�鶹Ӱ��

Motivated by the recently discovered high-𝑇𝑐 bilayer nickelate superconductor La3⁢Ni2⁢O7, 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, 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 (𝜋,𝜋,𝜋) 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 La3⁢Ni2⁢O7, 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 𝑥⁢𝑦 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/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. Our predictions can be tested by chemically doping La3⁢Ni2⁢O7.