Unique $d_{xy}$ Superconducting State in the Cuprate Member Ba$_{2}$CuO$_{3.25}$

Recent discovery of superconductivity at a transition temperature of $73$K in the doped layered compound Ba$_{2}$CuO$_{3+x}$ for $x\sim 0.2$ has generated a lot of interest. Experiments in this alternately stacked oxygen octahedral and chain layered structure reveal that a compression of the octahedra causes the Cu- {$d_{z^{2}}$} orbital to lie above the Cu- {$d_{x^2-y^2}$} orbital unlike in the well-known cuprate superconducting materials. Our first-principle calculations and low-energy Hamiltonian studies on the $x$ = 0.25 system reveal that this energy ordering results in the formation of $d_{z^2}$ dominated electron pockets. The strong nesting in the Fermi pockets leads to an AFM spin fluctuation mediated $d_{xy}$ wave superconducting state dominated by pairing among the $d_{z^{2}}$ orbitals. This is in contrast to the cuprate superconductors (e.g., YBCO) where both electron and hole pockets exist and the superconducting state with B$_{1g}$ symmetry is formed by the $d_{x^2-y^2}$ orbital electrons. Unlike the earlier reports, we find that inter-layer hybridization has an important contribution to the low-energy band structure and formation of the unconventional superconducting state.