Interplay between spin-orbit coupling and van Hove singularity in the Hund's metallicity of Sr$_2$RuO$_4$

We investigate the dynamical properties of Sr$_2$RuO$_4$ at zero and very low temperature using density functional theory plus dynamical mean-field theory with an exact diagonalization solver. By considering rotationally invariant local interaction, we examine how Hund's coupling and spin-orbit coupling affect the correlated nature of the system. In the absence of Hund's coupling, the system shows a Fermi liquid behavior over the entire range of temperatures we consider. We confirm that the Fermi liquid persists at zero temperature even with nonzero Hund's coupling; however, at sufficient temperatures Hund's coupling significantly reduces the Fermi liquid regime and the system evolves into a typical Hund's metal. At the bare electronic occupancy of Sr$_2$RuO$_4$ ($t_{2g}^4$), a stronger Hund's metallicity accompanies a larger long-time correlator. Remarkably, electron doping further destabilizes the Fermi liquid even though the long-time correlator and magnetic fluctuations decrease upon doping. This suppression of the Fermi liquid is driven by the van Hove singularity above the Fermi level in Sr$_2$RuO$_4$, combined with an enhanced Van Vleck susceptibility by spin-orbit coupling. Such findings point to the important role that electronic structure plays in the behavior of Hund's metals, in addition to magnetic fluctuations.