Altermagnetic Order: Multipole Magnetism

• Altermagnetic order is a class of collinear magnets with zero net dipole and symmetry-driven multipolar (d-, g-, i-wave) spin splitting.
• It is modeled via minimal two-band Hamiltonians that exhibit momentum-dependent anisotropy, enforcing nodal lines and sign-changing spin textures.
• Experimental detection employs techniques like spin-resolved ARPES, X-ray dichroism, and neutron diffraction to probe the unique multipole order.

Altermagnetic order defines a distinct class of collinear, compensated magnetic phases characterized by momentum-dependent spin splitting and a vanishing net magnetization, with physics strongly determined by higher-order multipolar symmetry rather than conventional dipolar order. Unlike ferromagnets (uniform dipole, net $M\neq0$) and classical antiferromagnets (opposite dipoles, $M=0$, Kramers-degenerate bands via $T$ combined with translation/inversion), altermagnets possess sublattices related by point-group or rotation–time-reversal symmetries, which break $T$ macroscopically but enforce compensation and multi-lobe, even-parity spin or orbital textures. This generic symmetry principle underlies characteristic $d$-, $g$-, or $i$-wave anisotropy in both real and momentum space, manifests in unconventional transport and spectroscopic responses, and extends to spin liquids, orbital, fractionalized, and topological altermagnetic phases. The emergence, detection, and applications of altermagnetism require specialized consideration of multipole order parameters, symmetry class, and crystallographic or even amorphous setting.

1. Symmetry Principles and Classification

Altermagnetic order arises in collinear magnets where the constituent spin densities (or orbital magnetic moments) are ferroically ordered in an even-parity higher multipole ($\ell$), such that time-reversal $T$ is broken macroscopically, but the net dipole vanishes: $M=0$. The crucial symmetry is that sublattices (or orbitals/sites) are mapped not by pure translation/inversion, but by a spatial rotation $R$ combined with $T$—for instance, $C_nT$ (n-fold rotation $\times$ $T$). This protects compensation:

• Real-space: On each atom/site, the spin or orbital density $\mathbf{S}(\mathbf{r})$ or $\mathbf{M}_\mathrm{orb}(\mathbf{r})$ expands in spherical harmonics $Y_{\ell m}(\hat{r})$ with

$$\mathbf{S}(\mathbf{r}) = \sum_{\ell, m} c_{\ell m} Y_{\ell m}(\hat{r})\;,$$

with zero $s$-wave ($\ell=0$) but nonzero $d$ ($\ell=2$), $g$ ($\ell=4$), $i$ ($\ell=6$), etc. The higher multipoles (quadrupole, octupole, ...) change sign between symmetry-related sites.

• Momentum space: The band splitting displays even-parity anisotropy, such as

$$\Delta(\mathbf{k}) = \Delta_0 [\cos(k_x a) - \cos(k_y a)]$$

for $d$-wave symmetry, with sign-changing domains and nodal lines enforced by symmetry.

The symmetry group of an altermagnet is typically written as

$$Z_2^{C_2T} \ltimes SO(2)_\mathrm{spin} \times [ (E \parallel H) + (C_2 \parallel G-H) ]$$

where $C_2$ is a spin rotation, $H$ a spatial subgroup, $G$ the full crystal group. This construction ensures compensation and an anisotropic spin density, with the leading nonzero multipole an irreducible representation of $G$ with even parity and no dipole component (e.g., $B_{2g}$ for $d_{x^2-y^2}$).

2. Electronic Structure and Microscopic Models

Altermagnetic band splitting is fundamentally nonrelativistic and not dependent on spin-orbit coupling (SOC), but on symmetry-protected exchange terms:

• Minimal 2-band (spin) lattice Hamiltonian:

$$H(\mathbf{k}) = -2t [\cos k_x a + \cos k_y a]\, 1 + \Delta_0 [\cos k_x a - \cos k_y a]\, \sigma_z$$

with eigenvalues $$E_{\uparrow,\downarrow}(\mathbf{k}) = \varepsilon_0(\mathbf{k}) \pm \Delta(\mathbf{k})$$. The symmetry-protected nodal lines (e.g., $k_x = \pm k_y$ for $d$-wave) are robust even without SOC.

• Higher angular-momentum textures (e.g., $g$-wave, $i$-wave) require extended symmetry groups: in octagonal or dodecagonal quasicrystals, the band splitting transforms as $$\Delta(\mathbf{k}) \propto \cos l \theta_{\mathbf{k}}$$ with $l=4$ ($g$-wave), $l=6$ ($i$-wave), yielding 8 or 12 nodal directions, respectively (Chen et al., 24 Jul 2025).
• In itinerant settings (e.g., Lieb lattice metal), the altermagnetic instability is realized as a finite-$l$ Pomeranchuk instability, driven by sublattice interference and particle-hole condensation at specific $\mathbf{k}$-space locations (Dürrnagel et al., 2024).
• Orbitally driven analogues use tight-binding models with complex hoppings and staggered loop currents, showing compensation of real-space orbital moments but $k$-space $d$-wave orbital-momentum locking (Pan et al., 1 Oct 2025).

3. Multicomponent Order Parameters and Multipole Quantification

Altermagnetic order is inadequately characterized by a dipole, requiring magnetic multipoles of rank $\ell \geq 2$:

• Multipolar expansion: Define the generalized local multipole density

$$\mathcal{M}^{k, p}_{i_1 \cdots i_k, q} = \int r_{i_1} \cdots r_{i_k} \varrho^p_q(\mathbf{r}) d^3 r$$

with $p=1$ for magnetization, $k$ spatial rank. Spherical tensor multipoles $w^{kpr}_t$ established by coupling spatial and spin irreps (Martinelli et al., 19 Dec 2025).

• Quantitative relation between band splitting and the multipole moments:

$$\Delta = \sum_{k, p, r, t} \delta^{kpr}_t\, w^{kpr}_t$$

For $d$-wave (octupole, rank-3) and $g$-wave (triakontadipole, rank-5) order, the splitting arises as a superposition—a multi-component order parameter (e.g., $Q_t$, $O_t$, $T_t$).

• Practical measures for quantifying altermagnetic spin splitting (e.g., full-Brillouin zone averages, max splitting, high-symmetry-path averages) are needed for robust, material-independent detection (Martinelli et al., 19 Dec 2025).

4. Detectors and Experimental Probes

Altermagnetic signatures require specialized probes sensitive to multipolar and momentum-space anisotropy:

• Spin-resolved ARPES: Directly images momentum-dependent band splitting (nodal surfaces, sign flips, multi-lobe texture). Spin-resolved photoemission mapping in MnTe and CrSb verified altermagnetic band structure and multipole order (Jaeschke-Ubiergo et al., 13 Mar 2025).
• X-ray dichroism (XMCD, XMLD): Measures local multipole moments via polarization-dependent absorption, with XMLD being sensitive to even-parity multipoles and their sign-alternation (Amin et al., 2024, Yamamoto et al., 25 Feb 2025).
• Neutron diffraction: Higher-order multipoles modulate extra Bragg peaks (“forbidden” positions); can directly image multipolar arrangement (Jungwirth et al., 28 Jun 2025).
• Quantum-impurity relaxometry: NV centers can probe anisotropic spin noise encoding the $k$-space pattern unique to altermagnets (Bittencourt et al., 6 Aug 2025).
• Magnetotransport: Spontaneous anomalous Hall effect persists in zero net $M$ due to Berry-curvature hot spots generated by multipolar band textures—sign and magnitude controlled via current direction or orientation of the Néel/altermagnetic vector (Bangar et al., 20 May 2025).

5. Material Platforms and Realizations

Altermagnetism is realized in both crystalline and non-crystalline (amorphous, quasicrystalline, organic) systems:

• Perovskite oxides (e.g., LaTiO$_3$, CaCrO$_3$, LaVO$_3$, LaMnO$_3$). GdFeO$_3$-type distortions, multi-orbital electronic structure, and collinear $q=0$ order suffice—no SOC is required for pronounced $d$-wave splitting; SOC enables anomalous Hall effect (Naka et al., 2024).
• MnTe (hexagonal, NiAs-type): G-type Néel order is compensated but sublattices are not related by translation/inversion; compensating orientation of the Néel vector and sixfold symmetry enables synthesis and control of spontaneous anomalous Hall effect, domain textures, and nanoscale vortices (Bangar et al., 20 May 2025, Amin et al., 2024, Yamamoto et al., 25 Feb 2025).
• Organic and amorphous systems: Non-alternant nanographene frameworks supporting $S=1$ moments (DBH-based) with C$_4$ rotations achieve zero net $M$ and $d$-wave NRSS (Ortiz et al., 5 Aug 2025), and amorphous lattices with directional orbitals replicate altermagnetic response without crystallographic symmetry (d'Ornellas et al., 11 Apr 2025).
• Quasicrystals: Octagonal and dodecagonal tilings stabilize $g$-wave ($C_8T$ symmetry, 8-fold nodes) and $i$-wave ($C_{12}T$, 12-fold nodes) altermagnetism, accessible via spin-resolved ARPES and multi-tip transport (Chen et al., 24 Jul 2025).
• Metal-organic frameworks: K[Co(HCOO)$_3$] is a chiral second-order topological insulator with $g$-wave splitting and chirality-locked spin-polarized hinge modes (Xie et al., 18 Aug 2025).

6. Fractionalization, Topology, and Nonlinear Effects

Altermagnetism admits generalizations into fractionalized, topological, and nonlinear regimes, often sharing symmetry with unconventional superconductors and spin liquids:

• Fractionalized altermagnets: Quantum fluctuations (Schwinger boson and SU(2) gauge theory) stabilize $Z_2$ spin liquids, pseudo-altermagnets (splitting without local order), and topological altermagnets with momentum-dependent band splitting in the absence of net $M$ (Sobral et al., 2024, Vijayvargia et al., 12 Mar 2025).
• Nonlinear Hall and magnetization: In $d$-wave altermagnets, linear and intrinsic second-order Hall signals vanish due to $C_4T$ symmetry; electric-field-induced Berry curvature dipole generates strong, tunable second-order Hall currents sensitive to underlying symmetry, providing electrical discrimination of order parameter (e.g., between $B_{1g}$ and $B_{2g}$) (Mukherjee et al., 16 Oct 2025, Pan et al., 1 Oct 2025).
• Topological magnon Hall: Altermagnetic symmetry partial-splits magnon bands; Dzyaloshinskii-Moriya interaction and external field lift residual degeneracies, yielding nontrivial magnon Chern number, protected edge modes, and thermal Hall response $\kappa_{xy} \propto T^4$ at low temperatures (Khatua et al., 23 Jul 2025).

7. Outlook and Applications

Altermagnetism offers distinctive functional advantages for spintronics due to time-reversal breaking with zero stray field, robust charge-to-spin conversion (giant spin-splitter angles), high anisotropy in transport, and compatibility with superconductors and topological phases. Nanoscale texture control (via microstructuring, field-cooling) enables ultra-dense memory, neuromorphic architectures, and proximity-coupled quantum devices. The multipolar framework provides clear experimental fingerprints (multi-lobe ARPES, higher-order Bragg peaks, nonlinear Hall signals) and unifies design principles across crystalline and amorphous platforms (Martinelli et al., 19 Dec 2025, Ortiz et al., 5 Aug 2025). Theoretical extensions into fractional, topological, and non-crystalline contexts further expand the landscape for “altertronics,” promising ultrafast, scalable, stray-field-free spintronic devices.