🔬 materials science

Effect of magnetic field on whirling-anti-whirling order in icosahedral-quasicrystal approximant

This paper theoretically demonstrates that applying a magnetic field along the (111) direction to the Au-SM-Tb icosahedral-quasicrystal approximant induces a simultaneous metamagnetic and topological transition in its whirling-anti-whirling magnetic order, leading to a topological Hall effect in the electrical conductivity.

Original authors: Shinji Watanabe, Tatsuya Iwasaki

Published 2026-02-04

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CC BY 4.0

Original authors: Shinji Watanabe, Tatsuya Iwasaki

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

Imagine a crystal not as a rigid, repeating grid like a brick wall, but as a complex, non-repeating mosaic that still holds a hidden, beautiful order. This is a quasicrystal. In this specific study, the researchers are looking at a "sibling" of this crystal called an approximant, which has a repeating pattern but shares the same local atomic neighborhoods.

Inside this crystal, there are tiny magnets (atoms of a rare-earth element called Terbium) arranged in clusters shaped like 20-sided dice (icosahedrons). The paper explores what happens when you spin these tiny magnets in a specific way and then poke them with a magnetic field.

Here is the breakdown of their findings using simple analogies:

1. The "Whirling" Dance

Normally, you might expect magnets to line up like soldiers in a straight row. But in this crystal, the magnets do something much more interesting. They form a "whirling" pattern.

Imagine a group of dancers standing in a circle. Instead of facing the center or the outside, they are all leaning and spinning in a coordinated swirl.

The Whirling State: In the center of the crystal's unit cell, the magnets swirl in one direction (like a clockwise vortex).
The Anti-Whirling State: In the corners of the unit cell, the magnets swirl in the exact opposite direction (a counter-clockwise vortex).

The researchers call this the "Whirling-Anti-Whirling" order. It's a delicate, non-collinear dance where the magnets aren't pointing in a straight line but are twisting in 3D space.

2. The Magnetic "Switch" (Metamagnetic Transition)

The researchers asked: What happens if we apply a strong external magnetic field to this crystal?

Think of the crystal's magnetic state as a ball sitting in a deep valley. The ball is stable there, representing the "whirling" dance. When they apply a magnetic field along a specific direction (the [111] direction, which is like a diagonal through the cube), they are essentially pushing the ball up the side of the valley.

The Tipping Point: At a specific strength of the magnetic field, the ball suddenly rolls over the edge and falls into a new valley. This is called a metamagnetic transition.
The New Dance: Once the field is strong enough, the magnets stop their original complex swirl. They flip some of their spins to align more with the external field, earning energy from the field. The result is a new, simpler magnetic state.

3. The Topological Twist

Here is the most fascinating part. The researchers found that when the magnets flip to this new state, they don't just change direction; they change their topology.

In physics, "topology" is like the difference between a coffee mug and a donut. You can stretch a donut into a mug without tearing it, but you can't turn a sphere into a donut without punching a hole.

Before the switch: The swirling magnets had a "topological charge" (a measure of how much they were twisted) of 3 or -3.
After the switch: The new state has a topological charge of 0.

The paper claims that this change from a twisted state to a non-twisted state happens at the exact same moment the magnets flip their direction. It's a double event: a magnetic flip and a topological reset happening simultaneously.

4. The Invisible Wind (Topological Hall Effect)

Why does this matter? The paper suggests that this swirling arrangement of magnets creates an "emergent fictitious magnetic field."

Imagine the electrons (tiny charged particles) flowing through the crystal like cars on a highway.

The Analogy: If the road is flat, cars drive straight. But if the road has a swirling, non-flat texture (caused by the twisted magnets), it acts like a sudden, invisible wind blowing across the road.
The Result: This "wind" pushes the cars (electrons) to the side, even though there is no real wind blowing. In physics terms, this creates a voltage perpendicular to the current, known as the Topological Hall Effect.

5. The Direction Matters

The researchers played with the angle of the magnetic field, like tilting a flashlight beam.

High Symmetry (The [111] direction): When the field points exactly along the crystal's main diagonal, the system is "confused" in a way that creates three different, equally stable states (like a three-way tie). Because there are three different versions of the "wind" blowing in different directions, they partially cancel each other out, but some effect remains.
Tilting the Field: If you tilt the magnetic field slightly away from that perfect diagonal, the "tie" is broken. The crystal picks just one specific state.
The Prediction: The paper concludes that if you apply the magnetic field anywhere between the diagonal ([111]) and the straight-up direction ([001]), you should be able to detect this "invisible wind" pushing electrons sideways. Specifically, they predict you will see this effect in the electrical conductivity measurements labeled $\sigma_{xy}$ and $\sigma_{yz}$ .

Summary

In short, the paper describes a crystal where magnets perform a complex swirling dance. When you apply a magnetic field, they suddenly switch to a different dance, and in doing so, they lose their "twist." This sudden change creates an invisible magnetic force that pushes electricity sideways. The researchers predict that by carefully adjusting the angle of the magnetic field, scientists could measure this effect in real experiments on these specific gold-aluminum-terbium crystals.

Technical Summary: Effect of Magnetic Field on Whirling-Anti-Whirling Order in Icosahedral-Quasicrystal Approximant

Problem Statement
The electronic and physical properties of quasicrystals (QCs) and their periodic approximant crystals (ACs) remain incompletely understood due to the inapplicability of the Bloch theorem. Recent experimental discoveries in rare-earth-based icosahedral QCs and their 1/1 ACs (specifically Au-SM-Tb, where SM=Al, Ga) have revealed a unique noncollinear magnetic order termed the "whirling-anti-whirling state." While the ground state of this system has been theoretically identified as a topological magnetic texture characterized by swirling moments on icosahedral (IC) clusters, the response of this state to external magnetic fields—particularly regarding field-direction dependence and the resulting topological properties—requires further theoretical investigation. Specifically, the emergence of an emergent fictitious magnetic field and its relation to the topological Hall effect in these systems under varying field orientations is the central problem addressed.

Methodology
The authors employ an effective spin model based on the point charge model for the crystalline electric field (CEF). The Hamiltonian includes nearest-neighbor ( $J_1$ ) and next-nearest-neighbor ( $J_2$ ) exchange interactions between rare-earth sites within and between ICs, alongside a Zeeman term for the external magnetic field ( $h$ ). The magnetic easy axis is determined by the CEF, defined by an anisotropy angle $\theta$ .

The study focuses on the 1/1 AC unit cell containing 24 Tb sites, specifically analyzing the case where $\theta = 90^\circ$ , which corresponds to the experimentally observed whirling-anti-whirling order. The authors perform exact-diagonalization calculations to determine the ground state energy and magnetic configurations under periodic boundary conditions. Key quantities calculated include:

Magnetization ( $m$ ): Per site along the field direction.
Topological Charge ( $n$ ): Defined by the solid angle subtended by magnetic moments on a single IC.
Total Chirality ( $\chi_T$ ): The vector sum of scalar chiralities over the 20 triangular surfaces of the IC, which acts as the emergent fictitious magnetic field.
Topological Hall Conductivity ( $\sigma_{\mu\nu}$ ): Proportional to the components of $\chi_T$ .

Key Contributions and Results

Simultaneous Metamagnetic and Topological Transitions:
Under a magnetic field applied along the (111) direction (the 3-fold axis), the system undergoes a metamagnetic transition at a critical field $h_M \approx 0.47$ (in units where $J_1=1.0$ and $J_2=8.0$ ). Below this threshold, the system remains in the whirling-anti-whirling state with zero net magnetization and a topological charge of $n = \pm 3$ (depending on the IC location) but zero total chirality ( $\chi_T = 0$ ). Above $h_M$ , the system transitions to a new state where magnetic moments at six sites on each IC flip to gain Zeeman energy. This new state is characterized by a topological charge of $n=0$ and a non-zero total chirality. Thus, the metamagnetic transition coincides with a topological transition.
Emergence of Fictitious Magnetic Field:
Post-transition, the noncoplanar arrangement of magnetic moments generates a finite total chirality vector. For the field along (111), the system exhibits three energetically degenerate ground states. The total chirality vectors for these states are $\chi_T = (-5.64, 0.00, 3.76)$ , $(0.00, 3.76, -5.64)$ , and $(3.76, -5.64, 0.00)$ . The presence of these non-zero components indicates the emergence of a fictitious magnetic field capable of inducing a topological Hall effect.
Field-Direction Dependence and Degeneracy:
The study systematically varies the field direction from the (001) axis (2-fold) to the (111) axis (3-fold).
- Along (001): The ground state is doubly degenerate. The $x$ and $y$ components of $\chi_T$ cancel out between the degenerate states, leaving a finite $z$ -component. This suggests a topological Hall effect primarily in the $\sigma_{xy}$ conductivity.
- Along (111): The ground state is triply degenerate. The components of $\chi_T$ partially cancel, but non-zero contributions remain for all directions, predicting topological Hall effects in $\sigma_{xy}$ , $\sigma_{yz}$ , and $\sigma_{zx}$ .
- Intermediate Angles ( $0^\circ < \theta_h < 54.7^\circ$ ): The degeneracy is lifted due to the lower symmetry of the field direction, resulting in a unique ground state with a fixed chirality vector $\chi_T = (-5.64, 0.00, 3.76)$ . In this regime, the topological Hall effect is expected to appear specifically in the electrical conductivities $\sigma_{xy}$ and $\sigma_{yz}$ .

Significance and Claims
The paper claims to provide a theoretical analysis of the magnetic-field-direction dependence on the whirling-anti-whirling order in icosahedral-quasicrystal approximants. The primary significance lies in demonstrating that the field-induced metamagnetic transition is intrinsically linked to a topological transition, resulting in the generation of an emergent fictitious magnetic field.

The authors conclude that the topological Hall effect is expected to be observable in the electrical conductivity of these materials. Specifically, for applied fields ranging from the (111) to the (001) direction, the lifting of ground-state degeneracy allows for the detection of the topological Hall conductivity components $\sigma_{xy}$ and $\sigma_{yz}$ . The authors note that since the whirling-anti-whirling order and metamagnetic behavior have already been experimentally observed in Au-SM-Tb (SM=Al, Ga) systems, the theoretical prediction of the topological Hall effect presented here offers a target for future experimental verification. The study does not propose new applications or experimental setups beyond suggesting that these effects should be detected in existing material systems.