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Recently, a 80 K superconductor was discovered in La$_3$Ni$_2$O$_7$ underhigh pressure. Density function theory (DFT) calculations identify$d_{x^2-y^2}$, $d_{z^2}$ as the active orbitals on the bilayer square latticewith a $d^{8-x}$ configuration of of Ni per site. One naive expectation is todescribe this system in terms of a two-orbital t-J model. However, we emphasizethe importance of Hund's coupling $J_H$ and the $x=0$ limit should be viewed asa spin-one Mott insulator. Especially, the significant Hund's coupling sharesthe inter-layer super-exchange $J_\perp$ of the $d_{z^2}$ orbital to the$d_{x^2-y^2}$ orbital, an effect that cannot be captured by conventionalperturbation or mean-field approaches. In this study, we first explore thelimit where the $d_{z^2}$ orbital is Mott localized, dealing with a one-orbitalbilayer t-J model focused on the $d_{x^2-y^2}$ orbital. Notably, we find thatstrong inter-layer pairing survives up to $x=0.5$ hole doping driven by thetransmitted $J_\perp$, which explains the existence of a high Tc superconductorin the experiment at this doping level. Next, we uncover the more realisticsituation where the $d_{z^2}$ orbital is slightly hole-doped and cannot besimply integrated out. We take the $J_H\rightarrow +\infty$ limit and propose atype II t-J model with four \textit{spin-half} singlon ($d^7$) states and three\textit{spin-one} doublon ($d^8$) states. Employing a parton mean-fieldapproach, we recover similar results as in the one-orbital t-J model, but nowwith the effect of the $J_\perp$ automatically generated. We propose futureexperiments to electron dope the system to further enhance $T_c$.

Type II t-J model and shared antiferromagnetic spin coupling from Hund's rule in superconducting La$_3$Ni$_2$O$_7$