Altermagnetic Insulators: Spin‐Split Magnets
- Altermagnetic insulators are magnets with compensated collinear order that exhibit momentum-dependent spin splitting due to non-inversion crystal symmetries.
- They encompass a diverse class including semiconducting, Mott, and higher-order topological insulators with distinct band gaps and spin textures.
- Research reveals that symmetry operations like rotations and mirrors enable novel spin transport, optical responses, and potential multiferroic functionality in these materials.
Altermagnetic insulators are insulating or semiconducting magnets with compensated collinear order, vanishing net magnetization, and momentum-dependent spin splitting permitted because opposite-spin sublattices are related by crystal rotations or mirrors rather than by inversion or translation. In current literature, the term spans ordinary semiconducting band insulators, correlation-driven Mott insulators, higher-order topological insulators, optically active multiferroics, tunnel-barrier materials, and disorder-localized states descended from metallic altermagnets. Representative systems include FeSO, FeSeO, FeO ( Cl, Br, I), LaOMnSe, rutile fluorides such as FeF0, MnF1, CoF2, and NiF3, LiFe4F5, and Janus chromium chalcogenides, while Ba6CoGe7O8 has been treated more cautiously as an insulating altermagnetic candidate (Wang et al., 5 Nov 2025, Wang et al., 14 Oct 2025, Wei et al., 2024, Samanta et al., 2024, Leiviskä et al., 31 Jan 2025, Ning et al., 10 Apr 2026).
1. Symmetry definition and distinction from ordinary antiferromagnets
The defining distinction is not compensation alone, but the symmetry operation connecting the two spin-opposed sublattices. In conventional collinear antiferromagnets, inversion- or translation-based antiunitary symmetries such as 9 or 0 enforce spin degeneracy throughout the Brillouin zone. In altermagnets, those protections are absent; the sublattices are instead connected by crystal rotations or mirrors, so zero net magnetization coexists with momentum-dependent spin splitting. This is stated explicitly for monolayer Fe1S2O and Fe3Se4O, where the spin-opposed Fe sublattices are connected by 5, and for La6O7Mn8Se9, where the opposite-spin Mn sublattices are related by a 0 rotation rather than by 1 or 2 (Wang et al., 5 Nov 2025, Wei et al., 2024).
This distinction is especially clear in materials families containing both conventional antiferromagnetic and altermagnetic members. In monolayer CrC3, CrSiS4, and the Janus compounds Cr5C6S7Se8 and Cr9Si0S1Se2, the non-Janus systems preserve 3 and remain spin degenerate, whereas the Janus systems break 4 and become altermagnetic because the compensated Néel sublattices are related by 5 rather than by inversion or translation (Ning et al., 10 Apr 2026). LiFe6F7 is presented in the same spirit: the ground-state A8 phase lacks 9 and 0, while opposite-spin Fe sites are related by 1, producing a 2-wave altermagnetic pattern in momentum space (Guo et al., 2023).
The literature also distinguishes several symmetry form factors. La3O4Mn5Se6 is described as a layered 7-wave altermagnet with nodal planes 8 and splitting maximal along the principal axes (Wei et al., 2024). Wurtzite MnTe and BiFeO9 are treated as insulating 0-wave altermagnets in nonlinear optical theory (Dong et al., 2024). Model Hamiltonians extend the same logic to 1-protected 2-wave and 3-protected 4-wave topological insulating phases (Gonzalez-Hernandez et al., 31 Jul 2025).
A recurrent misconception is that a compensated collinear insulator must be spin degenerate. The materials above are invoked precisely to show the opposite: compensation does not preclude spin splitting when the sublattice relation is rotational or mirror-like rather than inversion- or translation-like (Wang et al., 5 Nov 2025, Wei et al., 2024, Ning et al., 10 Apr 2026).
2. Material landscape and insulating regimes
Current proposals cover several distinct insulating regimes, from wide-gap rutile fluorides to narrow-gap ferroelectric altermagnets and correlated layered oxychalcogenides. The systems below illustrate the range of electronic structures already discussed in the literature.
| System or family | Insulating character | Distinctive insulating property |
|---|---|---|
| Fe5S6O, Fe7Se8O | Indirect-gap semiconductors, 9 eV and 0 eV | Mirror real Chern insulators |
| Fe1Cl2O, Fe3Br4O, Fe5I6O | PBE+7 gaps 8, 9, 0 eV | Spin-polarized valleys and corner modes |
| La1O2Mn3Se4 | Gap larger than 5 eV | Correlated layered 6-wave altermagnet |
| FeF7, MnF8, CoF9, NiF0 | Gaps 1 eV and 2–3 eV | Altermagnetic tunnel-barrier and magnonic platforms |
| Cr4C5S6Se7, Cr8Si9S00Se01 | Gaps 02 eV and 03 eV | Altermagnetic HOTIs with fractional corner charge |
| LiFe04F05 | Gaps 06 meV and 07 meV in A08 phases | Charge-order-mediated ferroelectric altermagnet |
These systems also span different structural motifs. Fe09S10O and Fe11Se12O are proposed as freestanding two-dimensional monolayers with a three-layer architecture, a tetragonal 13 lattice, optimized in-plane constants 14 Å and 15 Å, and no imaginary phonon frequencies (Wang et al., 5 Nov 2025). The related Fe16O monolayers are centrosymmetric 17 structures with 18 Å, 19 Å, and 20 Å for 21, large cohesive energies, no imaginary phonon modes, and ab initio molecular dynamics stability at 22 K (Wang et al., 14 Oct 2025). The Janus chromium chalcogenides derive from the CrSiTe23/CrGeTe24 motif and remain insulating after the Janus S/Se substitution that converts conventional antiferromagnets into altermagnets (Ning et al., 10 Apr 2026).
Correlated layered oxyselenides occupy a different part of this landscape. La25O26Mn27Se28 crystallizes in tetragonal 29, orders below 30 K, and is described as a correlated insulator of likely Mott origin with a 31-wave altermagnetic splitting reaching nearly 32 eV along 33-X(Y) (Wei et al., 2024). LiFe34F35 is narrower-gap and simultaneously ferroelectric: the high-symmetry A36 phase has a 37 meV gap, while the low-symmetry charge-ordered A38 phase has a 39 meV gap (Guo et al., 2023).
Rutile fluorides provide the most developed barrier-oriented examples. FeF40 is treated as an altermagnetic insulator with a 41 eV gap, zero net magnetization, and complex-band spin splitting that depends on transport direction (Chi et al., 2024). MnF42, CoF43, and NiF44 are studied as rutile altermagnetic insulators with calculated indirect gaps 45, 46, and 47 eV, respectively (Samanta et al., 2024). By contrast, Ba48CoGe49O50 is introduced explicitly as an “insulating altermagnetic candidate,” not as a definitively established altermagnet; its importance lies in interfacial spin Hall magnetoresistance rather than in direct band-structure confirmation (Leiviskä et al., 31 Jan 2025).
3. Correlated and Mott-insulating altermagnets
The strong-correlation regime changes the meaning of altermagnetic spectroscopy. In the square-lattice Hubbard model with nearest-neighbor hopping 51 and sublattice-dependent diagonal hopping 52, half filling and 53 produce a Mott insulator with Néel order, zero net magnetization, and altermagnetic momentum dependence. After a Schrieffer–Wolff transformation, the low-energy description becomes a projected 54-55 model, and the photoemission problem is no longer a one-electron band problem but the dynamics of a doped hole in an ordered antiferromagnet (Lanzini et al., 3 Jun 2025).
In that setting, the low-energy ARPES feature is a magnetic polaron interpreted as a spinon–holon bound state. The paper gives the characteristic confinement scaling
56
and derives a spin-dependent quasiparticle dispersion in which the altermagnetic part enters through 57 for spin-58 and 59 for spin-60. Tensor-network calculations confirm a strongly renormalized bandwidth, a spin-split coherent branch, and spin-dependent spectral-weight transfer. Hartree–Fock is reported to overestimate both the bandwidth and the spin splitting, so in correlated altermagnetic insulators the observable splitting is a renormalized many-body property rather than a direct transcription of a noninteracting band structure (Lanzini et al., 3 Jun 2025).
La61O62Mn63Se64 supplies an experimentally characterized correlated material counterpart. Its G-type order has propagation vector 65, the ordered moment points predominantly along 66, and the compound shows an insulating gap larger than 67 eV together with nearly 68 eV nonrelativistic spin splitting. Magnetic pair-distribution-function analysis further indicates a two-component magnetic state: a 3D long-range G-type component plus a short-range purely 2D intraplane antiferromagnetic component, with finite-range magnetic correlations persisting to at least 69 K (Wei et al., 2024).
The Mott-insulating theme also extends to bilayers. A later theory considers a two-dimensional bilayer Mott insulator with altermagnetic order, tunable layer polarization, and a counter-flow electric field that induces a polarization current and thereby drives a spin current in each layer. The polarization current is isotropic, but the spin current is strongly anisotropic and can be reversed by adjusting the photon energy (Sicheler et al., 9 Aug 2025). This suggests that insulating altermagnets need not be electronically passive even when charge motion is frozen at low energy.
4. Topological insulating phases and boundary states
A major recent development is the emergence of intrinsically altermagnetic topological insulators. In monolayer Fe70S71O and Fe72Se73O, each spin channel without SOC is treated as an effective spinless system with inversion 74, mirror 75, and time-reversal symmetry. The real Chern number is diagnosed through the inversion-parity formula
76
with the four TRIM 77, X, Y, and M. If mirror symmetry is ignored, each spin channel appears trivial. After mirror-sector decomposition,
78
the nontrivial sector is 79 with 80, while 81 is trivial, so each spin channel carries total 82 (Wang et al., 5 Nov 2025).
The boundary manifestation is higher-order. Square nanodisks built from Wannier-derived models exhibit four in-gap zero modes, two from spin up and two from spin down. Their charge density is sharply localized at the four corners, and the spin texture is spatially separated: the corners along the 83 axis host spin-up modes, whereas the corners along the 84 axis host spin-down modes. The paper names this a spin-corner coupling effect. The phase is reported to remain robust against SOC and against 85 uniaxial or biaxial compression, with uniaxial strain lifting the X/Y valley equivalence and splitting corner-state energies in a spin-selective manner (Wang et al., 5 Nov 2025).
Monolayer Fe86O (87 Cl, Br, I) generalizes the same motif. These systems are semiconducting altermagnetic insulators with 88, spin-polarized corner modes, and spin-polarized valleys at X and Y. In Fe89Br90O, for example, spin-up zero modes reside at left and right corners while spin-down zero modes appear at top and bottom corners; in Fe91Cl92O, SOC couples altermagnetism to ferroelasticity, giving an in-plane easy axis and a rectangular distortion with 93 Å (Wang et al., 14 Oct 2025).
A different crystalline topology appears in Janus chromium chalcogenides. For Cr94C95S96Se97 and Cr98Si99S00Se01, the 02-symmetric higher-order topology is characterized by
03
with 04 and therefore 05 for each spin channel. Triangular nanodisks show three degenerate in-gap corner states per spin without SOC and six in-gap corner states with SOC, consistent with a HOTI phase that survives weak spin-orbit coupling (Ning et al., 10 Apr 2026).
The model-Hamiltonian literature places these phases in a broader class of altermagnetic topological insulators protected by 06 or 07. In two dimensions, the relevant invariant is the spin Chern number,
08
while in three dimensions the topology is characterized by the spin Chern numbers on the 09 and 10 planes. Depending on parameters, the resulting phases support edges, corners, surfaces, and hinges (Gonzalez-Hernandez et al., 31 Jul 2025).
Not every topological altermagnetic proposal is fully bulk insulating. Monolayer Cr11BAl is presented as a 2D metallic 12-wave altermagnet that, with SOC and 13, develops local gaps of about 14–15 meV at symmetry-relevant crossings together with termination-dependent Dirac edge states. The same work does not establish a clean global bulk gap, so it is more precise to regard Cr16BAl as a metallic altermagnet in a proximate topological crystalline regime rather than as a confirmed altermagnetic insulator (Sattigeri et al., 12 Jun 2025).
5. Spin transport, optical response, and interfacial probes
One of the most concrete device roles for altermagnetic insulators is as tunnel barriers. In the rutile fluorides 17, transport inside the gap is controlled by spin-dependent complex bands, with longitudinal wave vector 18 and transmission scaling as 19. The momentum-resolved spin-filtering factor is defined as
20
and the corresponding local spin polarization is
21
Using a double-barrier spin-filter model, CoF22 and NiF23 are predicted to yield spin-filter TMR values of about 24–25 when the Fermi level is tuned close to the valence-band maximum (Samanta et al., 2024).
FeF26 extends the same idea through full complex-band analysis. The key result is anisotropic spin filtering: along FeF27[001], the current remains globally spin neutral but shows locally nonvanishing momentum-space spin polarization, whereas along FeF28[110] the current becomes globally spin polarized because 29 and 30 are no longer symmetry matched over the two-dimensional Brillouin zone. Prototype junctions RuO31(001)/FeF32/IrO33 and CrO34(110)/FeF35/IrO36 then produce TMR ratios of 37 and 38, respectively (Chi et al., 2024).
Insulating altermagnets also support magnonic Hall responses. A rutile-inspired spin model with Dzyaloshinskii–Moriya interaction predicts a spontaneous zero-field magnon thermal Hall effect in a collinear compensated altermagnet, together with a spin Nernst response. The thermal Hall conductivity is controlled by magnon Berry curvature and depends on Néel-vector orientation and on strain, with MnF39, CoF40, and NiF41 identified as candidate materials (Hoyer et al., 2024).
In nonlinear optics, crystal symmetry can separate spin and charge photocurrents by direction. For linearly or circularly polarized light, the shift and inject current tensors are
42
with spin-current response obtained by replacing the velocity operator with the anticommutator form involving 43. In wurtzite MnTe and BiFeO44, the relevant spin point group or magnetic point group routes spin and charge currents into different axes, so specific crystallographic directions carry a pure spin current even with SOC. The same work identifies a previously overlooked SOC-enabled linear-inject-current channel in BiFeO45 that may contribute to its bulk photovoltaic response (Dong et al., 2024).
Optical signatures also appear in the topological monolayers. Fe46S47O and Fe48Se49O exhibit pronounced linear dichroism and strong optical absorption, while Fe50Br51O shows valley linear dichroism with 52 close to 53 near X and close to 54 near Y, so 55-polarized light predominantly excites one valley and 56-polarized light the other (Wang et al., 5 Nov 2025, Wang et al., 14 Oct 2025).
Interfacial probes provide a complementary route. Pt on Ba57CoGe58O59 exhibits spin Hall magnetoresistance of order 60, specifically around 61 for some devices at 62 K and 63 T, with an approximately 64 anisotropy between current along 65 and 66. The authors treat this as evidence consistent with anisotropic altermagnetic ordering affecting interfacial spin transport, while explicitly stopping short of claiming definitive proof of altermagnetism from SMR alone (Leiviskä et al., 31 Jan 2025).
A more microscopic probe is proposed through NV-center relaxometry. In altermagnetic insulators, the longitudinal spin susceptibility acquires a directional diffusion term proportional to 67, and the distance-dependent magnetic-noise contrast measured by the impurity becomes a local signature distinguishing altermagnets from conventional antiferromagnets. The predicted contrast is distance dependent in the altermagnetic case but distance independent in the conventional antiferromagnetic case (Bittencourt et al., 6 Aug 2025).
6. Tunability, multiferroicity, disorder, and unresolved issues
The field places unusual emphasis on tunability. In Fe68S69O and Fe70Se71O, SOC changes magnetic anisotropy but does not close the gap; the reported easy axes are [010] for Fe72S73O with 74eV and [001] for Fe75Se76O with 77eV. Under 78 uniaxial or biaxial compression, the corner states remain intact, while uniaxial strain breaks 79 and produces valley polarization together with spin-selective corner-state splitting (Wang et al., 5 Nov 2025). In Fe80Cl81O, the strain response is even stronger: tensile strain along 82 favors AM83, tensile strain along 84 favors AM85, and the Néel vector can therefore be switched by 86 through ferroelastic coupling (Wang et al., 14 Oct 2025).
LiFe87F88 adds multiferroicity in a different way. Its A89 phase is presented as a 90-wave altermagnetic and charge-ordering-mediated ferroelectric material with Berry-phase polarization 91 and ferroelectric switching barrier 92 meV per Fe. Under biaxial compressive strain it transforms into a ferrimagnetic ferroelectric phase with 93, barrier 94 meV per Fe, and primitive-cell magnetization switching from 95 to 96 under polarization reversal. The same strained phase is proposed as a platform for spin-triplet excitonic-insulator physics because the top valence and bottom conduction bands have opposite spin character (Guo et al., 2023).
A broader design framework appears in the theory of spin-driven multiferroics derived from altermagnets. There, the directly polar phase is often a Kramers-degenerate collinear AFM insulator obtained from an altermagnetic parent through exchange striction. In the LiMnO97 family and strained RuF98-family compounds, the authors report polarization values exceeding 99, with values up to 00 in the fluorides, and magnetoelectric coupling one to two orders of magnitude stronger than in conventional SOC-driven multiferroics. At the same time, the polar phase is often not the intrinsic altermagnetic ground state, and stabilization can require strains above 01 or even 02 depending on the family (Cao et al., 2024).
Disorder introduces a separate route to insulating behavior. In a two-dimensional disordered altermagnet, numerical work finds a Kosterlitz–Thouless-type transition from an altermagnetic marginal metal to an insulator. For 03 and 04, the reported critical disorder is 05; for 06 and 07, it moves to 08. Across the transition, the characteristic spin anisotropy of the altermagnet persists but gradually fades away, so the disorder-driven insulator is not a topological altermagnetic insulator but a localized phase descended from an altermagnetic metal (Li et al., 14 Jul 2025).
The literature also retains several unresolved points. For Fe09S10O and Fe11Se12O, the topological diagnosis in the main text is symmetry-indicator based and supported by corner-state numerics, but no Wilson-loop analysis or low-energy analytic model is provided, and the SOC discussion relies on adiabatic continuity rather than on a direct invariant in the SOC-on magnetic space group (Wang et al., 5 Nov 2025). For Ba13CoGe14O15, the anisotropic SMR is presented as an “initial step” requiring quantitative theory and further experiments before it can be assigned unambiguously to an altermagnetic phase (Leiviskä et al., 31 Jan 2025). For Cr16BAl, the topological crystalline interpretation rests on SOC-opened local gaps and edge-state calculations without a demonstrated full bulk insulating gap (Sattigeri et al., 12 Jun 2025). For La17O18Mn19Se20, the case for altermagnetism is strong at the level of symmetry analysis plus DFT, but direct momentum-resolved experimental detection of the predicted spin splitting has not yet been reported (Wei et al., 2024).
Taken together, these works define altermagnetic insulators as a broad and internally diverse class rather than a single materials category. They include semiconducting band insulators, correlated Mott insulators, higher-order topological insulators, nonlinear-optical spin-current materials, multiferroic descendants of altermagnetic parents, and disorder-localized states that retain some memory of altermagnetic symmetry. What unifies them is the coexistence of compensated collinear order with crystal-symmetry-selected spin anisotropy, and the central research program is now to determine how robustly that symmetry survives in realistic synthesis, spectroscopy, interfaces, and finite-temperature operation.