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

Abstract

The three-band \emph{d-p} model is investigated by means of Variational Monte-Carlo (VMC) method with the BCS-like wave-function supplemented by the Gutzwiller and Jastrow correlators. The VMC optimization leads to $d$-$wave$ superconducting state with a characteristic dome-like shape of the order parameter for hole doping $\delta \lesssim 0.4$, in a 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 (DE-GWF) [cf. Phys. Rev. B \textbf{99}, 104511 (2019)]. Subsequently, the nature of the $d$-$wave$ pairing is investigated by means of recently proposed \emph{minimal-size real-space d-wave pairing operators} [Phys. Rev. B \textbf{100}, 214502 (2019)]. An emergence of the long-range superconducting ordering for both $d$ and $p$ orbitals is reported by analysing the corresponding off-diagonal pair-pair correlation functions. The dominant character of \emph{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 $\mathbf{q}=(\pi,\pi)$, short-range order, as expected. For the sake of completeness, the charge gap has been estimated, which for the parent compound takes the value $\Delta_{CG}\approx1.78\pm0.51\text{ eV}$, and agrees with values reported experimentally for the cuprates.