Density waves in low-pressure bilayer nickelates

The low-pressure phase diagram of La$_3$Ni$_2$O$_7$ provides an important reference for understanding its pressure-induced high-temperature superconductivity. While the spin-density-wave transition at $T_{\text{SDW}}\approx150$ K is increasingly well established, the origin of the second density-wave transition at $T_{\text{DW}}\approx130$ K has remained unresolved. Here, we perform unrestricted Hartree-Fock calculations to investigate the potential origin of the second transition. {Within the orthorhombic phase, the degeneracy between possible ordering wavevectors at $\boldsymbol{Q}_{Y}=(0,\pi)$ and at $\boldsymbol{Q}_{X}=(\pi,0)$ is lifted and the electronic system} develops a double-stripe spin-density wave with ordering vector $\boldsymbol{Q}_{Y}=(0,\pi)$. We identify that the pure double stripe spin state is unstable in La$_3$Ni$_2$O$_7$ towards a commensurate charge-density wave instability, which favors a spin-modulated double stripe order with intertwined charge and spin instabilities and establish the hierarchy of ordered states in La$_3$Ni$_2$O$_7$, providing an important link between its ambient-pressure and superconducting high-pressure phases. We further discuss our results in the context of available experimental literature and propose further experimental tests to elucidate the origin of the SDW/DW states in this system.