Ab initio effective Hamiltonians for cuprate superconductors

Ab initio low-energy effective Hamiltonians of two typical high-temperature copper-oxide superconductors, whose mother compounds are La$_2$CuO$_4$ and HgBa$_2$CuO$_4$, are derived by utilizing the multi-scale ab initio scheme for correlated electrons (MACE). The effective Hamiltonians obtained in the present study serve as platforms of future studies to accurately solve the low-energy effective Hamiltonians beyond the density functional theory. It allows further study on the superconducting mechanism from the first principles and quantitative basis without adjustable parameters not only for the available cuprates but also for future design of higher Tc in general. More concretely, we derive effective Hamiltonians for three variations, 1)one-band Hamiltonian for the antibonding orbital generated from strongly hybridized Cu $3d_{x^2-y^2}$ and O $2p_\sigma$ orbitals 2)two-band Hamiltonian constructed from the antibonding orbital and Cu $3d_{3z^2-r^2}$ orbital hybridized mainly with the apex oxygen $p_z$ orbital 3)three-band Hamiltonian consisting mainly of Cu $3d_{x^2-y^2}$ orbitals and two O $2p_\sigma$ orbitals. Differences between the Hamiltonians for La$_2$CuO$_4$ and HgBa$_2$CuO$_4$, which have relatively low and high critical temperatures, respectively, at optimally doped compounds, are elucidated. The main differences are summarized as i) the oxygen $2p_\sigma$ orbitals are farther(~3.7eV) below from the Cu $d_{x^2-y^2}$ orbital for the La compound than the Hg compound(~2.4eV) in the three-band Hamiltonian. This causes a substantial difference in the character of the $d_{x^2-y^2}-2p_\sigma$ antibonding band at the Fermi level and makes the effective onsite Coulomb interaction U larger for the La compound than the Hg compound for the two- and one-band Hamiltonians. ii)The ratio of the second-neighbor to the nearest transfer t'/t is also substantially different (~0.26) for the Hg and ~0.15 for the La compound in the one-band Hamiltonian.