Weak Coupling Studies on Pairing Mechanism and Relative Properties of Ruddlesden-Popper Phase Nickelate Superconductors*

The discovery of superconductivity in Ruddlesden-Popper (RP) phase layered nickelates under high pressure has opened a new avenue for exploring unconventional pairing mechanisms beyond cuprates and iron-based superconductors. In particular, La₃Ni₂O₇ exhibits a superconducting transition temperature (Tc) as high as 80 K at ~15 GPa, making it the second class of oxides that achieve liquid-nitrogentemperature superconductivity. Subsequent experiments have extended superconductivity to related compounds such as La₄Ni₃O₁₀ and La₅Ni₃O₁₁, as well as epitaxially grown thin films at ambient pressure. These findings have motivated extensive theoretical efforts to elucidate the microscopic pairing mechanism. This review summarizes recent progress from the perspective of weak-coupling theories, including random phase approximation (RPA), functional renormalization group (FRG), and fluctuation-exchange (FLEX) approaches. Density functional theory (DFT) calculations reveal that the low-energy degrees of freedom are dominated by Ni 3dz² and 3dx²-y2</sup> orbitals. In La₃Ni₂O₇, pressure-induced metallization of the bonding 3dz² band produces the γ pocket, enhancing spin fluctuations and stabilizing superconductivity. These fluctuations support superconductivity through interlayer 3dz² pairing characterized by an s± gap. Hole doping or substitution may restore the γ pocket and enable bulk superconductivity at ambient pressure. For La₄Ni₃O₁₀, theoretical calculations indicate predominantly s± pairing from interlayer 3dz² orbitals, with weaker strength than La₃Ni₂O₇, explaining its lower Tc and showing little sensitivity to band structure. In La₅Ni₃O₁₁, composed of alternating single-layer and bilayer units, superconductivity mainly arises from the bilayer subsystem, again dominated by 3dz² orbitals. Interestingly, the interplay between inter-bilayer Josephson coupling and the suppression of density of states leads to a dome-shaped Tc-pressure phase diagram, distinct from the monotonic behavior of La₃Ni₂O₇. Epitaxial (La,Pr)₃Ni₂O₇ thin films display superconductivity above 40 K at ambient pressure. Theory predicts doping-dependent pairing: s± symmetry is favored at low doping levels, while dxy pairing emerges at higher doping, in agreement with experimental indications of both nodeless and nodal gap behaviors. Beyond superconductivity, experiments have revealed spin-density-wave (SDW) order in bulk La₃Ni₂O₇ and La₄Ni₃O₁₀ at ambient pressure. Weak-coupling calculations confirm that these SDWs are driven by Fermi surface nesting and that their suppression under pressure gives rise to strong spin fluctuations which act as the glue for Cooper pairing. This highlights the intimate connection between the density-wave parent states and high-pressure superconductivity in nickelates. In summary, weak-coupling theories provide a unified framework for RP nickelates, highlighting the key role of 3dz² orbitals, interlayer pairing, and spin fluctuations. They suggest that pressure, doping, substitution, and epitaxial strain can optimize superconductivity and potentially achieve high-Tc phases at ambient pressure. Key challenges remain in clarifying orbital competition, the SDW-superconductivity interplay, and strong-correlation effects, requiring close collaboration between advanced experiments and multi-orbital many-body theory.