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

The discoveries of high-$T_c$ superconductivity (SC) in bulk bilayer La$_3$Ni$_2$O$_7$ under pressure and of ambient-pressure SC in compressively strained thin films provide an important platform for exploring unconventional SC. Experiments reveal orbital-dependent band renormalization and sizable interlayer magnetic correlations, and show that SC emerges in association with pressure- or strain-induced structural changes. However, the role of the $\\gamma$ band, the symmetyr of the superconducting gap, and the microscopic pairing mechanism remain unresolved. This article reviews the relevant experimental observations and theoretical developments, with a focus on the strong-coupling Hund-assisted interlayer-pairing scenario. Within this framework, the two active Ni $E_g$ orbitals play distinct roles. In the local strong-coupling limit, the nearly half-filled and heavily renormalized $3d_{z^2}$ orbitals experience a sizable antiferromagnetic exchange across the bilayer through the inner apical oxygen. This exchange is proposed to favor interlayer rung-singlet correlations and may produce an orbital-dependent pseudogap regime. The onsite Hund's coupling aligns the two $E_g$ spins and generates an effective interlayer exchange between the more itinerant $3d_{x^2-y^2}$ carriers. The resulting bilayer models favor an interlayer $s$-wave state and predict that electron doping strengthens pairing. The Hund-assisted scenario can accommodate either the presence or absence of a $\\gamma$ pocket, provided that the $3d_{z^2}$ sector retains substantial local-moment character and sizable interlayer spin correlations. We also compare this framework with weak- and intermediate-coupling theories and other strong-coupling proposals.