$t$-$J$ model for strongly correlated two-orbital systems: Application to bilayer nickelate superconductors

We derive a $t$-$J$ model applicable to strongly correlated two-orbital systems including bilayer nickelate superconductors. Using the Schrieffer-Wolff transformation, we exclude the doubly occupied states, which raise the onsite Coulomb energy, and derive the resulting spin interactions from the two-orbital Hubbard model. We also introduce effective interactions attributed to the interorbital Coulomb interaction. To adapt the effective model to bilayer nickelates that exhibit high-temperature superconductivity, we quantitatively evaluate the strengths of the spin interactions based on the hopping parameters in La$_3$Ni$_2$O$_7$. Considering the evaluated effective interactions, we propose a simplified $t$-$J$ model for bilayer nickelate superconductors.