Impact of Nonlocal Coulomb Repulsion on Superconductivity and Density-Wave Orders in Bilayer Nickelates

The recent discovery of high-temperature superconductivity in pressurized bilayer nickelate La$_3$Ni$_2$O$_7$ and its thin films has generated significant interest in uncovering the underlying pairing mechanisms and correlated electronic states. While earlier theoretical studies have mainly focused on onsite Coulomb interactions, the role of nonlocal Coulomb repulsion has remained largely unexplored. In this work, we systematically investigate the effects of nonlocal Coulomb interactions, in the presence of onsite interactions, on both superconducting and density-wave instabilities using the functional renormalization group (FRG) approach. We find that the interlayer intraorbital repulsion suppresses the interlayer intraorbital $s_{\pm}$-wave pairing and spin-density-wave (SDW) order, while promoting a transition to an interlayer interorbital $d_{x^2-y^2}$-wave pairing state and a mirror-symmetry-breaking charge order. Remarkably, the critical scale of the interorbital $d_{x^2-y^2}$-wave superconductivity is significantly lower than that of the intraorbital $s_{\pm}$-wave superconductivity, indicating that the former is unlikely to account for the observed high-$T_c$ superconductivity. Moreover, the interlayer interorbital repulsion suppresses this $d_{x^2-y^2}$-wave pairing but enhances the $s_{\pm}$-wave pairing through strengthened interlayer charge fluctuations. In addition, the intralayer nearest-neighbor repulsion favors an in-plane charge-density-wave (CDW) order with wave vector $(\pi,\pi)$. Our findings reveal the profound impact of nonlocal Coulomb repulsion and underscore the robustness of interlayer pairing rooted in the bilayer structure and multi-orbital nature, thereby advancing the understanding of the intricate correlation effects in bilayer nickelates.