Magnetic anisotropy from interligand hopping in strongly correlated insulators: application to the magnon spectrum of CrI$_3$

This paper proposes a method to calculate magnetic exchange interactions driven by interligand hopping and strong spin-orbit coupling in multi-orbital Mott insulators, applying it to CrI$_3$ to successfully reproduce experimental magnon spectra and explain how these interactions stabilize ferromagnetic order and induce nontrivial topological features.

The Gist

Imagine a tiny, two-dimensional world made of a single layer of atoms, specifically a material called CrI₃ (Chromium Iodide). In this world, the Chromium atoms are like little magnets (spins) that want to all point in the same direction, creating a strong magnetic field. Scientists have been trying to understand exactly how these magnets talk to each other to stay aligned, especially when the material is so thin that quantum effects take over.

This paper is like a detective story where the authors solve a mystery: Why do these magnets behave the way they do, and why do they have a specific "energy gap" (a safety buffer) that keeps them stable?

Here is the story broken down into simple concepts and analogies:

1. The Setting: A Dance Floor with a Twist

Think of the Chromium atoms as dancers on a floor. They are surrounded by Iodine atoms, which act like the "floor" or the "musicians" for the dance.

• The Problem: In the past, scientists thought the dancers (Chromium) only talked to each other directly. But in this material, the Iodine atoms are special because they have a strong "spin" (a quantum property called Spin-Orbit Coupling).
• The Twist: The Iodine atoms aren't just passive floorboards; they are active participants. They can catch a "step" (an electron) from one Chromium dancer, spin it around, and pass it to another Chromium dancer.

2. The New Mechanism: The "Relay Race"

The authors discovered a new way the magnets interact, which they call Interligand Hopping.

• Old View: Imagine two people (Chromium A and Chromium B) trying to pass a ball (an electron) to each other. Previously, we thought they had to throw it directly.
• New View: The authors realized the ball actually goes through a relay race. Chromium A throws the ball to a nearby Iodine atom. The Iodine atom spins the ball, changes its color (orbital state), and then passes it to a different Iodine atom, which finally hands it to Chromium B.
• Why it matters: This relay race allows the magnets to talk to each other over longer distances and in more complex ways than anyone thought possible. It's like realizing that to get a message across a crowded room, you don't just shout; you use a chain of friends who whisper, spin the message, and pass it along.

3. The "Hidden Symmetry" and the Missing Gap

The scientists wanted to explain a specific feature of the material: a Magnon Gap.

• The Analogy: Imagine a trampoline. If you jump on it, you can bounce up and down easily. But if the trampoline has a stiff spring in the middle, you need a minimum amount of energy just to get off the ground. That "minimum energy" is the Gap.
• The Mystery: Experiments showed a large gap (a stiff spring) at certain points in the material's energy map (called Dirac points). Previous theories tried to explain this using "Kitaev interactions" (a specific type of magnetic rule) or "DMI" (a twisting force between neighbors).
• The Discovery: The authors ran a super-complex simulation using their new "relay race" model. They found that:
  1. The "twisting force" (DMI) between distant magnets is actually zero because the Iodine atoms are arranged in a perfectly symmetrical way that cancels it out.
  2. The "Kitaev" rules are too weak to explain the big gap seen in experiments.
  3. The Real Hero: The gap is actually caused by Single-Ion Anisotropy. Think of this as a "personal preference" each magnet has. Because the Iodine atoms are hopping electrons around, they create a "dressing" around the Chromium magnets that forces them to stand up straight (pointing up or down) rather than lying flat. This preference creates the "stiff spring" (the gap) that stabilizes the whole system.

4. The Result: A Good Match, But Not Perfect

When the authors used their new model to predict how the waves of magnetism (magnons) move through the material:

• The Good News: Their prediction matched the experimental data almost perfectly for most of the energy spectrum. They successfully explained why the magnets stay aligned and why there is a gap at the center of the energy map.
• The Bad News: At the specific "Dirac points" (the corners of the energy map), their calculated gap was much smaller than what experiments showed.
• The Conclusion: While their model explains most of the physics, there is still a missing piece of the puzzle for those specific corners. It suggests that maybe the atoms are slightly distorted in reality (like a slightly wobbly dance floor) or that the interaction between the magnets and the vibrations of the atoms (phonons) is stronger than they calculated.

Summary in a Nutshell

This paper introduces a new way to calculate how magnets talk to each other in 2D materials. Instead of thinking of them as isolated islands, the authors show that the "island" (the Iodine atoms) is actually a busy highway where electrons hop, spin, and change states.

They found that this hopping creates a "personal preference" for the magnets to stand up, which stabilizes the material. While their model explains most of the behavior, it reveals that the "mystery" of the large energy gap at the corners of the map isn't solved by the usual suspects (twisting forces or Kitaev rules), but likely by a combination of this new hopping mechanism and subtle structural details we haven't fully captured yet.

The Takeaway: We now have a better map of how these quantum magnets interact, thanks to realizing that the "floor" (Iodine) is just as important as the "dancers" (Chromium).

Technical Summary

1. Problem Statement

The paper addresses the challenge of accurately calculating magnetic exchange interactions in Mott-Hubbard insulators, specifically two-dimensional van der Waals materials like CrI$_3$, where strong Spin-Orbit Coupling (SOC) exists on ligand ions (Iodine) rather than just the magnetic transition metal ions.

• The Discrepancy: While CrI$_3$ exhibits strong out-of-plane magnetic anisotropy and a large energy gap at the Dirac points in its magnon spectrum, existing theoretical models fail to explain the magnitude of the Dirac gap.
• Limitations of Previous Models:
  • Standard Goodenough-Kanamori (GK) rules account for isotropic Heisenberg exchange and some anisotropy but miss processes involving spin flips accompanied by orbital state changes in ligand holes.
  • Previous perturbative calculations (e.g., Ref. 21) included ligand SOC but neglected interligand hopping (hopping between p-orbitals of neighboring ligand ions).
  • Ab initio calculations often yield anisotropic exchange interactions that are too weak to explain the experimentally observed large gap at the Dirac points.
• The Gap: There is a need for a method that accounts for the delocalization of ligand holes via interligand hopping, which lowers local symmetry and generates new exchange paths, to accurately predict anisotropic interactions and the magnon spectrum.

2. Methodology

The authors propose a novel effective method to calculate exchange interactions at arbitrary separations by integrating out the ligand degrees of freedom within an extended $dp$ Hubbard model.

• Microscopic Model:
  • Hamiltonian: Uses a $dp$ model describing Cr $3d$ orbitals and I $5p$ orbitals.
  • Key Innovation: Unlike previous approaches, the interligand hopping ($H_{pp}$) is treated as part of the unperturbed Hamiltonian ($H_0$), while the metal-ligand hopping ($H_{dp}$) is treated as a perturbation. This allows for the summation of all possible exchange paths mediated by the ligand network.
  • Parameters: Hopping amplitudes ($t_{pd}$, $t_{pp}$) and on-site energies are derived from first-principles Density Functional Theory (DFT) calculations using maximally localized Wannier functions. Other parameters (Hubbard $U$, Hund's coupling $J_H$, crystal field splitting $\Delta_c$) are taken from literature and experimental data.
• Derivation Procedure:
  1. Band Structure Calculation: Compute the band structure of the ligand ($p$) electrons, including SOC and interligand hopping, to obtain Bloch states $|V_{k,n}\rangle$.
  2. Effective Hopping Amplitudes: Calculate effective $dd$ hopping amplitudes ($t_{dd}$) between transition metal sites mediated by the ligand band states. This involves summing over all intermediate ligand states.
  3. Spin Hamiltonian Mapping: Map the second-order perturbation theory of the effective $dd$ hopping to a spin Hamiltonian. This yields:
     • Isotropic Heisenberg exchange ($J$).
     • Symmetric anisotropic exchange ($\Gamma$).
     • Dzyaloshinskii-Moriya Interaction (DMI, $\mathbf{D}$).
     • Single-ion anisotropy (SIA) derived from long-range hopping processes.

3. Key Contributions

• Interligand Hopping Mechanism: The paper demonstrates that interligand hopping ($t_{pp}$) is a critical mechanism for generating magnetic anisotropy. It effectively lowers the local symmetry of the transition metal sites (even in an ideal, undistorted lattice), enabling exchange processes forbidden by standard GK rules.
• Effective Calculation Method: They developed a scalable procedure to calculate exchange interactions for arbitrary spin separations by utilizing the ligand band structure, avoiding the combinatorial explosion of counting individual exchange paths.
• Origin of Single-Ion Anisotropy (SIA): The authors show that SIA in CrI$_3$ arises from the "dressing" of Cr spins by ligand hole states. The delocalization of holes over the ligand network via $pp$-hopping induces an effective SOC on the Cr sites, leading to quadratic single-ion anisotropy ($A_c$).
• Symmetry Analysis: They rigorously analyzed the symmetry of the system, showing that while the monolayer symmetry allows for DMI, the specific inversion symmetry of the iodine sublattice (in the absence of trigonal distortions) forces the next-nearest-neighbor DMI to vanish.

4. Results

• Spin Model Parameters: Using first-principles parameters, the calculated spin Hamiltonian for monolayer CrI$_3$ yields:
  • Ferromagnetic nearest-neighbor exchange: $J_1 \approx -2.12$ meV.
  • Antiferromagnetic next-nearest-neighbor exchange: $J_2 \approx -0.2$ meV.
  • Significant Single-Ion Anisotropy: $A_c \approx -0.1$ meV (favoring out-of-plane spins).
  • Weak Kitaev interaction ($K_1 \approx 0.04$ meV) and negligible DMI.
• Magnon Spectrum:
  • The calculated magnon dispersion agrees well with experimental neutron scattering data for bulk CrI$_3$ regarding the acoustic and optical magnon bandwidths.
  • The gap at the $\Gamma$ point (zone center) is correctly reproduced ($\approx 0.3$ meV) and is identified as being driven primarily by the calculated single-ion anisotropy.
• The Dirac Gap Problem:
  • The calculated gap at the Dirac points (K point) is very small, significantly underestimating the large gap observed experimentally.
  • The authors conclude that neither the Kitaev interaction nor the next-nearest-neighbor DMI (which vanishes in their ideal model) is the source of the large experimental Dirac gap.
  • They suggest that the large experimental gap may require mechanisms not included in their static model, such as strong magnon-phonon coupling or lattice distortions not present in the idealized geometry.

5. Significance

• Theoretical Advancement: The work provides a robust framework for treating strongly correlated insulators with strong ligand SOC, moving beyond the limitations of local cluster approximations by explicitly including the delocalized nature of ligand holes.
• Clarification of Anisotropy Sources: It establishes that interligand hopping is a primary driver of magnetic anisotropy in van der Waals magnets, comparable in effect to lattice distortions. This challenges the view that anisotropy in these materials is solely due to structural distortions or direct metal-ligand SOC.
• Resolution of the $\Gamma$-Gap: The study successfully identifies single-ion anisotropy (induced by ligand hole delocalization) as the stabilizer of ferromagnetic order and the source of the zone-center magnon gap.
• Future Directions: The paper highlights that the origin of the large Dirac gap remains an open question, potentially pointing to dynamic effects (phonons) or subtle structural distortions. The proposed method is extendable to other systems like twisted bilayer CrI$_3$ and other ligand-SOC materials (e.g., CrGeTe$_3$), offering a path to understand complex magnetic textures like skyrmions and topological magnon edge states.