The discovery of high-temperature superconductivity (HTSC) in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ has aroused significant interest in exploring the pairing mechanism. Previous studies have proposed an effective $d_{x^2-y^2}$-orbital bilayer $t-J_{\parallel }-J_{\perp }$ model, in which the electrons of the $d_{x^2-y^2}$ orbital are charge carriers, which are subject to the intralayer antiferromagnetic (AFM) superexchange $J_{\parallel }$ and the large interlayer AFM superexchange $J_{\perp }$ typically stronger than $J_{\parallel }$, with $J_{\perp }$ transferred from the nearly half filled, and hence localized, $d_{z^2}$ orbital through the strong Hund's rule coupling. Here we study this model by the variational Monte Carlo (VMC) simulation and find a dominant interlayer $s$-wave pairing, in which the SC order parameters have a drastic improvement compared with those of the mean-field (MF)-type theories. In real materials, the Hund's coupling is finite, leading to reduced $J_{\perp }$, dictating that the MF-type theories have difficulty explaining the HTSC. However, our VMC calculations find that even for effective $J_{\perp }$ as weak as $J_{\perp }=J_{\parallel }$, the interlayer pairing is still considerably large and can be compared with the $T_c$ observed in experiments, which is very weak in MF-type theories. This result indicates the important role of the Gutzwiller projection in improving the $T_c$, which is ignored in the MF-type theories. In addition, our results show that suppressed interlayer hopping can promote interlayer pairing, which is consistent with the fact that the interlayer hopping of the $d_{x^2-y^2}$ orbital in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ is very weak. Our research offers a perspective for understanding the pairing mechanism of bilayer nickelates and provides a reference for recent ultracold-atom experiments in mixed-dimensional systems.

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• Superconductivity
• Unconventional superconductors
• Monte Carlo methods