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

This paper investigates the magnetic ground states of bilayer nickelates derived from a hypothetical $d^8$ parent state, demonstrating that the interplay of superexchange and Hund's coupling leads to distinct high-spin, low-spin, or spin-gapped phases, with the high-spin state proving more robust and highlighting the critical need to identify the specific spin state to understand the material's superconductivity.

The Gist

Imagine you have a new type of building block material called a "bilayer nickelate." Recently, scientists discovered that under certain conditions, this material can conduct electricity with zero resistance (superconductivity) at surprisingly high temperatures. This is a big deal because it could revolutionize how we transmit energy.

However, to understand how it works, scientists need to figure out what the tiny magnets inside the material are doing before the electricity starts flowing. This paper is like a detective story trying to solve the mystery of the "parent state"—the material's behavior when it's not yet superconducting.

Here is the breakdown of their findings using simple analogies:

The Setting: A Two-Layer Dance Floor

Think of the material as a two-story dance floor. On this floor, there are electrons (the dancers) moving around.

• The "X" Dancers: These dancers mostly move side-to-side on their own floor.
• The "Z" Dancers: These dancers are special; they love to jump between the top and bottom floors, holding hands with their partner directly across the gap.

The paper asks: How do these dancers pair up? The answer depends on two competing forces:

1. The "Hund's Coupling" (The Best Friend Rule): This force wants the dancers on the same spot to spin in the same direction, like best friends holding hands and marching in step.
2. The "Superexchange" (The Opposite Neighbor Rule): This force wants neighbors to spin in opposite directions, like a game of "opposites attract."

The Three Possible Outcomes

Depending on which force is stronger, the material settles into one of three distinct "mood states":

1. The "High Spin" State (The Marching Band)

If the "Best Friend Rule" is very strong, the dancers on the same spot lock arms and spin together.

• The Analogy: Imagine a marching band where every pair of drummers on the same beat is spinning in the same direction. They act like a single, strong magnet (Spin-1).
• The Result: This creates a very robust, strong magnetic order. It's like a solid wall of magnets that is hard to break.

2. The "Low Spin" State (The Silent Partners)

If the "Opposite Neighbor Rule" wins, specifically for the "Z" dancers who jump between floors, something interesting happens.

• The Analogy: The "Z" dancers jump between the floors and form a perfect, silent hug with their partner on the other side. They cancel each other out completely, becoming invisible to the magnetic world.
• The Result: The "Z" dancers disappear from the magnetic picture. Now, only the "X" dancers (who move side-to-side) are left doing the magnetic dance. This makes the whole system behave much simpler, almost like a single-layer material (similar to the famous cuprate superconductors).

3. The "Gapped Spin" State (The Frozen Silence)

If the forces are just right, the "Z" dancers form those silent hugs so strongly that the entire system stops moving magnetically.

• The Analogy: The dance floor freezes. Everyone is holding hands in pairs, but no one is spinning or moving. It's a quiet, non-magnetic state.
• The Result: There is no magnetism at all.

What Happens When You Add "Holes" (Doping)?

To get superconductivity, scientists usually "dope" the material, which means removing some electrons (creating "holes" or empty spots).

• The Finding: The authors used a computer simulation (Hartree-Fock method) to see what happens when they start removing dancers.
• The Result: The High Spin state (the marching band) is much tougher. It keeps its magnetic order even when you remove a lot of dancers. The Low Spin state (the simplified single-layer system) loses its magnetic order much more easily.

Why Does This Matter?

The paper concludes that figuring out which of these three states the real material is in is the key to understanding its superconductivity.

• If it's the Low Spin state, it behaves like the older cuprate superconductors we already know.
• If it's the High Spin state, it's a different beast entirely, behaving like a complex "Kondo lattice" (a specific type of magnetic interaction).

The authors don't say which one is definitely the winner in the real world yet. They simply say: "We need to run an experiment to see which 'mood' the material is actually in." If we know whether the internal magnets are marching in step (High Spin) or canceling each other out (Low Spin), we can finally understand the secret recipe for high-temperature superconductivity in these nickelates.

Technical Summary

Technical Summary: High Spin, Low Spin or Gapped Spins: Magnetism in the Bilayer Nickelates

Problem Statement
Following the discovery of high-temperature superconductivity (HTS) in bilayer nickelates (e.g., $La_3Ni_2O_7$), a critical theoretical question remains: how does the local quantum chemistry of these materials, specifically the $d^{7.5}$ valence configuration, differ from the $d^9$ configuration of cuprates, and how does this influence emergent magnetic and superconducting properties? Unlike cuprates, where low-energy physics is dominated by a single $d_{x^2-y^2}$ orbital, bilayer nickelates involve multiple $d$-orbitals ($d_{x^2-y^2}$ and $d_{3z^2-r^2}$) and significant Hund's coupling. The authors investigate the magnetic ground state of a hypothetical insulating $d^8$ parent compound (e.g., $La_2CeNi_2O_7$ or $La_3Ni_2O_{6.5}$) to determine whether the system favors a high-spin, low-spin, or spin-gapped configuration. Identifying this spin state is presented as a prerequisite for understanding the mechanism of superconductivity in the doped $d^{7.5}$ regime.

Methodology
The study employs a theoretical framework based on a bilayer two-orbital Hubbard model defined on a square lattice, incorporating the $d_{x^2-y^2}$ (denoted as $X$) and $d_{3z^2-r^2}$ (denoted as $Z$) orbitals. The Hamiltonian includes:

• In-plane hopping ($t_\parallel$) for the $X$ orbital.
• Interlayer hopping ($t_\perp$) for the $Z$ orbital.
• Onsite Hubbard repulsion ($U$).
• Ferromagnetic Hund's coupling ($J_H$) between orbitals on the same site.
• Crystal field splitting ($\Delta$).

The analysis proceeds in two main stages:

1. Insulating $d^8$ Limit ($x=0$): The system is analyzed as an effective two-orbital, spin-1/2 bilayer Heisenberg model with exchange couplings $J_\parallel$ (in-plane $X$), $J_\perp$ (interlayer $Z$), and $J_H$ (intra-site Hund's). The authors utilize Bond-Operator Mean-Field Theory (BOMFT) to capture quantum disordered phases (specifically interlayer singlets) and Schwinger-Boson Mean-Field Theory (SBMFT) for comparison. Additionally, Holstein-Primakoff (HP) spin-wave theory is used to analyze the spin-wave spectrum and Néel order parameters within the antiferromagnetic phases.
2. Hole-Doped Regime ($x > 0$): To compare the robustness of magnetic orders upon doping, the authors employ the Hartree-Fock (HF) approximation. They investigate the evolution of the Néel order parameter as a function of hole concentration $x$ for both low-spin and high-spin configurations, using parameters derived from Density Functional Theory (DFT) calculations.

Key Contributions and Results

• Magnetic Phase Diagram of the $d^8$ Parent: The interplay between superexchange ($J_\perp, J_\parallel$) and Hund's coupling ($J_H$) yields three distinct magnetic regimes:

  1. Low-Spin Phase (Spin-1/2): Occurs at small $J_H$. The $Z$-orbital spins form interlayer singlets (spin-gapped), while the $X$-orbital spins exhibit Néel order. This state is effectively described by a single-band bilayer Hubbard model upon doping.
  2. High-Spin Phase (Spin-1): Occurs at large $J_H$. Hund's coupling locks the spins in both orbitals into local triplets, forming an effective spin-1 moment per site. The system behaves as a bilayer spin-1 Heisenberg antiferromagnet.
  3. Spin-Gapped Phase: Occurs when both $J_\perp$ and $J_H$ are large relative to $J_\parallel$. The system forms interlayer singlets of the effective spin-1 moments, resulting in a non-magnetic, spin-gapped ground state.

• Quantitative Phase Boundaries: Using BOMFT, the authors map the phase boundary between the Néel-ordered phase and the interlayer singlet phase. They find that the critical interlayer coupling $J_{\perp,c}$ decreases as $J_H$ increases, approaching a value of $\approx 6.10$ in the $J_H \to \infty$ limit (consistent with Quantum Monte Carlo results for spin-1 models). The authors note that SBMFT overestimates the stability of the ordered phase, placing the true critical value between the BOMFT and SBMFT estimates.

• Robustness of Magnetism upon Doping: HF calculations reveal a stark contrast in the stability of magnetic order under hole doping:

  • The high-spin antiferromagnetic (AFM) order is significantly more robust, surviving up to higher hole concentrations ($x_c$) compared to the low-spin case.
  • The low-spin AFM order is suppressed more rapidly, resembling the behavior of overdoped cuprates where the $Z$-orbital degrees of freedom are gapped out, leaving a single-band system.
  • The transition nature differs: the low-spin case exhibits a first-order transition, while the high-spin case shows a complex behavior involving a discontinuous jump followed by a continuous decrease in the order parameter.

• Effective Low-Energy Models:

  • In the low-spin limit (small $J_H/J_\perp$), the doped system maps to a single-orbital bilayer Hubbard model.
  • In the high-spin limit (large $J_H$), the doped system is described as a bilayer Kondo lattice with ferromagnetic Kondo coupling, which can further simplify to a bilayer type-II $t-J$ model in the infinite-$J_H$ limit.
  • The spin-gapped state, when doped, exhibits superconductivity distinct from both high- and low-spin scenarios, characterized by strong interlayer pairing (as detailed in the authors' prior work).

Significance and Claims
The paper asserts that the nature of magnetism in the parent $d^8$ compound is the decisive factor determining the low-energy physics of the doped bilayer nickelates. The authors emphasize that:

1. Spin State Identification is Crucial: Determining whether the material realizes a high-spin, low-spin, or spin-gapped state is essential for understanding the pairing mechanism in the superconducting phase.
2. Competition vs. Glue: While the high-spin state provides a more robust magnetic order upon doping, the paper explicitly states it remains an open question whether this robustness acts as a stronger "pairing glue" (via spin fluctuations) or competes more strongly with superconductivity.
3. Experimental Guidance: The authors highlight that the spin state can be experimentally distinguished via magnetic susceptibility, neutron scattering, or X-ray scattering. They suggest that strain engineering (which tunes the ratio $J_\perp/J_\parallel$) offers a promising avenue to probe these distinct magnetic phases and potentially access the $d^8$ insulating limit in thin films.

The study concludes that the bilayer geometry plays a unique role in stabilizing the low-spin state via interlayer singlet formation, a feature not present in monolayer nickelates (which are known spin-1 insulators), thereby distinguishing the physics of bilayer nickelates from their monolayer counterparts and cuprates.