High spin, low spin or gapped spins: magnetism in the bilayer nickelates

Inspired by the recent discovery of high-temperature superconductivity in bilayer nickelates, we investigate the role of magnetism emerging from a hypothetical insulating $d^8$ parent state. We demonstrate that due to the interplay of superexchange and Hund’s coupling, the system can be in a high-spin, low-spin or spin-gapped state. The low-spin state has singlets across the bilayer in the $d_{z^2}$ orbital, with charge carriers in the $d_{x^2-y^2}$ orbital. Thus, at low energy scales, it behaves as an effective one band system when hole doped. By contrast, the high-spin state is a more robust, spin-1 antiferromagnet. Using Hartree-Fock methods, we find that for fixed interaction strength and doping, high-spin magnetism remains more robust than the low-spin counterpart. Whether this implies that the high spin state provides a stronger pairing glue, or more strongly competes with superconductivity remains an open question. Our analysis therefore underscores the importance of identifying the spin state for understanding superconductivity in nickelates.

💡 Research Summary

The paper addresses the magnetic landscape of bilayer nickelates that host high‑temperature superconductivity, focusing on a hypothetical insulating d⁸ parent compound (e.g., La₂CeNi₂O₇ or La₃Ni₂O₆.₅). Starting from a two‑orbital Hubbard model that includes the non‑degenerate d_{x²‑y²} (X) and d_{z²} (Z) orbitals on a bilayer square lattice, the authors incorporate the dominant in‑plane hopping t∥ for X, the inter‑layer hopping t⊥ for Z, an on‑site Hubbard repulsion U, and a ferromagnetic Hund’s coupling J_H between the two orbitals on the same Ni site. In the strong‑coupling limit (U/t∥, J_H/t∥ ≫ 1) the model reduces to an effective spin Hamiltonian with three exchange scales: J∥ = 4t∥²/U (in‑plane X‑X superexchange), J⊥ = 4t⊥²/U (inter‑layer Z‑Z superexchange), and J_H (Hund’s coupling).

Depending on the ratios of these three parameters, three distinct spin configurations emerge in the undoped d⁸ Mott insulator:

1. Spin‑gapped (spin‑0) inter‑layer singlet – realized when J_H is negligible but J⊥ ≫ J∥. The Z‑orbitals form singlets across the two layers, leaving the X‑orbitals essentially non‑magnetic. The system is a non‑magnetic insulator with a spin gap.

2. Low‑spin (spin‑½) antiferromagnet – occurs for small J_H and moderate J⊥. The X‑orbitals develop Néel order while the Z‑orbitals remain weakly correlated, yielding an effective single‑band (X‑only) description at low energies. This state is analogous to the cuprate parent antiferromagnet.

3. High‑spin (spin‑1) antiferromagnet – appears when Hund’s coupling dominates (J_H ≫ J∥, J⊥). The two electrons on each Ni site lock into a local triplet, producing a spin‑1 moment per site. The resulting bilayer spin‑1 Heisenberg model can support Néel order, with the inter‑layer exchange J⊥ now acting on the composite spin‑1 objects.

To map the phase boundaries quantitatively, the authors employ bond‑operator mean‑field theory (BOMFT), which treats the inter‑layer singlet as the vacuum and the triplet excitations as bosons. BOMFT predicts that the critical inter‑layer coupling J⊥^c diverges at J_H = 0 and decreases monotonically with increasing J_H, reaching J⊥^c ≈ 6.1 (in units of J∥) in the J_H → ∞ limit. This value is close to quantum Monte‑Carlo results (J⊥^c ≈ 7.15) and substantially lower than the Schwinger‑boson mean‑field estimate, indicating that BOMFT captures the physics of the J∥–J⊥–J_H model more faithfully.

Spin‑wave spectra are analyzed via Holstein‑Primakoff (HP) theory. For small J_H the Néel order resides almost entirely on the X‑orbitals (low‑spin regime). As J_H grows, Hund’s coupling transmits the inter‑layer exchange to the X‑orbitals while simultaneously enhancing the intra‑layer exchange on the Z‑orbitals, leading to a smooth crossover where the orbital‑resolved order parameters N_X and N_Z converge. In the large‑J_H limit the system behaves as a conventional spin‑1 antiferromagnet with the expected HP dispersion.

The impact of hole doping (moving from d⁸ toward the experimentally relevant d⁷·⁵ filling) is explored using a Hartree‑Fock (HF) decoupling of the Hubbard interaction. Assuming a sizable crystal‑field splitting Δ, doped holes are placed primarily in the X‑orbital. Calculations at U/t∥ = 4 show that the low‑spin AFM order disappears at a critical doping x_c ≈ 0.18, whereas the high‑spin AFM survives up to x_c ≈ 0.4–0.5, demonstrating that the spin‑1 antiferromagnet is considerably more robust against carrier injection. Both transitions are first‑order or weakly first‑order, a typical feature of HF treatments.

The authors argue that the identified spin state critically determines the low‑energy electronic model and, consequently, the pairing mechanism. In the low‑spin scenario the system reduces to a single‑band Hubbard model, suggesting that spin‑fluctuation mediated pairing akin to the cuprates could be operative. Conversely, the high‑spin state hosts a strong spin‑1 background that may suppress the same fluctuations, either weakening the pairing glue or competing with superconductivity. The spin‑gapped phase, lacking low‑energy magnetic excitations, would likely be detrimental to superconductivity. Therefore, experimental determination of the spin magnitude—via magnetic susceptibility, neutron scattering, or resonant X‑ray techniques—is essential for unraveling the origin of superconductivity in bilayer nickelates.

In summary, the paper provides a comprehensive theoretical framework that links superexchange, Hund’s coupling, and bilayer geometry to three possible magnetic ground states. It quantitatively maps the phase diagram, evaluates the stability of each state under hole doping, and highlights the profound implications of the spin configuration for the emergence of high‑temperature superconductivity in this new family of nickelate materials.