Competition between spin fluctuations in Ca$_{2-x}$Sr$_{x}$RuO$_{4}$ around $x=0.5$

We study the static susceptibilities for charge and spin sectors in paramagnetic states for Ca$_{2-x}$Sr$_{x}$RuO$_{4}$ in $0.5\leq x \leq 2$ within random phase approximation on the basis of an effective Ru $t_{2g}$ orbital Hubbard model. We find that several modes of spin fluctuation around $\boldq=(0,0)$ and $\boldq\sim(0.797\pi,0)$ are strongly enhanced for the model of $x=0.5$. This enhancement arises from the increase of the corresponding susceptibilities for the $d_{xy}$ orbital due to the rotation-induced modifications of the electronic structure for this orbital (i.e., the flattening of the bandwidth and the increase of the density of state near the Fermi level). We also find that the ferromagnetic spin fluctuation becomes stronger for a special model than for the model of $x=0.5$, while the competition between the modes of spin fluctuation at $\boldq=(0,0)$ and around $\boldq\sim (\pi,0)$ is weaker for the special model; in this special model, the van Hove singularity (vHs) for the $d_{xy}$ orbital is located on the Fermi level. These results indicate that the location of the vHs for the $d_{xy}$ orbital, which is controlled by substitution of Ca for Sr, is a parameter to control this competition. We propose that the spin fluctuations for the $d_{xy}$ orbital around $\boldq=(0,0)$ and $\boldq\sim (\pi,0)$ play an important role in the electronic states around $x=0.5$ other than the criticality approaching the usual Mott transition where all electrons are localized.