d-wave superconductivity in the virtual-electron pair quantum liquid

We find evidence that for zero spin density $m=0$, intermediate $U\mathrm{/}4t$ values, and a range $x\in (x_c,x_{\ast })$ of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature $T_c$ for a hole concentration range centered at $x_{op}=(x_c+x_{\ast })\mathrm{/}2$. It results from the effects of the residual interactions of the charge $c$ fermions and spin-neutral two-spinon $s1$ fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the $\eta$-spin SU(2) symmetry of the $SO(4) = [SU(2)\times SU(2)]/Z_2$symmetry,theU(1)symmetrybrokenat$T_c$isthe$c$ fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global $SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]\mathrm{/}{Z}_2$ symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range $x\in (x_c,x_{\ast })$ to a successful description of the universal properties of the hole-doped cuprate superconductors.