---
title: 'Altermagnets: Zero Net Magnetism & Spin Splitting'
url: https://www.emergentmind.com/topics/altermagnets
type: topic
---

# Altermagnets: Zero Net Magnetism & Spin Splitting

Altermagnets are a distinct class of magnetic materials combining fully compensated spin configurations (zero net magnetization) with robust, momentum-dependent nonrelativistic spin splitting and broken time-reversal symmetry. Defined by the absence of net magnetic moment yet exhibiting band spin splitting due to crystalline symmetries that interchange magnetic sublattices via rotations or improper operations rather than translations/inversions, altermagnets provide a symmetry-engineered bridge between conventional collinear antiferromagnets and ferromagnets. This unique symmetry lifts Kramers degeneracy in large portions of the Brillouin zone, enabling ferromagnet-like transport and optical responses—such as the anomalous Hall and Kerr effects, large spin currents, and crystal Nernst effects—without requiring net magnetization or strong spin–orbit coupling. Contemporary research has identified numerous material families exhibiting altermagnetic order, developed rigorous symmetry-based taxonomies, constructed minimal lattice and continuum models elucidating these symmetries, established connections to quantum geometry, and inspired application proposals in spintronics, multiferroics, and beyond.

## 1. Symmetry Foundations and Classification

Altermagnetism emerges from the interplay of time-reversal ($\mathcal T$) and spatial symmetries that relate magnetic sublattices via point group operations—rotations ($C_n$), roto-inversions, or glides—rather than translations or pure inversion. The defining features are:
- **Zero Net Magnetization:** $\sum_i \langle S_i \rangle = 0$ in the absence of spin–orbit coupling, distinguishing altermagnets from ferromagnets.
- **Broken PT Symmetry:** The combined parity-time ($\mathcal{PT}$) symmetry is always broken, lifting Kramers degeneracy at generic points in $\mathbf{k}$-space.
- **Sublattice Mapping by Rotation/Improper Symmetry:** For example, spins A and B map onto each other under a $C_4$ or $C_2$ rotation, but are not connected by translation or pure inversion as in a Néel antiferromagnet [2409.20456, 2406.02123, 2506.01823].
- **Classification:** Recent works classify altermagnets into three types [2409.20456]:
  - **Type I (M-type):** T-broken, P broken or preserved, net orbital moment may arise with SOC, 31 ferromagnetic point groups.
  - **Type II (S-type):** T-broken, fully compensated, no net moment even with SOC, 38 further magnetic point groups.
  - **Type III (A-type):** T-preserved, P-broken, "antipolar" altermagnets.

A further distinction separates **strong altermagnets** (spin splitting in the nonrelativistic limit) from **weak altermagnets**, where splitting appears only with finite SOC. The key criterion is the number of unbroken orthogonal spin rotation axes: strong for only a single axis (collinear), weak for three (e.g. up-up-down-down or cycloidal orders) [2409.20456].

## 2. Minimal Models and Band Structure

Minimal models for altermagnets are based on two (or more) sublattices related by rotation, with the prototypical Hamiltonian
\[
H(\mathbf{k}) = \varepsilon_0(\mathbf{k})\,\sigma_0 + \Delta(\mathbf{k})\,\sigma_z\,,
\]
where $\Delta(\mathbf{k})$ transforms as a $d$-, $g$-, or $i$-wave under the crystal point group and alternates sign under the relevant symmetry operation (e.g., $\Delta(k_x,k_y) = -\Delta(k_y,-k_x)$ for a $d$-wave altermagnet) [2406.02123, 2410.00909, 2411.11025, 2506.01823]. Collapse of spin degeneracy (even at zero SOC) appears everywhere except along nodal lines dictated by these symmetry constraints.

For interaction-induced altermagnetism, two-orbital (e.g., $d_{xz}$, $d_{yz}$) Hubbard models exhibit robust, tunable altermagnetic phases near van Hove singularities, supporting broad ranges of interaction strength and carrier doping, and enabling control of spin-charge conversion by chemical or mechanical means [2410.00909]. Supercell altermagnets extend this by allowing larger magnetic unit cells with propagation vectors $\mathbf{Q}\ne0$, further enriching the space of possible order parameter orientations and nodal patterns [2308.16662].

## 3. Momentum-dependent Spin Splitting and Quantum Geometry

The hallmark of altermagnetic order is a nonuniform, symmetry-enforced, momentum-dependent spin splitting in the absence of net moment. For example, in RuO$_2$ (rutile), the splitting arises as $\Delta(\mathbf{k}) \propto \cos k_x - \cos k_y$ ($d_{x^2-y^2}$ symmetry), while in hexagonal systems like MnTe, a $g$-wave form factor dominates [2412.18025, 2410.12789]. This spin splitting—protected and dictated by the spin-space group and the related real-space multipoles—undergirds the unique transport and optical characteristics of altermagnets.

Quantum geometric considerations further reveal that the quantum metric $g_{ab}(\mathbf{k})$ of the Bloch bands fundamentally favors and stabilizes altermagnetic instabilities, especially near the nodal manifolds where the band splittings vanish [2410.12789]. The quantum geometry directly connects to observable properties, such as the nonlinear Hall effect driven by Berry curvature dipoles accessible in low-symmetry or hybrid altermagnetic structures [2510.14899].

## 4. Observable Phenomena and Experimental Signatures

**Band- and transport signatures** uniquely distinguish altermagnets from both AFM and FM orders:
- **Spin-resolved ARPES:** Direct measurement of momentum-dependent spin splitting, with anistropic lobe structures signaling $d$, $g$, or $i$-wave symmetry [2406.02123, 2412.05377, 2411.11025].
- **Anomalous Hall and Kerr Effects:** The broken PT or $\mathcal T$ symmetry and uncompensated Berry curvature generate spontaneous Hall responses, Kerr/Faraday rotations, and related phenomena even in the absence of net magnetization [2412.05377, 2506.01823, 2412.02473].
- **Spin-splitting magnetoresistance:** Angular- and temperature-dependent magnetoresistance in heterostructures (e.g., RuO$_2$/Co) provides an all-electrical probe of the Néel vector orientation, with characteristic phase shift indicative of altermagnetic spin splitting [2412.18220].
- **Spin-current generation:** Nonrelativistic, symmetry-protected spin-polarized currents emerge under applied electric fields via the unique tensor structure $\alpha_{ik}L^aE_k$ [2506.01823].
- **Local signatures in STM:** The spatial structure of LDOS or in-gap bound states near point defects tracks the underlying nodal structure of the altermagnetic state, distinguishing $d$- and $g$-wave symmetries in STM conductance maps [2502.15606].
- **Magnon chirality:** Polarized neutron diffraction can directly measure momentum-dependent magnon band splitting and chirality anisotropy, providing the magnonic counterpart to spin-resolved ARPES [2410.10771].

## 5. Material Realizations and Structural Diversity

Altermagnetism is present in a wide class of magnetic compounds:
- **Binary chalcogenides and pnictides:** RuO$_2$, MnTe, CrSb, Mn$_5$Si$_3$ exhibit large nonrelativistic spin splitting and associated transport anomalies [2406.02123, 2411.11025, 2412.05377].
- **Correlated oxide perovskites:** Orthorhombic perovskites (LaTiO$_3$, CaCrO$_3$, LaVO$_3$, LaMnO$_3$, YFeO$_3$) support collinear C-type or G-type AFM with rotation-induced symmetry lowering, giving rise to characteristic $d$-wave splitting and cross-correlation phenomena between spin, charge, and lattice [2411.11025].
- **Quasicrystals and supercells:** Octagonal and dodecagonal quasicrystals can support $g$- and $i$-wave altermagnets with exotic nodal structures; supercell altermagnets such as MnSe$_2$ and $AX$B$_3$ (X=Co, BaMnO$_3$) allow for additional order parameter reorientation [2507.18408, 2308.16662].
- **Synthetic and engineered systems:** Artificially stacked ferromagnetic layers, rotated appropriately and coupled antiferromagnetically, can realize robust altermagnetic bands and responses (synthetic altermagnets) [2412.02473].
- **Multiferroic and switchable systems:** Antiferroelectric altermagnets (AFEAMs), such as CuWP$_2$S$_6$ or perovskite BiCrO$_3$ in AFE–AFM states, allow electrical switching between degenerate (altermagnetic) and non-degenerate (non-altermagnetic) band structures [2410.06071].

Table: Select material classes and key characteristics

| Material/Class           | Space Group           | Symmetry/Order      |
|--------------------------|-----------------------|---------------------|
| RuO$_2$, CrSb, MnTe      | P4$_2$/mnm, P6$_3$/mmc| $d$-, $g$-wave AM   |
| CaCrO$_3$, LaVO$_3$      | Pbnm (Perovskite)     | $d$-wave (C-type AFM)|
| MnSe$_2$                 | Pa$\bar{3}$           | Supercell $d$-wave  |
| CsCoCl$_3$, BaMnO$_3$    | Hexagonal, 3x cell    | Supercell $g$-wave  |
| CuWP$_2$S$_6$ (AFEAM)    | P2$_1$ (mono)         | Switchable AM       |

## 6. Landau-Ginzburg and Multipolar Theory

The order parameter for collinear altermagnets is the compensated Néel vector $\mathbf N$, transforming as a specific one-dimensional irrep of the crystal point group. The general Landau functional incorporates all invariants allowed by symmetry. The lowest allowed multipole—quadrupole, octupole, etc.—fixes the momentum dependence of the spin splitting and constrains all possible observable tensor responses. Higher-rank tensor couplings dictate whether effects such as linear magnetoelectricity, piezomagnetism, or nonlinear Hall responses are symmetry-allowed in a given compound [2412.18025, 2506.01823].

## 7. Device Concepts and Applications

Drawing from their unique combination of zero net moment, robust symmetry-protected spin splitting, and tunable magnetoelectric couplings, altermagnets are being advanced as a platform for:
- **Memory elements and spintronic logic:** Nonvolatile, zero-stray-field memory bits, spin valves, and logic gates exploiting electrical tuning, fast THz switching, and Hall/Kerr readout [2401.13069, 2412.05377, 2412.02473].
- **Multiferroic transducers:** AFEAM-based devices allow electric-field switching of spin polarization with low energy cost and sub-nanosecond speed [2410.06071].
- **Anomalous transport devices:** Frequency doublers, rectifiers, and high-efficiency spin current generators exploit the nonlinear and anisotropic Hall responses endemic to altermagnets [2510.14899].
- **Quantum sensing and imaging:** NV-center magnetometry and STM imaging can directly probe edge-induced magnetization or LDOS features at nanoscales [2406.07603, 2502.15606].
- **Topological magnonics and chiral excitation control:** Polarized neutron scattering provides direct access to magnonic signatures, permitting domain and multipolar order mapping [2410.10771].

In summary, altermagnets expand the taxonomy of magnetic order, enrich the symmetry-based design space for emergent transport and optical effects, and present a robust foundation for new generations of spintronic and multifunctional devices [2406.02123, 2506.01823].

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