Low-energy perspective of interacting electrons in the normal state of superconducting bilayer nickelate

Developing a low-energy model is essential for understanding unconventional superconductivity in bilayer nickelate La$_3$Ni$_2$O$_7$. Here, we analyze distinct low-energy scenarios of the normal state by downfolding the ab-initio determined band structure and applying the mean-field regime of rotational-invariant slave-boson theory. We compare models based on the single-site two-orbital, the two-site four-orbital and a proposed minimal cluster (MC) picture. The latter builds up on three adapted orbitals located on the sites of the basic Ni-O-Ni cluster across the bilayer. Intriguing interplay between the Hund coupling $J_{\rm H}$ and the interlayer exchange $J_{\perp}$ is encountered in the multiorbital multi-site problem. While the tendency for interlayer Ni-$d_{z^2}$ singlet formation is pronounced, a complete localization remains hindered by the coupling to the Ni-$d_{x^2-y^2}$ orbitals. The correlation physics in the MC picture is peculiar with respect to the effective interorbital/site exchange.