Spin-charge-orbital order in nickelate superconductors

The nickelate superconductors bilayer ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ and trilayer ${\mathrm{La}}_{4}{\mathrm{Ni}}_{3}{\mathrm{O}}_{10}$ exhibit remarkable high-temperature superconductivity under applied pressure, sparking considerable research attention. However, the nature of their spin, charge, and orbital order remains elusive. Here, supported by symmetry analysis and density functional theory calculations, we identify a double-stripe ground state of ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$. Such magnetic order is accompanied by a charge density wave in $Pmnm$ phase, intertwined with the ${d}_{{z}^{2}}^{1}{d}_{{x}^{2}\ensuremath{-}{y}^{2}}^{1}$ ($3{d}^{8}$) and ${d}_{{z}^{2}}^{1}$ ($3{d}^{7}$) orbital order. Furthermore, for trilayer ${\mathrm{La}}_{4}{\mathrm{Ni}}_{3}{\mathrm{O}}_{10}$, we demonstrate the existence of a long-range incommensurate spin density wave. Additionally, we point out that the ${\mathrm{Ni}}^{2+}, {\mathrm{Ni}}^{3+}$, and rare ${\mathrm{Ni}}^{4+}$ orders coexist in the ${\mathrm{La}}_{4}{\mathrm{Ni}}_{3}{\mathrm{O}}_{10}$ system due to interlayer antiferromagnetic frustration. Our work thus reveals a systematic spin-charge-orbital picture for ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{La}}_{4}{\mathrm{Ni}}_{3}{\mathrm{O}}_{10}$, paving the way for further understanding the mechanism of nickelate-based superconductivity.