Effective Bi-Layer Model Hamiltonian and Density-Matrix Renormalization Group Study for the High-T_c Superconductivity in La₃Ni₂O₇ under High Pressure

High-T_c superconductivity with possible T_c ≈ 80 K has been reported in the single crystal of La₃Ni₂O₇ under high pressure. Based on the electronic structure given by the density functional theory calculations, we propose an effective bi-layer model Hamiltonian including both 3d_z² and 3d_x²–y² orbital electrons of the nickel cations. The main feature of the model is that the 3d_z² electrons form inter-layer σ-bonding and anti-bonding bands via the apical oxygen anions between the two layers, while the 3d_x²–y² electrons hybridize with the 3d_z² electrons within each NiO₂ plane. The chemical potential difference of these two orbital electrons ensures that the 3d_z² orbitals are close to half-filling and the 3d_x²–y² orbitals are near quarter-filling. The strong on-site Hubbard repulsion of the 3d_z² orbital electrons gives rise to an effective inter-layer antiferromagnetic spin super-exchange J. Applying pressure can self dope holes on the 3d_z² orbitals with the same amount of electrons doped on the 3d_x²–y² orbitals. By performing numerical density-matrix renormalization group calculations on a minimum setup and focusing on the limit of large J and small doping of 3d_z² orbitals, we find the superconducting instability on both the 3d_z² and 3d_x²–y² orbitals by calculating the equal-time spin singlet pair–pair correlation function. Our numerical results may provide useful insights in the high-T_c superconductivity in single crystal La₃Ni₂O₇ under high pressure.