Magnetic configurations and excitations in high-$T_{c}$ multilayer nickelates

Abstract: We investigate the magnetic ground states and transverse spin excitations of bilayer and trilayer nickelates within a multi-orbital itinerant framework. For the bilayer system, although Hartree-Fock calculations slightly favor a double-stripe order, the calculated excitation spectrum of the single-stripe state, characterized by an anisotropic low-energy cone at $Q_{\text{BL}}$ and isotropic high-energy excitations near $Γ$, exhibits good qualitative agreement with recent RIXS and neutron scattering experiments. We further identify mirror-even optical interlayer modes at $Q_{\text{BL}}$ whose energies match the mirror-odd modes at $Γ$. For the trilayer system, both mirror-odd and mirror-even spin-density-wave states can be stabilized near $Q_{\text{TL}}$, with the mirror-odd state lower in energy in the parameter regime studied. The mirror-odd state hosts an additional nearly gapless mode dominated by the middle layer, while the mirror-even state contains only one acoustic branch together with two gapped optical modes. Comparison with available RIXS data favors the mirror-odd spin-density-wave scenario. Our results show that magnetic excitations provide a sensitive probe of the magnetic order and support a common itinerant origin of magnetism in multilayer nickelates.

• The paper demonstrates that single-stripe SDW order in bilayers aligns with observed neutron scattering and RIXS data.
• The study employs a multi-orbital itinerant framework with Hartree-Fock and RPA methods to analyze symmetry-dependent spin excitations.
• The paper identifies mirror-odd SDW order in trilayer nickelates with layer-selective features, supporting unified mechanisms for magnetism and superconductivity.

Multi-Orbital Itinerant Magnetism in High-$T_c$ Multilayer Nickelates

Background and Motivation

High-$T_c$ superconductivity in multilayer nickelates, specifically bilayer (La$_3$Ni$_2$O$_7$) and trilayer (La$_4$Ni$_3$O$_{10}$), has stimulated extensive investigation owing to their unconventional electronic structure and the close competition between superconductivity and spin-density-wave (SDW) order. Superconductivity notably emerges following structural transitions and is suppressed by pre-existing SDW phases. These nickelates with Ni $d^{7.5}$ and $d^{7.4}$ configurations feature both $T_c$0 and $T_c$1 orbital contributions to the low-energy electronic structure due to the presence of apical oxygens, in contrast to cuprates and infinite-layer nickelates. SDW order and its interplay with superconductivity remain central unresolved theoretical issues, motivating rigorous study within a multi-orbital itinerant framework to reconcile experimental findings and to elucidate the underlying mechanisms.

Model Formulation and Methodology

The study employs a multiorbital Hamiltonian incorporating both tight-binding and onsite Hubbard-Kanamori interaction terms, targeting the Ni $T_c$2 orbitals across all layers. Layer and orbital degrees of freedom allow for symmetry classification via mirror parity. For bilayer systems, magnetic instabilities are found to originate from Fermi-surface scattering between mirror-even bonding and mirror-odd antibonding sheets at $T_c$3. Analogous nesting-driven instabilities occur in trilayers between mirror-even and mirror-odd bands at $T_c$4. The Hartree-Fock method is applied to analyze SDW order energetics and texture, and transverse spin excitations are computed via RPA summation. The response is decomposed according to mirror and layer symmetries, enabling explicit differentiation of collective spin excitations in various symmetry sectors.

Results: Bilayer Nickelates

Hartree-Fock calculations show that the double-stripe (DS) order is slightly lower in energy than the single-stripe (SS) order. However, the excitation spectrum for the SS state exhibits an anisotropic Goldstone mode at $T_c$5 and isotropic high-energy excitations at $T_c$6, which aligns qualitatively with neutron scattering and RIXS experimental data (Chen et al., 5 May 2026, Chen et al., 2 Apr 2026). The DS state, while favored energetically, features additional gapped, asymmetric high-energy branches at $T_c$7 inconsistent with experiment. The SS order preserves certain mirror and PT symmetries, resulting in degenerate gapless cones in the excitation spectrum and distinctive constant-energy elliptical rings around $T_c$8.

Mirror-even optical interlayer modes appear at $T_c$9 with energies matching mirror-odd modes at $_3$0 for both magnetic configurations. Orbital-resolved spectra indicate the mirror-odd modes are dominated by $_3$1 orbitals, while mirror-even optical modes couple to both $_3$2 and $_3$3 contributions. The calculated spin-wave velocity anisotropy and symmetry-selected spectral weight distribution act as sensitive probes for discriminating magnetic orders. The inability of localized Heisenberg models to account for vanishing moments and in-plane order underscores the necessity of an itinerant formalism.

Results: Trilayer Nickelates

For trilayer nickelates, Hartree-Fock computations stabilize both mirror-odd and mirror-even SDW states near $_3$4, with the mirror-odd state favored in all realistic parameter regimes. The mirror-odd SDW features ordered moments on the outer layers and a nearly vanishing moment in the middle layer, consistent with experimental findings (Yang et al., 30 Jan 2026). The excitation spectrum hosts a Goldstone mode dominated by the outer layers and an additional nearly gapless mode concentrated in the middle layer—a consequence of dynamic coupling absent static exchange pinning.

Conversely, the mirror-even SDW aligns the middle layer antiferromagnetically with outer layers, resulting in an excitation spectrum with one Goldstone mode and two gapped optical modes near $_3$5. Available RIXS data (Chan et al., 6 Apr 2026) favor the mirror-odd scenario, which is theoretically robust due to opposite-mirror-parity nesting. The layer-selective excitation structure directly reflects the interaction symmetry and orbital-layer form factors inherent to the itinerant model.

Implications and Outlook

The results affirm that the magnetic states and their excitation spectra in multilayer nickelates are well-captured by a multi-orbital itinerant framework. The strong quantitative and qualitative correlation with spin-resolved experimental probes, notably RIXS and neutron scattering, substantiates the presence of single-stripe SDW order in bilayers and mirror-odd SDW with layer-selective features in trilayers. The coexistence of collective acoustic (Goldstone) modes and high-energy optical modes—tied to interlayer coupling strength—highlight effective itinerant mechanisms not accessible via localized spin models.

Theoretical implications extend to the unified treatment of magnetism and superconductivity. Both SDW order and unconventional superconducting pairing are shown to originate from Fermi-surface instabilities between bands of opposite mirror parity, supporting a scenario where low-energy electronic correlations drive both phenomena. Practically, the layer and orbital-resolved excitation spectra offer precise experimental avenues to identify the magnetic ground state and to disentangle competing orders.

Future research should incorporate charge fluctuations, lattice effects, and single-ion anisotropy to resolve energetic discrepancies and to further refine the theoretical description. The identification and tuning of high-energy optical modes via out-of-plane momentum selection may permit direct measurement of interlayer magnetic couplings. The framework established here invites further exploration of density wave orders and superconductivity in similarly structured correlated electron systems.

Conclusion

A comprehensive multi-orbital, itinerant analysis elucidates the magnetic configurations and excitation spectra of high-$_3$6 bilayer and trilayer nickelates. Single-stripe SDW order is consistent with experimental excitations in bilayers, while mirror-odd SDW with layer-selective gapless modes characterizes the trilayers. These findings validate the unified itinerant origin of magnetism and superconductivity, clarify the symmetry-driven nature of magnetic order, and provide detailed predictions for future experimental and theoretical inquiry into strongly correlated nickelate systems (2606.20533).