🔬 mesoscale physics

Complete Hierarchy of Nonrelativistic Odd-Parity Spin Splitting in Collinear Magnets

This paper establishes the complete group-theoretical classification of odd-parity spin splitting in collinear magnets, revealing previously unexplored high-order $h$ -, $k$ -, and $m$ -wave classes, and demonstrates their realization in specific materials alongside their distinct impacts on anomalous transport phenomena.

Original authors: Yichen Liu, Junxi Yu, Pu Zhang, Cheng-Cheng Liu

Published 2026-07-22

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CC BY 4.0

Original authors: Yichen Liu, Junxi Yu, Pu Zhang, Cheng-Cheng Liu

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 the invisible world inside a magnet not as a static block of metal, but as a bustling dance floor where tiny particles called electrons are constantly spinning and moving. In most magnets, these dancers are paired up perfectly, spinning in opposite directions so that their movements cancel out, leaving the crowd looking calm and balanced. But in a special class of magnets called "collinear magnets," the rules of the dance floor are a bit different. Here, the electrons can be separated into two distinct groups based on which way they spin, creating a kind of traffic jam where one lane moves faster than the other. This separation is called "spin splitting."

For a long time, scientists knew about two main ways this splitting could happen, which they named after the shapes of the waves they create: "p-wave" and "f-wave." Think of these like the simple, predictable patterns of a jump rope being swung in a circle (p-wave) or a more complex, figure-eight pattern (f-wave). These patterns act like a fingerprint, telling scientists exactly how the electrons are behaving. However, a big question remained: Could there be even more complex, stranger patterns hiding in the shadows? Could the electrons dance in shapes we hadn't even imagined yet? Understanding these patterns is crucial because they determine how electricity flows, how magnets interact with light, and could even lead to super-fast, energy-efficient computers in the future.

Now, a team of researchers has stepped onto the dance floor with a new map to answer that question. They have completed the "hierarchy" of these spin-splitting patterns, proving that while there are indeed more complex dances than the p-wave and f-wave, there is a hard limit to how complicated they can get. Using a mathematical toolkit called group theory, they discovered that beyond the known p-wave and f-wave, there are three new, higher-order dances allowed: the "h-wave," the "k-wave," and the "m-wave."

Imagine the p-wave as a simple circle, the f-wave as a cloverleaf, the h-wave as a star with five points, the k-wave as a star with seven points, and finally, the m-wave as a dazzling, nine-pointed star. The researchers found that this nine-pointed star is the absolute limit; no matter how you try to twist the rules, you cannot create a ten-pointed star or anything more complex in this specific type of magnet. They didn't just guess this; they built a complete catalog, mapping every possible crystal shape to the specific dance moves it allows. They even constructed simple computer models to show exactly how these new waves would look, confirming that the h-wave, k-wave, and m-wave are physically possible.

To see if these theoretical dances actually exist in the real world, the scientists went on a digital treasure hunt through a massive database of known materials. They found two promising candidates: a crystal called Fe2TeO6 and another called MgFe6Ge6. By simulating what happens when these materials are hit with a specific type of light—circularly polarized light—they showed that the light acts like a conductor, forcing the electrons to start dancing in these new, complex patterns. In the Fe2TeO6 crystal, the electrons would perform the five-pointed h-wave dance, while in the MgFe6Ge6 crystal, they would perform the seven-pointed k-wave dance. The simulations showed that these new dances would create unique patterns in the energy of the electrons and even in the waves of magnetism (called magnons) that ripple through the material.

One of the most exciting findings is that these new dances change how the material conducts electricity. The researchers calculated that the h-wave and k-wave dances allow the material to generate a "Hall effect," a phenomenon where electricity flows sideways when pushed, which is a key ingredient for new types of electronic devices. However, the most complex dance of all, the nine-pointed m-wave, is too symmetrical to allow this sideways flow. This means that while the m-wave is the most complex pattern possible, it might not be the most useful for certain electronic tricks.

The paper doesn't just stop at finding these patterns; it also provides a rulebook for how to create them. It explains that you need a specific type of magnetic order (the "parent" dance) and a specific external push (like the light) to trigger these new moves. While the researchers haven't found a solid material in the database that naturally does the nine-pointed m-wave dance yet, they suggest that scientists might be able to build it in the lab using cold atoms, where the rules of the dance floor can be tweaked with lasers. This work completes the picture of how electrons can split their spins in these magnets, turning a partial list of possibilities into a full, finished encyclopedia of magnetic dances.

Technical Summary: Complete Hierarchy of Nonrelativistic Odd-Parity Spin Splitting in Collinear Magnets

Problem Statement
Momentum-dependent nonrelativistic spin splitting serves as a symmetry fingerprint for collinear magnets, governing unconventional electronic, magnonic, and transport phenomena. While even-parity spin splitting (e.g., in altermagnets) has been extensively characterized, the classification of odd-parity counterparts remains incomplete. Existing literature has primarily focused on $p$ -wave ( $\ell=1$ ) and $f$ -wave ( $\ell=3$ ) splittings. A critical gap exists in understanding whether higher odd partial waves can occur in collinear magnets, if there is an upper bound to this hierarchy, how specific odd-parity form factors are selected by the interplay between parent Néel order and external symmetry-breaking fields, and which high-partial-wave classes support anomalous transport responses.

Methodology
The authors employ a group-theoretical framework to establish a symmetry-complete classification of odd-parity spin splitting. The methodology proceeds through three main stages:

Symmetry Classification: The study analyzes the transformation properties of the spin splitting $\Delta E(\mathbf{k}) = E_\uparrow(\mathbf{k}) - E_\downarrow(\mathbf{k})$ under crystallographic point-group operations. The authors distinguish the Néel-order representation ( $\Gamma_N$ ) from the spin-splitting representation ( $\Gamma_\Delta$ ). They derive a coupling rule where the induced spin-splitting representation is determined by the direct product of the Néel-order representation and the axial-vector representation ( $\Gamma_A$ ) of an external symmetry-breaking field (specifically, the helicity of circularly polarized light).
Basis Function Mapping: Using the derived coupling rules, the authors map crystallographic point-group irreducible representations to their lowest-order odd-parity basis functions. This allows for the identification of allowed partial-wave classes ( $\ell$ ) and their corresponding form factors.
Model Construction and Material Screening: To provide a microscopic foundation, the authors construct minimal four-band tight-binding models with two magnetic sublattices. These models utilize spin-independent, momentum-dependent sublattice-contrasting hopping terms coupled to a staggered exchange field to realize specific odd-parity classes. Finally, guided by the symmetry classification, the authors screen the MAGNDATA database to identify candidate materials that can host these states under circularly polarized light irradiation.

Key Contributions and Results

Complete Hierarchy of Partial Waves: The study establishes that the hierarchy of nonrelativistic odd-parity spin splitting in collinear magnets is finite and terminates at $\ell=9$ . In addition to the known $p$ -wave ( $\ell=1$ ) and $f$ -wave ( $\ell=3$ ) classes, the authors identify three new classes:
- $h$ -wave ( $\ell=5$ )
- $k$ -wave ( $\ell=7$ )
- $m$ -wave ( $\ell=9$ )
  The authors demonstrate that no one-dimensional odd-parity representation first appears above degree nine, establishing $\ell=9$ as the strict upper bound.
Coupling Rules and Form Factors: The paper provides a comprehensive two-step classification procedure (summarized in Tables I and II of the original text) that maps a given parent magnetic order and axial-field direction to a specific odd-parity form factor. Explicit basis functions are derived for all five classes, such as $k_x k_y k_z (k_x^2 - k_y^2)$ for the $h$ -wave and $k_x k_y k_z (3k_x^2 - k_y^2)(k_x^2 - 3k_y^2)$ for the $k$ -wave.
Minimal Lattice Models: The authors construct explicit tight-binding Hamiltonians that realize the $h$ -, $k$ -, and $m$ -wave splittings. These models clarify that the spin splitting arises from the interplay between a spin-independent, momentum-dependent sublattice contrast ( $d_z(\mathbf{k})$ ) and the staggered exchange field ( $J$ ). The models confirm that the long-wavelength limits of these lattice Hamiltonians reproduce the derived continuum basis functions.
Material Realizations: Through database screening, two candidate materials are identified for light-induced odd-parity phases:
- Fe $_2$ TeO $_6$ : A tetragonal $PT$-symmetric antiferromagnet that, under circularly polarized light, realizes an $h$ -wave phase.
- MgFe $_6$ Ge $_6$ : A hexagonal $PT$-symmetric antiferromagnet that realizes a $k$ -wave phase under similar conditions.
  The study shows that these materials exhibit characteristic spin splittings in both their electronic band structures and magnon spectra.
Anomalous Transport Properties: The paper analyzes the conditions for the anomalous Hall effect (AHE) in these high-partial-wave classes. The results indicate that while $p$ -, $f$ -, $h$ -, and $k$ -wave classes can support an anomalous Hall response (depending on the residual magnetic point group), the $m$ -wave class forbids it due to its high cubic symmetry ( $4'32'$ ). Calculations for the candidate materials show significant Berry curvature and finite anomalous Hall conductivity for the $h$ - and $k$ -wave phases.

Significance
The paper claims to complete the partial-wave hierarchy of odd-parity spin splitting in collinear magnets, moving beyond the previously limited scope of $p$ - and $f$ -waves. By establishing a rigorous symmetry classification and providing explicit microscopic models, the work offers a systematic framework for predicting and engineering high-partial-wave spin-splitting states. Furthermore, it delineates the symmetry criteria required for anomalous transport in these high-order classes, distinguishing between those that permit and those that forbid the anomalous Hall effect. The identification of specific material candidates ( $h$ - and $k$ -wave) and the proposal of cold-atom platforms for realizing the maximal $\ell=9$ ( $m$ -wave) class provide concrete pathways for experimental verification and future exploration of unconventional magnetism.

Technical Summary: Complete Hierarchy of Nonrelativistic Odd-Parity Spin Splitting in Collinear Magnets

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