Electronic structure, magnetic correlations, and superconducting pairing in the reduced Ruddlesden-Popper bilayer ${\mathrm{La}}_3{\mathrm{Ni}}_2{O}_6$ under pressure: Different role of $d_{3z^2-r^2}$ orbital compared with ${\mathrm{La}}_3{\mathrm{Ni}}_2{O}_7$

The recent discovery of superconductivity in bilayer ${\mathrm{La}}_3{\mathrm{Ni}}_2{O}_7$ (327-LNO) under pressure stimulated much interest in layered nickelates. However, superconductivity was not found in another bilayer nickelate system, ${\mathrm{La}}_3{\mathrm{Ni}}_2{O}_6$ (326-LNO), even under pressure. To understand the similarities and differences between 326-LNO and 327-LNO, using density functional theory and the random phase approximation (RPA), we systematically investigate 326-LNO under pressure. The large crystal-field splitting between the $e_g$ orbitals caused by the missing apical oxygen moves the $d_{3z^2-r^2}$ orbital farther away from the Fermi level, implying that the $d_{3z^2-r^2}$ orbital plays a less important role in 326-LNO than in 327-LNO. This also results in a smaller bandwidth for the $d_{x^2-y^2}$ orbital and a reduced energy gap for the bonding-antibonding splitting of the $d_{3z^2-r^2}$ orbital in 326-LNO, as compared to 327-LNO. Moreover, the in-plane hybridization between the $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals is found to be small in 326-LNO, while it is much stronger in 327-LNO. Furthermore, the low-spin ferromagnetic state is found to be the likely ground state in 326-LNO under high pressure. The weak interlayer coupling suggests that $s_{\pm }$-wave pairing is unlikely in 326-LNO. The robust in-plane ferromagnetic coupling also suggests that $d$-wave superconductivity, which is usually caused by antiferromagnetic fluctuations of the $d_{x^2-y^2}$ orbital, is also unlikely in 326-LNO. These conclusions are supported by our many-body RPA calculations of the pairing behavior. In addition, for the bilayer cuprate ${\mathrm{HgBa}}_2{\mathrm{CaCu}}_2{O}_6$, we find a strong self-doping effect of the $d_{x^2-y^2}$ orbital under pressure, with the charge of Cu being reduced by approximately 0.13 electrons from 0 GPa to 25 GPa. In contrast, we do not observe such a change in the electronic density in 326-LNO under pressure, establishing another important difference between the nickelates and the cuprates.