Type II $t-J$ model in superconducting nickelate Nd$_{1-x}$Sr$_x$NiO$_2$

The recent observation of superconductivity at relatively high temperatures in hole doped NdNiO$_2$ has generated considerable interest, particularly due to its similarity with the infinite layer cuprates. Building on the observation that the Ni$^{2+}$ ions resulting from hole doping are commonly found in the spin-triplet state, we introduce and study a variant of the $t-J$ model in which the holes carry S=1. We name this new model the Type II $t-J$ model. We find two distinct mechanisms for $d$ wave superconductivity. In both scenarios the pairing is driven by the spin coupling $J$. However, coherence is gained in distinct ways in these two scenarios. In the first case, the spin-one holes condense leading to a $d$ wave superconductor along with spin-symmetry breaking. Different orders including spin-nematic orders are possible. This scenario is captured by a spin one slave boson theory. In the second scenario, a coherent and symmetric $d$ wave superconductor is achieved from "Kondo resonance": spin one holes contribute two electrons to form large Fermi surface together with the spin 1/2 singly occupied sites. The large Fermi surface then undergoes $d$ wave pairing because of spin coupling $J$, similar to heavy fermion superconductor. We propose a three-fermion parton theory to treat these two different scenarios in one unified framework and calculate its doping phase diagram within a self consistent mean field approximation. Our study shows that a combination of "cuprate physics" and "heavy fermion physics" may emerge in the type II $t-J$ model.