Bilayer $t$-$J$-$J_{\perp }$ Model and Magnetically Mediated Pairing in the Pressurized Nickelate La${}_3$Ni${}_2$O${}_7$

The recently discovered nickelate superconductor La${}_3$Ni${}_2$O${}_7$ has a high transition temperature near 80 K under pressure, which offers additional avenues of unconventional superconductivity. Here with state-of-the-art tensor-network methods, we study a bilayer $t$-$J$-$J_{\perp }$ model for La${}_3$Ni${}_2$O${}_7$ and find a robust $s$-wave superconductive (SC) order mediated by interlayer magnetic couplings. Large-scale density matrix renormalization group calculations find algebraic pairing correlations with Luttinger parameter of $K_{\mathrm{S}\mathrm{C}}\simeq 1$. Infinite projected entangled-pair state method obtains a nonzero SC order directly in the thermodynamic limit, and estimates a strong pairing strength ${\bar{\mathrm{\Delta }}}_z\sim \mathcal{O}(0.1)$. Tangent-space tensor renormalization group simulations further determine a high SC temperature $T_c^{\ast }\mathrm{/}J\sim \mathcal{O}(0.1)$ and clarify the temperature evolution of SC order. Because of the intriguing orbital selective behaviors and strong Hund’s rule coupling in the compound, $t$-$J$-$J_{\perp }$ model has strong interlayer spin exchange (while negligible interlayer hopping), which greatly enhances the SC pairing in the bilayer system. Such a magnetically mediated strong pairing has also been observed recently in the optical lattice of ultracold atoms. Our accurate and comprehensive tensor-network calculations reveal robust SC order in the bilayer $t$-$J$-$J_{\perp }$ model and shed light on the high-$T_c$ superconductivity in the pressurized nickelate La${}_3$Ni${}_2$O${}_7$.