Magnetically Mediated Cross-Layer Pairing in Pressurized Trilayer Nickelate La$_4$Ni$_3$O$_{10}$

TL;DR

The paper tackles the problem of how superconductivity emerges in the pressurized trilayer nickelate La$_{4}$Ni$_{3}$O$_{10}$ by performing large-scale DMRG on a realistic two-orbital trilayer Hubbard model. It finds a strong orbital-selective mechanism: the $d_{x^2-y^2}$ orbital drives cross-layer $s$-wave pairing mediated by antiferromagnetic correlations and Hund's coupling, while the $d_{z^2}$ orbital provides AFM fluctuations and pairing channels that are frustrated by geometry and Pauli blocking. A key result is that a Hund's coupling $J_H \gtrsim 0.5$ eV is essential to stabilize quasi-long-range superconductivity in the cross-layer channel, with an optimal intermediate Coulomb interaction $U$ around $2$–$4$ eV where CDW competition is not yet dominant. Based on these insights, the authors propose an effective mixed-dimensional bilayer model that captures the essential physics and provides a unified framework linking trilayer and bilayer RP nickelates, offering guidance for engineering higher $T_c$ in these systems.

Abstract

The recently discovered trilayer nickelate superconductor La$_4$Ni$_3$O$_{10}$ under pressure has emerged as a promising platform for exploring unconventional superconductivity. However, the pairing mechanism remains a subject of active investigations. With large-scale density matrix renormalization group calculations on a realistic two-orbital trilayer Hubbard model, we elucidate the superconducting (SC) mechanism in this system. Our results reveal distinct magnetic correlations in the two different orbitals: while the $d_{z^2}$ orbital exhibits both interlayer and cross-layer antiferromagnetic (AFM) correlations, the $d_{x^2-y^2}$ orbital shows exclusively cross-layer AFM correlations, rendering a quasi-long-range SC order in the latter. We demonstrate that the Hund's rule coupling is essential for forming the SC order, and discuss the effects of kinetic AFM correlation and Hubbard repulsive $U$. Our findings motivate a further simplification of the trilayer Hubbard to an effective bilayer mixed-dimensional Hubbard model, providing a unified framework for understanding interlayer SC in both trilayer and bilayer nickelates.

Figures (8)

• Figure 1: (a) The two-orbital trilayer Hubbard model for La$_4$Ni$_3$O$_{10}$. The dominant hopping and interaction terms are shown, with $\mu=1,2,3$ labeling the layer number. The $d_{x^2-y^2}$ orbital has significant intra-layer hopping $t^{c\mu}_\parallel$, but its inter-layer hopping is negligible. The $d_{z^2}$ orbital has strong $t^d_\perp$ between the inner ($\mu=2$) and outer layers ($\mu=1,3$), and weak intra-layer ($t^{d\mu}_\parallel$) and cross-layer ($t^d_{\perp 13}$) hopping, while other hopping terms are negligibly small. The parameters $U$, $U'$, and $J_\mathrm{H}$ denote the intra-orbital Coulomb repulsion, inter-orbital Coulomb repulsion, and Hund's coupling, respectively. Diagrammatic illustration of (b) $d_{x^2-y^2}$ and (c) $d_{z^2}$ orbitals showing inter-layer and cross-layer AFM spin correlations (orange up-down arrows) and superconducting pairing ($\Delta$). The $d_{z^2}$ orbital exhibits both spin and pairing frustration stemming from its triangular geometry between the three layers, whereas the $d_{x^2-y^2}$ orbital largely evades such frustration owing to its spatially non-uniform electron distribution.
• Figure 1: Non-interacting band for the two-orbital trilayer model in two dimensions.
• Figure 2: Local properties of the two-orbital trilayer Hubbard model with $U=3.5$ eV and $J_\mathrm{H}=1.0$ eV. The charge density distribution of the (a) $d_{x^2-y^2}$ and (b) $d_{z^2}$ orbitals in the outer layer (red) and inner layer (blue). The interlayer (blue) and cross-layer (red) spin correlation for the (c) $d_{x^2-y^2}$ and (d) $d_{z^2}$ orbitals. Average values are shown with back dashed lines. Results obtained with bond dimensions $D=10000,15000,20000, 25000$ are displayed using progressively darker colors.
• Figure 2: Impact of cross-layer hopping $t^d_{\perp 13}$ on the two-orbital trilayer Hubbard model with fixed $U=3.5$ eV and $J_\mathrm{H}=1$ eV. (a) Luttinger parameter $K_\mathrm{SC}$ for cross-layer pairing $\Phi^{c\perp}_{1,3}$ (red) and the maximum value of $\rho^c_3(q)$ (black) in the outer layers of $d_{x^2-y^2}$ orbital. (b) Average charge densities $\bar{n}^\alpha_\mu$ for both $d_{z^2}$ (circle) and $d_{x^2-y^2}$ (square) orbitals. (c-d) Average interlayer and cross-layer spin correlation for (c) $d_{z^2}$ and (d) $d_{x^2-y^2}$ orbitals. All presented data correspond to the $D\to\infty$ limit.
• Figure 3: Correlation functions for the two-orbital trilayer Hubbard model with $U=3.5$ eV and $J_\mathrm{H}=1.0$ eV. (a) Cross-layer pairing correlation $\Phi^{\alpha\perp}_{1,3}(r)$ and (b) interlayer pairing correlation $\Phi^{\alpha\perp}_{1,2}(r)$ for $d_{x^2-y^2}$ (red) and $d_{z^2}$ (blue) orbitals. Intralayer pairing correlations for (c) $d_{x^2-y^2}$ and (d) $d_{z^2}$ orbitals, comparing outer (red) and inner (blue) layers. The SC correlation functions are presented for progressively increasing bond dimensions $D = 8000$, $10000$, $15000$, $20000$, $25000$ and the extrapolated $D \rightarrow \infty$ limit, with each curve depicted in sequentially darker colors. (e, g) Single-particle Green's functions and (f,h) intralayer spin correlation functions for the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals, showing only $D \rightarrow \infty$ extrapolated data (inner layer: blue; outer layer: red). The data involved in the fittings are marked in bright red or green, i.e., in the same color code as the fitting line.