Magnetically Mediated Cross-Layer Pairing in Pressurized Trilayer Nickelate La$_4$Ni$_3$O$_{10}$

Abstract: The recently discovered trilayer nickelate superconductor La$4$Ni$_3$O${10}$ under pressure has emerged as a promising platform for exploring unconventional superconductivity. However, the pairing mechanism remains a subject of active investigations. With large-scale density matrix renormalization group calculations on a realistic two-orbital trilayer Hubbard model, we elucidate the superconducting (SC) mechanism in this system. Our results reveal distinct magnetic correlations in the two different orbitals: while the $d_{z^2}$ orbital exhibits both interlayer and cross-layer antiferromagnetic (AFM) correlations, the $d_{x^2-y^2}$ orbital shows exclusively cross-layer AFM correlations, rendering a quasi-long-range SC order in the latter. We demonstrate that the Hund's rule coupling is essential for forming the SC order, and discuss the effects of kinetic AFM correlation and Hubbard repulsive $U$. Our findings motivate a further simplification of the trilayer Hubbard to an effective bilayer mixed-dimensional $t$-$J$ model, providing a unified framework for understanding interlayer SC in both trilayer and bilayer nickelates.