The newly discovered high-$T_c$ nickelate superconductor ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ has generated significant research interest. To uncover the pairing mechanism, it is essential to investigate the intriguing interplay between the two $e_g$, i.e., $d_{x^2-y^2}$ and $d_{z^2}$ orbitals. Here we conduct an infinite projected entangled-pair state (iPEPS) study of the bilayer $t\text{−}J$ model, directly in the thermodynamic limit and with orbitally selective parameters for $d_{x^2-y^2}$ and $d_{z^2}$ orbitals, respectively. The $d_{x^2-y^2}$ electrons exhibit significant intralayer hopping $t_{\parallel }$ (and spin couplings $J_{\parallel }$) as well as strong interlayer $J_{\perp }$ passed from the $d_{z^2}$ electrons. However, the interlayer $t_{\perp }$ is negligible in this case. In contrast, the $d_{z^2}$ orbital demonstrates strong interlayer $t_{\perp }$ and $J_{\perp }$, while the inherent intralayer $t_{\parallel }$ and $J_{\parallel }$ are small. Based on the iPEPS results, we find clear orbital-selective behaviors in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$. The $d_{x^2-y^2}$ orbitals exhibit robust superconductive (SC) order driven by the interlayer coupling $J_{\perp }$, while the $d_{z^2}$ band shows relatively weak SC order as a result of small $t_{\parallel }$ (lack of coherence) but large $t_{\perp }$ (strong Pauli blocking). Furthermore, by substituting rare-earth element Pm or Sm with La, we find an enhanced SC order, which opens up a promising avenue for discovering nickelate superconductors with even higher $T_c$.

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• Pairing mechanisms
• Superconductivity
• High-temperature superconductors
• Nickelates
• Strongly correlated systems
• Projected entangled pair states
• t-J model