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The intriguing interplay between the two $e_g$ orbitals in the high-$T_c$nickelate superconductor La$_3$Ni$_2$O$_7$ is studied via the density matrixrenormalization group (DMRG) and finite-$T$ thermal tensor-networkcalculations. We consider a bilayer $t$-$J$ model that incorporates both$d_{x^2-y^2}$ and $d_{z^2}$ orbitals, as well as inter-orbital hybridizationand ferromagnetic Hund's rule coupling. Our DMRG calculations reveal that bothorbitals exhibit algebraic superconductive (SC) pairing correlations in theground state, with the former being stronger than the latter. To confirm thedominant $d_{x^2-y^2}$-orbital SC order, we compute the pairing susceptibility$\chi_{\rm SC}$, and find a power-law divergence of $\chi_{\rm SC}$ below$T_c^* \simeq 0.03 t_c$, with $t_c$ being the intralayer hopping of the$d_{x^2-y^2}$ orbital. Regarding the $d_{z^2}$ orbitals, their finite-$T$$\chi_{\rm SC}$ results show weaker divergence and suggest lower $T_c^*$. Wedelve deeper into the intricate interplay between Hund's rule coupling andinter-orbital hybridization by tuning the two parameters, and reveal twodistinct SC phases dominated by Hund's rule and hybridization, respectively,separated by a non-SC regime. Our results treat the two $e_g$ orbitals in abilayer model on an equal footing, and clarify the essential roles of Hund'srule and hybridization in the formation of SC order in the pressurizednickelate La$_3$Ni$_2$O$_7$.

Roles of Hund's Rule and Hybridization in the Two-orbital Model for High-$T_c$ Superconductivity in the Bilayer Nickelate