Origin of the Heavy-Fermion behavior in Ca$_{2-x}$Sr$_{x}$RuO$_{4}$

We investigate the electronic states in Ca$_{2-x}$Sr$_{x}$RuO$_{4}$ for $0.5\leq x\leq2.0$ on the basis of the three-orbital Hubbard model in the Gutzwiller approximation. We take the dominant effects of the Ca-substitution into account as the changes of the hybridizations, whose squares are proportional to the nearest-neighbor hopping integrals of the Ru-$t_{2g}$-orbitals, between the Ru-$4d$- and O-$2p$-orbitals. In this presentation, we show the renormalization factors for the Ru $t_{2g}$-orbitals as a function of the intraorbital Coulomb interaction or an angle of the rotation of RuO$_{6}$-octahedra and discuss the origin of the heavy-fermion behavior around $x=0.5$. Our calculation suggests that moderately strong Coulomb interaction and the orbital-dependent modifications of the electronic structures for the Ru-$t_{2g}$-orbitals due to the Ca-substitution cause the heavy-fermion behavior in Ca$_{2-x}$Sr$_{x}$RuO$_{4}$.