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

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