Altermagnets: Zero Net Magnetism & Spin Splitting

• Altermagnets are magnetic materials with zero net magnetization that display robust, momentum-dependent spin splitting due to unique crystalline symmetries.
• Minimal lattice and continuum models reveal that rotated or improper symmetry operations lift Kramers degeneracy, enabling ferromagnet-like transport and optical effects.
• The distinctive properties of altermagnets offer promising applications in spintronics, memory devices, and quantum sensing through anomalous Hall, Kerr responses, and spin-current generation.

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 (Cheong et al., 2024, Bai et al., 2024, Mostovoy, 2 Jun 2025).
• Classification: Recent works classify altermagnets into three types (Cheong et al., 2024):
  • 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) (Cheong et al., 2024).

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) (Bai et al., 2024, Giuli et al., 2024, Naka et al., 2024, Mostovoy, 2 Jun 2025). 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 (Giuli et al., 2024). 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 (Jaeschke-Ubiergo et al., 2023).

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 (Schiff et al., 2024, Heinsdorf, 2024). 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 (Heinsdorf, 2024). 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 (Mukherjee et al., 16 Oct 2025).

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 (Bai et al., 2024, Tamang et al., 2024, Naka et al., 2024).
• 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 (Tamang et al., 2024, Mostovoy, 2 Jun 2025, Asgharpour et al., 2024).
• 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 (Chen et al., 2024).
• Spin-current generation: Nonrelativistic, symmetry-protected spin-polarized currents emerge under applied electric fields via the unique tensor structure $\alpha_{ik}L^aE_k$ (Mostovoy, 2 Jun 2025).
• 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 (Gondolf et al., 21 Feb 2025).
• Magnon chirality: Polarized neutron diffraction can directly measure momentum-dependent magnon band splitting and chirality anisotropy, providing the magnonic counterpart to spin-resolved ARPES (McClarty et al., 2024).

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 (Bai et al., 2024, Naka et al., 2024, Tamang et al., 2024).
• 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 (Naka et al., 2024).
• 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 (Chen et al., 24 Jul 2025, Jaeschke-Ubiergo et al., 2023).
• Synthetic and engineered systems: Artificially stacked ferromagnetic layers, rotated appropriately and coupled antiferromagnetically, can realize robust altermagnetic bands and responses (synthetic altermagnets) (Asgharpour et al., 2024).
• 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 (Duan et al., 2024).

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 (Schiff et al., 2024, Mostovoy, 2 Jun 2025).

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 (Cheong et al., 2024, Tamang et al., 2024, Asgharpour et al., 2024).
• Multiferroic transducers: AFEAM-based devices allow electric-field switching of spin polarization with low energy cost and sub-nanosecond speed (Duan et al., 2024).
• Anomalous transport devices: Frequency doublers, rectifiers, and high-efficiency spin current generators exploit the nonlinear and anisotropic Hall responses endemic to altermagnets (Mukherjee et al., 16 Oct 2025).
• Quantum sensing and imaging: NV-center magnetometry and STM imaging can directly probe edge-induced magnetization or LDOS features at nanoscales (Hodt et al., 2024, Gondolf et al., 21 Feb 2025).
• Topological magnonics and chiral excitation control: Polarized neutron scattering provides direct access to magnonic signatures, permitting domain and multipolar order mapping (McClarty et al., 2024).

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 (Bai et al., 2024, Mostovoy, 2 Jun 2025).