Superconducting properties of the hole-doped three-band d−p model studied with minimal-size real-space d -wave pairing operators

The three-band d-p model is investigated by means of a variational Monte Carlo (VMC) method with the BCS-like wave-function supplemented by the Gutzwiller and Jastrow correlators. The VMC optimization leads to a $d\text{\ensuremath{-}}\mathit{wave}$ superconducting state with a characteristic domelike shape of the order parameter for hole doping $\ensuremath{\delta}\ensuremath{\lesssim}0.4$, in good agreement with the experimental observations. Also, the off-diagonal pair-pair correlation functions, calculated within VMC, vindicates the results obtained very recently within the diagrammatic expansion of the Gutzwiller wave function method [cf. Zegrodnik, Biborski, Fidrysiak, and Spa\l{}ek, Phys. Rev. B 99, 104511 (2019)]. Subsequently, the nature of the $d\text{\ensuremath{-}}\mathit{wave}$ pairing is investigated by means of recently proposed minimal-size real-space d-wave pairing operators [Moreo and Dagotto, Phys. Rev. B 100, 214502 (2019)]. An emergence of the long-range superconducting ordering for both $d$ and $p$ orbitals is reported by analyzing the corresponding off-diagonal pair-pair correlation functions. The dominant character of d-wave pairing on $d$ orbitals is confirmed. Additionally, the trial wave function is used to investigate the magnetic properties of the system. The analysis of spin-spin correlation functions is carried out and shows antiferromagnetic $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$, short-range order, as expected. For the sake of completeness, the charge gap has been estimated, which for the parent compound takes the value ${\mathrm{\ensuremath{\Delta}}}_{\text{CG}}\ensuremath{\approx}1.78\ifmmode\pm\else\textpm\fi{}0.51\phantom{\rule{4.pt}{0ex}}\text{eV}$, and agrees with values reported experimentally for the cuprates.