The electronic states near the Fermi level of recently discovered superconductor ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ consist primarily of the Cu $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals. We investigate the electronic correlation effect and the orbital polarization of an effective two-orbital Hubbard model mimicking the low-energy physics of ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ in the hole-rich regime by utilizing the dynamical mean-field theory with the Lanczos method as the impurity solver. We find that the hole-overdoped ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ with $3d^8$ (${\mathrm{Cu}}^{3+}$) is in the orbital-selective Mott phase (OSMP) at half-filling, and the typical two-orbital feature remains in ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ when the electron filling approaches $n_e\sim 2.5$, which closely approximates to the experimental hole doping for the emergence of the high-$T_c$ superconductivity. We also obtain that the orbital polarization is very stable in the OSMP, and the multiorbital correlation can drive orbital polarization transitions. These results indicate that in hole-overdoped ${\mathrm{Ba}}_2{\mathrm{CuO}}_{4-\delta }$ the OSMP physics and orbital polarization, local magnetic moment, and spin or orbital fluctuations still exist. We propose that our present results are also applicable to ${\mathrm{Sr}}_2{\mathrm{CuO}}_{4-\delta }$ and other two-orbital cuprates, demanding an unconventional multiorbital superconducting scenario in hole-overdoped high-$T_c$ cuprates.

• Received 20 December 2019
• Revised 8 February 2021
• Accepted 1 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.214510