Spin-density wave and superconductivity in La$_4$Ni$_3$O$_{10}$ under ambient pressure

We investigate the spin-density wave (SDW) behavior and the potential for superconductivity (SC) in La$_4$Ni$_3$O$_{10}$ under ambient pressure using a multi-orbital random-phase approximation (RPA). Starting with a twelve-orbital tight-binding model derived from density functional theory (DFT) calculations, we explore the influence of Hubbard interactions on SDW formation. Our analysis reveals a stripe-like SDW characterized by an incommensurate wave vector, $Q\approx(\pm 0.7\pi,0)$, suggesting a possible density wave instability in agreement with recent experiments. This configuration is driven by nesting of outer-layer Ni $d_{z^2}$ orbitals and exhibits interlayer antiferromagnetic ordering between the top and bottom NiO layers, with the middle layer serving as a node. We demonstrate that the Hund's coupling $J_H$ is the primary driver of the observed SDW. While superconductivity is absent in the undoped system under ambient pressure, it becomes attainable with appropriate hole doping ($\delta=-0.4$), resulting in a SC gap structure similar to the high-pressure phase. Our study identifies the specific conditions for realizing the ambient pressure stripe density wave: $J_H>0.16U$. Additionally, when doping leads to sufficient nesting at (0,0), the system's magnetic fluctuations transition into a stable Neel-type antiferromagnetic state, analogous to the high-pressure case.