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Strong-coupling study of the pairing mechanism in pressurized La$_3$Ni$_2$O$_7$

Recently, the bilayer perovskite nickelate La$_3$Ni$_2$O$_7$ has been reported to exhibit high-temperature superconductivity near $80$ K under a moderate pressure of about $14$GPa. To investigate the underlying pairing mechanism and symmetry in this complex system, we propose and analyze a mixed spin-$1$ and spin-$\frac{1}{2}$ bilayer $t$-$J$ model in the strong coupling regime. This model explicitly incorporates the crucial role of strong Hund's coupling, which favors the formation of local spin-triplet states from the two onsite $E_g$ orbital electrons at half-filling. We further investigate the model using both slave-particle mean-field theory and the density matrix renormalization group method. Our simulation results reveal that the dominate pairing channel is the interlayer one in the $3d_{x^2-y^2}$ orbital. The Hund's coupling is shown to enhance superconductivity within a reasonable physical range. Moreover, electron doping strengthens superconductivity by increasing carrier density; in contrast, hole doping weakens superconductivity. These findings offer critical insights into the unconventional superconductivity of pressurized La$_3$Ni$_2$O$_7$ and underline the important role of orbital-selective behavior and Hund's rule.