Effective Hamiltonian for cuprate superconductors derived from multiscale ab initio scheme with level renormalization

Three types (three-band, two-band, and one-band) of effective Hamiltonians for ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}$ and three-band effective Hamiltonian for ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ are derived by improving the constrained-$GW$ approximation combined with the self-interaction correction ($\mathrm{c}GW\ensuremath{-}\text{SIC}$) formulated by Hirayama et al. [Phys. Rev. B 98, 134501 (2018)]. The improved treatment of the interband Hartree energy in the present paper turns out to be crucially important, because the solution of the present improved Hamiltonian shows excellent agreement with the experimental results, for instance, for the charge gap (2 eV) and antiferromagnetic ordered moment $(0.6\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\text{B}})$ of the mother compound of ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$, in sharp contrast to the estimates by the previous Hamiltonian, 4.5 eV and $0.8\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\text{B}}$, respectively. To our knowledge, this is the first simultaneous and quantitative reproduction of these quantities by $\mathit{ab}\phantom{\rule{0.28em}{0ex}}\mathit{initio}$ methods without introducing adjustable parameters. We also predict that the Mott gap and the magnetic ordered moment for ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}$ is about 0.7 eV and $0.4\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\text{B}}$, respectively, if the mother compound becomes available, indicating weaker electron correlations than ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$. Surprisingly, we find that while carriers are doped only in the highest antibonding band, only the Cu $3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ (O $2p$) carriers look doped in the electron (hole) doped side around the zero doping in the atomic orbital basis, implying that the Mott-Hubbard (single-band) and charge transfer (three-band) descriptions are both correct. The obtained Hamiltonians will serve to further clarify the electronic structures of these copper oxide superconductors and to elucidate the superconducting mechanism in an ab initio fashion.