🔬 materials science

Symmetry-enforced third-order nonlinear thermal Hall effects in altermagnets

This paper proposes a symmetry-enforced third-order nonlinear thermal Hall effect driven by Berry curvature in altermagnets, demonstrating that materials with out-of-plane Néel order (such as KV2Se2O and MnF2) exhibit a dominant third-order response with $\pi$ -periodic dependence, providing a novel diagnostic tool for identifying altermagnetism and determining Néel vector orientation.

Original authors: Yun-Mei Li, Jiacheng Yao, Hua Wang, Kai Chang

Published 2026-09-03

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CC BY 4.0

Original authors: Yun-Mei Li, Jiacheng Yao, Hua Wang, Kai Chang

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

Heat is often thought of as a simple, chaotic jumble of energy, but in the solid world of crystals, it can flow with surprising order. When scientists apply a temperature difference across a material, they expect heat to travel straight from the hot side to the cold side. However, under specific conditions involving the internal magnetic structure of a substance, this heat can be forced to turn sideways, flowing perpendicular to the temperature difference. This phenomenon is known as the thermal Hall effect. For decades, researchers have used this sideways heat flow to study materials that conduct electricity, but a major challenge has remained: how to detect this effect in materials that are electrical insulators, where no free electrons exist to carry the current. In these insulators, heat is carried by vibrations of the crystal lattice or by magnetic waves, making the measurement far more difficult. The question of how heat behaves in a newly discovered class of magnetic materials, called altermagnets, has been particularly elusive. These materials possess a unique magnetic order that combines features of both ferromagnets and antiferromagnets, yet they have long lacked a definitive experimental signature to confirm their existence and distinguish them from other magnetic states.

A team of researchers at Zhejiang University has now developed a theoretical framework to solve this puzzle, predicting a specific, powerful response in altermagnets that had been overlooked. By analyzing the behavior of both electrons and magnetic waves within these materials, the scientists discovered that the standard, linear version of the thermal Hall effect is often absent in altermagnets with a specific magnetic orientation. Instead, they found that a third-order effect dominates when the linear and second-order contributions vanish due to symmetry. In this scenario, the sideways heat flow does not simply grow in proportion to the temperature difference; rather, it grows in proportion to the cube of that difference. This means that if you double the temperature difference, the sideways heat flow increases eightfold, a behavior that is mathematically distinct from the linear responses seen in most other materials. The researchers identified that this unique effect is expected to occur in materials with specific magnetic symmetries, specifically when the magnetic spins are arranged in a particular pattern known as a d-wave or i-wave order with the magnetic axis pointing out of the material's plane.

To test their theory, the team applied their calculations to two real-world materials: a metal called KV2Se2O and an insulator called MnF2. In the metal, the effect arises from the movement of electrons, while in the insulator, it is driven by magnetic waves, known as magnons. The simulations showed that in both cases, the sideways heat current is nonzero and follows a very specific pattern. As the direction of the temperature gradient is rotated, the strength of the heat flow changes, but it repeats its pattern every 180 degrees. This 180-degree periodicity is a crucial fingerprint; it is different from the patterns seen in linear or second-order effects and serves as a clear diagnostic tool. The researchers calculated that this signal should be strong enough to be measured in a laboratory setting, provided the temperature gradient is applied with sufficient precision. They noted that for the insulator MnF2, the magnetic waves alone are responsible for the effect, while in the metal KV2Se2O, both electrons and magnetic waves contribute, provided certain magnetic interactions are present.

The study also clarified what does not happen in these materials, ruling out several possibilities that might have confused earlier experiments. The researchers showed that if the magnetic spins in these altermagnets are oriented within the plane of the material rather than pointing out, the dominant effect reverts to the standard linear thermal Hall effect, which is less distinctive. Furthermore, they demonstrated that without the specific magnetic splitting characteristic of altermagnets, the thermal Hall effect would vanish entirely. This distinction is vital because it means that observing this specific third-order response would not only confirm the existence of altermagnetism but also reveal the precise orientation of the magnetic spins within the crystal. The team emphasized that this approach works for both metals and insulators, offering a versatile method to probe materials that were previously difficult to study using electrical transport measurements.

The implications of this work extend beyond theoretical curiosity, offering a practical roadmap for experimentalists. The researchers proposed a method for detecting this effect using a three-terminal setup, where temperature differences are measured across a sample. Because the sideways heat flow depends on the cube of the temperature difference, scientists can distinguish this signal from background noise by simply varying the heating power and observing how the signal scales. The calculations suggest that the signal strength in materials like MnF2 is comparable to other known thermal effects, making it a realistic target for immediate experimental verification. By providing a clear, symmetry-enforced signature that is robust against the complexities of material imperfections, this work offers a definitive way to identify altermagnets. It transforms the search for these materials from a game of chance into a targeted investigation, where the presence of a specific heat flow pattern can confirm the unique magnetic nature of a substance and map the orientation of its internal magnetic structure.

Problem Statement
Altermagnets represent a novel class of collinear antiferromagnets characterized by nonrelativistic spin splitting in momentum space while maintaining zero net magnetization in real space. Despite numerous candidate materials, the existence of altermagnetism remains a subject of conflicting reports in both metallic and insulating systems. While angle-resolved photoemission spectroscopy (ARPES) and neutron scattering provide direct momentum-space probes, transport measurements offer complementary real-space information. However, electrical Hall measurements are limited to metallic systems with free carriers and fail in semiconductors and insulators. Furthermore, while the linear thermal Hall effect (THE) driven by Berry curvature is well-established, and recent works have explored second-order nonlinear responses, a proper theoretical framework for nonlinear THE that accounts for the spatial dependence of distribution functions and density matrices induced by temperature gradients has been lacking. Specifically, there is a need to identify symmetry conditions where higher-order thermal responses dominate, providing definitive signatures for altermagnetism across diverse material classes.

Methodology
The authors develop a theoretical framework based on semiclassical Boltzmann transport theory within the relaxation time approximation (RTA). This framework describes the transverse energy current up to the third order of the temperature gradient ( $\nabla T$ ), applicable to electrons, magnons, phonons, and their hybrid excitations.

Formalism: Starting from the wavepacket semiclassical theory and the Boltzmann equation, the authors derive expressions for the thermal current $J_Q^b$ expanded as a power series of the temperature gradient: $J_Q^b = \sum_l \kappa_{ba}^l (\partial_a T)^l$ . They explicitly calculate the coefficients for the first ( $\kappa^1$ ), second ( $\kappa^2$ ), and third ( $\kappa^3$ ) orders, incorporating the Berry curvature ( $\Omega$ ), group velocity ( $v$ ), and relaxation time ( $\tau$ ).
Symmetry Analysis: The authors perform a magnetic point group analysis to enumerate symmetry constraints that allow a dominant third-order response while forcing linear and second-order contributions to vanish. This involves identifying conditions where the Berry curvature distribution possesses specific parity properties (e.g., even parity in momentum space to suppress lower-order terms via integration).
Material Modeling:
- Electrons: For altermagnetic metals, a unified effective $k \cdot p$ Hamiltonian including Rashba-type spin-orbit coupling (SOC) is applied to planar $d$ -wave, $i$ -wave, and bulk $i$ -wave altermagnets.
- Magnons: For insulators, a general spin-spin interaction Hamiltonian including exchange interactions ( $J_{ij}$ ), dipole-dipole interactions (DDIs), and magnetic anisotropy is used. The Holstein-Primakoff transformation is employed to derive the magnon spectrum and Berry curvature.
Case Studies: The theory is applied to specific candidate materials: the $d$ -wave altermagnetic metal $KV_2Se_2O$ (using first-principles calculations for electronic bands) and the $d$ -wave altermagnetic insulator $MnF_2$ (using experimentally determined spin-spin interaction parameters).

Key Contributions and Results

Symmetry-Enforced Third-Order THE: The study identifies specific magnetic point groups ($mmm$, $m'm'm$ , $m'm'2$ , $2m'm'$ , $4'$ , $\bar{4}'$ , $4'/m$ , $4'mm'$, $4'/mmm'$, $6m'm'$ , $6/mm'm'$, $m\bar{3}m'$ ) that support a dominating third-order nonlinear THE while suppressing linear and second-order terms. This occurs in planar $d$ -wave, $i$ -wave, and bulk $i$ -wave altermagnets with out-of-plane Néel order.
Distinctive Signatures: The third-order thermal Hall conductivity ( $\kappa^3_{ba}$ ) exhibits a $\pi$ -periodic dependence on the direction of the temperature gradient. This contrasts with linear and second-order responses and arises from the $(v^a_{nk})^2$ term in the third-order expansion.
Material-Specific Predictions:
- $KV_2Se_2O$ (Metal): Both electrons and magnons contribute to a nonzero third-order nonlinear THE. The electronic contribution is calculated using first-principles bands, showing a strong dependence on temperature and gradient direction.
- $MnF_2$ (Insulator): Magnons alone yield a third-order nonlinear THE. The calculations confirm that with out-of-plane Néel order, the system satisfies the symmetry constraints for a dominant third-order response.
Orientation Dependence: The thermal response is highly sensitive to the Néel vector orientation. In-plane Néel order in these systems supports a linear THE, whereas out-of-plane order suppresses the linear term in favor of the third-order response. For $g$ -wave altermagnets, out-of-plane order leads to zero net THE, while in-plane order supports linear THE.
Experimental Feasibility: The authors estimate that for realistic parameters (temperature gradients of $10^3 - 10^4$ K/m and relaxation times of $10^{-6} - 10^{-5}$ s), the magnitude of the third-order signal in $MnF_2$ is comparable to first-order thermal Hall conductivities observed in previous works. The distinct cubic dependence of the transverse temperature difference on the longitudinal gradient ( $\Delta T_b \propto (\Delta T_a)^3$ ) is proposed as a clear experimental identifier.

Significance
The paper claims that these symmetry-guaranteed Berry-curvature phenomena provide a robust diagnostic tool for confirming altermagnetism in candidate materials, particularly those that are semiconductors or insulators where electrical Hall measurements are ineffective. By distinguishing between linear and third-order responses and analyzing the periodicity with respect to the temperature gradient direction, the proposed thermal transport measurements can not only verify the existence of altermagnetism but also determine the orientation of the Néel vector. This work fills a gap in the theoretical understanding of nonlinear thermal transport in magnetic systems and offers practical transport characteristics for identifying and characterizing altermagnetic materials.

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