Ab initio low-energy effective Hamiltonians of two typical high-temperature copper-oxide superconductors, whose mother compounds are ${\mathrm{La}}_2{\mathrm{CuO}}_4$ and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$, are derived by utilizing the multiscale 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 first principles and a quantitative basis without adjustable parameters not only for the available cuprates but also for future design of higher $T_{\mathrm{c}}$ in general. More concretely, we derive effective Hamiltonians for three variations: (1) a one-band Hamiltonian for the antibonding orbital generated from strongly hybridized Cu $3d_{x^2-y^2}$ and O $2p_{\sigma }$ orbitals, (2) a 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, and (3) a three-band Hamiltonian consisting mainly of Cu $3d_{x^2-y^2}$ orbitals and two O $2p_{\sigma }$ orbitals. Differences between the Hamiltonians for ${\mathrm{La}}_2{\mathrm{CuO}}_4$ and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$, which have relatively low and high critical temperatures $T_{\mathrm{c}}$, respectively, at optimally doped compounds, are elucidated. The main differences are summarized as follows: (i) the oxygen $2p_{\sigma }$ orbitals are farther ($\sim 3.7$ eV) below from the Cu $d_{x^2-y^2}$ orbital in the case of the La compound than the Hg compound ($\sim 2.4$ eV) 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 $t^{\prime }/t$ is also substantially different (0.26 for the Hg and 0.15 for the La compound) in the one-band Hamiltonian. Heavier entanglement of the two bands in the two-band Hamiltonian implies that the two-band rather than the one-band Hamiltonian is more appropriate for the La compound. The relevance of the three-band description is also discussed especially for the Hg compound.

• Received 1 June 2017
• Revised 11 April 2018

DOI:https://doi.org/10.1103/PhysRevB.98.134501