Intertwined Electron Pairing in the Bilayer Two-orbital Kanamori-Hubbard Model: a Unified Picture of Two Superconductivities in $\mathrm{L}{\mathrm{a}}_3\mathrm{N}{\mathrm{i}}_2{\mathrm{O}}_7$

The mechanism of superconductivity in $\mathrm{L}{\mathrm{a}}_3\mathrm{N}{\mathrm{i}}_2{\mathrm{O}}_7$ bulk and film superconductors remains actively debated. Here, we investigate the bilayer two-orbital Kanamori-Hubbard model for $\mathrm{L}{\mathrm{a}}_3\mathrm{N}{\mathrm{i}}_2{\mathrm{O}}_7$ using cellular dynamical mean-field theory. We discover two intertwined $s_{\pm }-$ wave superconductivities with distinct physical origins. We show that when the $d_{z^2}$ orbital is under-doped, electron pairing associated to Hund’s coupling $J_H$ prevails. As $d_{z^2}$ hole-doping ${\delta }_z$ increases, a second superconductivity, which is largely insensitive to $J_H$ but exhibiting a critical reliance on the $d_{z^2}$ - $d_{x^2-y^2}$ hybridization $V$, arises. These two primary pairing states exhibit comparable maximum transition temperatures $T_c$, and evolve from one to the other following a smooth $T_c$ versus ${\delta }_z$ relation. A stark particle-hole asymmetry is observed in the superconducting phase diagram, indicating the crucial role played by the $\gamma -$ band of $d_{z^2}$ orbital in pairing. Our results present a picture unifying the two possible pairing mechanisms in $\mathrm{L}{\mathrm{a}}_3\mathrm{N}{\mathrm{i}}_2{\mathrm{O}}_7$ superconductors. We discuss the implications of our findings to recent experiments.