Altermagnetic Phases: Symmetry Spin Splitting

• Altermagnetic phases are a class of collinear magnetic order exhibiting momentum-dependent, symmetry-enforced spin splitting with a compensated zero net magnetization.
• They feature distinct spin splitting mechanisms originating from non-relativistic local crystalline fields, leading to unique transport signatures such as the anomalous Hall effect at zero magnetization.
• Emerging material platforms in 3D oxides, 2D layers, and engineered heterostructures, alongside advanced theoretical models, underscore their potential for novel spintronic and topological applications.

Altermagnetic phases define a distinctive class of collinear magnetic order characterized by momentum-dependent, non-relativistic spin splitting and compensated (zero) net magnetization. Unlike conventional ferromagnetic (FM) or antiferromagnetic (AFM) systems, altermagnets maintain a collinear spin axis but exhibit an alternating spin–momentum locking protected and enabled by the underlying crystalline symmetries. Recent advances have established deep theoretical foundations and confirmed experimental signatures, including symmetry-driven spin splitting, unique transport phenomena, and the emergence of novel correlated and topological phases. Altermagnetism now encompasses three-dimensional crystals, two-dimensional layers, optical lattices, quasicrystals, heterostructures, and even metal–organic frameworks with chiral topological responses.

1. Symmetry and Classification of Altermagnetic Phases

Altermagnetism arises from the spontaneous breaking of time-reversal ($\mathcal{T}$) symmetry while retaining a collinear compensated spin structure and particular composite symmetries linking spatial and spin degrees of freedom. The formalism relies on "spin-group" theory, wherein symmetry operations are treated as pairs $[R_1 || R_2]$, with $R_1$ acting in spin space and $R_2$ in real space (Šmejkal et al., 2021, Tamang et al., 6 Dec 2024, Jungwirth et al., 28 Jun 2025).

The classification of collinear, non-relativistic magnets yields three distinct types:

• I. Ferromagnets: $[E || G]$, with $G$ the full Laue/crystallographic group; spin bands are non-degenerate everywhere.
• II. Conventional Antiferromagnets: $[E || G] + [C_2 || G]$; degeneracy is forced via translation or inversion, producing fully compensated, spin-degenerate bands.
• III. Altermagnets: $R_s^{(\mathrm{III})} = [E||H] + [C_2 || A][E||H]$, where $H$ is a halving subgroup (mapping same-spin sublattices) and $A$ is a rotation or improper operation exchanging opposite-spin sublattices. This coset decomposition ensures an alternating, momentum-dependent spin splitting while enforcing global compensation.

Momentum-space spin splitting then assumes even-parity "wave-like" forms (e.g., $d$-, $g$-, or $i$-wave), determined by the point group and the composite symmetry (such as $C_4 T$, $C_6 T$, or in quasicrystals, $C_8 T$, $C_{12} T$), and is quantified by an even integer winding number (e.g., $W=2,4,6$) (Šmejkal et al., 2021, Chen et al., 24 Jul 2025).

Altermagnetism is enforced not by spin–orbit coupling (SOC) but by non-relativistic local crystalline fields; Kramers degeneracy is lifted by purely symmetry-derived, momentum-dependent mechanisms.

2. Spin-Splitting Mechanisms and Electronic Structure

The spin splitting in altermagnets fundamentally differs from Zeeman or SOC-induced mechanisms. In collinear altermagnetic states, the local electric crystal field, present even in the paramagnetic phase, causes $k$-dependent splittings on different sublattices. Upon entering the altermagnetic phase, these splittings copy themselves in a spin-dependent fashion: dominant sublattice A states acquire spin-up polarization, and B, spin-down, or vice versa (Šmejkal et al., 2021).

The resulting energy bands

$$\Delta E(\mathbf{k}) = E_\uparrow(\mathbf{k}) - E_\downarrow(\mathbf{k})$$

show large, non-relativistic separation (on the order of hundreds of meV up to 1 eV), despite the total moment vanishing. Symmetry Analysis reveals "nodal surfaces" (momentum lines or planes where spin degeneracy is restored) dictated by composite operations such as $C_2T$ or, in quasicrystals, $C_8T$, $C_{12}T$ (Chen et al., 24 Jul 2025, Li et al., 3 Aug 2025). The momentum-dependent splitting may follow $d$-wave, $g$-wave, $i$-wave, $h$-wave, or higher angular symmetry, depending on lattice structure (Jungwirth et al., 28 Jun 2025, Li et al., 3 Aug 2025).

When SOC is included, originally protected nodal lines may gap, leading to mirror Chern bands and facilitating topological crystalline insulating behavior with Dirac edge states (Sattigeri et al., 12 Jun 2025). The spatial and momentum structure of the spin density reflects a ferroic order of anisotropic higher-partial-wave ($l>0$) components atop an antiparallel dipolar background.

3. Material Realizations and Experimental Evidence

Altermagnetism has now been identified—via first-principles density functional theory (DFT), symmetry analysis, and experiments—in a broad class of compounds (Šmejkal et al., 2021, Bernardini et al., 23 Jan 2024, Jeong et al., 9 May 2024, Jungwirth et al., 28 Jun 2025):

• 3D oxides: RuO$_2$ (high-$T_N$), KRu$_4$O$_8$, MnTe, CrSb, La$_2$CuO$_4$, BiFeO$_3$, La$_2$NiO$_4$, La$_3$Ni$_2$O$_7$
• 2D systems: Cr$_2$BAl (MBene), FeSe, RuO$_2$ thin films (Sattigeri et al., 12 Jun 2025)
• Chiral MOFs: K[Co(HCOO)$_3$], enabling chirality-locked topological transport (Xie et al., 18 Aug 2025)
• Quasicrystals: Ammann–Beenker ($C_8$) and Penrose ($C_{10}$) tilings, supporting $g$-wave and $h$-wave altermagnetism (Chen et al., 24 Jul 2025, Li et al., 3 Aug 2025)
• Optical lattices and Hubbard-type models with ultracold atoms (Das et al., 2023)
• Engineered heterostructures: Ultrathin epitaxially strained RuO$_2$/TiO$_2$ films exhibit novel metallic polar altermagnetic phases (Jeong et al., 9 May 2024)

Experimental probes include:

• Angle-resolved photoemission spectroscopy (ARPES, spin-resolved): Direct imaging of momentum-dependent spin splitting and nodal lines (Šmejkal et al., 2021, Vita et al., 27 Feb 2025, Jungwirth et al., 28 Jun 2025)
• Muon spin rotation, neutron scattering, XMCD: Confirmation of collinear compensated orders and spin textures
• Transport (AHE, Nernst, Hall, and spin torque measurements): Hall responses even at zero magnetization
• Optical SHG and Kerr/Faraday rotation: Symmetry fingerprinting of altermagnetic order
• Quantum oscillations (Shubnikov–de Haas): Discontinuous splitting of Landau quantization frequencies arising from momentum-dependent spin splitting (Li et al., 6 Jun 2024)

A systematic bottom-up design using spin clusters with explicit blends of ferromagnetic and antiferromagnetic correlations has been introduced to expand the catalog of models and candidates (Zhu et al., 12 Apr 2025).

4. Topological and Correlated Quantum Phases

Altermagnetism not only underpins novel electronic responses but also enables robust correlated and topological phases:

• Second-order topological insulators: Chiral altermagnetic MOFs host hinge-localized spin-polarized states, switchable via chiral lattice inversion, with sign-reversed Kerr, Faraday, and AHE in left/right enantiomers (Xie et al., 18 Aug 2025).
• Topological crystalline insulators with Dirac edge states: Demonstrated in 2D Cr$_2$BAl (Sattigeri et al., 12 Jun 2025).
• Quantum spin liquids and fractionalization: Altermagnetic local order can coexist with $\mathbb{Z}_2$ topological order and exotic excitations in Kitaev bilayer models, resulting in d-wave spin-split bands of neutral fermions (Vijayvargia et al., 12 Mar 2025).
• Spin-triplet excitonic insulators: In LiFe$_2$F$_6$, strong magnetoelectric coupling enables electrical control and switching of spin-triplet condensates (Guo et al., 2023).
• Quasicrystals: Noncrystallographic rotational symmetry (e.g., $C_8T$, $C_{12}T$) leads to $g$- and $i$-wave alternating spin splitting with eight- and twelve-fold nodal patterns in spectral functions and conductance (Chen et al., 24 Jul 2025, Li et al., 3 Aug 2025).

5. Novel Transport, Optical, and Spintronic Functionalities

The unique combination of collinear compensated order with large, symmetry-enforced, nonrelativistic spin splitting enables:

• Anomalous Hall and crystal Nernst effects at zero net magnetization.
• Chiral magnon modes and THz frequency dynamics.
• Electric-field manipulation of spin splitting on surfaces ("activation" of blind orientations) (Sattigeri et al., 2023).
• Optical switching: Ultrafast laser pulses allow direct quenching and recovery of the altermagnetic phase in layered materials, establishing the concept of "altertronics" (Vita et al., 27 Feb 2025).
• Sign-reversible transport: Lattice chirality inversion enables universal sign switching in anomalous Hall and optical responses (Xie et al., 18 Aug 2025).
• Dissipationless edge channel transport: Dirac edge states in topological crystalline altermagnets provide robust, gate-tunable spin current channels (Sattigeri et al., 12 Jun 2025).
• Highly-tunable spin-charge interconversion due to interaction-driven band splitting tunable by doping, strain, or pressure (Giuli et al., 1 Oct 2024).

Altermagnetism is also extendable to spin–orbital coupled phases ("spin–orbital altermagnetism"), facilitating complex spin–orbital textures in 3d systems and controlled via both intrinsic composite orders and extrinsic lattice effects (e.g., Jahn–Teller distortions) (Wang et al., 19 Sep 2025).

6. Theoretical Models, Methodologies, and Material Design Approaches

Sophisticated methodologies enable both prediction and design:

• Density-functional theory (DFT) and symmetry-based high-throughput screening for candidate discovery (Šmejkal et al., 2021, Bernardini et al., 23 Jan 2024)
• Group-theoretic spin-group taxonomy and Landau theory for rigorous phase classification (Šmejkal et al., 2021, Tamang et al., 6 Dec 2024, Jungwirth et al., 28 Jun 2025)
• Mean-field and many-body modeling: Hartree-Fock, functional renormalization, dynamical mean-field theory (DMFT), and rotationally invariant slave bosons capture interaction-driven and itinerant altermagnetic phases (Giuli et al., 1 Oct 2024, Dürrnagel et al., 18 Dec 2024, Das et al., 2023)
• Real-space and momentum-space symmetry analysis, with explicit construction of decorated aperiodic (quasicrystalline) lattices and spin–orbital Hamiltonians (Chen et al., 24 Jul 2025, Li et al., 3 Aug 2025, Wang et al., 19 Sep 2025)
• Design toolkit via spin clusters: Enables programmable construction of target altermagnetic orders with "designer" local and global symmetry (Zhu et al., 12 Apr 2025)

These methodologies also allow identification of key parameters (e.g., critical interaction strengths, strain regimes) for engineering phase transitions and controlling transport coefficients.

7. Outlook and Emerging Directions

Altermagnetism is rapidly evolving from a theoretical paradigm into an experimentally substantiated platform poised for technological exploitation:

• Materials scope: Ruddlesden–Popper phases, perovskites, layered transition metal dichalcogenides, MOFs, quasicrystals, 2D monolayers and van der Waals heterostructures, correlated oxides, and artificially engineered optical lattices.
• Device paradigms: Ultra-low-power memory and logic, THz and chiral magnonics, nonvolatile chirality-programmable interconnects, and "altertronics."
• Control handles: Electric field, strain, chemical doping, laser fluence, lattice chirality inversion, dimensional engineering, and supercell/twist-induced symmetry breaking.
• Frontiers: Integration with topological superconductivity, realization in neutral and charge–fractionalized systems (quantum spin liquids), transport in quasicrystal-based devices, and electrical and light-field-based switching of complex spin–orbital–momentum entangled phases.

Future work will focus on thorough experimental validation across new platforms, engineering of robust device architectures, elucidation of interplay with superconductivity or topological phases, and ongoing symmetry-driven material discovery. The rich physics of altermagnetism—spanning symmetry, electronic structure, topology, and quantum correlations—positions it as a key extension to the magnetic phase diagram and a cornerstone of next-generation quantum technology (Bai et al., 4 Jun 2024, Tamang et al., 6 Dec 2024, Jungwirth et al., 28 Jun 2025).