Altermagnetism: Collinear, Compensated Magnetic Order

Updated 2 January 2026

Altermagnetism is a newly recognized magnetic order featuring collinear spins with zero net magnetization and momentum-dependent even-parity splitting.
It arises from symmetry-breaking via crystal fields and exchange interactions, yielding distinct spin-split electronic bands without relying on traditional spin–orbit coupling.
Experimental signatures such as spin-resolved ARPES and neutron scattering underscore its potential for ultrafast, energy-efficient spintronic devices.

Altermagnetism is a recently recognized fundamental class of magnetic order, characterized by collinear arrangements of magnetic moments with strictly zero net magnetization and robust, nonrelativistic, time-reversal symmetry breaking. Unlike ferromagnets (with net moment and uniform spin splitting) or conventional Néel antiferromagnets (with compensated moments and globally degenerate bands enforced by translation or inversion), altermagnets exhibit compensated spin order whose distinct sublattices are related by real-space rotations, not by simple translation or inversion. This unique symmetry enables an even-parity, momentum-dependent spin splitting of the electronic band structure—typically of d-, g-, or i-wave form—in the absence of spin–orbit coupling. The principle mechanism behind altermagnetism is microscopic exchange and crystal field-induced symmetry breaking in spatially alternating spin or orbital environments. Altermagnets have been identified across correlated oxides, heavy fermion systems, itinerant metals, nanographenes, and quasicrystals, and are now regarded as a foundational platform for high-frequency, compensated spintronics with large device-design flexibility and tunable transport anisotropy.

1. Symmetry Principles and Classification

Altermagnetic order is defined by two features: (i) full compensation $\sum_i\langle\mathbf{S}_i\rangle = 0$ of spin (or total) angular momentum, and (ii) broken combined parity–time ( $\mathcal{PT}$ ) symmetry. In crystalline contexts, sublattices are related by a point-group rotation (e.g., $C_4$ or $C_6$ ), not by a Bravais translation or inversion, so that no operation alone maps one spin species onto the other (Cheong et al., 2024, Jungwirth et al., 28 Jun 2025, Tamang et al., 2024, Šmejkal et al., 2021).

Group-Theoretical Structure.

Altermagnets correspond to spin-space groups with structure $[E\|H]+[C_2\|G{-}H]$ , where $H$ is the halving subgroup of the crystallographic point group $G$ (preserving each sublattice), and $[C_2\|A]$ (with $A \in G{-}H$ ) generates the interchange of opposite-spin sublattices via a rotation (Šmejkal et al., 2021, Jungwirth et al., 28 Jun 2025, Cheong et al., 2024).

Types of Altermagnets.

M-type: Broken $\mathcal{T}$ , possible nonzero net magnetic moment (often purely orbital); 31 MPGs; exhibit linear anomalous Hall effect (AHE) and transverse piezomagnetism.
S-type: Broken $\mathcal{T}$ , net spin moment zero; 38 MPGs; even-order AHE (high odd-orders), transverse piezomagnetism.
A-type: Unbroken $\mathcal{T}$ , broken $\mathcal{P}$ symmetry; 21 MPGs; odd-order AHE (even-order current-induced magnetization) (Cheong et al., 2024).

The symmetry constraints dictate the number and type of protected nodal planes or lines in momentum space, giving rise to "planar" (d-wave), "g-wave", or "i-wave" multipolar anisotropy (Jungwirth et al., 28 Jun 2025).

Strong vs. Weak Altermagnets.

Strong: Spin splitting persists at $\lambda_{\mathrm{SOC}}=0$ , determined by the number of unbroken orthogonal spin-rotation symmetries $N_s$ : $N_s\le1$ .
Weak: Spin splitting exists only for $\lambda_{\mathrm{SOC}}\ne0$ , requiring $N_s\ge2$ (Cheong et al., 2024, Radaelli, 2024).

Table 1. Symmetry and Classification

Altermagnet Type	Broken Symmetries	Net Moment	Band Splitting Mechanism
M-type	T, (PT)	$\ne$ 0	Exchange+SOC (may be orbital)
S-type	T, (PT)	0	Exchange, even-parity ( $d$ , $g$ )
A-type	P (not T)	0	Parity-broken, requires SOC

2. Microscopic Models and Band Topology

Minimal Hamiltonian.

A prototypical band-theory model for collinear d-wave altermagnets is:

$H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\sigma_0 + \Delta(\mathbf{k})\,\sigma_z$

with $\varepsilon_0(\mathbf{k})$ a conventional tight-binding dispersion and $\Delta(\mathbf{k})$ an even-parity ( $d_{x^2-y^2}$ , for example) "spin splitting" that transforms according to a nontrivial irreducible representation of the point group:

$d$ -wave: $\Delta(\mathbf{k})\propto\cos k_x-\cos k_y$
$g$ -wave: $\propto \sin k_x \sin k_y [\cos k_x - \cos k_y]$
$i$ -wave: higher-degree harmonics (Jungwirth et al., 28 Jun 2025, Ortiz et al., 5 Aug 2025, Cheong et al., 2024)

Spin splitting is then strictly zero along nodal lines determined by the symmetry, with alternating lobes across the Brillouin zone (Tamang et al., 2024, Jungwirth et al., 28 Jun 2025).

Interaction-Driven Origin.

Strongly correlated origin is seen in Hubbard- and Kondo-lattice models, where the interplay of local exchange (Stoner mechanism) and van Hove singularities leads to altermagnetic ground states. In multiorbital systems, coexisting orbital order and staggered antiferromagnetism can stabilize robust altermagnetism with tunable band splitting and anisotropic spin transport (Giuli et al., 2024, Leeb et al., 2023, McClarty et al., 2024).

Quasicrystals and Amorphous Systems.

Altermagnetism extends to quasicrystals with high-fold (e.g., $C_8$ , $C_{12}$ ) rotations and to amorphous lattices, provided the real-space order parameter transforms according to a nontrivial, inversion-even one-dimensional irreducible representation of the point (or local) symmetry group. The resulting spectral and transport fingerprints include higher-fold nodal patterns, generic even in the absence of translation symmetry (Chen et al., 24 Jul 2025, Shao et al., 21 Aug 2025, d'Ornellas et al., 11 Apr 2025).

3. Experimental Signatures: Spectroscopy and Imaging

Momentum-Space Probes.

Spin-resolved ARPES: Directly visualizes momentum-dependent ( $d$ -, $g$ -, or $i$ -wave) spin splitting in the absence of net magnetization. This characterizes zero-moment but robustly spin-split Fermi surfaces in materials such as RuO $_2$ , MnTe, and CsV $_2$ Se $_2$ O (Jungwirth et al., 28 Jun 2025, Fu et al., 30 Dec 2025).
XMCD and XMLD: X-ray magnetic circular/linear dichroism in combination with photoemission electron microscopy achieves real-space imaging of altermagnetic order, domain walls, and multi-domain textures with full vectorial resolution (Amin et al., 2024).
Polarized Neutron Scattering: Directly measures chirality anisotropy in magnonic bands, revealing $d$ - or $g$ -wave splitting of magnon branches without external field, e.g., in MnTe (sixfold) or MnF $_2$ (fourfold) (McClarty et al., 2024).

Real-Space Microscopy.

STM/STS: Imaging of unidirectional charge order, symmetry-broken charging rings around defects, and quasiparticle interference patterns in altermagnets proves local rotational symmetry breaking and the existence of compensated spin split order (Fu et al., 30 Dec 2025).
SP-STM: Checkerboard maps with alternating sign of spin-polarized tunneling, despite global $M=0$ (Jungwirth et al., 28 Jun 2025).

Transport and Optical Probes.

Anomalous Hall, Nernst, and Magnetoresistance: The presence of non-relativistic, even-parity spin splitting enables large AHE/ANE/TMR responses even in zero net- $M$ systems (Tamang et al., 2024, Bai et al., 2024).

4. Physical Phenomena and Device-Level Impact

Altermagnets exhibit an array of macroscopic responses previously impossible in strictly compensated magnets:

Spin and Charge Transport.

Spin-Current Generation: Charge currents yield direction-dependent, pure spin currents (spin splitter torque), with spin polarization determined by $d$ -wave (or higher) anisotropy (Giuli et al., 2024, Šmejkal et al., 2022).
Giant/Tunnel Magnetoresistance (GMR/TMR): Magnetoresistance effects up to hundreds of percent are possible in altermagnet-based tunnel junctions, with zero stray fields and high scalability (Bai et al., 2024).

High-Frequency/THz Dynamics.

Magnon Chirality Splitting: Altermagnets support THz-scale non-degenerate magnon branches, facilitating chiral magnonics and ultrafast spin-torque oscillators (McClarty et al., 2024, Zhao et al., 2024).
Terahertz Magnetoelectronics: Intrinsic THz dynamics and anisotropic domain-wall stiffness permit ultrafast, energy-efficient memory and logic devices (Tamang et al., 2024, Šmejkal et al., 2022).

Magneto-Optical and Piezomagnetic Responses.

MOKE/XMCD: Most altermagnets, by symmetry, are also piezomagnetic and show strong Kerr rotation, with direct device-readout signatures (Radaelli, 2024).
Field-, Strain-, or Current-Driven Switching: The unique symmetry allows for control of domain textures, chirality, and Néel vector orientation using microstructuring, strain, or electrical gating (Amin et al., 2024, Bai et al., 2024).

5. Materials Landscape and Realizations

3D and 2D Materials.

Key experimentally confirmed candidates include:

RuO $_2$ (rutile): Collinear, $d$ -wave, $\Delta E \sim 1$ eV, AHE $\sim10^3\,\Omega^{-1}$ cm $^{-1}$ .
MnTe, CrSb: $g$ -wave, high $T_N$ (up to 700 K), spin splitting of 0.6–1.1 eV (Jungwirth et al., 28 Jun 2025, Chen et al., 24 Jul 2025).
FeF $_2$ , La $_2$ CuO $_4$ , V $_2$ Se $_2$ O: $d$ -wave, lower $T_N$ , applicable in superconducting heterostructures.
Organic frameworks (DBH-based COFs): Demonstrated purely $d$ -wave, compensated, spin-split order in carbon networks (Ortiz et al., 5 Aug 2025).

Heavy Fermion and Moiré Systems.

Itinerant $f$ -electron Kondo lattices with alternating next-nearest neighbor hopping (CeNiAsO, Ce $_4$ X $_3$ ) host robust $d$ -wave altermagnetism tunable by doping and pressure (Zhao et al., 2024). Twisted TMDs and van der Waals magnets further broaden the menu of controllable altermagnets (Bai et al., 2024).

Quasicrystals, Amorphous Systems, and Strongly Correlated Regimes.

Symmetry-guided approaches demonstrate robust altermagnetic order in 2D quasicrystals, amorphous networks, and correlated multipolar phases, with transport and spectral features beyond the reach of periodic lattice invariants (d'Ornellas et al., 11 Apr 2025, Shao et al., 21 Aug 2025, Chen et al., 24 Jul 2025).

6. Outlook and Theoretical Developments

The field continues to expand along multiple axes:

Quantum Criticality and Field Theory.

Effective nonlinear sigma models for altermagnetic order introduce unique Berry-phase derived "W-terms" into the action, resulting in quantum critical points with essential singularities and dynamical scaling controlled by competition between magnetic fluctuations and Coulomb interactions—with direct implications for deconfined quantum criticality (Lundemo et al., 7 May 2025).

Topological Phases and Berry Curvature.

The interplay of altermagnetic order and spin–orbit coupling produces topological band structures—Weyl nodes, nodal lines, quantized spin Hall states—embedded within a strictly compensated, collinear background (Jungwirth et al., 28 Jun 2025, McClarty et al., 2024).

Materials Design and Application Prospect.

Symmetry-based (tensorial, group-theoretical) screening provides systematic guidance for engineering strong and weak altermagnetic states, enabling designer materials for zero-stray-field memory, THz devices, caloritronics, and quantum logic with simultaneous robustness and high operational frequency (Cheong et al., 2024, Radaelli, 2024, Bai et al., 2024).

Generalized Paradigm.

Altermagnetism unifies and extends the concepts of collinear compensated magnetism, bridging conventional ferromagnetism and antiferromagnetism, enabling functional phases and phenomena—such as nonrelativistic spin Hall effect, pure spin currents, and topologically nontrivial chiral magnon modes—hitherto inaccessible in either conventional regime (Šmejkal et al., 2022).

References: (Cheong et al., 2024, Jungwirth et al., 28 Jun 2025, Ortiz et al., 5 Aug 2025, Amin et al., 2024, Giuli et al., 2024, Šmejkal et al., 2021, Radaelli, 2024, Chen et al., 24 Jul 2025, d'Ornellas et al., 11 Apr 2025, Fu et al., 30 Dec 2025, Bai et al., 2024, Das et al., 2023, Cheong et al., 2024, Zhao et al., 2024, McClarty et al., 2024, Tamang et al., 2024, Leeb et al., 2023, Lundemo et al., 7 May 2025, Shao et al., 21 Aug 2025, Šmejkal et al., 2022).