---
title: 'Altermagnets: Symmetry-Driven Spin Splitting'
url: https://www.emergentmind.com/topics/altermagnets-ams
type: topic
---

# Altermagnets: Symmetry-Driven Spin Splitting

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 [2401.08784][2409.00195][2412.10356].

## 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 [2603.19378][2606.21281]. This combination gives AMs some properties associated with ferromagnets, such as spin-split bands and anomalous transport, while retaining the compensated moment of antiferromagnets [2401.08784].

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 [2409.00195]. 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 [2409.00195][2309.02355][2502.21095].

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 [2606.21281][2509.14109].

## 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 \(\mathbf{k}\), altermagnetism becomes symmetry-allowed [2409.00195][2401.08784].

In spin-space-group language, a collinear AM can be written in the form
\[
G^{(s)} = H + \{\,M_z || \mathcal{R T}|\boldsymbol{\tau}\,\}H,
\]
with the crucial distinction that \(\mathcal{R}\neq \mathcal{P}\). When \(\mathcal{R}=\mathcal{P}\), the system has \(PT\) symmetry and remains spin-degenerate; when \(\mathcal{R}\) is a twofold rotation or screw axis instead, global spin degeneracy is not enforced [2511.00712]. In FeCuP\(_2\)S\(_6\), for example, the operations \(\{m_z||C_{2y}|0\,\tfrac{1}{2}\,0\}\) or \(\{m_z||C_{2y}\}\) are identified as AM-enabling, whereas pure translation or inversion lead back to ordinary AFM behavior [2511.00712].

The same logic appears in magnetic-space-group screening. For collinear magnets one may decompose the symmetry as
\[
G = H + A H,
\]
where \(H\) preserves spin orientations and \(AH\) flips them. If the spin-flipping coset contains \(PT\) or \(tT\), the material is a conventional AFM; otherwise it is an altermagnet candidate [2412.10356]. 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 [2502.21095].

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 [2401.08784].

## 3. Momentum-space structure and model descriptions

A common minimal form for the non-relativistic electronic structure is
\[
H(\mathbf{k}) = H_0(\mathbf{k}) + \mathbf{B}(\mathbf{k})\cdot \boldsymbol{\sigma},
\]
where \(\mathbf{B}(\mathbf{k})\) is an effective exchange field that depends on momentum and changes sign across the Brillouin zone [2409.00195]. 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
\[
H_{\mathbf{k}} = -2 t_0(\cos k_x a + \cos k_y a)\,\sigma_0 + 2 t_J (\cos k_x a - \cos k_y a)\,\sigma_z - \mu,
\]
which yields
\[
E_{\mathbf{k},\uparrow} - E_{\mathbf{k},\downarrow}
= 4 t_J (\cos k_x a - \cos k_y a).
\]
The splitting therefore vanishes on \(k_x=\pm k_y\) and changes sign under the interchange of \(k_x\) and \(k_y\), making explicit the \(d_{x^2-y^2}\)-type structure [2402.08263].

For 2D \(d\)-wave AMs with Rashba SOC, a continuum description used for Floquet transport is
\[
H_d(\mathbf{k})= \varepsilon k^2\sigma_0 +\lambda(k_x\sigma_y-k_y\sigma_x) + \varepsilon M_{\mathbf{k}}^{d}\,\sigma_z,
\]
with
\[
M_{\mathbf{k}}^{d} = M_{1g}(k_x^2-k_y^2) +2M_{2g} k_xk_y.
\]
This parametrizes the two standard tetragonal \(d\)-wave harmonics, \(B_{1g}\) and \(B_{2g}\), and makes the orientation of the altermagnetic order explicit [2606.21281].

Recent work also extends the taxonomy beyond nodal \(d\)-, \(g\)-, and \(i\)-wave forms. “Extended \(s\)-wave altermagnets” are defined by a staggered valley-spin order
\[
\Delta^{\text{sAM}}\,\tau^z\sigma^z,
\]
leading to
\[
H_{\text{sAM}} = H_c + \Delta^{\text{sAM}}\,\tau^z \sigma^z.
\]
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 [2508.20163].

## 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 \(MF_2\) (\(M=\) Mn, Co, Ni), the magnetic space group lacks \(PT\) and \(UT_{1/2}\), 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 \(\Gamma\)–M is reported as \(\approx 0.03\) eV in MnF\(_2\), \(\approx 0.44\) eV in CoF\(_2\), and \(\approx 0.21\) eV in NiF\(_2\), with indirect gaps of \(\sim 4.14\), \(\sim 4.36\), and \(\sim 4.30\) eV, respectively [2409.00195].

A distinct route is ferroelectricity-driven altermagnetism in FeCuP\(_2\)S\(_6\). 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\(+\)P \(\rightarrow\) AFE \(\rightarrow\) FE\(-\)P and via a paraelectric state are
\[
E_{B1} \approx 60\ \text{meV/f.u.},\quad
E_{B2} \approx 14\ \text{meV/f.u.},\quad
E_{B3} \approx 10\ \text{meV/f.u.},
\]
which makes electric-field switching a direct control knob for turning AM splitting on and off [2511.00712]. Interlayer sliding in the bilayer reverses the sign of the spin splitting without changing the Néel vector [2511.00712].

Out-of-plane electric fields and substrate asymmetry provide another route in 2D. In MnP(S,Se)\(_3\), breaking \(\mathcal{PT}\) while preserving spin-flipping mirrors converts a conventional AF into an AM with planar \(i\)-wave symmetry. For MnPSe\(_3\) under \(1\) eV/Å, the reported spin splitting reaches \(\sim 25\) meV [2309.02355]. In monolayer FeSe, inversion breaking from electric field or substrate asymmetry converts checkerboard AF order into a planar \(d_{x^2-y^2}\) altermagnet, with FeSe/STO calculations showing AM spin splittings up to \(\sim 100\) meV [2309.02355].

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\(_4\)Al\(_8\)**, in addition to **CrSb** and **RuO\(_2\)**, plus a dozen semiconducting AMs [2412.10356]. The reported spectral spin-splitting metric \(A_H\) is \(0.66\) eV for CrSb, \(0.47\) eV for CrSe, \(0.36\) eV for CaFe\(_4\)Al\(_8\), and \(0.28\) eV for RuO\(_2\) [2412.10356].

## 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\(_2\) and NiF\(_2\) barriers, the predicted spin-filter TMR is about \(150\)–\(170\%\) when the Fermi energy is tuned close to the valence-band maximum [2409.00195]. The mechanism is a spin- and momentum-dependent decay constant in the complex band structure, so that
\[
T_\sigma(\mathbf{k}_\parallel)\propto e^{-2\kappa_\sigma(\mathbf{k}_\parallel)d}
\]
and
\[
p(\mathbf{k}_\parallel)=\tanh\!\big[f(\mathbf{k}_\parallel)d\big]
\]
acquires a pronounced \(d\)-wave-like sign structure over the 2D Brillouin zone [2409.00195].

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\(_3\) find peak anomalous Hall conductivities of approximately \(202\) S/cm in the \(d\)-wave case and \(25\) S/cm in the \(i\)-wave case, with linear-plus-cubic and purely cubic dependence on the Néel vector, respectively [2502.21095].

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 \(PT\) symmetry in conventional AFMs. In monolayer V\(_2\)Se\(_2\)O, \(1\%\) tensile strain increases the Kerr response by several orders of magnitude, and in Janus Mn\(_2\)P\(_2\)S\(_3\)Se\(_3\) the calculated Kerr angle reaches about \(0.4^\circ\) [2505.24124][2309.02355]. 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 \(d\)-wave AM model and a RuO\(_2\) bilayer [2504.01640].

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 \(\Delta L = \pi/\mathrm{Re}(q_x)\) [2402.08263]. 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 \(1000\%\) in some weak-phase AM regimes and full tunability by rotating the Néel vector [2509.14109].

Periodic driving and dissipation further enlarge the response space. In 2D \(d\)-wave AMs with Rashba SOC, monochromatic Floquet driving induces out-of-plane magnetization, longitudinal AMR, and AHE, while bichromatic \(\omega\)–\(2\omega\) driving also activates transverse AMR through induced in-plane effective fields [2606.21281]. 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 [2603.19378].

## 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 [2401.08784]. At scale, a DFT+eDMFT workflow combining pre-screening, symmetry analysis, and spectral calculations identified metallic CrSe and CaFe\(_4\)Al\(_8\), together with CrSb and RuO\(_2\), and concluded that while altermagnets are abundant among magnetic materials, only a tiny fraction is metallic [2412.10356].

Several recent directions indicate that AMs now function as a broader symmetry platform rather than a single class of band structures. In ferroelastic CoTe\(_2\) 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 [2606.22186]. In 3D topological-insulator heterostructures, coupling to \(d_{x^2-y^2}\)- and \(d_{x^2-z^2}\)-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 [2512.03478]. Extended \(s\)-wave altermagnets generalize the concept further to fully gapped, spin-compensated, valley-exchange-protected states with isotropic spin splitting and pair-density-wave descendants [2508.20163].

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 [2409.00195][2511.00712][2412.10356]. 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.

Source: https://www.emergentmind.com/topics/altermagnets-ams