Altermagnetic Materials

• Altermagnetic materials are a symmetry-defined class of magnets that show zero net magnetization yet exhibit robust momentum-dependent spin splitting without relying on spin–orbit coupling.
• They occur in diverse materials such as TMDs, oxides, and synthetic heterostructures, with experimental ARPES and XMCD studies confirming d- and g-wave splitting patterns.
• Their unique band topology drives novel transport, magneto-optical, and topological effects, opening avenues for advanced spintronic and quantum device applications.

Altermagnetic materials are a symmetry-defined class of collinear (and certain non-collinear) magnets with vanishing macroscopic magnetization but robust, symmetry-protected momentum-dependent spin splitting of the electronic bands in the absence of relativistic spin–orbit coupling. This property distinguishes altermagnets from both conventional ferromagnets (which have a uniform exchange splitting and net magnetization) and conventional antiferromagnets (which are spin-degenerate at every k-point via inversion or translation symmetries). The unique band topology and spin group structure of altermagnets underpin a host of distinct physical phenomena, ranging from anomalous transport and magneto-optical effects to the emergence of topologically protected quasiparticles and tunable magnon dispersions.

1. Symmetry Principles and Band Topology of Altermagnets

In altermagnets, two (or more) magnetic sublattices carry antiparallel moments related not by a primitive translation or center inversion but by a pure spatial point-group operation—such as rotation or mirror—that does not restore Kramers degeneracy. The magnetic point group lacks any operation combining time reversal $\mathcal{T}$ and spatial inversion or translation that preserves the spin pattern, but retains a non-trivial “spin-exchanging” operation $R$ such that $R: (r, \sigma) \rightarrow (Rr , -\sigma)$.

Formally, for a magnetic crystal Hamiltonian $H$ and combined operation $S \mathcal{T}$,

• For a ferromagnet, no such spatial symmetry exists, and $$\varepsilon_{\uparrow}(k) \neq \varepsilon_{\downarrow}(k)$$ everywhere.
• For a conventional antiferromagnet, $S$ is a translation or inversion, ensuring Kramers-like degeneracy.
• In altermagnets, $S$ is a non-inverting operation (e.g., $C_2$, $C_4$, mirror), allowing $$\varepsilon_{\uparrow}(k) \neq \varepsilon_{\downarrow}(k)$$ over generic $k$, but enforcing $$\int_\text{BZ} [\varepsilon_{\uparrow}(k) - \varepsilon_{\downarrow}(k)] d^3k = 0$$.

The prototypical spin splitting in altermagnets adopts higher even-parity angular patterns (frequently $d$-wave and $g$-wave), changing sign under specific point-group rotations or mirrors and vanishing along symmetry-imposed nodal planes or lines in the Brillouin zone. For example, in tetragonal systems (e.g., rutile RuO$_2$), the splitting $f(k)\propto k_xk_y$ (pure $d$-wave), and in hexagonal lattices (e.g., CoNb$_4$Se$_8$), $f(k) \sim k_x k_y (k_x^2 - k_y^2) k_z$ ($g$-wave) (Regmi et al., 2024, Mostovoy, 2 Jun 2025, Day-Roberts et al., 5 Jan 2026).

2. Materials Classes and Symmetry Realizations

Altermagnetism emerges in a diverse range of crystal families, unified by the presence of symmetry operations that exchange spin sublattices:

• Layered Intercalated Transition Metal Dichalcogenides (TMDs):
  • CoNb$_4$Se$_8$: P6$_3$/mmc, MSG P6$_3'$/m'$m'$c, g-wave symmetry, spin splitting $\sim$50–80 meV, $T_N$ = 168 K (Regmi et al., 2024).
  • Co$_{1/4}$TaSe$_2$: direct ARPES observation of $g$-wave splitting up to 250 meV, $T_N$ = 178 K (Sprague et al., 18 Aug 2025, Day-Roberts et al., 5 Jan 2026).
• Wurtzite and NiAs-type Chalcogenides:
  • WZ-MnSe: P6$_3$mc, A-type collinear AFM, d-wave splitting $\sim$50 meV, $T_N > 800$ K (Grzybowski et al., 2023).
  • MnTe: hexagonal NiAs, $g$-wave splitting, ultrafast THz dynamics, bulk altermagnetic XMCD confirmed (Yamamoto et al., 25 Feb 2025).
• Ruddlesden-Popper and Perovskite Oxides:
  • La$_2$NiO$_4$, La$_3$Ni$_2$O$_7$, BiFeO$_3$, etc.: collinear AFMs with symmetry-protected nodal planes; non-relativistic spin splittings 100–300 meV (Bernardini et al., 2024).
  • Perovskites (e.g., LaTiO$_3$, CaCrO$_3$): nonzero cross-correlation phenomena (electrical spin splitting, AHE under SOC) arising from GdFeO$_3$-type lattice distortions (Naka et al., 2024).
• Rutile-type Oxides and Fluorides:
  • RuO$_2$, MnF$_2$, CoF$_2$, NiF$_2$: d-wave altermagnetism, large SOC-free spin filtering effects (Samanta et al., 2024, Chen et al., 2 Nov 2025).
• Inverse Lieb Lattice Systems:
  • ILL materials (e.g., Sr$_2$CrO$_2$Cr$_2$OAs$_2$): d-wave AM phase stabilized for $d^{2-3}$, $d^{5}$ configurations, robust up to $T_N$ ≈ 600 K, magnonic chiral splitting set by $J_{2a}-J_{2b}$ anisotropy (Chang et al., 6 Aug 2025).
• Quasicrystals:
  • Octagonal/dodecagonal (Ammann-Beenker, Stampfli tilings): realization of $g$- and $i$-wave altermagnetism with anomalous eight-/twelve-fold nodal structures (Chen et al., 24 Jul 2025).
• Synthetic and Engineered Altermagnets:
  • Artificial heterostructures: two orthogonally oriented ferromagnetic layers, or twisted bilayer van der Waals magnets, leading to designer $d$-wave spin textures, gate-tunable Berry curvature and Hall responses (Asgharpour et al., 2024).

A comprehensive survey with specific crystalline examples and spin splitting magnitudes is presented in (Gao et al., 2023, Bernardini et al., 2024, Day-Roberts et al., 5 Jan 2026).

3. Electronic Structure, Spin Splitting, and Excitations

The defining feature of altermagnets is the non-relativistic, $k$-dependent exchange splitting: $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\sigma_0 + \lambda(\mathbf{k})\sigma_z$$ with $\lambda(\mathbf{k})$ even under inversion, odd under the spin-exchanging rotation or mirror, and integrating to zero over the BZ.

• Magnitude and Dispersion: DFT and ARPES experiments observe splittings on the Fermi surface up to 100–300 meV for layered TMD altermagnets (Sprague et al., 18 Aug 2025), and 10–50 meV in perovskites and wurtzite phases (Grzybowski et al., 2023, Bernardini et al., 2024).
• Nodal Topology: The spin splitting vanishes along symmetry-determined planes or lines, resulting in characteristic star-shaped or petal-like nodal patterns (Day-Roberts et al., 5 Jan 2026, Chen et al., 24 Jul 2025).
• Magnetic excitations: Magnon spectra computed for d-wave AMs on inverse Lieb lattices display large chiral splittings, directly tied to second neighbor exchange anisotropy; up to 60% of the magnon bandwidth in Sr$_2$CrO$_2$Cr$_2$OAs$_2$ (Chang et al., 6 Aug 2025).

4. Transport, Magneto-Optical, and Topological Phenomena

Altermagnets enable a host of transport and optical phenomena regulated by their symmetry:

• Anomalous Hall Effect (AHE): Collinear altermagnets exhibit zero intrinsic AHE in the easy-axis Néel configuration; however, spin canting or symmetry lowering (e.g., via field or SOC) can switch on sizable AHE, quantified by the Berry curvature concentration near lifted nodal surfaces (Regmi et al., 2024, Yang et al., 23 Feb 2025, Naka et al., 2024).
• Crystal Hall, Nernst, and Thermal Hall Effects: The intrinsic (Berry curvature) component is sharply peaked at (pseudo)nodal surfaces, with angular harmonics controlled by Néel orientation; room-temperature conductivities $\sigma_{xy} = 10$–$100$ S\,cm$^{-1}$, $\alpha_{xy} = 0.1$–$1$ A K$^{-1}$ m$^{-1}$ (Yang et al., 23 Feb 2025).
• Ultrafast Spin Dynamics: Altermagnets exhibit large separation between momentum relaxation ($\tau_\text{e-e}\sim10$ fs) and spin polarization decay ($\tau_S\sim1$ ps), enabling robust, long-lived optically driven spin polarization, unlike conventional magnets (Weber et al., 2024).
• Spin Filtering and Tunnel Magnetoresistance: Rutile-type AM insulators (CoF$_2$, NiF$_2$) have spin- and $k$-resolved evanescent decay rates supporting near-100% spin polarization in tunneling, with double-barrier TMR ratios of 150–170% (Samanta et al., 2024).
• Topological Phases: Altermagnetic symmetry allows the realization of bipolarized (spin-polarized) Weyl semimetals and quantum crystal valley Hall effects in 2D materials (e.g., Fe$_2$WTe$_4$, Fe$_2$MoZ$_4$), with strain and Néel vector manipulation converting between QCVH and Chern-insulator phases (Tan et al., 2024).

5. Experimental Probes and Detection

Direct and indirect diagnosis of altermagnetic order leverages several advanced techniques:

• ARPES and Spin-ARPES: Direct momentum-resolved observation of $g$-wave (sixfold) spin splitting at the Fermi surface has been achieved in Co$_{1/4}$TaSe$_2$ (Sprague et al., 18 Aug 2025).
• X-Ray Magnetic Circular Dichroism (XMCD): Transmission-mode XMCD imaging resolves bulk altermagnetic domains, with nanoscale contrast and amplitude in agreement with DFT predictions (e.g., $\sim1.8\%$ peak in MnTe) (Yamamoto et al., 25 Feb 2025).
• Momentum Density Spectroscopy: Spin-polarized positron annihilation (2D-ACAR) and magnetic Compton scattering probe $d$- or $g$-wave spin splitting in the bulk, yielding clear, symmetry-reflective momentum profiles (Chen et al., 2 Nov 2025).
• Magneto-Optical Kerr and Faraday Effects: SOC-induced gyrotropic and birefringent responses proportional to allowed magnetic multipoles in the altermagnetic phase (Mostovoy, 2 Jun 2025).

6. Synthetic Altermagnets and Design Rules

Proposals for engineered altermagnetism utilize synthetic bilayers of anisotropic ferromagnetic films with orthogonally oriented moments. Tuning the in-plane anisotropy and interlayer hopping realizes d-wave spin splitting textures with zero net magnetization and gate-switchable Hall responses. Candidate platforms include Moiré-twisted bilayer magnetic van der Waals materials and rutile-type oxide thin films (Asgharpour et al., 2024).

Design criteria based on symmetry, exchange interaction hierarchy, and $d$-electron filling (favoring $d^{2-3}$ or $d^5$) are summarized for the inverse Lieb and perovskite lattices (Chang et al., 6 Aug 2025, Bernardini et al., 2024).

7. Outlook, Classifications, and Applications

The theoretical framework for altermagnetic order encompasses both collinear and non-collinear magnets, provided their magnetic point group admits zero net moment but breaks combined inversion/time-reversal ($P\mathcal{T}$) or translation/$\mathcal{T}$ symmetry, mapping sublattices only under pure rotations or mirrors. The corresponding magnetic multipole expansions account for ferromagnet-like responses (Hall, Kerr, Faraday) in a compensated context (Mostovoy, 2 Jun 2025, Singh et al., 2024).

Recent AI-accelerated discovery pipelines have substantially expanded the known library to include $i$-wave altermagnets and new candidate semiconductors and metals, validated by DFT (Gao et al., 2023).

Technological implications:

• Spintronic devices with zero stray fields, nonvolatile AFM memory, ultrafast (THz) dynamics.
• High-sensitivity magnetic sensors, spin caloritronics, and Hall-based logic without stray field interference.
• Topological quantum devices exploiting symmetry-protected band crossings and valley polarizations in 2D and layered materials (Tan et al., 2024, Yang et al., 23 Feb 2025).

Ongoing research is focused on further refining the classification of altermagnets beyond $d/g/i$-wave harmonics, engineering heterostructures with interfacial altermagnetic proximity, and exploiting exotic non-linear Hall and spin transport phenomena (Bernardini et al., 2024, Singh et al., 2024, Chen et al., 24 Jul 2025).

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Key References:

• First experimental ARPES demonstration: (Sprague et al., 18 Aug 2025)
• Theoretical and computational materials libraries: (Day-Roberts et al., 5 Jan 2026, Gao et al., 2023)
• Perovskite/oxide symmetry and models: (Bernardini et al., 2024, Naka et al., 2024)
• Magnonic and topological phenomena: (Chang et al., 6 Aug 2025, Tan et al., 2024)
• Experimental nanodomain imaging: (Yamamoto et al., 25 Feb 2025)
• Spin filtering applications: (Samanta et al., 2024)
• Synthetic and engineered altermagnets: (Asgharpour et al., 2024)
• Phenomenological frameworks: (Mostovoy, 2 Jun 2025)