Altermagnetic Properties

• Altermagnetic Properties are defined as a class of compensated, collinear magnetic order that supports momentum-dependent spin splitting at zero net magnetization.
• The research outlines symmetry foundations, theoretical models, and microscopic realizations that explain nonrelativistic spin splitting and related cross-correlation effects.
• Key findings include unconventional transport responses such as the anomalous Hall effect, piezomagnetism, and high-frequency magnon dynamics with promising spintronic applications.

Altermagnetic Properties

Altermagnetism defines a symmetry-distinct class of compensated, collinear magnetic order that supports momentum-dependent spin splitting in the electronic band structure at zero net magnetization. Unlike conventional ferromagnetism (which features a uniform magnetization and globally broken time-reversal symmetry), or Néel antiferromagnetism (which preserves combined time-reversal and translation or inversion and maintains exact band spin degeneracy), altermagnets are characterized by sublattices connected through non-primitive spatial operations (rotations, mirrors, glides) that, in combination with time reversal, relate opposite spin orientations. The resulting "^^^^0^^^^"—typically d-, g-, or i-wave—spin structures give rise to unconventional phenomena, including nonrelativistic (exchange-driven) spin splitting, anomalous Hall and Nernst effects, high-frequency magnonics, edge magnetization, and piezomagnetism in zero net moment systems. The landscape of altermagnetic physics spans a broad range of crystalline, synthetic, amorphous, and topologically nontrivial materials.

1. Symmetry Foundations and Classification

Altermagnetic order emerges in systems with fully compensated spin textures (∑_i⟨S_i⟩=0) but with macroscopic PT (parity×time-reversal) symmetry broken. The essential criterion is that symmetry-related sublattices—often not related by translation or inversion—are mapped onto one another by point-group operations (C_n, mirror, glide) possibly combined with time-reversal. This enables a finite, generally anisotropic momentum-dependent spin splitting ΔE(k), even in the absence of spin–orbit coupling. The magnetic space (spin) group is lowered such that only a halved subgroup H of the original G_crystal is preserved in real space, while the other coset elements require a 180° spin rotation (C_2T) for symmetry (Jungwirth et al., 28 Jun 2025).

The main symmetry-based classification comprises three types (Cheong et al., 2024):

• Type I (M-type): PT-broken, T-broken, nonzero orbital magnetization M ≠ 0 from SOC (31 point groups).
• Type II (S-type): PT-broken, T-broken, zero net M (38 point groups); exhibits pure compensated spin order yet finite spin splitting.
• Type III (A-type): PT-broken, T-preserved, typically noncollinear multiferroics; M = 0 (21 point groups).

The strong/weak dichotomy further distinguishes altermagnets with nonrelativistic spin splitting from those where splitting only occurs with finite SOC, determined by the number of unbroken orthogonal spin-rotation operations (Cheong et al., 2024).

In general, the order parameter is a ferroic higher-order magnetic multipole (octupole, hexadecapole, etc.) that is even under inversion but odd under point-group elements mapping between sublattices. In k-space, the splitting term is Δℓ(k)=Δ_0fℓ(k), with f_ℓ a d- (two-fold), g- (four-fold), or i-wave (six-fold) basis (Jungwirth et al., 28 Jun 2025, Gao et al., 2023).

2. Theoretical Models and Microscopic Realizations

Minimal altermagnetic models are based on multiband, multi-sublattice (or multi-orbital) Hamiltonians with exchange-coupled spins:

• Generic two-band model:

$$H(\mathbf{k}) = \epsilon_0(\mathbf{k})\,\mathbb{I} + \Delta_\ell(\mathbf{k})\,\sigma_z$$ with $$\Delta_\ell(\mathbf{k}) = \text{d-, g-, i-wave basis function}$$, and $\sigma_z$ acts in spin space (Gao et al., 2023, Jungwirth et al., 28 Jun 2025).

• Perovskite altermagnets:

Multi-orbital Hubbard models on distorted $ABX_3$ lattices (octahedral tilts/gliads, C- or G-type AFM) generate sublattice-dependent, anisotropic inter-orbital hopping, leading to nonrelativistic spin splitting and, with SOC, cross-correlation effects such as the anomalous Hall effect (Naka et al., 2024).

• Itinerant mechanisms:

Two-orbital (e.g., $d_{xz}, d_{yz}$) Hubbard models at moderate doping near van Hove singularities show robust altermagnetic order over a wide U/t and doping range (Giuli et al., 2024).

• Synthetic and amorphous systems:

Bilayer synthetic altermagnets—two rotated, anisotropic FM layers coupled with opposite magnetizations—yield d-wave band splitting with zero net magnetization (Asgharpour et al., 2024). Amorphous altermagnets at the mean-field level demonstrate spontaneous local C₄T symmetry breaking, preserving all generic altermagnetic features without long-range crystal order (d'Ornellas et al., 11 Apr 2025).

• Quasicrystals:

Self-consistent mean-field calculations in octagonal/dodecagonal tilings produce stable g- and i-wave altermagnetic orders with global C₈T and C₁₂T protection, respectively, manifesting as eight- and twelve-fold nodal structures in spin-resolved spectral maps (Chen et al., 24 Jul 2025).

3. Band Structure, Spin Splitting, and Spectroscopic Signatures

The central band-structure hallmark is spin degeneracy lifting—without net moment—along high-symmetry k-paths determined by the magnetic point group:

• d-wave: $\Delta_d(\mathbf{k}) \propto k_x^2-k_y^2$
• g-wave: $\Delta_g(\mathbf{k}) \propto k_x k_y (k_x^2-k_y^2)$
• i-wave: $$\Delta_i(\mathbf{k}) \propto k_x k_y k_z (k_x^2-k_y^2)$$

Experimental ARPES and spin-resolved ARPES directly image such spin splittings, e.g., ΔE ~ 0.1–0.8 eV in metallic and insulating altermagnets such as Nb₂FeB₂, NaFeO₂, and MnTe (Gao et al., 2023, Chilcote et al., 2024, Bangar et al., 20 May 2025). Nonrelativistic splitting persists in zero SOC, vanishing only on the symmetry-imposed nodal lines or planes (Jungwirth et al., 28 Jun 2025, Sattigeri et al., 2023). In quasicrystals, ℓ-fold star patterns in $A_\uparrow(\mathbf{p})-A_\downarrow(\mathbf{p})$ map directly onto the order parameter symmetry (Chen et al., 24 Jul 2025).

Optical signatures include:

• Finite Kerr/Faraday rotation at zero field (Asgharpour et al., 2024).
• Spin-polarized neutron scattering resolves domain populations and magnon chirality anisotropy tied to the multipolar order (McClarty et al., 2024).
• STM/STS can image local, oscillatory edge magnetization, distinguishing it from rapidly-decaying antiferromagnetic cases (Hodt et al., 2024).

4. Transport, Cross-Correlation, and Magnonic Phenomena

Altermagnetic order produces a suite of unconventional transport and magnonic responses:

• Anomalous Hall effect (AHE):

Arises from Berry curvature hotspots induced by band crossings voided by SOC. Notably, finite spontaneous σ_xy ~ 10–100 Ω⁻¹ cm⁻¹ at room temperature in RuO₂, MnTe, and synthetic models (Tamang et al., 2024, Gao et al., 2023, Bangar et al., 20 May 2025, Asgharpour et al., 2024). The AHE can exhibit high-order angular harmonics (e.g., 3φ in hexagonal MnTe) directly reflecting the symmetry of the underlying altermagnetic state (Bangar et al., 20 May 2025, Naka et al., 2024).

• Piezomagnetism and piezomagnetic control:

Nonzero third-rank axial piezomagnetic tensors (e.g., Q_xyz in MnTe) allow stress-induced magnetization, directly probing and controlling antiferromagnetic domain populations (Aoyama et al., 2023). In 2D altermagnets and valley-compensated chalcogenides, strain produces strong valley polarization, a linear piezomagnetic response, and valley-selective optical and transport features (Fan et al., 12 Dec 2025).

• Spin currents and torques:

Nonrelativistic (T-odd) spin Hall effect: charge current converts into a crystal-axis-selective spin current, j_i^a = α_{ik} E_k (Mostovoy, 2 Jun 2025). Quantified spin-splitter ratios (up to ~9%) are tunable via doping/strain in two-orbital models (Giuli et al., 2024). Directional (nonlocal) magnon transport and field-free spin Seebeck effects are demonstrated in d-wave LuFeO₃ (Galindez-Ruales et al., 20 Aug 2025).

• Magnon chirality and THz dynamics:

Altermagnetic order splits magnon branches into oppositely chiral pairs, measurable via polarized-neutron inelastic scattering. The magnon splitting and chirality map are robust to small SOC-induced gaps (McClarty et al., 2024). Magnetization dynamics occur at THz frequencies, enabling ultrafast, robust spintronic functionality (Tamang et al., 2024, Galindez-Ruales et al., 20 Aug 2025).

5. Material Realizations and Engineering

A wide range of strongly and weakly correlated materials—including perovskite oxides, pnictides, fluorides, antimonides, transition-metal chalcogenides, and engineered heterostructures—host robust altermagnetic order (Naka et al., 2024, Fan et al., 12 Dec 2025, Gao et al., 2023, Bangar et al., 20 May 2025). Prototypical systems include:

Material	Type	Key Features
α–MnTe	g-wave	NiAs, A-type AF, ΔE_spin ~ 0.1 eV
RuO₂	d-wave	Rutile, BZ spin splitting ~0.1–0.2 eV
CaCrO₃, LaVO₃	d-wave	Perovskites, tunable AHE/χ_xy
Nb₂FeB₂, Ta₂FeB₂	g-wave	AI-discovered, σ_xz ~ 100 Ω⁻¹ cm⁻¹
Fe₂MoTe₄	d-wave	2D, valley-contrasted, ΔE ~ 0.7 eV
Synthetic AM	d-wave	Bilayer, AHE, spin current, M=0

Device integration leverages MBE growth, buffer engineering for phase control (e.g., WZ–MnSe on CdSe/GaAs), or heterostructuring to realize desired order (Grzybowski et al., 2023, Jungwirth et al., 28 Jun 2025). AI-guided screening accelerates new material discovery, now yielding i-wave altermagnets for the first time (Gao et al., 2023).

6. Surfaces, Interfaces, Topology, and Future Perspectives

• Surface orientation:

Altermagnetic spin splitting projects onto surfaces only for select crystal terminations; "blind" surfaces annihilate splitting by merging contributions of k-points with opposite sign (Sattigeri et al., 2023). Electric field gating can activate AM at such blind interfaces, providing a route to engineered functionality.

• Interface magnetization:

Large, oscillatory edge/interface magnetization profiles arise from spin-asymmetric Friedel oscillations (decaying as x^–½) (Hodt et al., 2024), in striking contrast to conventional AFM. This enables nanoscale edge-memory and spin-filter devices with zero stray field in the bulk.

• Topological and multifold phenomena:

Altermagnets host nodal lines, Weyl nodes, chiral/Dirac fermion points, and topological anomalous responses, especially in i-wave or noncollinear cases (Gao et al., 2023, Chen et al., 24 Jul 2025). Integration with topological insulators may enable novel proximitized Dirac spin splitting and chiral edge transport (Sattigeri et al., 2023).

• Valley and piezospintronic functionalities:

Layered and 2D altermagnets with strong valley-contrasting Berry curvature and dichroism (e.g., Fe₂MoTe₄) allow strain-induced valley-selective transport, piezomagnetism, and optically addressable spin/valley memories (Fan et al., 12 Dec 2025).

• Synthetic and amorphous altermagnets:

The altermagnetic phase is not restricted to periodic order; local (e.g., C₄T) symmetry in synthetic bilayers or amorphous networks replicates anisotropic spin splitting and cross-coupled transport (d'Ornellas et al., 11 Apr 2025, Asgharpour et al., 2024).

7. Outlook and Device Implications

Altermagnets unify and generalize the functional advantages of both ferromagnets (readout, AHE, magnonics) and antiferromagnets (stray-field-free, THz dynamics, robustness), establishing a design framework for next-generation spintronic, magnonic, and valleytronic devices. Recent work enables tailored material discovery (AI, symmetry-guided GNNs (Gao et al., 2023)), precise engineering of domain populations and response via strain, gating, and interface design, and integration with topological states for quantum-computational or chiral transport effects (Chen et al., 24 Jul 2025, Sattigeri et al., 2023, Naka et al., 2024). Open avenues include the development of high-T_N g- and i-wave altermagnets, 2D and heterostructured platforms, and piezo- and valleytronic actuators and memories, further expanding the functional material basis for spin-based quantum information technologies.