2D Altermagnets: Symmetry-Driven Spin Splitting

• Two-dimensional altermagnets are compensated collinear magnets that exhibit nonrelativistic spin splitting driven by specific in-plane symmetry operations, distinguishing them from conventional antiferromagnets.
• They are classified into d-, g-, and i-wave categories based on even-parity spin-momentum locking as revealed through detailed spin-group symmetry analysis.
• Their unique electronic, magnonic, and optical responses enable innovative applications in spintronics, multiferroics, and superconductivity through tunable material platforms.

Two-dimensional altermagnets are collinear, compensated magnets with zero net magnetization whose electronic bands are split between opposite spins in momentum space by a nonrelativistic mechanism, rather than by spin–orbit coupling. In the 2D limit this phenomenon is symmetry-constrained more strongly than in bulk, because inversion-related, translation-related, $m_z$-related, or $C_{2z}$-related spin sublattices restore spin degeneracy. Consequently, 2D altermagnetism requires opposite-spin sublattices to be connected instead by proper or improper in-plane rotations or mirrors encoded in spin-group or spin-space-group symmetries. The resulting even-parity spin-momentum locking is commonly organized into $d$-, $g$-, and $i$-wave classes and has become the basis for a broad literature on electronic, magnonic, optical, and multiferroic responses in atomically thin systems (Sødequist et al., 2024, Zeng et al., 2024, Zeng et al., 15 Jan 2026).

1. Symmetry foundations and classification

The modern description of 2D altermagnetism is formulated in spin-group language, where a symmetry operation is written as $[R_{\rm spin}\Vert R_{\rm real}]$, with separate actions on spin and real space. For collinear magnets, the nontrivial altermagnetic structure is captured by

$R_3=[E\Vert H]+[C_2\Vert G-H],$

where $G$ is the full layer group and $H$ is a halving subgroup connecting sites of the same spin. The coset $G-H$ exchanges the two collinear spin sublattices through a real-space operation accompanied by a spin flip. This distinguishes altermagnets from conventional antiferromagnets, in which opposite-spin sublattices are connected by inversion or pure translation and therefore remain spin-degenerate in the absence of SOC (Zeng et al., 2024, Zeng et al., 15 Jan 2026).

In strictly two dimensions, the exclusion rules are sharper. Operations of the forms $C_{2z}$0, $C_{2z}$1, $C_{2z}$2, and $C_{2z}$3 enforce $C_{2z}$4, so they are incompatible with nonrelativistic altermagnetic splitting. A recurring misconception is therefore that any compensated collinear monolayer with broken time reversal is an altermagnet; the symmetry criteria show that this is not sufficient (Sødequist et al., 2024, Zeng et al., 15 Jan 2026).

Spin-group analysis yields seven nontrivial 2D spin-layer groups capable of supporting altermagnetism: $C_{2z}$5

$C_{2z}$6

These extend beyond the conventional five spin Laue groups used in earlier classifications. Around $C_{2z}$7, the spin texture has an even winding number $C_{2z}$8, with $C_{2z}$9 corresponding to $d$0-wave, $d$1 to $d$2-wave, and $d$3 to $d$4-wave altermagnetism (Zeng et al., 2024, Zeng et al., 15 Jan 2026).

2. Band structure, order parameters, and hidden multipoles

At minimal level, the nonrelativistic electronic structure can be written as

$d$5

with $d$6 but $d$7 for real-space operations $d$8. On a square lattice, a standard $d$9-wave form factor is $g$0, giving the familiar $g$1 structure

$g$2

This produces nodal lines where the connecting symmetry leaves $g$3 invariant, and spin splitting elsewhere in the Brillouin zone (Zeng et al., 15 Jan 2026, Zeng et al., 2024).

Microscopic orbital content is decisive. In an antiferromagnetic square lattice, a single-orbital model remains spin-degenerate, whereas interwoven dual-orbital configurations lift Kramers degeneracy and generate $g$4-wave or $g$5-wave altermagnetic states. In the orbital-engineered square-lattice framework, the spin splitting originates from orbital anisotropy in same-spin hopping channels; this was used to motivate $g$6-wave altermagnetism in $g$7-TCNX monolayers with $g$8 topology (Che et al., 24 May 2026).

The hidden order carried by 2D altermagnets is not exhausted by dipolar language. Tani and Zülicke showed that a generic minimal three-site model carries a non-zero magnetic-octupole order characterized by the indicator $g$9. In a four-site model representative of monolayer FeSe, the magnetic-octupole order vanishes globally, while a magnetic-hexadecapole order appears instead through the combined degree of freedom $i$0, where $i$1 is a sublattice-isospin variable. In that case, altermagnetic band splitting arises through the interplay of spin and sublattice isospin rather than through a pure octupolar channel (Tani et al., 24 Jul 2025).

3. Collective excitations and interaction-driven phenomena

Two-dimensional altermagnets also possess nontrivial bosonic spectra. In the checkerboard Heisenberg model with antiferromagnetic nearest-neighbor exchange and two inequivalent diagonal exchanges, the quadratic spin-wave Hamiltonian yields two chiral magnon branches,

$i$2

and the long-wavelength form

$i$3

Nonlinear spin-wave theory further shows that the $i$4 correction vanishes at the $i$5-point, so quantum fluctuations do not open a magnon gap, and that the ground-state energy up to order $i$6 is insensitive to the altermagnetic coupling $i$7 (Cichutek et al., 15 Jul 2025).

Cichutek, Kopietz, and Rückriegel established a more specific consequence of this anisotropic spectrum: spontaneous three-magnon decay at $i$8. In their 2D altermagnetic model, convexity of the magnon dispersion permits one magnon to decay into three, with a low-$i$9 rate proportional to $[R_{\rm spin}\Vert R_{\rm real}]$0, a direction-dependent prefactor maximal along the Brillouin-zone diagonals, and a chirality selection rule such that, for fixed $[R_{\rm spin}\Vert R_{\rm real}]$1, only one magnon branch can decay. This implies finite linewidths along the diagonals but not along the crystal axes at lowest order (Cichutek et al., 27 Feb 2025).

The same electronic and magnonic asymmetries feed into pairing problems. In a minimal three-site microscopic model without SOC, spin-split electron bands and nondegenerate magnon bands both inherit a $[R_{\rm spin}\Vert R_{\rm real}]$2-wave form. Integrating out magnons yields a same-spin pairing interaction, and the dominant superconducting instability is spin-polarized and chiral $[R_{\rm spin}\Vert R_{\rm real}]$3-wave, with a critical temperature that can be enhanced strongly by tuning the chemical potential. This identifies altermagnetic band geometry as a direct input into magnon-mediated superconductivity (Brekke et al., 2023).

4. Excitons, Berry-phase optics, and polarization–spin locking

Excitonic structure in 2D altermagnets has been formulated in spin-space-group terms. For spin-polarized valleys $[R_{\rm spin}\Vert R_{\rm real}]$4 and $[R_{\rm spin}\Vert R_{\rm real}]$5 related by $[R_{\rm spin}\Vert R_{\rm real}]$6, band representations can be classified into two cases. In Case 1, the conduction–valence coupling produces bright $[R_{\rm spin}\Vert R_{\rm real}]$7-like excitons; in Case 2, it produces bright $[R_{\rm spin}\Vert R_{\rm real}]$8-like excitons. The exciton state is written as

$[R_{\rm spin}\Vert R_{\rm real}]$9

and $R_3=[E\Vert H]+[C_2\Vert G-H],$0 together with the exciton energy $R_3=[E\Vert H]+[C_2\Vert G-H],$1 follows from the Bethe–Salpeter equation. In the isotropic limit, hydrogenic $R_3=[E\Vert H]+[C_2\Vert G-H],$2 and $R_3=[E\Vert H]+[C_2\Vert G-H],$3 states are recovered; in the actual $R_3=[E\Vert H]+[C_2\Vert G-H],$4 symmetry, $R_3=[E\Vert H]+[C_2\Vert G-H],$5, $R_3=[E\Vert H]+[C_2\Vert G-H],$6, and $R_3=[E\Vert H]+[C_2\Vert G-H],$7 transform as $R_3=[E\Vert H]+[C_2\Vert G-H],$8, $R_3=[E\Vert H]+[C_2\Vert G-H],$9, and $G$0, respectively (Cao et al., 6 Jun 2025).

The optical selection rules are correspondingly alternating. In Case 1, only the $G$1 exciton at $G$2 is bright for $G$3-polarized light, while only the $G$4 exciton at $G$5 is bright for $G$6-polarized light. In Case 2, the bright states are $G$7 in $G$8 and $G$9 in $H$0 for $H$1-polarization, with the orthogonal pairing for $H$2-polarization. Strain transfers directly from the electronic valleys to the excitons: the cited model gives valley shifts up to $H$3 for $H$4 uniaxial strain in $H$5, and the resulting split resonances can be used to generate spin- and valley-polarized photocurrent (Cao et al., 6 Jun 2025).

A distinct Berry-phase effect is $H$6-wave polarization–spin locking. In tetragonal $H$7-wave altermagnets, the electronic polarization in spin channel $H$8 is

$H$9

and symmetry forces the spin-up and spin-down polarizations to be perpendicular when the parity-eigenvalue condition

$G-H$0

is satisfied. First-principles calculations identified monolayer $G-H$1 ($G-H$2) as candidate materials, with spin-up and spin-down electrons accumulating at orthogonal edges. The same framework implies a spin-driven ferroelectricity in which rotating the Néel vector rotates the polarization pattern by $G-H$3 (Liu et al., 22 Feb 2025).

5. Materials platforms and routes to control

High-throughput and symmetry-guided searches have established multiple material classes. A C2DB-based screening found seven 2D altermagnets among roughly $G-H$4 magnetic monolayers, with four $G-H$5-wave examples in $G-H$6: $G-H$7, $G-H$8, $G-H$9, and $C_{2z}$00. The first three are experimentally known van der Waals materials in bulk form, and the first-principles analysis reported nonrelativistic valence-band splitting up to $C_{2z}$01 in $C_{2z}$02 and $C_{2z}$03, about $C_{2z}$04 in $C_{2z}$05, together with an $C_{2z}$06-wave example in 2H-$C_{2z}$07 (Sødequist et al., 2024).

A complementary “multi-component altermagnet” construction uses subgroup orbits to generate candidate crystal structures and then enumerates collinear compensated spin patterns by coset decomposition. This produced previously unreported 2D altermagnets such as $C_{2z}$08, $C_{2z}$09, and $C_{2z}$10, all with phonon-stable spectra in the reported calculations (Peng et al., 22 Feb 2025). Spin-layer-group classification has likewise added monolayer $C_{2z}$11 and $C_{2z}$12 as predicted 2D altermagnets (Zeng et al., 2024).

Ferroelectric and multiferroic routes provide an electrical control knob. In VOX$C_{2z}$13 and VSX$C_{2z}$14 monolayers, lattice distortion converts a ferroelectric antiferromagnet with $C_{2z}$15 into a ferroelectric altermagnet with $C_{2z}$16; in $C_{2z}$17, the reported spin splitting reaches about $C_{2z}$18 along $C_{2z}$19, reversing sign under ferroelectric switching, with calculated barriers of $C_{2z}$20–$C_{2z}$21 per V atom (Zhu et al., 8 Apr 2025). In $C_{2z}$22 and $C_{2z}$23, reversing ferroelectric polarization simultaneously flips the electronic spin splitting and magnon chirality splitting; the calculated electronic splittings are about $C_{2z}$24 and $C_{2z}$25, respectively (Wang et al., 28 Apr 2025).

Van der Waals multiferroics add nonsymmorphic control. Zhao et al. showed that monolayer and bilayer $C_{2z}$26 become altermagnetic when opposite-spin ferroelectric sublattices are connected by a screw axis rather than by pure translation or inversion. In the antiferroelectric monolayer, the maximum splitting along $C_{2z}$27 is about $C_{2z}$28; in bilayers it is about $C_{2z}$29 for one stacking type and $C_{2z}$30 for another. Interlayer sliding can suppress, restore, or reverse the sign of the spin splitting by changing the spin-space-group structure (Zhao et al., 1 Nov 2025).

Strain is another symmetry-selective tuning parameter. A strain-resolved framework for orthorhombic pentagonal altermagnets classified responses into Type I, Type II, and Type III according to whether strain preserves, reconstructs, or destroys the altermagnetic spin-momentum locking. In a screened set of $C_{2z}$31 monolayers generated from 20 pentagonal templates, 94 dynamically stable altermagnetic candidates were identified, with $C_{2z}$32-$C_{2z}$33, $C_{2z}$34-$C_{2z}$35, and $C_{2z}$36 as representative materials for the three response types (Wang et al., 26 Jun 2026).

6. Transport signatures, optical probes, and experimental realization

Transport observables are strongly symmetry-filtered. For intrinsic in-plane anomalous Hall conductivity $C_{2z}$37, the spin-layer-group analysis shows that only two of the seven nontrivial 2D groups permit a nonzero response: the $C_{2z}$38-wave group $C_{2z}$39 and the $C_{2z}$40-wave group $C_{2z}$41. Treating the Néel vector $C_{2z}$42 as an order parameter, the allowed invariant expansions are linear-plus-cubic in the $C_{2z}$43-wave case and purely cubic in the $C_{2z}$44-wave case. First-principles calculations on bilayer $C_{2z}$45 gave peak values $C_{2z}$46 for the $C_{2z}$47-wave stacking and $C_{2z}$48 for the $C_{2z}$49-wave stacking, with sign reversal under $C_{2z}$50 (Sheoran et al., 28 Feb 2025).

Magneto-optical probes provide complementary access. In ferroelectric altermagnets, the Kerr angle tracks the electrically switchable spin splitting: in VOI$C_{2z}$51, the calculated Kerr rotation reaches $C_{2z}$52–$C_{2z}$53 near $C_{2z}$54–$C_{2z}$55, while in $C_{2z}$56 and $C_{2z}$57 the reported values are $C_{2z}$58–$C_{2z}$59 in the visible range, reversing sign when the ferroelectric polarization is reversed (Zhu et al., 8 Apr 2025, Wang et al., 28 Apr 2025). For magnons, the predicted spontaneous decay channels imply anisotropic zero-temperature linewidths observable by inelastic neutron scattering or Brillouin-light scattering, especially along Brillouin-zone diagonals (Cichutek et al., 27 Feb 2025).

A decisive development is the reported experimental realization of genuine 2D altermagnetism in epitaxial CrSb ultrathin films on $C_{2z}$60. Unit-cell-thin films are ferrimagnetic because interfacial symmetry breaking produces a local moment imbalance; above a critical thickness of $C_{2z}$61 unit cell, the key spin-space-group symmetries $C_{2z}$62 and $C_{2z}$63 are restored, and the net moment collapses from about $C_{2z}$64 per cell at $C_{2z}$65 UC to about $C_{2z}$66 per cell at $C_{2z}$67 UC. Scanning tunneling spectroscopy shows the disappearance of a Kondo resonance present at $C_{2z}$68 UC, and ARPES reveals a momentum-dependent splitting of about $C_{2z}$69 along $C_{2z}$70 but degeneracy along $C_{2z}$71, matching the expected altermagnetic anisotropy (Li et al., 14 Oct 2025).

Taken together, these results establish 2D altermagnets as a symmetry-defined class of compensated magnets in which nonrelativistic spin splitting survives dimensional reduction and reorganizes electronic, optical, and bosonic structure. The field now spans symmetry classification, microscopic model building, materials screening, multiferroic control, and an initial experimental realization, with the central organizing principle remaining the same: zero net moment does not imply spin degeneracy when the opposite-spin sublattices are connected by the appropriate spin-space symmetry rather than by inversion, translation, $C_{2z}$72, or $C_{2z}$73.