Superconductivity in bilayer La$_3$Ni$_2$O$_7$: A review focusing on the strong-coupling Hund's rule assisted pairing mechanism

Abstract: Discovery of high-$T_c$ superconductivity (SC) in the bilayer nickelate series La$<em>3$Ni$_2$O$_7$ have attracted substantial interest, providing a new platform for exploring unconventional SC. Certain experimental evidence has pointed to a correlated electronic nature, which is the driving force responsible for its high critical temperature ($T_c$). This work reviews the SC in La$_3$Ni$_2$O$_7$, with a particular focus on theoretical understanding of its pairing mechanism driven by this strong-coupling, Hund-assisted scenario. The electronic landscape is governed by two $E_g$-orbitals within the bilayer structure of NiO$_2$ planes. The $3d</em>{z<sup>2}$ orbital is nearly half-filled and exhibits a stronger localized character, while the $3d_{x<sup>2-y<sup>2}$ is approximately quarter-filled and remains highly itinerant. The localized $3d_{z<sup>2}$ orbitals experience robust interlayer hybridization, mediated by the $2p_z$ orbitals of the inner apical oxygen atoms. This hybridization generates a strong interlayer antiferromagnetic (AFM) exchange. In the strong coupling regime, Hund's rule coupling aligns the spins of the two $E_g$ orbitals on the same nickel site. The strong interlayer AFM exchange is effectively transferred to the itinerant $3d_{x<sup>2-y<sup>2}$ orbital, generating an effective coupling $J_{\perp}$ within this orbital. This mechanism is captured by a minimal strong-coupling bilayer $t$-$J$-$J_{\perp}$ model for the $3d_{x<sup>2-y<sup>2}$ band. Driven by $J_{\perp}$, $3d_{x<sup>2-y<sup>2}$ electrons can form interlayer Cooper pairs, leading to an extended $s$-wave pairing SC with high $T_c$. Meanwhile, the strongly localized $3d_{z<sup>2}$ electrons tend to form interlayer rung singlets. Due to a lack of phase coherence, these singlets do not directly participate in the SC condensate, but instead give rise to a pseudogap phase.