The discovery of high-$T_c$ superconductivity in pressurized bilayer nickelate ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ presents a new arena for exploring unconventional pairing mechanisms. A pivotal yet unresolved issue is the specific role of the $3d_{z^2}$ orbital of Ni. While its interlayer superexchange antiferromagnetic coupling is widely considered crucial for superconductivity, the role of its itinerancy remains undetermined. Early studies showed that the superconductivity is accompanied by the emergence of a small Fermi pocket of $3d_{z^2}$ orbitals. However, recent experiments show controversial results on the role of the $3d_{z^2}$ Fermi pocket on superconductivity. Motivated by these experimental results, we investigate an effective bilayer two-orbital model for ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ using density-matrix renormalization group on a minimal one-dimensional geometry. By systematically varying the $3d_{z^2}$ orbital filling from $1/12$ doping to half filling, we observe a pronounced suppression of superconducting correlations near half filling. Our results demonstrate the itinerancy of the $3d_{z^2}$ orbital is favorable for pairing in the bilayer two-orbital model for ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$. Moreover, we observe that the pairing correlation is enhanced in regions where charge fluctuations are large, suggesting a competition between charge order and superconductivity in the model.

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• Doping effects
• Superconductivity
• Nickelates
• Density matrix renormalization group
• Lattice models in condensed matter