Superconductivity in the three-band model of cuprates: Variational wave function study and relation to the single-band case

The $d$-$wave$ superconductivity is analyzed within the three-band $d$-$p$ model with the use of the diagrammatic expansion of the Guztwiller wave function method (DE-GWF). The determined stability regime of the superconducting state appears in the range of hole doping $\delta\lesssim 0.35$, with the optimal doping close to $\delta\approx 0.19$. The pairing amplitudes between the $d$-orbitals due to copper and $p_x/p_y$ orbitals due to oxygen are analyzed together with the hybrid $d$-$p$ pairing. The $d$-$d$ pairing between the nearest neighboring atomic sites leads to the dominant contribution to the SC phase. Moreover, it is shown that the decrease of both the Coulomb repulsion on the copper atomic sites ($U_d$) and the charge transfer energy between the oxygen and copper atomic sites ($\epsilon_{dp}$) increases the pairing strength as it moves the system from the strong to the intermediate-correlation regime, where the pairing is maximized. Such a result is consistent with our analysis of the ratio of changes in the hole content at the $d$ and $p$ orbitals due to doping, which, according to experimental study, increases with the increasing maximal critical temperature [cf. Nat. Commun. 7, 11413 (2016)]. Furthermore, the results for the three-band model are compared to those for the effective single-band picture and similarities between the two approaches are discussed. For the sake of completeness, the normal-state characteristics determined from the DE-GWF approach are compared with those resulting from the Variational Quantum Monte Carlo method with inter-site correlations included through the appropriate Jastrow factors.