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

Abstract

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.

Figures (7)

• Figure 1: Schematic diagram for the two orbital's $t-J-J_{H}$ model, the yellow orbital is $d_{x^2 - y^2}$ with $t_{\parallel}$ and $J_{\parallel}$, and the blue one is d$_{z^2}$ orbital with $t_{z\perp}$ and $J_{z}$. The red arrow represents the Hund coupling which transfers $J_{z}$ to $d_{x^2 - y^2}$ orbital and makes an effective $J_{\perp}$ of $d_{x^2 - y^2}$ orbital.
• Figure 2: Pairing parameters calculated by VMC and SBMF. The picture (a) is the pairing parameters $\Delta_{\perp}$ versus electron filling for s-wave SC with $J_{\perp}/J_{\parallel}=0.5,0.75,1,2$ by VMC calculation, and insert shows pairing parameters $\Delta_{\parallel}$. The picture (b) is SBMF solution of $\Delta_{\perp}$ at $J_{\perp}/J_{\parallel}=0.75,1,2$lu2023bilayertJ, and insert shows the MF solution of $\Delta_{\perp}$ at $J_{\perp}/J_{\parallel}=0.75,1,2$ as a comparison.
• Figure 3: Correlation functions $G_z$ and $G_{xy}$ changing with electron filling at $J_{\perp}/J_{\parallel}=0.5,0.75,1,2.$
• Figure 4: Energy gain per site ($E_{normal}-E_{SC}$) $|\Delta E|/t$ of s-wave and d-wave pairing versus electron filling at (a) $J_{\perp}/J_{\parallel}=0.5$, (b) $J_{\perp}/J_{\parallel}=0.75$ and (c) $J_{\perp}/J_{\parallel}=1$.
• Figure 5: Pairing parameters $\Delta_{\perp}$ versus electron filling at different interlayer hopping $t_{\perp}/t_{\parallel}=0.05,0.1,0.2$ at $J_{\perp}/J_{\parallel}=1$.