Daily Overview: The highlight of today’s paper focuses on the theoretical establishment of a $d$-wave superconducting mechanism in the five-layer bulk superconductor La$_6$Ni$_5$O_{12}. Based on an effective five-layer Hubbard model constructed via density functional theory and variational Monte Carlo calculations, the spin-singlet $d_{x^2-y^2}$-wave superconducting state is shown to be energetically dominant, while extended $s$-wave pairing is strongly suppressed. This theoretical framework predicts the characteristics of a clean nodal superconductor, including a V-shaped structure in the tunneling density of states, a low-temperature linear depletion of the superfluid density, a decrease in the Knight shift, and a spin-lattice relaxation rate following the scaling behavior $1/T_1T\propto T^2$. Furthermore, the study indicates that infinite-layer LaNiO$_2$ exhibits a robust $d$-wave tendency, whereas three-layer La$_4$Ni$_3$O_8$ lies near the boundary between $d$-wave superconductivity and a non-superconducting state at nominal filling. These results collectively suggest a prevalent $d_{x^2-y^2}$-wave pairing trend within the reduced Ruddlesden-Popper nickel oxide family and establish La$_6$Ni$_5$O_{12}$ as a key platform for probing pairing symmetry in nickelates. arXiv submission processing window: 2026-10-16 00:00 to 2026-10-16 00:00 UTC.

1. $d$-wave superconductivity in quintuple-layer bulk superconductor La$_6$Ni$_5$O$_{12}$

Summary: This study investigates the pairing symmetry of the five-layer bulk superconductor La$_6$Ni$_5$O$_{12}$ by employing an effective five-layer Hubbard model constructed within density functional theory and variational Monte Carlo methods. Calculations indicate that the spin-singlet $d_{x^2-y^2}$-wave superconducting state is energetically dominant, whereas extended $s$-wave pairing is strongly suppressed. Based on the optimized state, the tunneling density of states is predicted to exhibit a V-shape, the superfluid density shows a linear low-temperature depletion, the Knight shift decreases, and the spin-lattice relaxation rate follows $1/T_1T\propto T^2$, characteristics that collectively point to a clean nodal superconductor. Furthermore, this framework reveals that infinite-layer LaNiO$_2$ possesses a robust $d$-wave tendency, while three-layer La$_4$Ni$_3$O$_8$ lies near the boundary between $d$-wave superconductivity and a non-superconducting state at nominal filling. These results establish La$_6$Ni$_5$O$_{12}$ as a platform for probing pairing symmetry in nickelates and suggest a prevalent $d_{x^2-y^2}$-wave pairing trend within the reduced Ruddlesden-Popper nickelate family.