Hierarchical structure of primary and hybridization-induced superconducting correlations in bilayer nickelates

Abstract

High-pressure superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$, with a transition temperature approaching 80 K, has stimulated intense debate regarding its microscopic origin. Although an $s_{\pm}$ gap symmetry has been widely proposed, the electronic degrees of freedom responsible for pairing remain unsettled. Here we investigate a bilayer two-orbital Hubbard model using the variational Monte Carlo method and reveal a hierarchical pairing structure in bilayer nickelates. The primary pairing interaction originates from the bonding--antibonding splitting of the Ni $3d_{z^2}$ orbitals, while orbital hybridization redistributes superconducting correlations to the $d_{x^2-y^2}$ channel despite its weak intrinsic pairing interaction. This distinction between the origin of pairing and resulting superconducting correlations explains why the two orbital channels exhibit comparable long-range correlations. The resulting $s_{\pm}$ state is robust against changes in Fermi-surface topology. These results reconcile apparently competing theoretical scenarios and provide a comprehensive understanding, highlighting the distinctive role of orbital hybridization in multilayer correlated superconductors.

Figures (6)

• Figure 1: Noninteracting band structure of the bilayer two-orbital Hubbard model for $\Delta E/t_{\perp}=0.50$. High-symmetry points are labeled as $\Gamma(0,0)$, M$(\pi,\pi)$, and X$(\pi,0)$. The four bands originate from the $z^2$ and $x^2-y^2$ orbitals with interlayer hybridization. The Fermi level is indicated by the horizontal dashed line. The bonding and antibonding $z^2$ bands are denoted as $\gamma$ and $\delta$, respectively. The quantities $E^{\gamma}_{\mathrm{edge}}$ and $E^{\delta}_{\mathrm{edge}}$ denote the band edges of the $\gamma$ and $\delta$ bands measured from the Fermi level, defined as the extrema closest to $E_F$. The energy difference $E^{\delta}_{\mathrm{edge}}-E^{\gamma}_{\mathrm{edge}}$ remains unchanged as $\Delta E/t_{\perp}$ is varied.
• Figure 2: Interaction-driven reorganization of orbital occupancy and Fermi-surface topology. (a) $z^2$ orbital occupancy $n_{z^2}$ as a function of $\Delta E/t_\perp$. Open symbols (dashed line) denote the noninteracting case, while filled symbols (solid line) include the effect of Coulomb interactions. In the interacting case, $n_{z^2}$ is strongly renormalized and becomes nearly pinned over a broad parameter range, indicating an interaction-induced redistribution of charge between orbitals. The horizontal dashed line marks half filling of the $z^2$ orbital. (b--d) Evolution of the Fermi-surface sheets for representative values of $\Delta E/t_\perp$: (b) $\Delta E/t_\perp=0.50$ (non-superconducting regime), (c) $\Delta E/t_\perp=0.65$, and (d) $\Delta E/t_\perp=1.20$. Solid lines correspond to the interacting case and dashed lines to the noninteracting case. The $\alpha$, $\beta$, and $\gamma$ sheets are defined in (b). With increasing $\Delta E/t_\perp$, the $\gamma$ sheet shrinks substantially and eventually disappears, while the $\beta$ sheet is strongly reshaped.
• Figure 3: Variational superconducting gap parameters as a function of $\Delta E/t_\perp$. The interlayer $z^2$ gap $\tilde{\Delta}_{zz}$ becomes finite for $\Delta E/t_\perp \gtrsim 0.65$ and increases gradually with increasing $\Delta E/t_\perp$, indicating that the primary pairing interaction develops in the bonding--antibonding $z^2$ channel. In contrast, the interlayer $x^2-y^2$ gap $\tilde{\Delta}_{xx}$ remains negligible over the entire parameter range, demonstrating that pairing is not generated primarily in the $x^2-y^2$ orbital.
• Figure 4: Long-range superconducting correlation functions as a function of $\Delta E/t_\perp$. $P_{zz}$ and $P_{xx}$ denote the pairing correlations in the $z^2$ and $x^2-y^2$ channels, respectively. $P_{xx}$ exceeds $P_{zz}$ over a broad parameter range and exhibits a contrasting dependence on $\Delta E/t_\perp$. This behavior indicates that, although the effective pairing interaction is strongest in the $z^2$ channel, orbital hybridization redistributes superconducting correlations to the $x^2-y^2$ channel.
• Figure 5: Orbital- and band-resolved density of states near the Fermi level as a function of $\Delta E/t_\perp$. Open blue squares represent the $z^2$ density of states associated with the $\gamma$ band, $D^{\gamma}_{z^2}$, while filled blue squares denote the total $z^2$ density of states $D_{z^2}$ obtained by summing over all bands. Red circles show the $x^2-y^2$ density of states $D_{x^2-y^2}$, which mainly originates from the $\alpha$ and $\beta$ bands. As $\Delta E/t_\perp$ increases, $D^{\gamma}_{z^2}$ decreases as the $\gamma$ Fermi-surface sheet shrinks and disappears, whereas $D_{z^2}$ remains finite due to the hybridization-induced $z^2$ component on the $\alpha$ and $\beta$ bands. The evolution of these quantities correlates with the behavior of the superconducting correlation functions shown in Fig. \ref{['PSC']}.