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

This study employs a multi-orbital itinerant framework to investigate magnetic ground states and spin excitations in bilayer and trilayer nickelates, demonstrating that calculated excitation spectra for specific stripe and spin-density-wave orders show strong qualitative agreement with experimental RIXS and neutron scattering data, thereby supporting a common itinerant origin of magnetism in these high-$T_c$ materials.

The Gist

Imagine a world built from tiny, stacked Lego bricks made of a special material called nickel. Scientists have recently discovered that when they stack these bricks in specific ways (two layers or three layers) and squeeze them with pressure, they can conduct electricity without any resistance at all. This is called superconductivity, and it's a holy grail of physics.

However, right before these materials become superconductors, they act like a different kind of magnet. They form a "Spin Density Wave" (SDW), which is a fancy way of saying the tiny magnetic arrows inside the atoms line up in a specific, repeating pattern. The big mystery is: What does this pattern actually look like?

This paper is like a detective story where the authors use a computer simulation to figure out the shape of these magnetic patterns and then check if their "theoretical fingerprints" match the "crime scene photos" taken by real experiments.

Here is the breakdown of their investigation:

The Two Suspects: Single-Stripe vs. Double-Stripe

For the two-layer (bilayer) nickel material, the scientists looked at two main suspects for the magnetic pattern:

1. The Double-Stripe: Imagine a checkerboard where every square has a magnetic arrow.
2. The Single-Stripe: Imagine a checkerboard where the arrows exist on some squares, but every other square is completely empty (spinless).

The Energy Test:
When the authors ran their math to see which pattern is the most "comfortable" (lowest energy) for the atoms, the Double-Stripe won. It seemed like the natural choice.

The "Sound" Test (The Twist):
But here's the catch. In physics, you can also listen to how the material "vibrates" when you poke it with energy. These vibrations are called spin excitations.

• The authors calculated what the "sound" of the Double-Stripe would look like. It didn't match the real-world photos (from experiments like RIXS and neutron scattering).
• They then calculated the "sound" of the Single-Stripe. Even though it wasn't the most "comfortable" energy-wise, its vibration pattern looked exactly like the real-world experiments. It had a specific shape: a cone of low-energy vibrations in one direction and a smooth, round pattern in another.

The Verdict for Two Layers:
The paper concludes that nature is tricky. Even though the Double-Stripe is energetically "easier," the real material is likely the Single-Stripe. The authors suggest that some other invisible force (perhaps related to the movement of electric charge) is helping to hold this Single-Stripe pattern in place, even though the basic math didn't predict it perfectly.

The Three-Layer Mystery

Next, they looked at the three-layer (trilayer) material. This is like a sandwich with a top slice, a middle slice, and a bottom slice.

They tested two scenarios for how the magnetic arrows align across these three slices:

1. Mirror-Odd: The top and bottom slices have arrows pointing in opposite directions (like a mirror image), but the middle slice is completely empty (no magnetic arrow).
2. Mirror-Even: The top and bottom are the same, but the middle slice has an arrow pointing the opposite way, sandwiched between them.

The Energy Test:
The math said the Mirror-Odd (empty middle) state is the winner. It has lower energy.

The "Sound" Test:

• Mirror-Odd: Because the middle slice is empty and not "pinned" by its neighbors, it acts like a loose drum skin. It creates a special, very soft vibration (a nearly gapless mode) that is dominated by that middle layer.
• Mirror-Even: Because the middle slice has an arrow and is tightly held by the top and bottom, it's stiff. It only produces standard vibrations, with no special soft mode.

The Verdict for Three Layers:
When the authors compared their "sound" predictions to real experimental data, the Mirror-Odd scenario (the one with the empty middle layer) matched perfectly. The data showed only one type of high-energy vibration, which fits the Mirror-Odd story.

The Big Picture

The main takeaway from this paper is that magnetic vibrations are a perfect fingerprint.

Even if two magnetic patterns look similar on paper, they "sing" differently. By listening to these vibrations, scientists can tell exactly how the atoms are arranged. The paper argues that both the two-layer and three-layer materials likely share a common origin: their magnetism comes from the way electrons move freely (itinerant) between layers, rather than being stuck in place.

In short: The authors used a computer to predict how these magnetic materials should "hum." They found that the "hum" of the Single-Stripe (for two layers) and the Mirror-Odd (for three layers) matches what we see in real life, even if the basic energy math suggested other options might be better. This proves that listening to the material's vibrations is the best way to solve the mystery of its magnetic structure.

Technical Summary

Technical Summary: Magnetic Configurations and Excitations in High-Tc Multilayer Nickelates

Problem Statement
Recent discoveries of high-temperature superconductivity in bilayer ($La_3Ni_2O_7$) and trilayer ($La_4Ni_3O_10$) nickelates under pressure have sparked intense interest, yet the underlying magnetic mechanisms remain debated. Unlike cuprates, these materials possess nominal $Ni$ $d^{7.5}$ and $d^{7.4}$ configurations with significant contributions from both $d_{x^2-y^2}$ and $d_{z^2}$ orbitals due to apical oxygens. While superconductivity emerges upon the suppression of Spin Density Wave (SDW) orders, the precise nature of these SDW ground states is unclear. Experimental data from Resonant Inelastic X-ray Scattering (RIXS) and neutron scattering suggest complex magnetic structures, but theoretical consensus is lacking. Specifically, it is unresolved whether the magnetic order in bilayers follows a single-stripe (SS) or double-stripe (DS) configuration, and whether trilayer systems exhibit mirror-odd or mirror-even SDW states. Furthermore, localized Heisenberg models struggle to explain the observed layer-dependent moments and in-plane ordering, suggesting a need for an itinerant framework that accounts for orbital and layer symmetries.

Methodology
The authors employ a multi-orbital itinerant framework based on a tight-binding Hamiltonian including onsite Hubbard-Kanamori interactions ($U, U', J_H, J_P$).

• Models: For the bilayer system, they utilize a tight-binding model fitted to previous literature; for the trilayer, they use an ARPES-fitted model. Both models incorporate layer ($\ell$) and orbital ($\mu = x, z$) degrees of freedom, allowing classification by mirror parity.
• Ground State Calculation: Hartree-Fock (HF) mean-field theory is applied to determine the magnetic ground states. The authors introduce spin-density wave ansatzes with specific phase parameters ($\phi$) to distinguish between SS and DS configurations in bilayers, and mirror-odd versus mirror-even configurations in trilayers. Self-consistent equations are solved to determine the magnetic moments and energy per site.
• Excitation Spectrum: Transverse spin excitations are calculated using the Random Phase Approximation (RPA) within the self-consistent SDW state. The authors compute the transverse spin susceptibility $\chi^{+-}_{RPA}(q, \omega)$, projecting the response onto specific layer and orbital channels (e.g., mirror-odd, mirror-even, outer-layer, middle-layer) to identify distinct spectral features.

Key Contributions and Results

1. Bilayer Nickelates ($La_3Ni_2O_7$):

• Ground State Energetics: HF calculations indicate that the double-stripe (DS) order is slightly energetically favored over the single-stripe (SS) order. The energy difference increases with interaction strength ($U, J$).
• Excitation Spectra vs. Experiment: Despite the energetic preference for DS, the calculated excitation spectrum of the SS state shows better qualitative agreement with recent RIXS and neutron scattering data.
  • The SS state exhibits an anisotropic low-energy cone at the ordering vector $Q_{BL} = (0.5, 0.5)\pi$ and isotropic high-energy excitations near $\Gamma$.
  • The DS state, conversely, shows a highly anisotropic dispersion and an additional intense feature bending downward along $Q_{BL}-X$, which does not match experimental observations.
  • The SS state preserves specific mirror symmetries ($M_{11}$) absent in the DS state, leading to degenerate gapless cones, whereas the DS state exhibits cone splitting due to broken $PT$ symmetry.
• Optical Modes: The authors identify mirror-even optical interlayer modes at $Q_{BL}$ with energies matching the mirror-odd modes at $\Gamma$.

2. Trilayer Nickelates ($La_4Ni_3O_10$):

• Competing SDW States: Both mirror-odd and mirror-even SDW states can be stabilized near the ordering vector $Q_{TL} \approx (0.62, 0.62)\pi$.
  • The mirror-odd state (driven by scattering between mirror-even $\alpha$ and mirror-odd $\beta'$ pockets) is energetically lower in the studied parameter regime. It features a layer pattern $(M, 0, -M)$, where the middle layer has a vanishing static moment.
  • The mirror-even state (driven by same-mirror-parity scattering) favors a configuration with finite moments on all layers, antiferromagnetically coupled between the middle and outer layers.
• Distinct Excitation Signatures:
  • Mirror-odd SDW: Hosts two nearly gapless modes and one gapped optical mode. Crucially, it features a nearly gapless mode dominated by the middle layer, which remains dynamically coupled but not exchange-pinned due to the lack of a static moment.
  • Mirror-even SDW: Contains only one gapless (Goldstone) mode and two gapped optical modes. The middle-layer fluctuation is exchange-pinned, preventing the emergence of a second gapless mode.
• Experimental Consistency: Comparison with available RIXS data favors the mirror-odd scenario, as experiments typically identify only one optical mode, consistent with the mirror-odd prediction.

Significance and Claims
The paper argues that magnetic excitations serve as a sensitive probe for identifying the true magnetic order in multilayer nickelates, often providing clearer distinctions than ground-state energy calculations alone.

• Itinerant Origin: The results support a unified itinerant description where both SDW order and superconductivity arise from the same low-energy electronic structure, specifically driven by scattering between Fermi pockets of opposite mirror parity.
• Resolution of Structural Debates: The study suggests that the experimentally realized state in bilayer nickelates is likely the single-stripe order, despite HF calculations favoring the double-stripe order, implying that additional charge degrees of freedom or lattice effects (not included in the current HF treatment) may be necessary to stabilize the SS phase energetically.
• Layer-Dependent Physics: For trilayers, the identification of a middle-layer-dominated gapless mode in the mirror-odd state provides a theoretical basis for distinguishing between competing magnetic ground states experimentally.
• Optical Modes: The high-energy optical spin-wave modes are identified as a characteristic consequence of strong interlayer coupling, offering a direct measure of effective interlayer magnetic coupling.

The authors conclude that their multi-orbital itinerant framework successfully captures the essential features of magnetic ground states and excitations in these systems, resolving ambiguities regarding the magnetic structure through the analysis of excitation spectra.