---
title: '2D Altermagnets: Symmetry-Driven Spin Splitting'
url: https://www.emergentmind.com/topics/two-dimensional-altermagnets
type: topic
---

# 2D Altermagnets: Symmetry-Driven Spin Splitting

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 [2401.05992, 2405.03557, 2601.10183].

## 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 [2405.03557, 2601.10183].

In strictly two dimensions, the exclusion rules are sharper. Operations of the forms \([C_2\Vert P]\), \([C_2\Vert \tau]\), \([C_2\Vert m_z]\), and \([C_2\Vert C_{2z}]\) enforce \(\varepsilon(\uparrow,\mathbf k)=\varepsilon(\downarrow,\mathbf k)\), 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 [2401.05992, 2601.10183].

Spin-group analysis yields seven nontrivial 2D spin-layer groups capable of supporting altermagnetism:
\[
{}^2 2/{}^{2}m_x,\quad {}^2 m^{2}m^{1}m,\quad {}^2 4/{}^{1}m,\quad {}^2 4/{}^{1}m^{2}m^{1}m,
\]
\[
{}^1 4/{}^{1}m^{2}m^{2}m,\quad {}^1\overline{3}{}^{2}m,\quad {}^1 6/{}^{1}m^{2}m^{2}m.
\]
These extend beyond the conventional five spin Laue groups used in earlier classifications. Around \(\Gamma\), the spin texture has an even winding number \(N\), with \(N=2\) corresponding to \(d\)-wave, \(N=4\) to \(g\)-wave, and \(N=6\) to \(i\)-wave altermagnetism [2405.03557, 2601.10183].

## 2. Band structure, order parameters, and hidden multipoles

At minimal level, the nonrelativistic electronic structure can be written as
\[
H(\mathbf k)=\varepsilon_0(\mathbf k)\sigma_0+\Delta(\mathbf k)\sigma_z,
\]
with \(\Delta(-\mathbf k)=\Delta(\mathbf k)\) but \(\Delta(g\mathbf k)=-\Delta(\mathbf k)\) for real-space operations \(g\in G-H\). On a square lattice, a standard \(d\)-wave form factor is \(f(\mathbf k)=\cos k_x-\cos k_y\), giving the familiar \(k\cdot p\) structure
\[
H(\mathbf k)\approx \varepsilon_0+\alpha(k_x^2+k_y^2)+\Delta_0(k_x^2-k_y^2)\sigma_z.
\]
This produces nodal lines where the connecting symmetry leaves \(\mathbf k\) invariant, and spin splitting elsewhere in the Brillouin zone [2601.10183, 2405.03557].

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 \(d\)-wave or \(g\)-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 \(g\)-wave altermagnetism in \(M\)-TCNX monolayers with \(mcm\) topology [2605.25128].

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 \(\sigma(k_x^2-k_y^2)\). 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 \(\tau\,\sigma\,(q_x^2-q_y^2)\), where \(\tau\) 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 [2507.18020].

## 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,
\[
\omega_{\mathbf k}^{(0,p)}=4JS\,\epsilon_{\mathbf k}-4pSJ_2'\sin(k_xa)\sin(k_ya),
\]
and the long-wavelength form
\[
\omega_{\mathbf k}^{\pm}\simeq c|\mathbf k|\pm \frac{k_xk_y}{m}.
\]
Nonlinear spin-wave theory further shows that the \(1/S\) correction vanishes at the \(\Gamma\)-point, so quantum fluctuations do not open a magnon gap, and that the ground-state energy up to order \(1/S^2\) is insensitive to the altermagnetic coupling \(J_2'\) [2507.11218].

Cichutek, Kopietz, and Rückriegel established a more specific consequence of this anisotropic spectrum: spontaneous three-magnon decay at \(T=0\). In their 2D altermagnetic model, convexity of the magnon dispersion permits one magnon to decay into three, with a low-\(k\) rate proportional to \(k^5\), a direction-dependent prefactor maximal along the Brillouin-zone diagonals, and a chirality selection rule such that, for fixed \(\mathbf k\), only one magnon branch can decay. This implies finite linewidths along the diagonals but not along the crystal axes at lowest order [2502.19815].

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 \(d\)-wave form. Integrating out magnons yields a same-spin pairing interaction, and the dominant superconducting instability is spin-polarized and chiral \(p\)-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 [2308.08606].

## 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 \(X\) and \(Y\) related by \(\{T\Vert C_{4z}\}\), band representations can be classified into two cases. In Case 1, the conduction–valence coupling produces bright \(s\)-like excitons; in Case 2, it produces bright \(p\)-like excitons. The exciton state is written as
\[
|\tau,S\rangle=\sum_{\mathbf k}A^S_{\tau,\mathbf k}\,c^\dagger_{\tau,c,\mathbf k}c_{\tau,v,\mathbf k}|GS\rangle,
\]
and \(A^S_{\tau,\mathbf k}\) together with the exciton energy \(\Omega^S_\tau\) follows from the Bethe–Salpeter equation. In the isotropic limit, hydrogenic \(s\) and \(p\) states are recovered; in the actual \(2mm\) symmetry, \(1s\), \(p_x\), and \(p_y\) transform as \(\Gamma^R_1\), \(\Gamma^R_3\), and \(\Gamma^R_4\), respectively [2506.05753].

The optical selection rules are correspondingly alternating. In Case 1, only the \(1s\) exciton at \(X\) is bright for \(x\)-polarized light, while only the \(1s\) exciton at \(Y\) is bright for \(y\)-polarized light. In Case 2, the bright states are \(2p_y\) in \(X\) and \(2p_x\) in \(Y\) for \(x\)-polarization, with the orthogonal pairing for \(y\)-polarization. Strain transfers directly from the electronic valleys to the excitons: the cited model gives valley shifts up to \(0.15\,{\rm eV}\) for \(5\%\) uniaxial strain in \( {\rm V}_2{\rm Se}_2{\rm O}\), and the resulting split resonances can be used to generate spin- and valley-polarized photocurrent [2506.05753].

A distinct Berry-phase effect is \(d\)-wave polarization–spin locking. In tetragonal \(d\)-wave altermagnets, the electronic polarization in spin channel \(\sigma\) is
\[
\mathbf P^\sigma=-\frac{e}{(2\pi)^2}\int_{\rm BZ} d^2k\,\mathbf A^\sigma(\mathbf k),
\]
and symmetry forces the spin-up and spin-down polarizations to be perpendicular when the parity-eigenvalue condition
\[
X_{-,\sigma}+Y_{-,\sigma}\equiv 1 \pmod 2
\]
is satisfied. First-principles calculations identified monolayer \({\rm Cr}_2X_2{\rm O}\) (\(X={\rm Se,Te}\)) 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 \(90^\circ\) [2502.16103].

## 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 \(1700\) magnetic monolayers, with four \(d\)-wave examples in \(P2_1'/c'\): \({\rm RuF}_4\), \({\rm VF}_4\), \({\rm AgF}_2\), and \({\rm OsF}_4\). 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 \(240\,{\rm meV}\) in \({\rm RuF}_4\) and \({\rm OsF}_4\), about \(50\,{\rm meV}\) in \({\rm VF}_4\), together with an \(i\)-wave example in 2H-\({\rm FeBr}_3\) [2401.05992].

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 \({\rm Cr}_2{\rm Si}_2{\rm S}_3{\rm Se}_3\), \({\rm Fe}_2{\rm P}_2{\rm S}_3{\rm Se}_3\), and \({\rm V}_2{\rm O}_2{\rm BrI}_3\), all with phonon-stable spectra in the reported calculations [2502.16217]. Spin-layer-group classification has likewise added monolayer \({\rm MnTeMoO}_6\) and \({\rm VP}_2{\rm H}_8({\rm NO}_4)_2\) as predicted 2D altermagnets [2405.03557].

Ferroelectric and multiferroic routes provide an electrical control knob. In VOX\(_2\) and VSX\(_2\) monolayers, lattice distortion converts a ferroelectric antiferromagnet with \([C_2\Vert t]\) into a ferroelectric altermagnet with \([C_2\Vert M]\); in \({\rm VOI}_2\), the reported spin splitting reaches about \(40\,{\rm meV}\) along \(S\!-\!\Gamma\!-\!S'\), reversing sign under ferroelectric switching, with calculated barriers of \(\sim0.10\)–\(0.22\,{\rm eV}\) per V atom [2504.06258]. In \({\rm CrPS}_3\) and \({\rm V}_2{\rm I}_2{\rm O}_2{\rm BrCl}\), reversing ferroelectric polarization simultaneously flips the electronic spin splitting and magnon chirality splitting; the calculated electronic splittings are about \(\pm 30\,{\rm meV}\) and \(\pm 25\,{\rm meV}\), respectively [2504.19585].

Van der Waals multiferroics add nonsymmorphic control. Zhao et al. showed that monolayer and bilayer \({\rm FeCuP}_2{\rm S}_6\) 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 \(\Gamma\!-\!S\) is about \(35\,{\rm meV}\); in bilayers it is about \(30\,{\rm meV}\) for one stacking type and \(12\,{\rm meV}\) for another. Interlayer sliding can suppress, restore, or reverse the sign of the spin splitting by changing the spin-space-group structure [2511.00712].

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 \(3330\) monolayers generated from 20 pentagonal templates, 94 dynamically stable altermagnetic candidates were identified, with \(\alpha\)-\({\rm CoS}_2\), \(\alpha\)-\({\rm CoP}_2\), and \({\rm FeSSe}\) as representative materials for the three response types [2606.27896].

## 6. Transport signatures, optical probes, and experimental realization

Transport observables are strongly symmetry-filtered. For intrinsic in-plane anomalous Hall conductivity \(\sigma_{xy}\), the spin-layer-group analysis shows that only two of the seven nontrivial 2D groups permit a nonzero response: the \(d\)-wave group \({}^2 2/{}^2 m\) and the \(i\)-wave group \({}^1\overline{3}{}^2 m\). Treating the Néel vector \(\mathbf N\) as an order parameter, the allowed invariant expansions are linear-plus-cubic in the \(d\)-wave case and purely cubic in the \(i\)-wave case. First-principles calculations on bilayer \({\rm MnPSe}_3\) gave peak values \(\max|\sigma_{xy}|\approx 202\,{\rm S/cm}\) for the \(d\)-wave stacking and \(\approx 25\,{\rm S/cm}\) for the \(i\)-wave stacking, with sign reversal under \(\mathbf N\to-\mathbf N\) [2502.21095].

Magneto-optical probes provide complementary access. In ferroelectric altermagnets, the Kerr angle tracks the electrically switchable spin splitting: in VOI\(_2\), the calculated Kerr rotation reaches \(10^{-4}\)–\(10^{-3}\,{\rm rad}\) near \(1\)–\(2\,{\rm eV}\), while in \({\rm CrPS}_3\) and \({\rm V}_2{\rm I}_2{\rm O}_2{\rm BrCl}\) the reported values are \(|\theta_K|\sim 0.1\)–\(0.3\,{\rm mrad}\) in the visible range, reversing sign when the ferroelectric polarization is reversed [2504.06258, 2504.19585]. 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 [2502.19815].

A decisive development is the reported experimental realization of genuine 2D altermagnetism in epitaxial CrSb ultrathin films on \({\rm Bi}_2{\rm Te}_3\). Unit-cell-thin films are ferrimagnetic because interfacial symmetry breaking produces a local moment imbalance; above a critical thickness of \(7/4\) unit cell, the key spin-space-group symmetries \([C_2\Vert C_{6z}t]\) and \([C_2\Vert M_z]\) are restored, and the net moment collapses from about \(0.94\,\mu_B\) per cell at \(1\) UC to about \(0.02\,\mu_B\) per cell at \(7/4\) UC. Scanning tunneling spectroscopy shows the disappearance of a Kondo resonance present at \(1\) UC, and ARPES reveals a momentum-dependent splitting of about \(0.55\,{\rm eV}\) along \(\Gamma\!-\!M\) but degeneracy along \(\Gamma\!-\!K\), matching the expected altermagnetic anisotropy [2510.12344].

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, \(m_z\), or \(C_{2z}\).

Source: https://www.emergentmind.com/topics/two-dimensional-altermagnets