Effective Hamiltonian for superconducting Ni oxides Nd$_{1-x}$Sr$_x$NiO$_2$.

We derive the effective single-band Hamiltonian in the NiO$_2$ planes for nikelate superconducting compounds Nd$_{1-x}$Sr$_x$NiO$_2$. We first implement the first-principles calculation to study electronic structures of nickelates with the Heyd-Scuseria-Ernzerhof hybrid density functional, and derive a three-band Hubbard model for the Ni-O $pd\sigma$ bands of Ni$^+$ $3d_{x^2-y^2}$ and O$^{2-}$ $2p_{x/y}$ orbitals in the NiO$_2$ planes. For the three-band Hubbard model, we perform the exact diagonalization studies of finite clusters Ni$_5$O$_{16}$, and map low-energy spectra onto the effective one-band $t$-$t'$-$J$ model Hamiltonian with $t=260$~meV, $t'=-38$~meV and $J=26$~meV. The undoped NiO$_2$ plane is a Hubbard Mott insulator and the doped holes primarily locate on Ni$^+$ sites. The physics of the NiO$_2$ plane is a doped Mott insulator described by the one-band $t$-$t'$-$J$ model, further away from a conventional Fermi liquid than the superconducting cuprates. The ``self-doping'' effect and heavy fermion behavior of electron pockets of Nd$^{3+}$ $5d$ character in Nd$_{1-x}$Sr$_x$NiO$_2$ are also discussed.