DFT prediction.
While the agreement with DFT might indicate that SrRu$_2$O$_6$ is in the weak coupling regime,
our DMFT studies provide evidence from the mass enhancement and local moment formation that indicate correlation effects play a significant role.
The local moment per Ru site is about a factor 2 larger than the ordered moment at low temperatures and remains finite in the whole temperature range investigated.
Our theoretical N{\'e}el temperature $\sim 700$~K is in reasonable agreement with experimental observations.
Due to a small lattice distortion, the degenerate $t_{2g}$ manifold is split and the quasiparticle weight is renormalized significantly in the $a_{1g}$ state,
while correlation effects in $e_g'$ states are about a factor of 2--3 weaker.
SrRu$_2$O$_6$ is a unique system in which localized and itinerant electrons coexist with the proximity to an orbitally-selective Mott transition within the $t_{2g}$ sector.
