Variation Monte Carlo Study on the bilayer $t-J_{\parallel}-J_{\perp}$ model for La$_3$Ni$_2$O$_7$

The discovery of high-temperature superconductivity (HTSC) in La$_3$Ni$_2$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 $J_{\perp}\approx 2J_{\parallel}$, with the latter 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 of 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 La$_3$Ni$_2$O$_7$ is very weak. Our research offers a new perspective for understanding the pairing mechanism of bilayer nickelates and provides a reference for recent ultra-cold atom experiments in mixed-dimensional systems.