Magnetism and superconductivity in bilayer nickelate

The discovery of high-temperature superconductivity in bilayer nickelate La$_{3}$Ni$_{2}$O$_{7}$ necessitates a minimal theoretical model that unifies the superconducting phase with the spin-density-wave (SDW) phase without external pressure or strain. We propose a model where half-filled $d_{z^{2}}$ local moments interact with itinerant $d_{x^{2}-y^{2}}$ electrons via strong Hund's coupling $J_H$, which reduces to a bilayer type-II t-J model in the large $J_H$ limit. Using iDMRG calculations on an $L_y=4, L_z=2$ cylinder, we demonstrate that the competition between double-exchange ferromagnetism and in-plane superexchange generates period-4 stripe-like SDW order-a feature absent in one-orbital t-J model with only $d_{x^2-y^2}$ orbital. Furthermore, increasing the interlayer exchange coupling suppresses magnetic order and stabilizes interlayer s-wave superconductivity. These results identify the type-II t-J model as a minimal framework for capturing the interplay of magnetism and superconductivity in bilayer nickelates.