Spin density wave in the bilayered nickelate La$_3$Ni$_2$O$_{7-\delta}$ at ambient pressure

The recent discovery of high-temperature superconductivity in high-pressurized La$_3$Ni$_2$O$_{7-\delta}$ has garnered significant attention. Using density functional theory, we investigate the magnetic properties of La$_3$Ni$_2$O$_{7-\delta}$ at ambient pressure. Our calculations suggest that with $\delta=0$, the double spin stripe phase is favored as the magnetic ground state. Oxygen vacancies may effectively turn nearest Ni spins into \textit{charge} sites. Consequently, with moderate $\delta$ values, our theoretical magnetic ground state exhibits characteristics of both double spin stripe and spin-charge stripe configurations, providing a natural explanation to reconcile the seemingly contradictory experimental findings that suggest both the configurations as candidates for the spin-density-wave phase. With higher $\delta$ values, we anticipate the ground state to become a spin-glass-like noncollinear magnetic phase with only short-range order. The oxygen vacancies are expected to significantly impact the magnetic excitations and the transition temperatures $T_{SDW}$. Notably, the magnetic ordering also induces concomitant charge ordering and orbital ordering, driven by spin-lattice coupling under the low symmetry magnetic order. We further offer a plausible explanation for the experimental observations that the measured $T_{SDW}$ appears insensitive to the variation of samples and the lack of direct evidence for long-range magnetic ordering.