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

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.

💡 Research Summary

In this work the authors address the long‑standing question of what drives superconductivity in the high‑pressure trilayer nickelate La₄Ni₃O₁₀, a material that exhibits a critical temperature of roughly 20–30 K under compression. They construct a realistic three‑dimensional Hubbard model that explicitly includes the two e_g orbitals of nickel, dₓ²₋ᵧ² and d_z², on each of the three NiO₂ layers. All hopping amplitudes (in‑plane, inter‑layer, and cross‑layer) are taken from density‑functional‑theory calculations, and the average electron filling per orbital is set to 2/3, corresponding to the nominal Ni 3d⁷·³³ configuration. The interaction part follows the Kanamori form with intra‑orbital repulsion U≈3.5 eV, Hund’s coupling J_H≈1 eV and inter‑orbital repulsion U′=U−2J_H.

To solve this strongly correlated problem the authors employ large‑scale density‑matrix renormalization group (DMRG) simulations on a cylinder of length L=48 and width W=3 (three layers). They exploit both Abelian charge conservation and non‑Abelian spin symmetries, reaching a maximum bond dimension of roughly 9 000 multiplets (≈2.5×10⁴ states) and achieving truncation errors below 3×10⁻⁵. Correlation functions are measured in the central half of the system and extrapolated to the infinite‑bond‑dimension limit.

The first set of results concerns charge distribution. Both orbitals show a modest excess of electrons on the outer layers compared with the inner layer, and the d_z² orbital carries slightly more charge overall. A charge‑density‑wave modulation with wave vector k≈0.58π is clearly visible in the dₓ²₋ᵧ² sector, while a weaker replica appears in d_z² due to hybridization.

Spin correlations reveal a striking orbital selectivity. The d_z² orbital exhibits antiferromagnetic (AFM) correlations both between the inner and outer layers (inter‑layer) and directly between the two outer layers (cross‑layer). The cross‑layer AFM correlation is slightly stronger, producing a triangular geometry that is magnetically frustrated. By contrast, the dₓ²₋ᵧ² orbital has negligible direct inter‑layer hopping, yet develops a sizable cross‑layer AFM correlation (≈−0.077) mediated by the Hund’s coupling to d_z². This orbital‑specific AFM pattern matches recent neutron and Raman experiments that report AFM coupling between the outer Ni layers.

Superconducting (SC) pairing correlations are then examined. The authors define a spin‑singlet pair operator for any two sites on any two layers and compute the distance‑dependent correlation functions for intra‑layer, inter‑layer, and cross‑layer channels. The most important finding is that the cross‑layer pairing in the dₓ²₋ᵧ² orbital, Φ_c⊥¹,³(r), decays as a power law with a Luttinger exponent K_sc≈1.75, i.e., below the critical value 2 required for quasi‑long‑range order in a one‑dimensional system. All other channels for this orbital either decay exponentially or have K_sc>2, indicating only short‑range fluctuations. The d_z² orbital shows power‑law decay in all channels but with exponents K_sc>2 (ranging from ≈2.3 to ≈4.0), meaning it contributes only fluctuating pairs and does not develop long‑range order.

Single‑particle Green’s functions and spin‑spin correlations corroborate these conclusions. In the dₓ²₋ᵧ² outer layers both G(r) and the spin correlator decay exponentially with correlation lengths ξ≈4–9 lattice spacings, indicating a finite single‑particle gap and gapped spin excitations. The d_z² outer layers also show exponential decay (ξ≈6), while the inner d_z² layer displays a very short ξ≈1.3 followed by a plateau, reflecting its more localized character.

A systematic study of the Hund’s coupling reveals that a threshold J_H≈0.5 eV is required for the cross‑layer AFM correlation and the associated SC pairing to become dominant; below this value the pairing exponent exceeds 2 and the SC tendency disappears. Increasing U strengthens AFM correlations but simultaneously suppresses charge mobility, reducing the magnitude of the SC correlations.

Based on these observations the authors propose a drastic simplification: the essential physics can be captured by a single‑orbital bilayer “mixed‑dimensional” Hubbard model (mix‑D) in which only the dₓ²₋ᵧ² orbital on the two outer layers is retained, with an effective inter‑layer pairing interaction mediated by the eliminated d_z² degrees of freedom. This reduced model reproduces the cross‑layer s‑wave pairing found in the full three‑layer calculation and places La₄Ni₃O₁₀ on the same theoretical footing as the n=2 nickelate La₃Ni₂O₇, where a similar inter‑layer pairing mechanism has been proposed.

In summary, the paper provides compelling numerical evidence that (i) the dₓ²₋ᵧ² orbital is the primary host of quasi‑long‑range superconductivity in pressurized La₄Ni₃O₁₀, (ii) this superconductivity is driven by cross‑layer antiferromagnetic correlations between the outer Ni layers, (iii) a sufficiently strong Hund’s coupling is essential to transmit the AFM correlation from the d_z² orbital to the dₓ²₋ᵧ² sector, and (iv) the full three‑layer problem can be faithfully reduced to an effective bilayer mixed‑dimensional Hubbard model. These insights resolve the long‑standing debate over orbital dominance in nickelate superconductivity and suggest concrete routes—such as enhancing Hund’s coupling or tuning inter‑layer hopping—to further raise the critical temperature in this family of materials.