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Computational symmetry hierarchy of time-reversal-even and -odd spin Hall conductivity tensors in altermagnets

This paper establishes a computational symmetry framework that separates the spin Hall conductivity in altermagnets into time-reversal-even and -odd channels, deriving magnetic-point-group tensor constraints and a design atlas to predict and prescreen symmetry-allowed responses across diverse magnetic crystals.

Original authors: Dameul Jeong, Seoung-Hun Kang, Young-Kyun Kwon

Published 2026-09-29

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CC BY 4.0

Original authors: Dameul Jeong, Seoung-Hun Kang, Young-Kyun Kwon

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

In the world of modern electronics, a fundamental challenge is how to move information without moving matter. Traditional computers rely on the flow of electric charge, but this generates heat and consumes significant energy. A promising alternative lies in spintronics, a field that seeks to use the intrinsic "spin" of electrons—a quantum property that makes them behave like tiny, spinning magnets—to carry data. One of the most powerful tools in this arena is the spin Hall effect. Imagine a river of electricity flowing through a material; under the right conditions, the spinning electrons are pushed to the sides, creating a separate stream of pure spin current. This phenomenon allows engineers to convert electrical signals into magnetic ones and back again, a capability essential for faster, more efficient memory and logic devices. However, predicting exactly how a specific material will behave is notoriously difficult. The rules that govern these electron flows depend on the material's internal symmetry, a complex geometric arrangement of atoms and magnetic moments. When a material is magnetic, these rules become even more intricate, as the magnetic order adds a new layer of constraints that can either enable or block the flow of spin.

A team of researchers has now mapped out these hidden rules for a special class of magnetic materials known as altermagnets. These materials are unique because they possess a magnetic order that cancels itself out on a large scale, much like a conventional magnet, yet they still exhibit strong spin-splitting effects usually reserved for magnets with a net magnetic field. The researchers focused on six representative compounds, including iron antimonide and manganese fluoride, to understand how their internal symmetries dictate the behavior of the spin Hall effect. They discovered that the response of these materials can be split into two distinct channels: one that remains unchanged when time is reversed and another that flips its sign. This separation is crucial because the two channels obey different rules. The first channel, which behaves like the spin Hall effect in non-magnetic crystals, is governed entirely by the crystal's geometric shape. The second channel, however, is sensitive to the magnetic order and can be reshaped by specific magnetic symmetries that have no counterpart in ordinary crystals.

The study reveals that for the first channel, the crystal structure alone is enough to predict which components of the spin current can exist. If the atoms are arranged in a certain way, specific spin currents are allowed; if the arrangement changes, different currents become possible. This part of the response is driven by the collective behavior of all the electrons in the material, acting like a sea of particles responding to the underlying atomic lattice. In contrast, the second channel is far more delicate. It is controlled by the magnetic symmetry of the material, specifically by operations that combine a spatial movement, such as a screw-like rotation or a glide, with the reversal of time. These operations can act as gatekeepers, allowing certain spin currents to flow while blocking others, or even creating new pathways that do not exist in the crystal structure alone. The researchers found that in some materials, these magnetic rules can completely reorganize the available spin currents, creating a pattern that is entirely different from what the crystal shape would suggest.

To visualize how these currents arise, the team looked at the behavior of electrons at the atomic level. They found that the first channel is strongest where energy bands of electrons come close together and mix, a process driven by the interaction between the electron's spin and its motion. These "avoided crossings" act as hotspots where the spin current is generated. The second channel, however, tells a different story. Its behavior is determined by the specific texture of the electron spins near the surface of the material's energy states. Instead of a smooth background, this channel is shaped by highly anisotropic patterns, where the spin current flows strongly in some directions and is suppressed in others. These patterns are a direct fingerprint of the magnetic symmetry, revealing how the material's internal magnetic order sculpts the flow of spin.

By converting these complex symmetry rules into a set of linear constraints, the researchers created a computational framework that can predict the spin Hall conductivity of a material before performing expensive and time-consuming calculations. They applied this method not only to their six representative compounds but also to a database of sixty-two other spin-split magnetic materials. The results showed that while many materials share the same crystal structure, their magnetic symmetries can lead to vastly different spin transport properties. For instance, in one material, the magnetic rules compressed the possible spin currents down to a single independent parameter, while in another, they allowed for a full partition of the tensor space, where the two channels complement each other to cover all possible directions of spin flow. This work provides a clear design rule for future materials: if one wants to engineer a specific type of spin current, one must look beyond the crystal shape and consider the magnetic symmetry, particularly the presence of these time-reversal combined operations. The study confirms that the full picture of spin transport in these advanced materials cannot be inferred from crystal symmetry alone; the magnetic order is an active participant that can fundamentally reshape the flow of information.

Technical Summary: Computational Symmetry Hierarchy of Time-Reversal-Even and -Odd Spin Hall Conductivity Tensors in Altermagnets

Problem Statement

Predicting the full spin Hall conductivity (SHC) tensor, $\sigma^k_{ij}$ , in magnetic crystals is computationally challenging because crystallographic and magnetic symmetries constrain the response tensor differently. While general symmetry classifications exist, inferring the complete tensor structure before performing dense first-principles calculations remains difficult, particularly for altermagnets. Altermagnets are a distinct class of magnetic materials characterized by collinear, compensated magnetic order combined with momentum-dependent spin splitting, often arising from nonsymmorphic symmetries (e.g., screw rotations and glide mirrors).

The core problem addressed is the lack of a unified framework to separate and predict the time-reversal-even ( $T$ -even) and time-reversal-odd ( $T$ -odd) components of the SHC tensor in these systems. The $T$ -even channel is traditionally associated with Berry-curvature physics and Fermi-sea responses, while the $T$ -odd channel is sensitive to magnetic order and Fermi-surface properties. It is unclear how magnetic symmetries, specifically antiunitary operations involving time reversal, restructure the $T$ -odd tensor compared to the $T$ -even tensor, which is often inherited from crystallographic symmetry.

Methodology

The authors developed a computational framework that combines symmetry analysis with first-principles calculations to establish a hierarchy of SHC constraints in altermagnets.

Symmetry Decomposition: The SHC tensor is explicitly decomposed into $T$ -even and $T$ -odd channels based on the parity of the response coefficient under the reversal of the ordered magnetic state ( $M \to TM$ ).
- $T$ -even: Invariant under time reversal. The authors demonstrate that for this channel, antiunitary operations ( $Tg$ ) impose the same selection rules as their unitary counterparts ( $g$ ). Thus, the allowed tensor components are determined by the parent crystallographic point group.
- $T$ -odd: Changes sign under time reversal. Antiunitary operations impose independent constraints that differ from unitary operations. The tensor form must be derived from the full magnetic point group (MPG), including the antiunitary coset.
Linear Constraint Formulation: The 27 components of the rank-three SHC tensor are vectorized. Each symmetry operation (unitary and antiunitary) is represented as a $27 \times 27$ matrix acting on this vector. The allowed tensor spaces are identified by solving the resulting linear constraints ( $\rho_\sigma(g)\vec{\sigma} = \pm \vec{\sigma}$ ) using singular-value decomposition.
Material Set: The framework was applied to six representative altermagnets spanning distinct crystallographic classes and symmetry types:
- FeSb $_2$ (Orthorhombic, $mmm$)
- MnO $_2$ and MnF $_2$ (Tetragonal, $4/mmm$)
- CuF $_2$ (Monoclinic, $2/m$ )
- FeF $_3$ (Trigonal, $\bar{3}m$ )
- MnSe (Hexagonal, $6mm$)
First-Principles Validation: Density Functional Theory (DFT) calculations were performed using Quantum ESPRESSO with DFT+U corrections. SHC tensors were evaluated using Wannier interpolation (WannierBerri) on dense $k$ -point grids. The authors analyzed momentum-resolved Berry-curvature-like quantities ( $\Omega^{even/odd}_{ij}$ ) to link tensor components to microscopic origins (SOC-driven avoided crossings vs. near-Fermi-level spin textures).
Database Extension: To test transferability, the same MPG analysis was applied to a database of 62 spin-split collinear antiferromagnets (from Guo et al.), generating a canonical "tensor atlas" for various magnetic point groups.

Key Results

1. Symmetry Hierarchy and Tensor Partitioning

The study establishes a clear hierarchy:

$T$ -even Channel: Governed by crystallographic symmetry. The allowed components match those of nonmagnetic systems with the same point group. For example, in FeSb $_2$ ($mmm$), six independent components are allowed for both channels.
$T$ -odd Channel: Governed by magnetic symmetry. Antiunitary operations (specifically those combining time reversal with nonsymmorphic elements like screws or glides) can drastically alter the tensor structure.
- CuF $_2$ Case: This material exhibits the most distinct behavior. The $T$ -even channel allows 13 independent components, while the $T$ -odd channel allows 14. Crucially, the two sets are complementary: the 13 $T$ -even components and 14 $T$ -odd components have no overlap and together exhaust all 27 entries of the SHC tensor. This demonstrates that nonsymmorphic antiunitary symmetries can activate unconventional SHC channels forbidden in the $T$ -even sector.
- MnSe Case: Magnetic symmetry strongly compresses the $T$ -odd sector, reducing it to only 4 allowed components with a single independent parameter, whereas the $T$ -even sector retains 6 components.

2. Microscopic Origins

$T$ -even Response: The momentum-resolved analysis confirms that the $T$ -even SHC is driven by spin-orbit coupling (SOC)-induced avoided crossings and interband coherence. The signal is localized near these band degeneracies, consistent with a Fermi-sea response.
$T$ -odd Response: The $T$ -odd SHC arises from anisotropic spin-current matrix elements near the Fermi level. The momentum distribution exhibits sign-alternating structures, nodal lines, and lobes dictated by the magnetic symmetry, rather than smooth background contributions.

3. Database-Level Atlas

The extension to 62 materials reveals that while many type-I magnetic space groups (where the antiunitary coset is empty) share identical $T$ -even and $T$ -odd tensor patterns, type-III groups (containing antiunitary operations) frequently distinguish the two channels. The authors provide a "magnetic-point-group tensor atlas" (Supplementary Note S3) that maps specific MPG classes to their allowed tensor patterns and independent parameter counts.

Significance and Claims

The paper claims to provide a computational prescreening route for identifying symmetry-allowed spin Hall channels in altermagnets before undertaking expensive first-principles transport calculations.

Design Rules: The framework offers specific design limits. For instance, it identifies that CuF $_2$ allows a complete 13/14 tensor partition, while MnSe allows a one-parameter compression of the $T$ -odd tensor.
Separation of Concerns: It clarifies that for screening conventional spin-charge conversion ( $T$ -even), crystallographic class is sufficient. However, for targeting magnetic-symmetry-controlled responses ( $T$ -odd), the ordered-state symmetry must be included from the outset.
Transferability: The results demonstrate that materials sharing a crystallographic class do not necessarily share the same $T$ -odd tensor once magnetic sublattice symmetry is included. The framework allows researchers to determine which symmetry levers (e.g., moment direction, presence of nonsymmorphic operations) are worth tuning to enhance specific SHC channels.

The authors emphasize that this work does not propose new experimental devices but rather establishes a rigorous theoretical and computational foundation for understanding and predicting the tensorial nature of spin transport in the emerging class of altermagnetic materials.