Spin point group symmetry and classification of non-relativistic spin splitting in non-collinear magnetic structures: Identification of high-order spin splitting types (l=5,7, and 9)

This paper establishes a comprehensive classification of non-relativistic spin splitting in coplanar and non-coplanar magnetic structures by tabulating 1,249 spin point groups and identifying previously overlooked high-order splitting types (l=5, 7, and 9), exemplified by the material LaMnAu5.

The Gist

Imagine a crystal as a bustling city where electrons are the citizens moving through the streets. Usually, in a perfectly symmetrical city, if you look at a citizen moving north, you'd see an identical twin moving south. They are "degenerate," meaning they have the exact same energy and behavior.

However, in magnetic materials, the "magnetic field" acts like a traffic cop that forces these twins to behave differently. One might speed up while the other slows down. This is called spin splitting.

For a long time, scientists thought they understood all the rules of this traffic cop, but only for cities where the magnetic "citizens" (spins) were lined up in neat, straight rows (collinear magnets). This paper says, "Wait a minute! What about the chaotic, messy cities where the spins are pointing in all different directions (non-collinear)?"

Here is a breakdown of what the authors did, using simple analogies:

1. The Great Census (The Spin Point Groups)

The authors started by creating a massive, comprehensive census of all possible ways these magnetic "traffic cops" can organize themselves.

• The Old Map: Previous maps only listed 598 types of magnetic organizations.
• The New Map: The authors realized that many real-world materials have a special "time-reversal" rule (like a magic mirror that flips time) that the old maps ignored. By adding this rule, they expanded the census to 1,249 unique types of magnetic organizations.
• The Database: They put this entire new map into a free online tool called SPGENPOS. Think of it as a giant library where any scientist can look up a specific magnetic material and instantly see its "ID card" and rules.

2. The Traffic Rules (Spin Splitting)

Once they had the list of 1,249 organizations, they asked: "How does the traffic cop split the electron twins in each of these cities?"

They looked at the "shape" of the split. Imagine the energy difference between the twins as a hill.

• Low-order splits (Simple hills): In many materials, the hill is simple, like a gentle slope (called s-wave) or a simple curve (called p-wave).
• High-order splits (Complex hills): The authors discovered that in these messy, non-collinear cities, the hills can be incredibly complex. They found shapes that twist and turn in ways never seen before, corresponding to mathematical orders 5, 7, and 9.
  • They named these after musical notes or wave types: h-wave (order 5), j-wave (order 7), and ℓ-wave (order 9).
  • The Surprise: In the old, neat cities (collinear), the most complex hill you could ever find was order 6. But in these new, messy cities, the hills can get much more twisted (up to order 9), unless the city has a specific symmetry that forbids it (like order 8, which they found is impossible).

3. The "Magic Mirror" (Time Reversal)

A key discovery was about a rule called Time Reversal (TR).

• Old Belief: Scientists thought that if a material had this "magic mirror" rule, the traffic cop would be forced to keep the twins identical (no splitting).
• New Discovery: The authors proved that in these messy, non-collinear cities, the magic mirror does not stop the splitting. The twins can still be separated, even with the mirror present. This opens the door to finding many more materials with these unique properties.

4. Finding a Real-World Example

Theory is great, but does it exist in nature?
The authors went hunting in a database of known materials. They found one specific material, LaMnAu5 (a compound of Lanthanum, Manganese, and Gold).

• This material is a "coplanar" city (spins lie flat on a plane).
• It perfectly matches the rules for the h-wave (order 5) splitting.
• This is the first time a real material has been identified with this specific, high-order "twisted hill" shape.

Summary

Think of this paper as the authors drawing a new, much larger map of the magnetic universe. They found that the "rules of the road" for electrons are much more varied and complex than we thought. They discovered that in disordered magnetic structures, electrons can split in incredibly complex patterns (up to order 9), and they even found a real-life example of one of these rare patterns.

They didn't invent a new device or cure a disease in this paper; they simply provided the theoretical foundation and the map that other scientists can now use to find materials with these special properties for future technologies.

Technical Summary

Technical Summary: Spin Point Group Symmetry and Classification of Non-Relativistic Spin Splitting

Problem Statement
The development of magnetic materials exhibiting non-relativistic spin splitting in electronic bands is a central objective for spintronics applications. While the concept of altermagnetism has successfully identified compensated collinear magnetic states with even-parity spin splitting (e.g., $d$-, $g$-, and $i$-waves, corresponding to $\ell=2, 4, 6$), the landscape of non-collinear magnetic structures remains less explored. Previous theoretical frameworks often overlooked the full symmetry of magnetic structures containing antitranslations (time-reversal combined with translation), particularly those belonging to magnetic space groups of type IV. Furthermore, there has been debate regarding the existence of higher-order odd-parity spin splittings (such as $h$-waves with $\ell=5$) in non-centrosymmetric systems, with some studies arguing such states are impossible. This work addresses the need for a comprehensive symmetry analysis of non-relativistic spin splitting in both coplanar and non-coplanar magnetic structures, specifically aiming to identify all allowed splitting types, including high-order terms ($\ell=5, 7, 9$).

Methodology
The authors employ spin-point-group (SpPG) theory to analyze the symmetry constraints on the effective Zeeman field $\mathbf{B}(\mathbf{k})$, which determines the spin splitting of electronic bands in the absence of spin-orbit coupling (SOC). The methodology proceeds in three main stages:

1. Enumeration of Spin Point Groups (SpPGs): The authors systematically enumerate all non-equivalent SpPGs. They extend previous work by Litvin, which listed 598 "non-trivial" SpPGs (where the spin-only subgroup is trivial), by considering the direct product of these non-trivial groups with intrinsic spin-only subgroups ($P_{SO}$). Crucially, they include cases where the spin-only subgroup is augmented by the time-reversal operation ($1'$), a necessary step for describing structures with type IV magnetic space groups (which possess antitranslations). This results in a total of 1249 distinct SpPGs, categorized into six subsets based on spin arrangement (collinear, coplanar, non-coplanar) and the presence of time-reversal symmetry.
2. Database Construction (SPGENPOS): The enumerated SpPGs, along with their full lists of symmetry operations, are tabulated and made available via the Bilbao Crystallographic Server under the database name SPGENPOS. This database utilizes an extended Hermann-Mauguin notation and a numerical labeling scheme to uniquely identify each group.
3. Symmetry Analysis of Spin Splitting: Using the program STENSOR (also hosted on the Bilbao Crystallographic Server), the authors analyze the symmetry-allowed tensor forms of the effective Zeeman field $\mathbf{B}(\mathbf{k})$ for each of the 1249 SpPGs. The analysis focuses on the lowest-order non-vanishing term in the Taylor expansion of $\mathbf{B}(\mathbf{k})$ around the Brillouin zone center ($\mathbf{k}=0$). The order of this term, denoted by $\ell$, defines the "wave type" of the spin splitting (e.g., $\ell=0$ is $s$-wave, $\ell=1$ is $p$-wave, etc.).

Key Contributions and Results

• Comprehensive Classification: The paper provides the first complete enumeration of 1249 non-equivalent SpPGs, significantly extending previous lists by incorporating time-reversal symmetry and antitranslations. This fills a gap where approximately 40% of known commensurate magnetic structures (those with type IV MSGs) were previously excluded from the spin-point-group framework.
• Identification of High-Order Splitting Types: Contrary to the collinear case, where the lowest-order spin splitting is limited to $\ell \le 6$ and time-reversal symmetry forbids splitting entirely, the authors find that non-collinear structures allow a much broader range of splitting behaviors. Specifically, they identify symmetry-allowed spin splittings with lowest-order monomials ranging from $\ell=0$ to $\ell=9$, with the exception of $\ell=8$.
  • New High-Order Types: The study explicitly identifies and provides the functional forms for high-order odd-parity spin splittings: $h$-waves ($\ell=5$), $j$-waves ($\ell=7$), and $\ell$-waves ($\ell=9$). These occur in specific coplanar and non-coplanar SpPGs, often requiring the presence of time-reversal symmetry in the point group.
  • Correction of Previous Assumptions: The results disprove arguments linking $h$-waves ($\ell=5$) strictly to 5-fold rotational symmetry, demonstrating that such splittings can arise in tetragonal systems without 5-fold axes.
• Material Identification: The authors performed a preliminary survey of the MAGNDATA database to identify real materials exhibiting these predicted symmetries. They identified LaMnAu$_5$ as a real-world example of a coplanar tetragonal compound with an SpPG symmetry that permits $\ell=5$ ($h$-wave) non-relativistic spin splitting. The analysis confirms that the spin space group of LaMnAu$_5$ supports this high-order splitting type.
• Parity and Symmetry Constraints: The analysis clarifies that while time-reversal symmetry ($1'$) in the point group forbids spin splitting in collinear systems, it does not necessarily forbid it in non-collinear systems. In coplanar structures with TR, spin splitting is restricted to odd-parity modes, whereas in non-coplanar structures, both even and odd parity modes can exist depending on the specific SpPG.

Significance and Claims
The paper claims to establish a rigorous symmetry framework for understanding non-relativistic spin splitting in non-collinear magnetic structures. By providing the SPGENPOS database and the STENSOR analysis, the authors offer tools to predict spin-splitting behaviors in magnetic materials without relying on specific band structure calculations.

The primary significance lies in the identification of previously overlooked high-order spin splitting types ($\ell=5, 7, 9$) and the demonstration that these can exist in specific non-centrosymmetric magnetic structures. The authors modestly state that these results point to a "relatively unexplored class of materials" with non-relativistic spin splitting. They suggest that this expanded classification could help broaden the range of magnetic states with potential relevance for applications or fundamental interest, particularly by challenging the notion that certain high-order splittings are impossible in crystalline lattices. The work does not propose new experimental protocols or specific device applications but rather provides the theoretical symmetry foundation necessary for such future developments.