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

Motivated by the recently discovered high-Tc bilayer nickelate superconductor La3Ni2O7, we comprehensively research a bilayer 2 x 2 x 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, where U is the Hubbard strength and W the bandwidth. We have observed numerous states, such as A-AFM, Stripes, G-AFM, and C-AFM. At 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 (pi, pi, pi) with antiferromagnetic couplings both 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 x 2 x 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, even though both techniques are based on quite different procedures. Additionally, a state with q/pi = (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. Our predictions can be tested by chemically doping La3Ni2O7.