🔬 materials science

Prediction of Magnetic Topological Materials Combining Spin and Magnetic Space Groups

This paper proposes a new scheme combining spin space groups (SSGs) and magnetic space groups (MSGs) to predict electronic band topology in collinear magnets, enabling the identification of topological materials that are invisible to traditional MSG-based analysis and successfully identifying 26 such materials, including FePSe $_3$ , from a high-throughput screening of 488 compounds.

Original authors: Liangliang Huang, Yuanze Song, Houhao Wang, Zhixian Yu, Xiangang Wan, Feng Tang

Published 2026-10-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Liangliang Huang, Yuanze Song, Houhao Wang, Zhixian Yu, Xiangang Wan, Feng Tang

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

Matter is rarely just a collection of atoms sitting still; often, the tiny magnetic spins of electrons within a material align in specific patterns, turning the material into a magnet. When these magnetic patterns combine with the way electrons move through the crystal lattice, strange and useful things can happen. Electrons might flow without resistance, or they might carry a specific kind of twist that makes them immune to impurities. Scientists call these "topological" materials, and they are the holy grail for next-generation electronics because they could lead to faster, more efficient computers and sensors. For decades, researchers have used a set of rules based on the symmetry of a crystal's shape to predict which materials would behave this way. However, these rules worked best for materials that were not magnetic, or for magnets where the magnetic order was simple and well-understood. When the magnetic order gets complicated, or when the interaction between the electron's spin and its motion becomes significant, the old rules often fail, leaving scientists unable to predict if a material will be a topological wonder or just an ordinary insulator.

A team of researchers at Nanjing University has now bridged this gap by developing a new method to predict the behavior of magnetic materials with much greater accuracy. They realized that to understand these materials, one must look at them through two different lenses simultaneously. The first lens ignores the subtle interaction between an electron's spin and its motion, treating the magnetic order as a perfect, idealized symmetry. The second lens includes this interaction, which slightly distorts the symmetry and changes the material's properties. By connecting these two views, the researchers created a comprehensive map that can diagnose the topological nature of nearly any magnetic material, even when the magnetic order is complex. Their work not only explains why some materials behave the way they do but also predicts the existence of new, previously unknown magnetic topological materials.

The core of this new approach lies in how the researchers handle the transition from a material with no magnetic order to one with a specific magnetic pattern. In the absence of magnetic interactions, a material's symmetry is described by a standard set of rules. When the material becomes magnetic, the symmetry changes. The researchers introduced a new framework that treats the magnetic state as a step in a hierarchy. First, they consider the material's symmetry when the subtle spin-motion interaction is turned off. In this state, the material belongs to a vast family of symmetry groups that describe how the magnetic spins are arranged. Then, they turn the interaction back on. This action lowers the symmetry further, narrowing the possibilities down to a specific, more restrictive group. The researchers calculated the mathematical connections between these two states for every possible magnetic arrangement. This allowed them to see how the electronic bands—the energy levels electrons can occupy—evolve as the material transitions from the idealized state to the real, physical state.

Using this new framework, the team performed a massive survey of known magnetic materials. They examined 488 materials whose magnetic structures had already been determined by experiments. In 26 of these cases, the standard methods, which only look at the final, real-world symmetry, would have labeled the material as ordinary and topologically trivial. However, the new method, which accounts for the material's underlying idealized symmetry, revealed that these 26 materials actually possess a hidden topological nature. In these materials, the topological features are protected by the idealized symmetry and only become slightly distorted, rather than completely destroyed, when the real-world interactions are included. One specific example is a material called FePSe3. Under standard analysis, it appears to be a simple magnet. But the new method shows it hosts special lines of electrons that should exist on its surface, a feature that standard analysis misses. These surface states are robust and could be useful for electronic devices, even though the material is not a perfect topological insulator in the traditional sense.

The researchers did not stop at analyzing known materials; they also used their method to predict new ones. They started with 1,483 chemical compounds that are known to exist but have not yet been confirmed to be magnetic. By simulating the formation of magnetic order in these compounds and applying their new symmetry rules, they predicted the ground-state magnetic structures for 224 of them. From this group, they identified 24 distinct materials that are likely to be magnetic topological materials. This list includes three materials that have already been confirmed by other experiments: Co3Sn2S2, MnBi2Te4, and Fe3Sn2. The fact that their method correctly identified these known materials serves as a strong validation of their approach. More importantly, it gave them confidence in the 21 new predictions, which include materials like LiCoO2 and GaNi3. For these new candidates, the researchers calculated the specific direction of the magnetic spins and the resulting electronic structure, finding that they should host exotic states like Weyl semimetals, where electrons behave as if they have no mass, or axion insulators, which have unique electrical responses.

The power of this work lies in its ability to see what was previously invisible. For years, scientists relied on a single set of symmetry rules to classify magnetic materials. If a material's symmetry was too low or too complex, it was often dismissed as topologically uninteresting. This new approach recognizes that the "ideal" symmetry of the magnetic order, even if slightly broken by real-world effects, still leaves a fingerprint on the material's electronic structure. It is like looking at a sculpture through a slightly foggy window; the old method might only see a blurry shape and assume it is a simple block, while the new method understands the underlying form of the sculpture and knows exactly what it is, even through the fog. By combining the idealized view with the realistic view, the researchers have expanded the known pool of magnetic topological materials significantly.

The study provides a clear path forward for experimentalists. Instead of guessing which materials might be useful, researchers can now use this new classification scheme to screen databases of known compounds and identify the most promising candidates for synthesis. The team has made their computational data and software tools available to the scientific community, allowing others to apply these rules to any magnetic material they encounter. This opens the door to discovering a new generation of materials that could revolutionize spintronics, the field of electronics that uses electron spin rather than just charge. By understanding how the magnetic order dictates the flow of electrons, scientists can now design materials with specific, desired properties, moving from accidental discovery to deliberate engineering. The work confirms that the interplay between magnetism and topology is far richer than previously thought, and that by looking at the problem through the right lens, we can find hidden treasures in the periodic table.

Technical Summary: Prediction of Magnetic Topological Materials Combining Spin and Magnetic Space Groups

Problem Statement
Magnetic topological materials exhibit fascinating physical properties, such as the Weyl semimetal phase and the quantum anomalous Hall effect. While symmetry indicators and topological quantum chemistry based on space groups (SGs) or magnetic space groups (MSGs) have enabled efficient large-scale searches for topological materials, a significant gap remains in the experimental verification of theoretically predicted magnetic topological materials. A primary limitation of existing frameworks is their reliance on MSGs, which serve as the exact symmetry group for magnets with commensurate order but may fail to capture specific electronic band degeneracies, particularly in the presence of spin-orbit coupling (SOC).

Recently, spin space groups (SSGs) have been introduced to describe the symmetry of magnetic materials in the absence of SOC, leading to the discovery of altermagnetism. However, a routine and efficient approach to predict electronic band topology by combining SSGs with their MSG subgroups—specifically addressing the evolution of topology from negligible SOC (SSG symmetry) to finite SOC (MSG symmetry)—has not been established. The challenge lies in diagnosing how band topology changes as symmetry lowers from SSG to MSG determined by specific magnetic moment (MM) directions, and identifying materials that retain topological features protected by SSG symmetry even when they appear trivial under MSG.

Methodology
The authors propose a comprehensive scheme that integrates SSGs and their MSG subgroups to diagnose band topology in collinear magnets. The methodology involves three main components:

Symmetry Indicator Construction: The authors established symmetry indicators for all 1,421 collinear SSGs and their 3,420 resulting MSG subgroups. This involved computing the little-group (co-)irreducible representations (irreps) at all high-symmetry points (HSPs), compatibility relations between k-points, and site-symmetry group irreps for all Wyckoff positions.
Compatibility Relations and Weak-SOC Limit: A critical theoretical advancement is the derivation of compatibility relations from SSG irreps to MSG irreps. These relations allow for the quantitative characterization of SOC strength. Specifically, if the symmetry data ( $n_{k}^{MSG}$ ) calculated with realistic SOC matches the data derived from the SSG ( $n_{k}^{SSG}$ ) via compatibility relations (assuming negligible SOC), the system is considered to satisfy the "weak-SOC limit." In this limit, the topological classification for all possible MM directions can be exhaustively mapped using only first-principles data calculated without SOC.
Software Implementation: The authors implemented these theories into a Mathematica package, TopoSSGtoMSG.L, which can routinely diagnose the electronic band topology of any collinear magnet by combining SSG and MSG analyses with first-principles wavefunctions.

Key Contributions and Results

High-Throughput Analysis of MAGNDATA:
The scheme was applied to 488 collinear magnets from the MAGNDATA database with experimentally determined magnetic structures.
- Identification of "Hidden" Topology: The study identified 46 materials that are diagnosed as topological by SSG but trivial by MSG. Among these, 26 materials satisfy the weak-SOC limit.
- Case Study (FePSe3): The material FePSe3 was showcased as a representative example. Under SSG symmetry (negligible SOC), it hosts Dirac nodal lines (DNLs) and protected boundary states. When SOC is included, the MSG framework predicts a trivial gap. However, the SSG-MSG scheme reveals that the nodal lines become gapped only by a tiny amount, leaving robust surface states and large Berry curvature (anomalous Hall conductivity $\approx -150 \, \Omega^{-1}\text{cm}^{-1}$ ) that are invisible to standard MSG analysis.
- Mn2AlB2: The authors demonstrated the ability to classify topology for all MM directions using Mn2AlB2. The material transitions from a nodal-line semimetal (SSG) to a Dirac semimetal or magnetic topological insulator (MSG) depending on the MM direction and SOC strength.
Prediction from Paramagnetic Structures (ICSD):
The authors developed a workflow to predict ground-state collinear magnetic structures and their topological phases directly from paramagnetic crystal structures in the Inorganic Crystal Structure Database (ICSD).
- Workflow: Starting from stoichiometric paramagnetic structures in space groups No. 166 and No. 221, they constructed candidate magnetic structures constrained by maximal collinear SSG subgroups. Total energies were calculated without SOC to determine the ground-state SSG and magnetic order. Subsequently, SOC was included to determine the lowest-energy MM direction and the corresponding MSG-based topology.
- Results: From 1,483 initial structures, 224 converged to stable magnetic structures. This process identified 24 distinct magnetic topological materials.
- Validation: The workflow successfully recovered three experimentally confirmed materials: the magnetic Weyl semimetal Co $_3$ Sn $_2$ S $_2$ , the magnetic topological insulator MnBi $_2$ Te $_4$ , and the magnetic Weyl nodal-line semimetal Fe $_3$ Sn $_2$ . The calculated magnetic moment magnitudes for these materials showed good agreement with experimental values.
- New Predictions: The study predicted 21 new magnetic topological materials, including the axion insulator LiCoO $_2$ and the Weyl semimetal GaNi $_3$ , providing detailed band structures, surface states, and Fermi arc predictions.

Significance
The paper demonstrates that combining SSG and MSG approaches offers superior predictive power compared to methods relying on either group alone. By utilizing the symmetry-reduction hierarchy from SSG to MSG, the authors provide a framework that:

Captures Weak-SOC Physics: It identifies materials where topological features protected by SSG symmetry persist even when SOC opens small gaps, a regime often missed by standard MSG analysis.
Enables Efficient Screening: It allows for the exhaustive classification of topology for all possible magnetic moment directions without requiring computationally expensive first-principles calculations with SOC for every direction, provided the weak-SOC limit holds.
Accelerates Discovery: The workflow for predicting magnetic structures from paramagnetic parents significantly expands the pool of candidate magnetic topological materials, facilitating future experimental realization.

The authors conclude that their high-throughput results extend the database of magnetic topological materials and provide a robust theoretical tool for understanding the interplay between lattice symmetry, magnetism, SOC, and topology.

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