  We employ correlated density-functional theory methods (DFT + Hubbard $U$) to
investigate the spin-density wave state of the bilayer Ruddlesden-Popper (RP)
nickelate La$_{3}$Ni$_{2}$O$_{7}$ which becomes superconducting under pressure.
We predict that the ground state of this bilayer RP material is a single
spin-charge stripe phase with in-plane up$^\prime$/up/down$^\prime$/down
diagonal stripes with up$^\prime$/down$^\prime$ being low spin (formally
Ni$^{3+}$: $d^7$) and up/down being high spin (formally Ni$^{2+}$: $d^8$). The
main feature of this solution (that is insulating even at $U=0$) is the
dominant role of $d_{x^{2}-y^{2}}$ bands around the Fermi level, which would
become doped with the introduction of electrons via oxygen vacancies. In spite
of the similarity with cuprates in terms of the dominant role of
$d_{x^{2}-y^{2}}$ bands, some differences are apparent in the magnetic ground
state of La$_{3}$Ni$_{2}$O$_{7}$: the antiferromagnetic out-of-plane coupling
within the bilayer (linked to the $d_{z^2}$ orbitals forming a
spin-singlet-like configuration) is found to be the dominant one while in-plane
interactions are reduced due to the stripe order of the ground state. With
pressure, this striped magnetic ground state remains similar in nature but the
increase in bandwidth quickly transitions La$_{3}$Ni$_{2}$O$_{7}$ into a
metallic state with all the activity close to the Fermi level involving, to a
large extent, $d_{x^2-y^2}$ orbitals. This is reminiscent of the cuprates and
may provide key insights into how superconductivity arises in this material
under pressure.
