d-wave superconductivity in Hubbard model on the square lattice perturbed by weak 3D uniaxial anisotropy

The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate $U/4t$ values and a hole-concentration range $x\in (x_c,x_*)$ the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The lower critical concentration $x_c$ involves the Ginzburg number Gi and is approximately given by $x_c\approx {\rm Gi}+x_0\approx 0.05$. Here $x_0<{\rm Gi}$ is a small critical hole concentration that marks a sharp quantum phase transition from a Mott-Hubbard insulator with long-range antiferromagnetic order for $x<x_0$ to an Anderson insulator with short-range incommensurate spiral spin order for $x\in (x_0,x_c)$. The value of the critical hole concentration $x_*$ depends on $U/4t$ and is given by $x_*\approx 0.27$ for $U/4t\approx 1.525$. The long-range d-wave superconducting order emerges below a critical temperature $T_c$ for a hole concentration range centered at $x_{op}= (x_c+x_*)/2\approx 0.16$. It results from the effects of the residual interactions of the charge $c$ and spin-neutral two-spinon $s1$ fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. The spin subsystem provides through such interactions the energy needed for the effective pairing coupling between the $c$ fermions of the virtual-electron pair configurations.