What Does the Single-Particle Spectrum Imply on the Pairing Nature and Pairing Mechanism in La${}_3$Ni${}_2$O${}_7$?

The pairing mechanism of the bilayer nickelates La${}_3$Ni${}_2$O${}_7$ remains a hotly-debated open question. Existing strong-coupling theories are divided into class favoring intralayer d-wave pairing and that favoring interlayer s-wave pairing, with the latter further divided into $d_{z^2}$ orbital dominated mechanism driven by orbital hybridization and $d_{x^2-y^2}$ orbital dominated mechanism driven by Hund’s rule. Recent angle-resolved-photoemission-spectrum (ARPES) and scanning-tunneling-microscope (STM) combinedly reveal a nodeless full pairing gap with low anisotropy, supporting the s-wave pairing. Here we propose that the pairing gap along the Brillouin zone (BZ) diagonal can serve as a useful probe of pairing mechanism. Symmetry analysis suggests that orbital hybridization vanishes along the BZ diagonal, rendering that the pairing gaps on the $\gamma$- and $\alpha \mathrm{/}\beta$- pockets reflect the $d_{z^2}$- and $d_{x^2-y^2}$- orbital pairing strength respectively. Under the $d_{z^2}$ orbital dominated pairing mechanism driven by orbital hybridization, gap nodes are inevitable on the $\alpha$- and $\beta$- pockets along the BZ diagonal, which conflicts with the full gap revealed by ARPES and the U-shaped dI/dV curve observed by STM. The Hund’s rule driven pairing mechanism instead leads to a full pairing gap, which well fits the ARPES and STM results. Furthermore, through a random-phase-approximation based calculation, we show that the weak-coupling theory, which tends to yield a $d_{z^2}$-orbital dominated pairing, also leads to nodes or near-nodes on the $\alpha$- and $\beta$- pockets along the BZ diagonal, conflicting with experiments. This analysis clarifies the dominant role of $d_{x^2-y^2}$ orbital in the pairing and establishes the Hund’s rule driven pairing mechanism as the most relevant one in La${}_3$Ni${}_2$O${}_7$.