Magnetic phase diagram of a two-orbital model for bilayer nickelates varying doping

Motivated by the recently discovered high-Tc bilayer nickelate superconductor La₃Ni₂O₇, we comprehensively research a bilayer 2×2×2 cluster for different electronic densities n by using the Lanczos method. We also employ the random-phase approximation to quantify the first magnetic instability with increasing Hubbard coupling strength, also varying n. Based on the spin structure factor $S(q)$, we have obtained a rich magnetic phase diagram in the plane defined by n and $U/W$, at fixed Hund coupling. We have observed numerous states, such as A-AFM, Stripes, G-AFM, and C-AFM. For half-filling $n=2$ (two electrons per Ni site, corresponding to N = 16 electrons), the canonical superexchange interaction leads to a robust G-AFM state (π,π,π) with antiferromagnetic couplings in plane and between layers. By increasing or decreasing electronic densities, ferromagnetic tendencies emerge from the ``half-empty'' and ``half-full'' mechanisms, leading to many other interesting magnetic tendencies. In addition, the spin-spin correlations become weaker both in the hole or electron doping regions compared with half-filling. At $n = 1.5$ (or $N=12$), density corresponding to La₃Ni₂O₇, we obtained the ``Stripe 2'' ground state (antiferromagnetic coupling in one in-plane direction, ferromagnetic coupling in the other, and antiferromagnetic coupling along the z-axis) in the 2×2×2 cluster. In addition, we obtained a much stronger AFM coupling along the z-axis than the magnetic coupling in the $xy$ plane. The random-phase approximation calculations with varying n give very similar results as Lanczos. Meanwhile, a state with q/π = (0.6, 0.6, 1) close to the E-phase wavevector is found in our RPA calculations by slightly reducing the filling to $n=1.25$, possibly responsible for the E-phase SDW recently observed in experiments.