Type II t-J model in charge transfer regime in bilayer La₃Ni₂O₇ and trilayer La₄Ni₃O₁₀

Recent observations of an 80 K superconductor in La₃Ni₂O₇ under high pressure have attracted significant attention. Recent experiments indicate that La₃Ni₂O₇ may be in the charge transfer regime, challenging the previous models based purely on the Ni dx²-y² and dz² orbitals. In this study, we propose a low energy model that incorporates doped holes in the oxygen p orbitals. Given that the parent nickel state is in the 3d⁸ configuration with a spin-one moment, doped hole only screens it down to spin-half, in contrast to the Zhang-Rice singlet in cuprate. We dub the single hole state as Zhang-Rice spin-half and build an effective model which includes three spin-one states (d⁸) and two Zhang-Rice spin-half states (d⁸ L). At moderate pressure around $20$ GPa, the dominated oxygen orbital is an in-plane Wannier orbital with the same lattice symmetry as the dx²-y² orbital. The resulting model reduces to the bilayer type II t-J model previously proposed in the Mott-Hubbard regime. Notably, the hopping between the in-plane p orbitals of the two layers is still suppressed. Density matrix renormalization group (DMRG) simulation reveals a pairing dome with the optimal hole doping level at x=0.4~0.5, distinct from the hole doped cuprate where optimal doping occurs around $x=0.19$. Further increasing pressure initially raises the critical temperature (Tc) until reaching an optimal pressure beyond which the pz orbital of oxygen becomes favorable and superconductivity is diminished. This shift from in-plane p orbital to pz orbital may elucidate the experimentally observed superconducting dome with varying pressure. As an extension, we also suggest a trilayer version of the type II t-J model as the minimal model for pressured La₄Ni₃O₁₀, which is distinct from the models in the Mott-Hubbard regime.