🔬 mesoscale physics

Residual orbital magnetization governs the anomalous Hall effect in altermagnets

This paper demonstrates that residual orbital magnetization, generated by the interplay of local crystal fields and spin-orbit coupling, intrinsically governs the anomalous Hall effect in altermagnets through the generalized Středa relation, challenging the view that their small remanent magnetization is irrelevant to transport phenomena.

Original authors: Yufei Zhao, Yiyang Jiang, Kamal Das, Chao-Xing Liu, Binghai Yan

Published 2026-06-25

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CC BY 4.0

Original authors: Yufei Zhao, Yiyang Jiang, Kamal Das, Chao-Xing Liu, Binghai Yan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Ghost" Magnet

Imagine you have a team of dancers (electrons) moving on a stage. In most magnets, the dancers all spin in the same direction, creating a strong, obvious magnetic pull. In antiferromagnets (a specific type of magnetic material), the dancers are paired up: one spins left, the other spins right. They cancel each other out perfectly, so the whole group looks like it has zero magnetism.

For a long time, scientists thought these "zero-magnetism" materials couldn't do something called the Anomalous Hall Effect (AHE). The AHE is like a traffic jam where cars (electrons) are forced to swerve to the side, creating a sideways electric current. Usually, you need a strong magnetic field to cause this swerving.

However, researchers recently found that certain materials called altermagnets (like a specific type of Manganese Telluride, or MnTe) do show this sideways swerving, even though they have almost no net magnetism. They also noticed a tiny, almost invisible "remanent magnetization" (a tiny leftover magnetic wobble), but they dismissed it as too small to matter. They thought it was just a tiny accident or a side effect, not the cause of the swerving.

This paper says: "Stop dismissing that tiny wobble! It's actually the key."

The Main Discovery: The "Speedometer" vs. The "Speed"

The authors use a mathematical rule (called the Středa relation) to explain why the tiny wobble matters.

Think of the Anomalous Hall Effect (the sideways swerving) not as being caused by how strong the magnet is, but by how fast the magnetism changes as you tweak the energy of the electrons.

The Old View: "The magnet is too weak (like a tiny 1% battery), so it can't push the electrons."
The New View: "It doesn't matter if the battery is small. What matters is how quickly the battery level changes when you turn a knob."

The authors show that even a tiny magnetic moment (the "wobble") can create a huge sideways current if it changes rapidly enough. It's like a tiny, sensitive steering wheel on a race car: even a tiny turn of the wheel (a small change in magnetism) can make the car swerve wildly.

How It Works: The "Spinning Top" Analogy

The paper digs into why this tiny wobble exists in the first place. They look at the atoms inside the material, specifically the Manganese (Mn) atoms surrounded by Tellurium (Te) atoms, forming a shape like a soccer ball (an octahedron).

The Setup: Imagine a spinning top (an electron) inside a cage (the crystal structure).
The Conflict: The cage is slightly tilted (due to the crystal field), and the top is spinning (spin).
The Twist: There is a subtle interaction called Spin-Orbit Coupling (SOC). You can think of this as a "glue" that tries to lock the direction of the spin to the shape of the cage.

Because the cage is tilted in a specific way, the "glue" forces the spinning top to lean slightly out of its flat plane.

In normal magnets, neighbors push each other to lean (like a crowd doing a wave).
In these altermagnets, the authors show that the lean happens locally. You don't need neighbors to push; the shape of the cage itself, combined with the "glue," forces the spin to tilt.

This tilt creates two things:

A tiny spin wobble: The electron spins slightly up or down.
An orbital wobble: The electron's path around the nucleus also tilts.

The "Molecule" vs. The "City"

The authors used two different ways to prove this:

The "Molecule" Model (The Single House): They looked at just one single atom and its immediate neighbors. They found that even in isolation, the shape of the cage forces the spin to tilt. This proves the effect is intrinsic (built-in), not caused by outside noise or defects.
The "City" Model (The Whole Neighborhood): They then looked at the whole crystal lattice. They found that while the single atom creates a "tilt," the movement of electrons between atoms (hopping) is what actually generates the big sideways current (the Hall Effect).

The Crucial Insight: The "tilt" (the tiny magnetism) is the thermodynamic property (the state of the system) that governs the current. Even if the tilt is tiny, it dictates how the electrons flow.

The "Three-Step" Dance

One of the coolest findings is how this tilt depends on the angle of the spin.

If you rotate the spin direction, the amount of tilt doesn't just go up and down smoothly.
It follows a three-step pattern (mathematically, a sine wave with three peaks).
This means the material has a "preference" for the spin to point in three specific directions. If you try to force it elsewhere, it snaps back. This explains why the material behaves the way it does in experiments.

Summary

The Problem: Scientists saw a tiny magnetic wobble in these special magnets and ignored it, thinking it was too small to cause the big sideways electric current they observed.
The Solution: The paper proves that the change in this tiny wobble is what drives the current. The wobble is not an accident; it is the engine.
The Mechanism: The shape of the atomic cage forces the electron spins to lean slightly, creating this wobble without needing help from neighboring atoms.
The Takeaway: In these "unconventional" magnets, the tiny leftover magnetism is the boss. It controls the traffic flow of electricity, and understanding it is essential to understanding how these materials work.

Technical Summary: Residual Orbital Magnetization in Altermagnets

Problem Statement
Altermagnets are a class of collinear, compensated magnets characterized by vanishing net magnetization yet exhibiting momentum-dependent spin splitting and the anomalous Hall effect (AHE). Representative materials, such as $\alpha$ -MnTe, display a clear AHE alongside a small remanent magnetization (typically $\sim 10^{-5} \mu_B$ per Mn atom). Historically, this remanent magnetization has been dismissed as an extrinsic background, a stoichiometric artifact, or a negligible higher-order residue (e.g., from Dzyaloshinskii-Moriya interactions) that is irrelevant to the Hall response. This dismissal relies on the empirical relation $\rho_H = R_0H + R_sM$ , which suggests the Hall response scales with the magnitude of the net magnetization. However, this view overlooks the possibility that the small remanent moment is intrinsically linked to the AHE mechanism.

Methodology
The authors employ a dual theoretical approach to investigate the relationship between remanent magnetization and the AHE in altermagnets:

Lattice Model (Momentum Space): An effective four-band model for a $g$ -wave altermagnet on a hexagonal lattice is constructed. This model includes crystal field terms, staggered Néel exchange ( $J$ ), and spin-orbit coupling (SOC). The authors calculate the intrinsic anomalous Hall conductivity ( $\sigma_{xy}^{AH}$ ) using the Kubo-Berry curvature formula and the orbital magnetization ( $M_z$ ) using the modern theory of orbital magnetization, which separates the quantity into "trivial" (wave-packet self-rotation) and "topological" (Berry curvature) components.
Molecular Model (Real Space): To elucidate the microscopic origin of the magnetization, the authors analyze an isolated MnTe $_6$ octahedral molecule. They rotate the octahedron to a trigonal configuration ( $D_{3d}$ symmetry) to mimic the local environment in MnTe. The local Hamiltonian includes crystal field splitting, in-plane exchange coupling, and atomic SOC. Exact diagonalization is performed to trace the evolution of energy levels and spin states.

Key Contributions and Results

The Generalized Středa Relation: The paper establishes that the intrinsic AHE is governed not by the magnitude of the remanent magnetization, but by its derivative with respect to the chemical potential. Using the generalized Středa relation, $\sigma_{xy} = -e (\partial M_z / \partial \mu)$ , the authors demonstrate that even a tiny orbital magnetization ( $M_z$ ) can generate a significant AHE if it varies sufficiently with the chemical potential. A sensitivity estimate shows that a variation of $10^{-5} \mu_B$ by 1% over 1 meV corresponds to an AHC of $\sim 0.1$ S/cm, consistent with experimental observations in MnTe.
Decoupling of Magnitude and Response: Calculations on the lattice model reveal that the total orbital magnetization ( $M_z$ ) and its components ( $M_z^{tri}$ and $M_z^{topo}$ ) can be much larger than the net moment, and $M_z$ can even accidentally vanish at specific chemical potentials while $\sigma_{xy}^{AH}$ remains finite (exhibiting peaks). This contradicts the empirical assumption that a small moment implies a negligible Hall effect.
Microscopic Mechanism of Spin Canting: The molecular model reveals that the net out-of-plane magnetization arises intrinsically from the interplay between the local crystal field and SOC, without requiring exchange interactions between neighboring spins (such as Dzyaloshinskii-Moriya interaction).
- In the absence of SOC, the $d^5$ high-spin configuration yields zero net orbital and spin moment.
- The inclusion of SOC mixes spin channels, inducing a spontaneous out-of-plane spin canting ( $S_z$ ) and an on-site orbital moment ( $M_z$ ).
- This canting follows a strict third-harmonic dependence, $M_z(\phi) = A_3 \sin(3\phi)$ , where $\phi$ is the in-plane spin angle. The prefactor scales as $A_3 \propto \lambda^3 \Delta_{cf}^3 \Delta_{ex}^{-6}$ , indicating the effect is a third-order perturbative result in SOC.
Sublattice Locking: In altermagnets, the two spin sublattices experience opposite crystal fields due to the specific symmetry of the lattice (unlike ordinary antiferromagnets). The "crystal field–sublattice locking" ensures that despite the reversal of the spin angle and crystal field between sublattices, the orbital magnetization $M_z$ remains invariant (sublattice-even). This generates a finite net orbital moment that coexists with the in-plane Néel order.

Significance and Claims
The paper argues that the weak out-of-plane ferromagnetism observed in altermagnets is not an accidental byproduct or an extrinsic defect, but an intrinsic thermodynamic property essential for understanding their transport phenomena.

Redefining the Role of Remanence: The authors claim that the residual orbital and spin magnetization governs the anomalous transport in unconventional antiferromagnets (including altermagnets and noncollinear antiferromagnets) via the Středa relation.
Mechanism Clarification: The work clarifies that the net magnetization arises from local crystal field and SOC interplay within a single atomic environment, rather than solely from intersite exchange interactions.
Implications for Control: The specific $\sin(3\phi)$ dependence implies that an external magnetic field can flip the out-of-plane magnetization and spontaneously rotate the in-plane spins by $\pm 60^\circ$ . This suggests a hysteretic mechanism for controlling the Néel vector, linking the remanent ferromagnetization directly to the switching of the Néel order and the anomalous Hall response.

The authors conclude that the dismissal of small remanent magnetization in the literature is incomplete; rather, this residual moment is the bulk observable that carries the Hall response through its chemical-potential dependence.