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The electronic states near the Fermi level of recently discoveredsuperconductor Ba$_2$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 effectand the orbital polarization of an effective two-orbital Hubbard modelmimicking the low-energy physics of Ba$_2$CuO$_{4-\delta}$ in the hole-richregime by utilizing the dynamical mean-field theory with the Lanczos method asthe impurity solver. We find that the hole-overdoped Ba$_2$CuO$_{4-\delta}$with $3d^8$ (Cu$^{3+}$) is in the orbital-selective Mott phase (OSMP) athalf-filling, and the typical two-orbital feature remains inBa$_2$CuO$_{4-\delta}$ when the electron filling approaches $n_e\sim 2.5$,which closely approximates to the experimental hole doping for the emergence ofthe high-$T_c$ superconductivity. We also obtain that the orbital polarizationis very stable in the OSMP, and the multiorbital correlation can drive orbitalpolarization transitions. These results indicate that in hole-overdopedBa$_2$CuO$_{4-\delta}$ the OSMP physics and orbital polarization, localmagnetic moment, and spin or orbital fluctuations still exist. We propose thatour present results are also applicable to Sr$_2$CuO$_{4-\delta}$ and othertwo-orbital cuprates, demanding an unconventional multiorbital superconductingscenario in hole-overdoped high-$T_c$ cuprates.

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