Altermagnets: Symmetry-Driven Spin Splitting

• Altermagnets are collinearly ordered magnets with zero net magnetization that display momentum-dependent spin splitting due to symmetry-enforced exchange fields.
• They differ from ferromagnets and conventional antiferromagnets as their band splitting arises from rotational, mirror, or screw symmetry operations rather than spin–orbit coupling.
• Applications of altermagnets include tunable spintronic devices, magneto-optical effects, and superconducting junctions, enabled by their unique electronic structure and symmetry control.

Altermagnets (AMs) are a recently recognized class of collinear magnets that combine zero net magnetization with strong, momentum-dependent spin splitting of electronic bands. In this sense they are distinct from both conventional ferromagnets, which have a nonzero macroscopic magnetization, and conventional collinear antiferromagnets, which typically remain spin-degenerate in the absence of spin–orbit coupling. Across recent work, AMs are defined by compensated real-space order, broken time-reversal symmetry, and band splittings that are enforced by crystal and magnetic symmetries rather than by relativistic spin–orbit mechanisms (Smolyanyuk et al., 2024, Samanta et al., 2024, Wan et al., 2024).

1. Definition and distinguishing characteristics

Altermagnets are collinearly ordered magnets with zero net magnetization but finite spin splitting at a given momentum $\mathbf{k}$. A standard expression of the defining band-structure property is

$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$

even though the spin moments on different sublattices cancel in real space (Zhao et al., 19 Mar 2026, Yarmohammadi et al., 19 Jun 2026). This combination gives AMs some properties associated with ferromagnets, such as spin-split bands and anomalous transport, while retaining the compensated moment of antiferromagnets (Smolyanyuk et al., 2024).

The central contrast with conventional collinear antiferromagnets is symmetry. In standard AFMs without SOC, combined symmetries such as $PT$ or $UT_{1/2}$ enforce global spin degeneracy, whereas AMs break $PT$ and $UT_{1/2}$ in such a way that a non-relativistic, momentum-dependent spin splitting appears (Samanta et al., 2024). By contrast with ferromagnets, the splitting in AMs is not approximately uniform in $\mathbf{k}$-space. It changes sign across the Brillouin zone, often in patterns with $d$-wave, $g$-wave, or $i$-wave character (Samanta et al., 2024, Mazin et al., 2023, Sheoran et al., 28 Feb 2025).

This sign-changing structure is often summarized as “alternating magnetism”: spin polarization is locally strong in momentum space but integrates to zero globally. That is why AMs can support spin-polarized transport without a macroscopic magnetic moment or stray fields (Yarmohammadi et al., 19 Jun 2026, Sahoo et al., 17 Sep 2025).

2. Symmetry foundations

The symmetry content of altermagnetism is more restrictive than simple moment compensation. Several works formulate the distinction by asking which operations relate opposite-spin sublattices. If compensation is protected by inversion or translation combined with time reversal, Kramers-like degeneracy survives and the system is a conventional antiferromagnet. If compensation is instead enforced by rotations, mirrors, screws, or related spatial operations that do not restore spin degeneracy at the same $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$0, altermagnetism becomes symmetry-allowed (Samanta et al., 2024, Smolyanyuk et al., 2024).

In spin-space-group language, a collinear AM can be written in the form

$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$1

with the crucial distinction that $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$2. When $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$3, the system has $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$4 symmetry and remains spin-degenerate; when $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$5 is a twofold rotation or screw axis instead, global spin degeneracy is not enforced (Zhao et al., 1 Nov 2025). In FeCuP$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$6S$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$7, for example, the operations $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$8 or $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$9 are identified as AM-enabling, whereas pure translation or inversion lead back to ordinary AFM behavior (Zhao et al., 1 Nov 2025).

The same logic appears in magnetic-space-group screening. For collinear magnets one may decompose the symmetry as

$PT$0

where $PT$1 preserves spin orientations and $PT$2 flips them. If the spin-flipping coset contains $PT$3 or $PT$4, the material is a conventional AFM; otherwise it is an altermagnet candidate (Wan et al., 2024). In two dimensions, spin-layer-group analysis further restricts which symmetries can host AM order and which can support responses such as an in-plane anomalous Hall effect (Sheoran et al., 28 Feb 2025).

A recurrent consequence is methodological: because moment compensation in AMs is symmetry-driven, identification can in principle be performed directly from the magnetic crystal structure, without first computing the electronic bands (Smolyanyuk et al., 2024).

3. Momentum-space structure and model descriptions

A common minimal form for the non-relativistic electronic structure is

$PT$5

where $PT$6 is an effective exchange field that depends on momentum and changes sign across the Brillouin zone (Samanta et al., 2024). This is formally similar to exchange splitting in a ferromagnet, but with a symmetry-enforced anisotropy that is absent in the ferromagnetic case.

A canonical square-lattice AM model is

$PT$7

which yields

$PT$8

The splitting therefore vanishes on $PT$9 and changes sign under the interchange of $UT_{1/2}$0 and $UT_{1/2}$1, making explicit the $UT_{1/2}$2-type structure (Das et al., 2024).

For 2D $UT_{1/2}$3-wave AMs with Rashba SOC, a continuum description used for Floquet transport is

$UT_{1/2}$4

with

$UT_{1/2}$5

This parametrizes the two standard tetragonal $UT_{1/2}$6-wave harmonics, $UT_{1/2}$7 and $UT_{1/2}$8, and makes the orientation of the altermagnetic order explicit (Yarmohammadi et al., 19 Jun 2026).

Recent work also extends the taxonomy beyond nodal $UT_{1/2}$9-, $PT$0-, and $PT$1-wave forms. “Extended $PT$2-wave altermagnets” are defined by a staggered valley-spin order

$PT$3

leading to

$PT$4

These states are fully gapped, spin-compensated, and spin-polarized, with the compensation enforced through valley-exchange symmetries rather than conventional crystallographic spin-group operations (Dürrnagel et al., 27 Aug 2025).

4. Materials platforms and routes to control

The materials landscape now spans metals, insulators, van der Waals systems, ferroelastics, and electrically tunable multiferroics. In rutile fluorides $PT$5 ($PT$6 Mn, Co, Ni), the magnetic space group lacks $PT$7 and $PT$8, and DFT without SOC shows spin-degenerate bands on glide-invariant planes but clear spin splitting away from them. The splitting near the valence-band maximum along $PT$9–M is reported as $UT_{1/2}$0 eV in MnF$UT_{1/2}$1, $UT_{1/2}$2 eV in CoF$UT_{1/2}$3, and $UT_{1/2}$4 eV in NiF$UT_{1/2}$5, with indirect gaps of $UT_{1/2}$6, $UT_{1/2}$7, and $UT_{1/2}$8 eV, respectively (Samanta et al., 2024).

A distinct route is ferroelectricity-driven altermagnetism in FeCuP$UT_{1/2}$9S$\mathbf{k}$0. In monolayer and bilayer forms, the AFE–AFM state has a screw-axis-based spin space group and exhibits momentum-dependent spin splitting, whereas the FE–AFM state is spin-degenerate. The transition barriers reported for FE$\mathbf{k}$1P $\mathbf{k}$2 AFE $\mathbf{k}$3 FE$\mathbf{k}$4P and via a paraelectric state are

$\mathbf{k}$5

which makes electric-field switching a direct control knob for turning AM splitting on and off (Zhao et al., 1 Nov 2025). Interlayer sliding in the bilayer reverses the sign of the spin splitting without changing the Néel vector (Zhao et al., 1 Nov 2025).

Out-of-plane electric fields and substrate asymmetry provide another route in 2D. In MnP(S,Se)$\mathbf{k}$6, breaking $\mathbf{k}$7 while preserving spin-flipping mirrors converts a conventional AF into an AM with planar $\mathbf{k}$8-wave symmetry. For MnPSe$\mathbf{k}$9 under $d$0 eV/Å, the reported spin splitting reaches $d$1 meV (Mazin et al., 2023). In monolayer FeSe, inversion breaking from electric field or substrate asymmetry converts checkerboard AF order into a planar $d$2 altermagnet, with FeSe/STO calculations showing AM spin splittings up to $d$3 meV (Mazin et al., 2023).

Correlation-based high-throughput screening has also expanded the metallic AM set. A DFT+embedded-DMFT workflow applied to over 2,000 magnetic materials identified two previously unreported metallic AMs, CrSe and CaFe$d$4Al$d$5, in addition to CrSb and RuO$d$6, plus a dozen semiconducting AMs (Wan et al., 2024). The reported spectral spin-splitting metric $d$7 is $d$8 eV for CrSb, $d$9 eV for CrSe, $g$0 eV for CaFe$g$1Al$g$2, and $g$3 eV for RuO$g$4 (Wan et al., 2024).

5. Responses, probes, and device concepts

The combination of compensated magnetization and spin-split bands makes AMs relevant to transport, optics, superconductivity, and topological response. In spin-filter magnetic tunnel junctions using altermagnetic insulating CoF$g$5 and NiF$g$6 barriers, the predicted spin-filter TMR is about $g$7–$g$8 when the Fermi energy is tuned close to the valence-band maximum (Samanta et al., 2024). The mechanism is a spin- and momentum-dependent decay constant in the complex band structure, so that

$g$9

and

$i$0

acquires a pronounced $i$1-wave-like sign structure over the 2D Brillouin zone (Samanta et al., 2024).

The anomalous Hall effect has become a symmetry-sensitive diagnostic. For 2D altermagnets, symmetry analysis shows that only two of the seven nontrivial spin layer groups exhibit an unconventional in-plane AHE, and first-principles calculations on bilayer MnPSe$i$2 find peak anomalous Hall conductivities of approximately $i$3 S/cm in the $i$4-wave case and $i$5 S/cm in the $i$6-wave case, with linear-plus-cubic and purely cubic dependence on the Néel vector, respectively (Sheoran et al., 28 Feb 2025).

Optical detection is likewise symmetry-selective. A strain-mediated magneto-optical protocol predicts that uniaxial strain activates linear optical and Kerr signatures unique to AMs while preserving $i$7 symmetry in conventional AFMs. In monolayer V$i$8Se$i$9O, $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$00 tensile strain increases the Kerr response by several orders of magnitude, and in Janus Mn$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$01P$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$02S$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$03Se$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$04 the calculated Kerr angle reaches about $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$05 (Sun et al., 30 May 2025, Mazin et al., 2023). Ultrafast pumping introduces another probe: the linear polarization direction of a femtosecond pulse controls which spin species is preferentially photo-excited, allowing pump-polarization-controlled post-pump spin polarization in both a $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$06-wave AM model and a RuO$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$07 bilayer (Eskandari-asl et al., 2 Apr 2025).

Superconducting hybrid structures exploit the same momentum-selective spin physics. In a 90°-rotated AM–SC–AM junction, crossed Andreev reflection dominates nonlocal transport in the strong AM phase, and the conductances oscillate with superconducting length with a Fabry–Pérot-like period $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$08 (Das et al., 2024). In a four-terminal Josephson junction with Rashba SOC, a field-free transverse Josephson diode effect and transverse anomalous Josephson effect are predicted, with TJDE efficiency exceeding $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$09 in some weak-phase AM regimes and full tunability by rotating the Néel vector (Sahoo et al., 17 Sep 2025).

Periodic driving and dissipation further enlarge the response space. In 2D $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$10-wave AMs with Rashba SOC, monochromatic Floquet driving induces out-of-plane magnetization, longitudinal AMR, and AHE, while bichromatic $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$11–$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$12 driving also activates transverse AMR through induced in-plane effective fields (Yarmohammadi et al., 19 Jun 2026). In dissipative 2D AMs, an imaginary staggered exchange field generates non-Hermitian Chern transitions, exceptional points, and corner-selective hybrid skin–topological modes controlled by boundary sublattice termination (Zhao et al., 19 Mar 2026).

6. Identification, screening, and frontier extensions

Because altermagnetism is symmetry-governed, both direct classification and automated screening have become central. One practical outcome is an open-access code that checks whether a symmetry-compensated collinear magnetic material is antiferro- or altermagnetic directly from the crystal structure, without computing the electronic structure (Smolyanyuk et al., 2024). At scale, a DFT+eDMFT workflow combining pre-screening, symmetry analysis, and spectral calculations identified metallic CrSe and CaFe$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$13Al$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$14, together with CrSb and RuO$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$15, and concluded that while altermagnets are abundant among magnetic materials, only a tiny fraction is metallic (Wan et al., 2024).

Several recent directions indicate that AMs now function as a broader symmetry platform rather than a single class of band structures. In ferroelastic CoTe$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$16 monolayers, a universal magnetoelastic mechanism reverses the sign of nonrelativistic magnon spin conductivity across ferroelastic domains, leading to sign reversals of the spin Seebeck and spin Nernst conductivities without magnetic fields or Berry curvature (Cai et al., 20 Jun 2026). In 3D topological-insulator heterostructures, coupling to $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$17- and $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$18-type AM exchange fields yields hybrid-order and second-order topological phases with hinge modes whose localization and propagation direction are controlled by the relative exchange strengths (Subhadarshini et al., 3 Dec 2025). Extended $$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$19-wave altermagnets generalize the concept further to fully gapped, spin-compensated, valley-exchange-protected states with isotropic spin splitting and pair-density-wave descendants (Dürrnagel et al., 27 Aug 2025).

A persistent misconception is that zero net magnetization implies spin-degenerate bands. The accumulated literature directly contradicts that equivalence: compensated collinear order can coexist with large exchange-driven band splitting provided the symmetry relating opposite-spin sublattices is rotational, mirror-based, screw-like, or valley-exchanging rather than inversion- or translation-based (Samanta et al., 2024, Zhao et al., 1 Nov 2025, Wan et al., 2024). This symmetry distinction is the organizing principle behind the present understanding of altermagnets, their materials realization, and their emerging transport, optical, superconducting, topological, and magnonic phenomena.