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Understanding the pairing mechanism in bilayer nickelate superconductorsconstitutes a fascinating quest. Using density matrix renormalization group for$T=0$ and thermal tensor networks for $T>0$ properties, along with densityfunctional theory calculations, we investigate the intriguing interplay betweenthe Hund's rule coupling and interorbital hybridization that explains thepressure-dependent high-$T_c$ superconductivity in bilayer nickelates. Bystudying a two-orbital model, we identify three distinct superconductive (SC)regimes: hybridization dominant, Hund's rule dominant, and the hybrid-Hundsynergistic SC regimes. In these SC regimes, both $d_{x^2-y^2}$ and $d_{z^2}$orbitals exhibit algebraic pairing correlations with similar Luttingerparameters $K_{\rm SC}$. In particular, the former always exhibits a muchstronger amplitude than the latter, with a distinctly higher SC characteristictemperature $T_c^*$. Below this temperature, the pairing susceptibilitydiverges as $\chi_{\rm SC}(T) \sim 1/T^{2-K_{\rm SC}}$. With realistic modelparameters, we find the pressurized La$_3$Ni$_2$O$_7$ falls into the Hund'srule dominated SC regime. As hybridization further enhances under pressure, itleads to significant interorbital frustration and in turn suppresses the SCcorrelations, explaining the rise and fall of high-$T_c$ superconductivityunder high pressure [J. Li, et al., arXiv:2404.11369 (2024)]. Our results offera comprehensive understanding of the interlayer pairing in superconductingLa$_3$Ni$_2$O$_7$.

Hund's Rule, Interorbital Hybridization, and High-$T_c$ Superconductivity in the Bilayer Nickelate