Type II t-J model in charge transfer regime in bilayer La$_3$Ni$_2$O$_7$ and trilayer La$_4$Ni$_3$O$_{10}$

Recent observations of an 80 K superconductor in La$_3$Ni$_2$O$_7$ under high pressure have attracted significant attention. Recent experiments indicate that La$_3$Ni$_2$O$_7$ may be in the charge transfer regime, challenging the previous models based purely on the Ni $d_{x^2-y^2}$ and $d_{z^2}$ 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 $3d^{8}$ 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 ($d^8$) and two Zhang-Rice spin-half states ($d^8 L$). At moderate pressure around $20$ GPa, the dominated oxygen orbital is an in-plane Wannier orbital with the same lattice symmetry as the $d_{x^2-y^2}$ 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 $x=0.4\sim0.5$, distinct from the hole doped cuprate where optimal doping occurs around $x=0.19$. Further increasing pressure initially raises the critical temperature ($T_c$) until reaching an optimal pressure beyond which the $p_z$ orbital of oxygen becomes favorable and superconductivity is diminished. This shift from in-plane $p$ orbital to $p_z$ 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 La$_4$Ni$_3$O$_{10}$, which is distinct from the models in the Mott-Hubbard regime.