Simulating altermagnets using mumax+

This paper demonstrates the simulation of altermagnets in the newly released micromagnetic package mumax+ by introducing a dedicated magnet class that successfully reproduces analytical domain wall profiles, anisotropic magnon dispersion, and spin-transfer-torque-driven skyrmion dynamics, leveraging its object-oriented design to accurately handle multi-sublattice magnetostatic fields.

The Gist

Imagine you are trying to build a digital city to understand how tiny magnets behave. For a long time, scientists had two main types of "magnetic citizens" in their simulations: Ferromagnets (like fridge magnets, where everyone points the same way) and Antiferromagnets (where neighbors point in opposite directions, canceling each other out perfectly).

But recently, physicists discovered a third, mysterious type of citizen called an Altermagnet. These are tricky! They act like antiferromagnets (canceling out so they don't attract metal), but they also have a hidden "superpower" that breaks symmetry in a very specific, wavy pattern (called d-wave symmetry).

The problem? The existing software tools used to simulate these magnetic cities were like old maps. They could draw ferromagnets and antiferromagnets, but when they tried to draw altermagnets, they made a critical mistake: they calculated the "invisible pressure" (magnetostatic fields) wrong. It was like trying to predict traffic flow in a city but ignoring the wind.

This paper introduces a major upgrade to a simulation tool called mumax+. The authors have built a brand-new "Altermagnet" module that fixes these errors and allows scientists to finally simulate these exotic materials correctly.

Here is a breakdown of what they did, using some everyday analogies:

1. The New "City Planner" (The Software Upgrade)

Think of the simulation software as a LEGO set. Previously, you could only build with standard bricks (ferromagnets) or simple alternating bricks (antiferromagnets).

• The Innovation: The authors added a new, special type of brick that represents the d-wave altermagnet.
• Why it matters: In the old software, if you tried to build an altermagnet, the software would assume the "wind" (magnetic fields) was zero because the magnets canceled out. But in reality, altermagnets create subtle, complex wind patterns that matter. The new mumax+ software correctly calculates these winds, even when the magnets are packed tightly together in the same "cell" of the simulation.

2. Test Drive #1: The "Traffic Jam" (Domain Walls)

To prove their new bricks work, they simulated a Domain Wall. Imagine a long line of people where half are facing North and half are facing South. The "wall" is the messy transition zone where they turn around.

• The Theory: Because the "wind" (exchange interaction) is stronger in some directions than others for altermagnets, the transition zone shouldn't be a perfect, symmetrical turn. It should be lopsided, creating a tiny net movement.
• The Result: The simulation showed exactly this lopsided turn. The digital "people" turned in a way that matched the complex math equations perfectly. It was like checking a new car's suspension by driving over a bump and seeing if it bounced exactly as the engineers predicted.

3. Test Drive #2: The "Sound Wave" (Magnon Dispersion)

Next, they looked at Magnons. Think of these as sound waves traveling through a crowd of people.

• The Twist: In normal materials, sound travels the same speed in all directions. But in altermagnets, because of their special "d-wave" shape, sound travels faster in some directions and slower in others.
• The Result: The simulation showed that these "sound waves" split into two different paths (like a fork in the road) depending on the angle. When they rotated the simulation grid, the paths merged and split again exactly as the math predicted. This proved the software understands the "directional flavor" of the new material.

4. Test Drive #3: The "Drifting Skater" (Skyrmion Hall Effect)

Finally, they simulated a Skyrmion. Imagine a tiny, stable whirlpool or a spinning top made of magnetic spins.

• The Problem: When you push a regular skater, they go straight. But in altermagnets, some researchers claimed that if you push a Skyrmion, it drifts sideways (the Hall Effect).
• The Surprise: The authors discovered that in previous simulations, this sideways drift was actually a glitch—a numerical error caused by the simulation grid being too "pixelated."
• The Fix: Using their new, high-precision mumax+ tool, they showed that as you make the simulation grid finer (less pixelated), the "ghost" sideways drift disappears. They proved that the drift wasn't a real physical effect in this specific setup, but a mistake in how the old software handled the math.

The Big Picture

This paper is like a mechanic saying, "We fixed the engine."

• Before: Trying to simulate altermagnets was like trying to navigate a city with a map that had missing streets. You might get there, but your route would be wrong.
• Now: mumax+ has a complete, accurate map. It allows scientists to design new spintronic devices (like faster, more efficient computer memory) using altermagnets without worrying that their computer models are lying to them.

In short, the authors have updated the "operating system" for magnetic simulations, ensuring that this exciting new class of materials can be studied with the precision it deserves.

Technical Summary

1. Problem Statement

Altermagnets are a newly identified class of magnetic materials characterized by compensated magnetic moments (like antiferromagnets) but broken time-reversal symmetry (like ferromagnets), often exhibiting $d$-wave symmetry in their spin ordering. While theoretical studies using ab initio methods and atomistic DFT exist, there is a critical gap in micromagnetic modeling tools.

• Current Limitation: Existing methods often model altermagnets by coupling two separate ferromagnetic layers. This approach fails to correctly calculate magnetostatic (stray) fields because the sublattices are treated as spatially distinct rather than occupying the same simulation cell.
• Consequence: In altermagnets, stray fields are not always negligible (unlike in ideal antiferromagnets), leading to inaccurate simulations of domain walls, skyrmions, and magnon dynamics.
• Need: A dedicated micromagnetic solver capable of handling multi-sublattice systems within a single cell to accurately capture both exchange anisotropy and magnetostatic interactions.

2. Methodology

The authors implemented a new Altermagnet class within mumax+, a GPU-accelerated, finite-difference micromagnetic simulation package. The implementation leverages the object-oriented design of mumax+, which allows multiple sublattices to reside in the same simulation cell.

Key Technical Implementations:

• Anisotropic Exchange Field: The core physics added is an anisotropic exchange interaction arising from broken symmetry in the local environment.
  • The exchange stiffness is defined by a symmetric, positive-definite matrix $\mathbf{A}$.
  • The exchange field $\mathbf{H}_{ex}$ is calculated as $A_{ij} \partial_i \partial_j \mathbf{m}(s)$, where $\mathbf{m}(s)$ is the magnetization of sublattice $s$.
  • Rotation Handling: Since the eigenbasis of $\mathbf{A}$ may not align with the simulation grid, the authors implemented a rotation matrix $R(\phi)$ to transform the exchange tensor.
  • Sublattice Interchange: For $d$-wave altermagnets, the eigenvalues of $\mathbf{A}$ ($A_1, A_2$) are automatically swapped between the two sublattices to reflect the $90^\circ$ rotation of the spin order.
• Numerical Schemes:
  • Diagonal terms ($\partial^2/\partial x^2$, $\partial^2/\partial y^2$) use standard second-order central differences.
  • Off-diagonal mixed derivative terms ($\partial^2/\partial x \partial y$) were implemented using a specific second-order stencil to handle the cross-terms introduced by the rotated exchange matrix.
• Verification Strategy: The implementation was validated against three analytical models derived from Lagrangian descriptions and linearized Landau-Lifshitz-Gilbert (LLG) equations.

3. Key Contributions

1. New Physics Module: Introduction of the Altermagnet class in mumax+, enabling the simulation of $d$-wave altermagnets with correct magnetostatic field calculations in multi-sublattice systems.
2. Analytical Verification:
   • Static Domain Walls: Verified the non-vanishing net magnetization in domain walls caused by unequal exchange stiffness ($A_1 \neq A_2$) between sublattices.
   • Magnon Dispersion: Validated the lifting of degeneracy in magnon branches due to anisotropic exchange, showing dependence on the angle $\phi$ between the exchange eigenbasis and the simulation grid.
   • Skyrmion Hall Effect: Investigated the "Skyrmion Hall Effect" in altermagnets, distinguishing between physical effects and numerical artifacts (rounding errors) related to grid cell size.
3. Open Source Tool: The code is integrated into mumax+ (v1.2+), available under GPLv3, facilitating reproducible research in altermagnetism.

4. Results

The paper presents three specific test cases demonstrating high fidelity between simulation and theory:

• Static Domain Wall Profiles:

  • Simulated a Bloch domain wall separating out-of-plane domains.
  • Result: The simulation reproduced the analytical line profiles for the Néel vector and net magnetization with high precision.
  • Deviation: Maximal deviation was 0.05% for the Néel vector and 3% for the net magnetization (which is of order $10^{-4}$).
  • Observation: Confirmed that unequal exchange stiffness leads to different domain wall widths for each sublattice, resulting in a net magnetic moment.

• Magnon Dispersion Relation:

  • Simulated 1D magnon propagation and performed spatiotemporal Fourier transforms.
  • Result: The simulation correctly reproduced the splitting of magnon branches ($\omega_+$ and $\omega_-$) predicted by theory.
  • Anisotropy: Confirmed that when the exchange eigenbasis is rotated by $45^\circ$ relative to the grid ($\phi=45^\circ$), the degeneracy is restored (no splitting), whereas other angles show distinct branches.
  • Deviation: Maximal deviation of 1.5% for frequency gaps and 2.5% for sublattice intensity ratios. Discrepancies at high $k$ (small wavelengths) were attributed to the breakdown of continuum micromagnetic theory.

• Skyrmion Motion (Skyrmion Hall Effect):

  • Simulated the motion of a Néel skyrmion under spin-transfer torque.
  • Result: The study confirmed that the transverse velocity (Skyrmion Hall effect) observed in previous studies was largely a numerical artifact scaling with the simulation cell size.
  • Scaling: As the cell size decreased, the transverse velocity vanished, aligning with the theoretical prediction that the effect is a discretization error rather than a physical phenomenon in this specific context.

5. Significance

• Correcting Magnetostatics: This work resolves a fundamental flaw in previous altermagnet simulations by ensuring stray fields are calculated correctly within a single-cell multi-sublattice framework.
• Enabling Complex Dynamics: The tool allows researchers to study complex topological textures (like skyrmions) and spin-wave dynamics in altermagnets, which are crucial for future spintronic and magnonic applications.
• Extensibility: The successful integration demonstrates the flexibility of the mumax+ architecture. The authors note that this framework can easily be extended to other exchange symmetries (e.g., $f$-wave, $g$-wave) and 3D simulations in future releases.
• Community Impact: By providing a verified, open-source tool, the paper lowers the barrier to entry for numerical studies in the rapidly emerging field of altermagnetism.