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Symmetry and Minimal Hamiltonian of Nonsymmorphic Collinear Antiferromagnet MnTe

This paper establishes a minimal effective Hamiltonian for the altermagnet $\alpha$ -MnTe using group theory and first-principles calculations, revealing that spin-orbit coupling plays a dominant role in its spin splitting and demonstrating how electron transport characteristics vary with high-symmetry points and Néel vector orientations.

Original authors: Koichiro Takahashi, Hong-Fei Huang, Jie-Xiang Yu, Jiadong Zang

Published 2026-10-30

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CC BY 4.0

Original authors: Koichiro Takahashi, Hong-Fei Huang, Jie-Xiang Yu, Jiadong Zang

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, controlling the flow of electricity usually requires magnets. Traditional magnets have a north and a south pole, creating a net magnetic field that can be felt and measured. However, a newer class of materials called antiferromagnets behaves differently. In these substances, the tiny magnetic moments of atoms point in opposite directions, canceling each other out so that the material has no overall magnetic field. For decades, scientists struggled to use these materials for technology because their hidden magnetic order was difficult to detect or manipulate. Recently, a specific type of antiferromagnet known as an "altermagnet" has emerged as a promising candidate. These materials possess a unique property: even though they have no net magnetism, their internal structure splits electrons based on their spin, a quantum property that can be thought of as a tiny internal compass. This splitting allows them to generate electrical effects usually reserved for strong magnets, opening the door to faster, more stable electronic devices.

The material at the center of this new research is manganese telluride, a compound made of manganese and tellurium atoms arranged in a specific crystal pattern. While scientists knew this material exhibited interesting magnetic behaviors, the precise rules governing how its electrons move were not fully understood. The researchers set out to build a simplified, yet accurate, mathematical model to describe the behavior of electrons near the material's energy limit, known as the Fermi level. They focused on two specific locations within the material's internal map, called the Gamma point and the A point, where the electrons are most active. By combining advanced computer simulations with the rules of symmetry—the geometric patterns that the crystal structure follows—they constructed a minimal model that captures the essential physics without unnecessary complexity.

A key discovery in this work challenges a common assumption in the field. Many researchers had believed that spin-orbit coupling, a relativistic effect where an electron's motion interacts with its spin, was too weak to matter in these types of magnetic materials. The team found that in manganese telluride, this interaction is actually the dominant force driving the splitting of electron energy levels. The strength of this effect comes from the tellurium atoms, which are heavy and create a strong interaction. Without accounting for this force, the model fails to explain the material's behavior. The researchers showed that when they included this interaction, the electrons near the Gamma point split into distinct paths based on their spin direction, creating a landscape where electrons with different spins travel differently.

The study also revealed that the direction in which the material's internal magnetic order points changes the rules of the road for these electrons. When the magnetic order aligns in one direction, the electrons form four distinct pockets of activity, arranged symmetrically around the center. In this configuration, the spin direction of the electrons flips depending on which quadrant they occupy. However, when the magnetic order points in a different direction, the landscape changes dramatically. The electrons condense into just two pockets, and remarkably, all the electrons in these pockets point their internal compasses in the same direction. This means that even though the material as a whole has no net magnetism, the current flowing through it can be fully spin-polarized, carrying a pure stream of electrons with a specific orientation.

At the other location, the A point, the behavior is more subtle. Here, the splitting of electron energy levels is much smaller, occurring only in specific, less symmetric areas of the material's map. The researchers found that the effects at this point are fragile and easily disturbed by temperature or impurities. This suggests that while the material shows promise for spin-based electronics, the most robust and useful effects come from the Gamma point, where the spin splitting is large and stable. The team provided a set of parameters that describe these behaviors, allowing other scientists to predict how the material will respond to electric currents and magnetic fields.

These findings provide a clear roadmap for understanding how altermagnets work. By establishing a model that connects the material's atomic structure to its electronic properties, the researchers have created a tool for future experiments. They demonstrated that the interplay between the crystal's shape and the heavy tellurium atoms creates a unique environment where spin and charge are tightly linked. This understanding is crucial for designing next-generation devices that could switch information faster and with less energy than current technologies. The work confirms that manganese telluride is a fertile ground for exploring new magnetic phenomena, offering a playground where theoretical predictions can be tested against real-world measurements.

Technical Summary: Symmetry and Minimal Hamiltonian of Nonsymmorphic Collinear Antiferromagnet MnTe

Problem Statement
$\alpha$ -MnTe, an A-type collinear antiferromagnet, has recently emerged as a prototypical "altermagnet"—a class of materials characterized by zero net magnetization yet exhibiting significant spin splitting in their electronic bands. While altermagnets are theoretically predicted to host unique spin transport phenomena (such as intrinsic anomalous Hall effects and spin-splitting torques) distinct from conventional antiferromagnets and ferromagnets, the microscopic mechanisms governing these properties in MnTe remain unclear. A primary challenge lies in the complex electronic structure of hexagonal $\alpha$ -MnTe: the valence band maximum (VBM) is located near both the $\Gamma$ and $A$ points in the Brillouin zone, and these regions exhibit distinctly different behaviors depending on the orientation of the Néel vector and the presence of spin-orbit coupling (SOC). Furthermore, while SOC is often considered negligible in non-relativistic altermagnets, its specific role in MnTe—particularly given the heavy Te atoms—requires rigorous investigation to understand the origin of spin splitting and anisotropic transport.

Methodology
The authors employ a multi-faceted approach combining first-principles calculations, group representation theory, and tight-binding modeling to construct a minimal effective Hamiltonian for $\alpha$ -MnTe.

First-Principles Calculations: Density Functional Theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP) with Projector Augmented Wave (PAW) pseudopotentials. The study utilized the Generalized Gradient Approximation (GGA-PBE) and included Hubbard $U$ corrections ( $U=4.0$ eV, $J=0.9$ eV) on Mn(3d) orbitals to account for strong correlation. Spin-orbit coupling was explicitly included for antiferromagnetic phases with Néel vectors aligned along the $[11\bar{2}0]$ and $[1\bar{1}00]$ directions.
Wannier Function Projection: To bridge DFT results with analytical models, the authors projected the electronic states onto localized Wannier functions, specifically focusing on Te(5p) orbitals, as Mn states were found to be far from the Fermi level.
Group Representation Theory: The effective Hamiltonians were derived using the representation theory of space groups, spin groups, and magnetic groups. The authors analyzed the symmetry constraints imposed by the nonsymmorphic space group $P6_3/mmc$ (No. 194) and the specific magnetic point groups associated with different Néel vector orientations.
Tight-Binding Modeling: A tight-binding Hamiltonian was constructed including Te-Te hopping, Te-Mn $p-d$ hybridization, and on-site SOC. The Schrieffer-Wolff transformation was applied to integrate out the Mn $d$ -orbitals, yielding an effective Hamiltonian solely in terms of Te $p$ -electrons.
Parameter Fitting: The coefficients of the derived effective Hamiltonians were fitted to the DFT band structures using a conjugate gradient method, ensuring the models accurately reproduce the band dispersion and spin splitting near the $\Gamma$ and $A$ points.

Key Contributions and Results

Derivation of Minimal Effective Hamiltonians: The paper successfully derives analytical $k \cdot p$ effective Hamiltonians for $\alpha$ -MnTe near the $\Gamma$ and $A$ points. These models are valid for two distinct in-plane Néel vector orientations: $[11\bar{2}0]$ and $[1\bar{1}00]$ .
Dominant Role of Spin-Orbit Coupling (SOC): Contrary to the general assumption that SOC is insignificant in altermagnets, the authors demonstrate that SOC is the dominant mechanism driving spin splitting in MnTe.
- At the $\Gamma$ Point: In the absence of SOC, the valence band is spin-degenerate. Upon including SOC, the symmetry is reduced, leading to significant spin splitting. For the Néel vector along $[11\bar{2}0]$ , the splitting creates four hole pockets with alternating spin polarization in the $k_x$ - $k_y$ plane. For the $[1\bar{1}00]$ orientation, the splitting results in two hole pockets that are fully spin-polarized along the $z$ -direction, despite the material being a collinear antiferromagnet.
- At the $A$ Point: The behavior is more complex due to the nonsymmorphic nature of the lattice. Without SOC, the bands exhibit specific degeneracies related to sublattice and spin mixing. With SOC, a small gap opens between $p_x$ and $p_y$ orbitals, and the spin splitting is primarily governed by non-relativistic altermagnetic features (bulk $g$ -wave splitting) rather than strong SOC-induced splitting, though SOC still plays a role in symmetry reduction.
Anisotropic Fermi Surfaces: The study reveals that the Fermi surface topology is highly sensitive to the Néel vector orientation.
- When the VBM is at $\Gamma$ , the $[11\bar{2}0]$ orientation yields four mirror-symmetric hole pockets with alternating spins, while the $[1\bar{1}00]$ orientation yields two pockets with identical spin polarization.
- When the VBM is at $A$ , the Fermi surfaces exhibit a hexagram structure with bulk $g$ -wave spin splitting, but the energy window for these features is narrow (< 0.01 eV), making them susceptible to thermal fluctuations.
Symmetry Analysis: The authors provide detailed character tables and irreducible representations for the relevant magnetic groups, clarifying how the combination of time-reversal symmetry breaking and nonsymmorphic lattice translations dictates the allowed terms in the Hamiltonian.

Significance
The authors claim that this work provides critical insights into the electron transport mechanisms in altermagnetic MnTe. By establishing a model playground based on realistic orbitals and rigorous symmetry analysis, the paper:

Clarifies the physical origin of spin splitting in MnTe, highlighting the essential role of SOC in high-symmetry points, which contrasts with the non-relativistic nature of ideal altermagnets.
Demonstrates that the transport properties (such as anisotropic conductivity and potential spin-polarized currents) are fundamentally dependent on the specific orientation of the Néel vector and the location of the Fermi level relative to the $\Gamma$ or $A$ points.
Provides a validated minimal Hamiltonian that can serve as a foundation for future theoretical and experimental studies on electron, spin, and orbital transport in MnTe and other transition metal chalcogenides with similar structures.

The paper concludes that while the non-relativistic altermagnetic features are present (particularly at the $A$ point), the strong SOC of Te atoms is indispensable for generating the large spin splitting observed at the $\Gamma$ point, which is likely the dominant factor in the material's observed spintronic phenomena.

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