Altermagnets: Symmetry-Driven Spin Splitting

Updated 17 January 2026

Altermagnets are collinear magnetic materials with zero net magnetization that feature large, momentum-dependent spin splitting due to symmetry-protected operations.
Their electronic structure displays sign-changing g-/d-wave patterns and nondegenerate bands, distinguishing them from traditional ferromagnets and antiferromagnets.
Validated by ARPES, neutron diffraction, and advanced modeling, altermagnets offer promising applications in next-generation quantum and spintronic devices.

Altermagnet (AM)

Altermagnets (AMs) are a symmetry-distinct class of collinear magnetic materials characterized by zero net magnetization yet exhibiting large, nonrelativistic, momentum-dependent spin splitting of their electronic bands. Unlike conventional ferromagnets (FMs), which have a uniform exchange field leading to spin-splitting and a net macroscopic magnetization, and unlike antiferromagnets (AFMs), which exhibit collinear moment compensation and band Kramers degeneracy enforced by inversion or translation plus time-reversal, AMs derive their unique properties from spin group symmetries—nonsymmorphic or rotational operations relating antiparallel sublattices without inversion or translation. As a consequence, AMs combine features such as high spin splitting, fully compensated order (no magnetic stray fields), and pronounced anisotropic, sign-changing momentum-space textures, establishing a new paradigm for spintronic materials (Regmi et al., 2024, Smolyanyuk et al., 2024, Ding et al., 2024, Wan et al., 2024).

1. Symmetry Principles and Theoretical Foundations

Altermagnets arise in crystals where the symmetry operation that pairs opposite-spin sublattices is a proper rotation or mirror (possibly nonsymmorphic) rather than pure translation or inversion combined with time-reversal (Smolyanyuk et al., 2024, Ding et al., 2024, Wan et al., 2024). A prototypical AM breaks time-reversal (𝒯), inversion (𝒫), and t𝒯, so the bands are not Kramers degenerate except on high-symmetry nodal planes. Formally, the Hamiltonian reads

$\mathcal{H}(\mathbf{k}) = \epsilon_0(\mathbf{k}) \mathbb{1} + \Delta(\mathbf{k}) \sigma_z,$

where $\Delta(\mathbf{k})$ is an odd function under the protecting point-group operations and vanishes along enforced nodal lines or planes. The momentum dependence of $\Delta(\mathbf{k})$ admits various representations, such as $d$ -wave [ $(k_x)^2 - (k_y)^2$ ], $g$ -wave, or higher harmonics, depending on lattice symmetry (Ding et al., 2024, López-Alcalá et al., 16 Dec 2025).

Crucially, the spin-splitting in AMs is:

Large and nonrelativistic: Due to exchange, often $\Delta E(\mathbf{k})$ exceeds spin–orbit effects.
Sign-changing in $\mathbf{k}$ : The splitting reverses sign under certain rotations, forming "harmonic petals" in the Brillouin zone (e.g., sixfold $g$ -wave in CrSb).
Zero net magnetization: The spatial integral of the magnetic moment vanishes.

In group-theoretical terms, AMs are identified by the following:

Zero net moment enforced by a group operation $g$ with character $\chi(g) = -1$ .
Non-degenerate bands at generic $\mathbf{k}$ because for some $g$ with $\chi(g)=+1$ , no operation fulfills $M_i \to -M_i$ —a definitive criterion for AMs (Smolyanyuk et al., 2024).

2. Realizations: Crystal Structures and Experimental Systems

Material realization of altermagnetism is connected to specific crystallographic and magnetic order criteria:

Sublattice order: AMs generally feature two interleaved, symmetry-inequivalent magnetic sublattices.
Noncentrosymmetric pairing: The sublattices are related by a non-inversion operation, such as: a 2-fold or 6-fold rotation (e.g., $C_6$ in CrSb (Ding et al., 2024)), mirror plane (e.g., in certain 2D dichalcogenides (Regmi et al., 2024)), screw axis, or nontrivial glide reflection.
Electronic bands: Large, momentum-dependent spin splitting not tied to net $M$ .

Notable AM compounds and their structural and physical properties include:

Material	Space Group	Magnetic Order	Band Splitting	$T_N$ (K)	Representative Symmetry	Source
CrSb	P6₃/mmc (No. 194)	$A$ -type c-axis AFM	$g$ -wave, $\sim$ 0.93 eV	$\sim$ 700	$C_2 \\| C_{6z}t$ , $C_2 \\| M_z$	(Ding et al., 2024, Peng et al., 2024)
CoNb $_4$ Se $_8$	P6₃/mmc (No. 194)	A-type AFM	$g$ -wave, 0.05–0.12 eV	168	Triangular (2 $\times$ 2) Co order	(Regmi et al., 2024)
RuO $_2$ , CaFe $_4$ Al $_8$	See refs	Collinear AM	d-/g-wave, 0.3–0.7 eV	300+	Glide/screw sym	(Wan et al., 2024)
Cr-based MOFs	P4bm, engineered	2D AFM	$g$ - or $d$ -wave, 40–84 meV	—	Ligand-tunable	(López-Alcalá et al., 16 Dec 2025)
FeCuP $_2$ S $_6$	2D vdW, SSG controlled	ML/Bilayer AM	50 meV	—	Nonsymmorphic SSG	(Zhao et al., 1 Nov 2025)

Angle-resolved photoemission spectroscopy (ARPES), neutron diffraction, and scanning tunneling microscopy have confirmed AM order and band splitting in these families (Regmi et al., 2024, Ding et al., 2024, Peng et al., 2024).

3. Electronic Structure: Spin-Split Bands and Symmetry-Protected Textures

The defining electronic signature is a k-dependent, often sign-changing spin splitting. For example, in CrSb: $\Delta(\mathbf{k}) = \Delta_0 \sin(3\varphi_{\mathbf{k}})$ producing six $g$ -wave nodes and petals in the Fermi surface ("star-of-David" structure) with giant splitting up to 0.93 eV near $E_F$ (Ding et al., 2024).

In CoNb $_4$ Se $_8$ , the symmetry-allowed nodal planes (including $\Gamma$ and vertical mirror planes) enforce vanishing $\Delta E(\mathbf{k})$ , but off-nodal planes the splitting reaches $\sim$ 100 meV and alternates sign, consistent with $g$ -wave classification (Regmi et al., 2024).

The AM state is robust over wide parameter ranges, confirmed by DFT+DMFT calculations, lattice Monte Carlo solutions for the Ising–Kondo model, and mean-field studies of t–U–V Hubbard models (Dong et al., 1 Jul 2025, Zhao et al., 10 Jan 2026, Wan et al., 2024).

4. Magnetotransport, Optical, and Nonlinear Spintronic Signatures

AMs manifest distinctive transport and optical properties stemming from their k-dependent spin splitting:

Anisotropic magnetoresistance (AMR): The AMR is set by the anisotropy of spin-split Fermi surfaces. In CrSb, non-saturating, positive MR up to 52.6% at 7 T, 6 K is observed, with extended-Kohler scaling capturing all MR(T, H) data using a single T-dependent parameter (Peng et al., 2024).
Nonlinear Hall resistivity: In CrSb, the multi-band Fermi surface leads to strongly nonlinear $\rho_{yx}(H)$ , with no anomalous Hall effect, governed by magneto-crystalline symmetry (Peng et al., 2024).
Spin-polarized current injection: Altermagnetic electrodes create spin-polarized thermionic injection and Schottky currents without net M, owing to the spin-contrasting Fermi surface topology (Ang, 2023).
Supercurrent nonreciprocity (Josephson diode effect): Clean Josephson junctions with AM interlayers exhibit large supercurrent rectification ( $\sim50\%$ efficiency), tunable by AM strength and orientation (Boruah et al., 1 Apr 2025).
Optical and magneto-optical effects: AMs under strain break key rotation/mirror symmetries, enabling distinctive linear magneto-optical responses (e.g., Kerr rotation). Calculations predict angles $\theta_K$ up to 60 mdeg under moderate strain in CrSb, with optical absorption peak splitting $\sim$ 0.1–0.2 eV (Sun et al., 30 May 2025). In contrast, AFMs remain strictly silent due to persistent PT symmetry.
Ultrafast and optospintronic phenomena: The light polarization controls the spin orientation of photo-excited electrons and directly probes k-resolved spin splitting—a unique attribute enabling ultrafast optical switches and access to nonequilibrium spin textures (Eskandari-asl et al., 2 Apr 2025).

5. Model Hamiltonians and Theoretical Approaches

Microscopic modeling of AMs reveals their essential physics independently of SOC, ferromagnetic order, or Dzyaloshinskii–Moriya interactions:

Minimal lattice Hamiltonian:

$H = \sum_{\mathbf{k}} \epsilon_0(\mathbf{k}) \mathbb{1} + \Delta(\mathbf{k}) \sigma_z$

$g$ - and $d$ -wave form factors: In 2D, $\Delta(\mathbf{k}) \propto (k_x^2-k_y^2)$ ( $d$ -wave), or $(k_x k_y)((k_x)^2-(k_y)^2)$ ( $g$ -wave) for square and tetragonal/honeycomb lattices, respectively (López-Alcalá et al., 16 Dec 2025).
Kondo, Hubbard, and extended models: Altermagnetic order is stable in the Ising–Kondo lattice with staggered next-nearest-neighbor hopping, in single-orbital extended Hubbard models, and in 2D MOF frameworks through ligand and frontier-orbital engineering (Zhao et al., 2024, Dong et al., 1 Jul 2025, López-Alcalá et al., 16 Dec 2025, Zhao et al., 10 Jan 2026).
Mean-field, lattice Monte Carlo, and DFT+DMFT: All confirm the robustness of AM over wide regions of $U$ , doping, Kondo coupling $J$ , and lattice tuning. Nonmagnetic impurity scattering and the spatial structure of induced magnetization clouds enable real-space confirmation of $d$ -wave symmetry (Zhao et al., 10 Jan 2026).

6. Spintronics and Multiferroic Device Implications

The distinctive symmetry and k-space spin splitting of AMs enable a spectrum of next-generation spintronic functionalities:

Spin valves and tunnel junctions: Direction-selective, high polarization, and absence of stray field; fast operation due to large exchange splitting (Ding et al., 2024, Regmi et al., 2024).
Field-free superconducting diodes: AM interlayers in Josephson junctions yield large and tunable supercurrent rectification without external fields or Rashba SOC (Boruah et al., 1 Apr 2025).
Electrically switchable logic: In van der Waals systems like FeCuP $_2$ S $_6$ , ferroelectric switching and interlayer sliding toggle the AM band splitting and associated anomalous Hall conductivity, facilitating high-density, nonvolatile memory (Zhao et al., 1 Nov 2025).
Nonlinear charge–spin conversion: Linear (d-wave) or cubic (g-wave) spin currents in 2D MOFs; chiral magnonics from altermagnetic order enable ultrafast and low-power spin–orbit-free spintronic applications (López-Alcalá et al., 16 Dec 2025).
Optical and quantum edge devices: Higher-order topological corner modes induced by AM on the Lieb lattice—gapless Dirac edge states are gapped by in-plane AM, producing robust, well-localized corner zero modes absent in FM or FIM (Huo et al., 19 Dec 2025).
Ultrafast optoelectronic control: Femtosecond laser pulses enable coherent switching and reading of spin textures by exploiting polarization-dependent excitation of nonequilibrium carriers (Eskandari-asl et al., 2 Apr 2025).

7. Experimental Characterization, Materials Discovery, and Outlook

Unambiguous experimental detection of AMs employs complementary strategies:

Band-resolved probes: Spin-ARPES and spin-resolved STM can visualize momentum-dependent spin splitting and sublattice-resolved densities of states (Regmi et al., 2024, Wan et al., 2024).
Transport and Hall measurements: Anisotropic MR, absence of anomalous Hall effect (when symmetry forbids it), and multi-band scaling in $\rho_{yx}$ provide indirect but robust signatures (Peng et al., 2024).
Magneto-optical response: Strain-induced Kerr rotation and dichroic absorption are universal, noninvasive fingerprints distinguishing AMs from AFMs (Sun et al., 30 May 2025).
First-principles and high-throughput search: Automated symmetry classification (e.g., via pymatgen or custom tools) and massive DFT+eDMFT workflows expedite AM discovery; statistical analysis highlights design rules favoring certain lattices and 3d/4d elements (Wan et al., 2024, Smolyanyuk et al., 2024).

AMs are theoretically predicted and experimentally validated in a diverse array of transition metal compounds, heavy-fermion systems, 2D MOFs, and engineered van der Waals heterostructures. The coexistence of high tunability, large spin splitting without net magnetization, and straightforward integration into complex device architectures positions altermagnets as an essential platform for fundamental studies of symmetry-driven magnetism and the development of next-generation quantum, spintronic, and multifunctional devices.