Electronic correlation driven orbital polarization transitions in the orbital-selective Mott compound Ba$_2$CuO$_{4-\delta}$.

The electronic states near the Fermi level in the recently discovered superconductor Ba$_2$CuO$_{4-\delta}$ consist primarily of the Cu $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals. In this {\it Letter} we investigate the electronic correlation effect and orbital polarization of an effective two-orbital Hubbard model mimicking the low-energy physics of the compressed compound Ba$_2$CuO$_{4}$ for a hole doping $x$=1 by utilizing the dynamical mean-field theory with the Lanczos method as the impurity solver. We find that the system is in the orbital-selective Mott phase (OSMP) when the Hund's rule coupling $J_H$=$U/4$, or at the edge of an OSMP when $J_H$=$U/8$, which suggests that a local magnetic moment and spin fluctuations still exist in Ba$_2$CuO$_{4-\delta}$ with $3d^8$ (Cu$^{3+}$) configuration. We also observe the correlation driven orbital polarization transitions for both $J_H$=$U/8$ and $U/4$. Our results are also applicable to Sr$_2$CuO$_{4-\delta}$ and other two-orbital cuprates, demanding an unconventional multiorbital superconducting scenario in hole-overdoped high-$T_C$ cuprates.