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We study the static susceptibilities for charge and spin sectors inparamagnetic 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 modelof $x=0.5$. This enhancement arises from the increase of the correspondingsusceptibilities for the $d_{xy}$ orbital due to the rotation-inducedmodifications of the electronic structure for this orbital (i.e., theflattening of the bandwidth and the increase of the density of state near theFermi level). We also find that the ferromagnetic spin fluctuation becomesstronger for a special model than for the model of $x=0.5$, while thecompetition 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 Fermilevel. 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 tocontrol this competition. We propose that the spin fluctuations for the$d_{xy}$ orbital around $\boldq=(0,0)$ and $\boldq\sim (\pi,0)$ play animportant role in the electronic states around $x=0.5$ other than thecriticality approaching the usual Mott transition where all electrons arelocalized.

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