---
title: 'Altermagnetic Insulators: Spin‐Split Magnets'
url: https://www.emergentmind.com/topics/altermagnetic-insulators
type: topic
---

# Altermagnetic Insulators: Spin‐Split Magnets

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 Fe$_2$S$_2$O, Fe$_2$Se$_2$O, Fe$_2X_2$O ($X=$ Cl, Br, I), La$_2$O$_3$Mn$_2$Se$_2$, rutile fluorides such as FeF$_2$, MnF$_2$, CoF$_2$, and NiF$_2$, LiFe$_2$F$_6$, and Janus chromium chalcogenides, while Ba$_2$CoGe$_2$O$_7$ has been treated more cautiously as an insulating altermagnetic candidate [2511.03602][2510.12748][2410.14542][2409.00195][2501.19198][2604.08907].

## 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 $\mathcal{PT}$ or $t\mathcal{T}$ 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 Fe$_2$S$_2$O and Fe$_2$Se$_2$O, where the spin-opposed Fe sublattices are connected by $\mathcal{C}_{4z}\mathcal{T}$, and for La$_2$O$_3$Mn$_2$Se$_2$, where the opposite-spin Mn sublattices are related by a $90^\circ$ rotation rather than by $\mathcal{PT}$ or $t\mathcal{T}$ [2511.03602][2410.14542].

This distinction is especially clear in materials families containing both conventional antiferromagnetic and altermagnetic members. In monolayer CrC$X_3$, CrSiS$_3$, and the Janus compounds Cr$_2$C$_2$S$_3$Se$_3$ and Cr$_2$Si$_2$S$_3$Se$_3$, the non-Janus systems preserve $\mathcal{PT}$ and remain spin degenerate, whereas the Janus systems break $\mathcal{PT}$ and become altermagnetic because the compensated Néel sublattices are related by $\mathcal{M}_{\bar{1}10}$ rather than by inversion or translation [2604.08907]. LiFe$_2$F$_6$ is presented in the same spirit: the ground-state A$+$ phase lacks $\{C_2^\perp\|I\}$ and $\{C_2^\perp\|t\}$, while opposite-spin Fe sites are related by $\{C_2^\perp\|C_{4z}t\}$, producing a $d$-wave altermagnetic pattern in momentum space [2312.13911].

The literature also distinguishes several symmetry form factors. La$_2$O$_3$Mn$_2$Se$_2$ is described as a layered $d_{x^2-y^2}$-wave altermagnet with nodal planes $k_x=\pm k_y$ and splitting maximal along the principal axes [2410.14542]. Wurtzite MnTe and BiFeO$_3$ are treated as insulating $g$-wave altermagnets in nonlinear optical theory [2412.09216]. Model Hamiltonians extend the same logic to $\mathcal{C}_{4z}\mathbb{T}$-protected $d$-wave and $\mathcal{C}_{3z}\mathbb{T}$-protected $f$-wave topological insulating phases [2507.23173].

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 [2511.03602][2410.14542][2604.08907].

## 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 |
|---|---|---|
| Fe$_2$S$_2$O, Fe$_2$Se$_2$O | Indirect-gap semiconductors, $0.72$ eV and $0.26$ eV | Mirror real Chern insulators |
| Fe$_2$Cl$_2$O, Fe$_2$Br$_2$O, Fe$_2$I$_2$O | PBE+$U$ gaps $2.924$, $2.765$, $2.386$ eV | Spin-polarized valleys and corner modes |
| La$_2$O$_3$Mn$_2$Se$_2$ | Gap larger than $1.5$ eV | Correlated layered $d_{x^2-y^2}$-wave altermagnet |
| FeF$_2$, MnF$_2$, CoF$_2$, NiF$_2$ | Gaps $3.5$ eV and $4.14$–$4.36$ eV | Altermagnetic tunnel-barrier and magnonic platforms |
| Cr$_2$C$_2$S$_3$Se$_3$, Cr$_2$Si$_2$S$_3$Se$_3$ | Gaps $1.879$ eV and $1.293$ eV | Altermagnetic HOTIs with fractional corner charge |
| LiFe$_2$F$_6$ | Gaps $311$ meV and $474$ meV in A$+$ phases | Charge-order-mediated ferroelectric altermagnet |

These systems also span different structural motifs. Fe$_2$S$_2$O and Fe$_2$Se$_2$O are proposed as freestanding two-dimensional monolayers with a three-layer architecture, a tetragonal $P4/mmm$ lattice, optimized in-plane constants $a=b=4.00$ Å and $4.05$ Å, and no imaginary phonon frequencies [2511.03602]. The related Fe$_2X_2$O monolayers are centrosymmetric $P4/mmm$ structures with $a=4.035$ Å, $4.067$ Å, and $4.123$ Å for $X=\mathrm{Cl,Br,I}$, large cohesive energies, no imaginary phonon modes, and ab initio molecular dynamics stability at $300$ K [2510.12748]. The Janus chromium chalcogenides derive from the CrSiTe$_3$/CrGeTe$_3$ motif and remain insulating after the Janus S/Se substitution that converts conventional antiferromagnets into altermagnets [2604.08907].

Correlated layered oxyselenides occupy a different part of this landscape. La$_2$O$_3$Mn$_2$Se$_2$ crystallizes in tetragonal $I4/mmm$, orders below $T_N\approx166$ K, and is described as a correlated insulator of likely Mott origin with a $d_{x^2-y^2}$-wave altermagnetic splitting reaching nearly $0.7$ eV along $\Gamma$-X(Y) [2410.14542]. LiFe$_2$F$_6$ is narrower-gap and simultaneously ferroelectric: the high-symmetry A$+$ phase has a $311$ meV gap, while the low-symmetry charge-ordered A$+$ phase has a $474$ meV gap [2312.13911].

Rutile fluorides provide the most developed barrier-oriented examples. FeF$_2$ is treated as an altermagnetic insulator with a $3.5$ eV gap, zero net magnetization, and complex-band spin splitting that depends on transport direction [2409.03415]. MnF$_2$, CoF$_2$, and NiF$_2$ are studied as rutile altermagnetic insulators with calculated indirect gaps $4.14$, $4.36$, and $4.30$ eV, respectively [2409.00195]. By contrast, Ba$_2$CoGe$_2$O$_7$ 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 [2501.19198].

## 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 $t$ and sublattice-dependent diagonal hopping $t_+$, half filling and $U\gg t$ 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 $t$-$J$ model, and the photoemission problem is no longer a one-electron band problem but the dynamics of a doped hole in an ordered antiferromagnet [2506.03263].

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
\[
E_{\rm FSA}\sim (J/t)^{2/3},
\]
and derives a spin-dependent quasiparticle dispersion in which the altermagnetic part enters through $\cos k_1$ for spin-$\uparrow$ and $\cos k_2$ for spin-$\downarrow$. 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 [2506.03263].

La$_2$O$_3$Mn$_2$Se$_2$ supplies an experimentally characterized correlated material counterpart. Its G-type order has propagation vector $\mathbf{k}=(0,0,0)$, the ordered moment points predominantly along $c$, and the compound shows an insulating gap larger than $1.5$ eV together with nearly $0.7$ 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 $400$ K [2410.14542].

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 [2508.06938]. 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 Fe$_2$S$_2$O and Fe$_2$Se$_2$O, each spin channel without SOC is treated as an effective spinless system with inversion $\mathcal{P}$, mirror $\mathcal{M}_z$, and time-reversal symmetry. The real Chern number is diagnosed through the inversion-parity formula
\[
(-1)^{\nu_R} = \prod_i (-1)^{\lfloor n_-^{(\Gamma_i)} / 2 \rfloor},
\]
with the four TRIM $\Gamma$, X, Y, and M. If mirror symmetry is ignored, each spin channel appears trivial. After mirror-sector decomposition,
\[
\mathcal{H}_\sigma=\mathcal{H}_\sigma^+\oplus \mathcal{H}_\sigma^-,
\]
the nontrivial sector is $M_z=+1$ with $\nu_R^+=1$, while $M_z=-1$ is trivial, so each spin channel carries total $\nu_R=1$ [2511.03602].

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 $x$ axis host spin-up modes, whereas the corners along the $y$ 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 $-2\%$ uniaxial or biaxial compression, with uniaxial strain lifting the X/Y valley equivalence and splitting corner-state energies in a spin-selective manner [2511.03602].

Monolayer Fe$_2X_2$O ($X=$ Cl, Br, I) generalizes the same motif. These systems are semiconducting altermagnetic insulators with $\nu_R^\uparrow=\nu_R^\downarrow=1$, spin-polarized corner modes, and spin-polarized valleys at X and Y. In Fe$_2$Br$_2$O, for example, spin-up zero modes reside at left and right corners while spin-down zero modes appear at top and bottom corners; in Fe$_2$Cl$_2$O, SOC couples altermagnetism to ferroelasticity, giving an in-plane easy axis and a rectangular distortion with $a-b\sim0.1$ Å [2510.12748].

A different crystalline topology appears in Janus chromium chalcogenides. For Cr$_2$C$_2$S$_3$Se$_3$ and Cr$_2$Si$_2$S$_3$Se$_3$, the $C_3$-symmetric higher-order topology is characterized by
\[
Q_s^{(3)} = -\frac{|e|}{3}[K_1^{(3)}]\pmod{e},
\]
with $[K_1^{(3)}]=-2$ and therefore $Q_s^{(3)}=e/3$ 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 [2604.08907].

The model-Hamiltonian literature places these phases in a broader class of altermagnetic topological insulators protected by $\mathcal{C}_{4z}\mathbb{T}$ or $\mathcal{C}_{3z}\mathbb{T}$. In two dimensions, the relevant invariant is the spin Chern number,
\[
\mathrm{SCN}=\frac{1}{2}(c_1^\uparrow-c_1^\downarrow),
\]
while in three dimensions the topology is characterized by the spin Chern numbers on the $k_z=0$ and $k_z=\pi$ planes. Depending on parameters, the resulting phases support edges, corners, surfaces, and hinges [2507.23173].

Not every topological altermagnetic proposal is fully bulk insulating. Monolayer Cr$_2$BAl is presented as a 2D metallic $d_{x^2-y^2}$-wave altermagnet that, with SOC and $\mathbf{N}\parallel[001]$, develops local gaps of about $10$–$26$ 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 Cr$_2$BAl as a metallic altermagnet in a proximate topological crystalline regime rather than as a confirmed altermagnetic insulator [2506.10782].

## 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 $MF_2$, transport inside the gap is controlled by spin-dependent complex bands, with longitudinal wave vector $k_z=q+i\kappa$ and transmission scaling as $e^{-2\kappa d}$. The momentum-resolved spin-filtering factor is defined as
\[
f(\mathbf{k}_{\parallel})=\kappa_{\downarrow}(\mathbf{k}_{\parallel})-\kappa_{\uparrow}(\mathbf{k}_{\parallel}),
\]
and the corresponding local spin polarization is
\[
p(\mathbf{k}_{\parallel})=\tanh[f(\mathbf{k}_{\parallel})d].
\]
Using a double-barrier spin-filter model, CoF$_2$ and NiF$_2$ are predicted to yield spin-filter TMR values of about $150$–$170\%$ when the Fermi level is tuned close to the valence-band maximum [2409.00195].

FeF$_2$ extends the same idea through full complex-band analysis. The key result is anisotropic spin filtering: along FeF$_2$[001], the current remains globally spin neutral but shows locally nonvanishing momentum-space spin polarization, whereas along FeF$_2$[110] the current becomes globally spin polarized because $\kappa_\uparrow(\mathbf{k}_{\parallel})$ and $\kappa_\downarrow(\mathbf{k}_{\parallel})$ are no longer symmetry matched over the two-dimensional Brillouin zone. Prototype junctions RuO$_2$(001)/FeF$_2$/IrO$_2$ and CrO$_2$(110)/FeF$_2$/IrO$_2$ then produce TMR ratios of $216\%$ and $3956\%$, respectively [2409.03415].

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 MnF$_2$, CoF$_2$, and NiF$_2$ identified as candidate materials [2405.05090].

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
\[
J_c = \sigma_{abc} E_a(\omega)E_b(-\omega), \qquad
J_c = \eta_{abc} E_a(\omega)E_b(-\omega),
\]
with spin-current response obtained by replacing the velocity operator with the anticommutator form involving $\sigma^\alpha$. In wurtzite MnTe and BiFeO$_3$, 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 BiFeO$_3$ that may contribute to its bulk photovoltaic response [2412.09216].

Optical signatures also appear in the topological monolayers. Fe$_2$S$_2$O and Fe$_2$Se$_2$O exhibit pronounced linear dichroism and strong optical absorption, while Fe$_2$Br$_2$O shows valley linear dichroism with $\eta(\mathbf{k})$ close to $-1$ near X and close to $+1$ near Y, so $y$-polarized light predominantly excites one valley and $x$-polarized light the other [2511.03602][2510.12748].

Interfacial probes provide a complementary route. Pt on Ba$_2$CoGe$_2$O$_7$ exhibits spin Hall magnetoresistance of order $2\times10^{-4}$, specifically around $2.5\times10^{-4}$ for some devices at $2$ K and $1.9$ T, with an approximately $2:1$ anisotropy between current along $\langle100\rangle$ and $\langle110\rangle$. 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 [2501.19198].

A more microscopic probe is proposed through NV-center relaxometry. In altermagnetic insulators, the longitudinal spin susceptibility acquires a directional diffusion term proportional to $D_2^2k^4\cos^2(2\phi_k)$, 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 [2508.04788].

## 6. Tunability, multiferroicity, disorder, and unresolved issues

The field places unusual emphasis on tunability. In Fe$_2$S$_2$O and Fe$_2$Se$_2$O, SOC changes magnetic anisotropy but does not close the gap; the reported easy axes are [010] for Fe$_2$S$_2$O with $\Delta E_{\mathrm{001-010}}=798\,\mu$eV and [001] for Fe$_2$Se$_2$O with $\Delta E_{\mathrm{001-010}}=-777\,\mu$eV. Under $-2\%$ uniaxial or biaxial compression, the corner states remain intact, while uniaxial strain breaks $\mathcal{C}_{4z}\mathcal{T}$ and produces valley polarization together with spin-selective corner-state splitting [2511.03602]. In Fe$_2$Cl$_2$O, the strain response is even stronger: tensile strain along $x$ favors AM$_y$, tensile strain along $y$ favors AM$_x$, and the Néel vector can therefore be switched by $90^\circ$ through ferroelastic coupling [2510.12748].

LiFe$_2$F$_6$ adds multiferroicity in a different way. Its A$+$ phase is presented as a $d$-wave altermagnetic and charge-ordering-mediated ferroelectric material with Berry-phase polarization $P=15.1\,\mu\mathrm{C/cm}^2$ and ferroelectric switching barrier $9.9$ meV per Fe. Under biaxial compressive strain it transforms into a ferrimagnetic ferroelectric phase with $P=13.4\,\mu\mathrm{C/cm}^2$, barrier $41.5$ meV per Fe, and primitive-cell magnetization switching from $-2\,\mu_B$ to $+2\,\mu_B$ 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 [2312.13911].

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 LiMnO$_2$ family and strained RuF$_4$-family compounds, the authors report polarization values exceeding $1.0\,\mu\mathrm{C/cm}^2$, with values up to $1.8\,\mu\mathrm{C/cm}^2$ 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 $4\%$ or even $6\%$ depending on the family [2412.20347].

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 $t_J=0.3t$ and $\mu=-2t$, the reported critical disorder is $W_c=4.13t\pm0.08t$; for $t_J=2.0t$ and $\mu=-2t$, it moves to $W_c=11.27t\pm0.33t$. 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 [2507.10762].

The literature also retains several unresolved points. For Fe$_2$S$_2$O and Fe$_2$Se$_2$O, 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 [2511.03602]. For Ba$_2$CoGe$_2$O$_7$, 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 [2501.19198]. For Cr$_2$BAl, the topological crystalline interpretation rests on SOC-opened local gaps and edge-state calculations without a demonstrated full bulk insulating gap [2506.10782]. For La$_2$O$_3$Mn$_2$Se$_2$, 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 [2410.14542].

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.

Source: https://www.emergentmind.com/topics/altermagnetic-insulators