Bulk g-Wave Altermagnets

• Bulk g-wave altermagnets are thermodynamically stable, collinear, spin-compensated phases characterized by a momentum‐dependent exchange field and an even-parity l=4 form factor.
• They are distinguished by four symmetry-enforced nodal surfaces, placing them between conventional antiferromagnets and ferromagnets in both spectroscopy and symmetry classification.
• Experimental studies on materials like CrSb, CoF₃, and G-type BiFeO₃ reveal unique electronic, magnon, and surface transport responses that elucidate the practical impact of g-wave symmetry.

Bulk g-wave altermagnets are thermodynamically stable, collinear, spin-compensated magnetic phases whose band spin splitting transforms with g-wave symmetry rather than as a uniform exchange field. In periodic crystals this denotes an even-parity $l=4$ form factor with four spin-degenerate nodal surfaces crossing $\Gamma$, whereas in the octagonal quasicrystalline setting the same label is used by analogy with orbital harmonics for a $C_8T$-protected eight-sector sign-changing pattern. In both settings, zero net magnetization coexists with nonrelativistic spin-split electronic structure, placing these phases between conventional antiferromagnets and ferromagnets in symmetry and spectroscopy (Jungwirth et al., 2024, Chen et al., 24 Jul 2025).

1. Classification and defining criteria

Altermagnets form the third collinear class beside ferromagnets and conventional antiferromagnets. In the spin-group formulation, ferromagnets have $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$, conventional antiferromagnets have $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$, and altermagnets have $$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$, with $\mathbf{H}$ a halving subgroup of the crystallographic Laue group $\mathbf{G}$ and $A$ a proper or improper rotation but not inversion. This structure permits spin splitting at generic $\mathbf{k}$ while preserving zero net magnetization, because spin-up and spin-down states are related at different momenta rather than at the same momentum (Jungwirth et al., 2024, Jaeschke-Ubiergo et al., 2023).

Within the even-parity altermagnetic hierarchy, d-wave, g-wave, and i-wave phases are distinguished by the symmetry of the spin-splitting form factor and by the number of nodal surfaces through $\Gamma$0. The review literature states that d-wave, g-wave, and i-wave altermagnets have 2, 4, and 6 nodal surfaces crossing $\Gamma$1, respectively (Jungwirth et al., 2024). In the three-dimensional classification developed for G-type $\Gamma$2, the spin-splitting function $\Gamma$3 transforms under a one-dimensional inversion-even irrep of the crystallographic Laue group, and the phase is called g-wave when the lowest even $\Gamma$4 containing that irrep is $\Gamma$5 (Urru et al., 25 May 2025).

The octagonal quasicrystal literature uses the same label in a distinct symmetry setting. There, “g-wave” refers to an eightfold angular structure in the spin-resolved spectral function difference $\Gamma$6, antisymmetric under $\Gamma$7 and vanishing along $\Gamma$8. The phase breaks $\Gamma$9 and $C_8T$0 separately but preserves $C_8T$1, and the g-wave designation is tied to this non-crystallographic global rotation symmetry rather than to Bloch-band point-group harmonics (Chen et al., 24 Jul 2025).

2. Symmetry mechanisms and order parameters

The defining symmetry mechanism of a bulk g-wave altermagnet is a momentum-dependent exchange field controlled by a higher-rank magnetic multipole. In CrSb, a Landau-type description identifies the order parameter as a rank-5 magnetic multipole moment

$C_8T$2

with Néel vector along $C_8T$3. The corresponding bulk spin-splitting term near $C_8T$4 is

$C_8T$5

so the g-wave texture is encoded directly in the $C_8T$6-space form factor (Sorn et al., 17 Jun 2026).

A complementary real-space perspective is provided by atomic altermagnetism. In MnTe and CrSb, first-principles spin-density decompositions reveal a ferroically ordered even-parity $C_8T$7 atomic form factor on the magnetic site, labeled $C_8T$8. In MnTe, the staggered $C_8T$9-wave dipolar component satisfies $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$0, while the g-wave component is ferroic,

$\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$1

establishing a direct-space counterpart of the momentum-space g-wave spin splitting (Jaeschke-Ubiergo et al., 13 Mar 2025).

Specific bulk models make this symmetry explicit. In CoF$\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$2, the spin group is identified as $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$3, and the effective altermagnetic term is

$\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$4

which is even under inversion, vanishes on symmetry-dictated nodal manifolds, and produces nonrelativistic spin splitting without spin–orbit coupling (Tagani, 2024). In the octagonal quasicrystal model, the local order parameter is

$\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$5

and the full mean-field Hamiltonian preserves the composite antiunitary symmetry $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$6 rather than any crystal translation symmetry (Chen et al., 24 Jul 2025).

3. Crystalline and quasicrystalline bulk realizations

The current literature spans bulk periodic crystals, supercell altermagnets, rare-earth magnets, multiferroics, and model quasicrystals. Representative cases are summarized below.

System	Structural or symmetry setting	Bulk g-wave hallmark
CrSb (Sorn et al., 17 Jun 2026)	Hexagonal, $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$7, Néel vector $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$8	Rank-5 MMM and $\mathbf{R}_s^{\rm I}=[E\parallel \mathbf{G}]$9
CoF$$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$0 (Tagani, 2024)	Rhombohedral $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$1, spin group $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$2	Bulk spin splitting up to $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$3 meV
G-type BiFeO$$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$4 (Urru et al., 25 May 2025)	Rhombohedral $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$5, multiferroic	$$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$6 of $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$7, lowest even $$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$8
TbPt$$\mathbf{R}_s^{\rm II}=[E\parallel \mathbf{G}] + [C_2\parallel \mathbf{G}]$$9Al$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$0 (Oishi et al., 12 Sep 2025)	Trigonal $$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$1, magnetic point group $$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$2	Spin Laue group $$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$3
CsCoCl$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$4, RbCoBr$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$5, BaMnO$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$6(2H) (Jaeschke-Ubiergo et al., 2023)	Hexagonal supercell altermagnets	Four transposing mirror planes, eight spin-polarization sectors
Ammann–Beenker octagonal quasicrystal (Chen et al., 24 Jul 2025)	2D quasicrystalline bulk with global $$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$7	$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$8-protected g-wave spectral and transport textures

CrSb and MnTe are the canonical bulk crystalline g-wave altermagnets in the recent literature. CrSb is used repeatedly as the prototype bulk g-wave system in multipolar, quantum-geometric, surface-projection, and ultrafast studies (Sorn et al., 17 Jun 2026, Chakraborti et al., 30 Apr 2026, Zhou et al., 18 Sep 2025). MnTe appears both as a canonical g-wave altermagnet in the atomic-spin-density analysis and as one of the hexagonal g-wave materials in the broader review of nodal magnetic phases (Jaeschke-Ubiergo et al., 13 Mar 2025, Jungwirth et al., 2024).

CoF$$\mathbf{R}_s^{\rm III}=[E\parallel \mathbf{H}] + [C_2\parallel A][E\parallel \mathbf{H}]$$9 provides a centrosymmetric rhombohedral realization with explicit density-functional support. In that case the optimized structure remains $\mathbf{H}$0 over the studied $\mathbf{H}$1 range, the magnetic ground state is a collinear antiferromagnet with moments along $\mathbf{H}$2, and the maximum spin splitting reaches $\mathbf{H}$3 meV for $\mathbf{H}$4 eV (Tagani, 2024). G-type $\mathbf{H}$5 extends the class to a multiferroic oxide: its rhombohedral $\mathbf{H}$6 phase combines G-type antiferromagnetism, altermagnetic spin splitting up to $\mathbf{H}$7 eV on generic $\mathbf{H}$8, and a symmetry classification that places the spin-splitting function in the $\mathbf{H}$9 irrep of $\mathbf{G}$0, with lowest even $\mathbf{G}$1 (Urru et al., 25 May 2025).

TbPt$\mathbf{G}$2Al$\mathbf{G}$3 adds a rare-earth example. It crystallizes in the NdPt$\mathbf{G}$4Al$\mathbf{G}$5-type trigonal structure, orders antiferromagnetically at $\mathbf{G}$6 K with propagation vector $\mathbf{G}$7, and has collinear Tb$\mathbf{G}$8 moments of $\mathbf{G}$9 along the $A$0 axis. Comparison of its magnetic point group with the nontrivial spin Laue group classifies it as a bulk g-wave altermagnet (Oishi et al., 12 Sep 2025).

The supercell framework generalizes g-wave altermagnetism beyond $A$1 magnetic structures. In CsCoCl$A$2, RbCoBr$A$3, and BaMnO$A$4(2H), the magnetic unit cell is three times larger than the chemical one, yet the magnetic space group is type III rather than type IV, so nonrelativistic spin splitting survives. Their spin point group

$A$5

produces four transposing mirror planes and an eight-sector g-wave spin texture (Jaeschke-Ubiergo et al., 2023).

4. Electronic structure, nodal topology, and relativistic evolution

In bulk crystalline g-wave altermagnets, the nonrelativistic electronic hallmark is a sign-changing spin splitting with symmetry-enforced nodal manifolds. In CoF$A$6, the model Hamiltonian predicts no spin splitting at $A$7 and none along $A$8-L-F, corresponding to the $A$9 plane, while sizable splitting develops for $\mathbf{k}$0, with maximum magnitude near $\mathbf{k}$1. The DFT calculations show that the valence-band splitting is somewhat larger than the conduction-band splitting and reaches $\mathbf{k}$2 meV (Tagani, 2024).

CrSb provides the most explicit bulk g-wave band description. In a centrosymmetric tight-binding model used to analyze quantum geometry, the altermagnetic term is

$\mathbf{k}$3

with

$\mathbf{k}$4

The resulting spin-projected Fermi surface in the $\mathbf{k}$5–$\mathbf{k}$6 plane has six nodal points and a six-lobe sign pattern characteristic of g-wave symmetry on the hexagonal Brillouin-zone cut, even though the underlying three-dimensional classification remains that of a bulk g-wave altermagnet (Chakraborti et al., 30 Apr 2026).

For $\mathbf{k}$7, the focus is the full three-dimensional nodal topology. The spin-splitting function

$\mathbf{k}$8

exhibits both symmetry-enforced and continuity-enforced nodal surfaces. The $\mathbf{k}$9 cut shows six nodal lines through $\Gamma$00, but the full 3D reconstruction demonstrates that only four distinct nodal surfaces cross $\Gamma$01, which is the criterion for bulk g-wave rather than i-wave classification (Urru et al., 25 May 2025).

The relativistic problem is subtler because SOC and Néel-vector orientation can reduce the symmetry of the dominant spin component. In centrosymmetric CrSb and in noncentrosymmetric wurtzite MnTe, the dominant component retains g-wave character in the relativistic regime only when the Néel vector is aligned along $\Gamma$02; in ferroelectric wurtzite MnTe the same conclusion requires both the Néel vector and the inversion-breaking electric field to be along $\Gamma$03. In CrSb with $\Gamma$04, the dominant $\Gamma$05 component remains the g-wave multipole $\Gamma$06, while the subdominant $\Gamma$07 and $\Gamma$08 components are d-wave-like. For in-plane Néel vectors, the dominant component reduces to d-wave in CrSb and to p-wave or mixed odd–even structures in ferroelectric wurtzite MnTe, with accidental nodal surfaces replacing part of the nonrelativistic nodal-plane structure (Gong et al., 22 May 2026).

5. Collective excitations and field-driven bulk responses

Bulk g-wave altermagnetism extends beyond electron bands into magnon spectra and real-space textures. In easy-axis CrSb, the magnon branch splitting is

$\Gamma$09

which has g-wave symmetry and is unaffected by the easy-axial anisotropy. Each magnon branch carries a fixed, momentum-independent magnetic moment $\Gamma$10. In easy-plane MnTe, by contrast, the raw magnon splitting is not strictly g-wave, but the branch magnetic moments form a g-wave pattern in momentum space, and the generalized observable

$\Gamma$11

has g-wave symmetry in both easy-axis and easy-plane cases (Kravchuk et al., 7 Apr 2025).

The bulk transport selection rules are unusually restrictive. In the spin-conserving two-band theory of higher-wave magnets, only the $\Gamma$12-th order nonlinear transverse spin current proportional to $\Gamma$13 survives when the number of nodes is $\Gamma$14. For g-wave altermagnets, which have four nodes, the only allowed response is a third-order spin current. In two dimensions the nonzero component is $\Gamma$15, and in three dimensions it is $\Gamma$16; linear and quadratic spin-current responses vanish in the ideal bulk theory (Ezawa, 2024).

Centrosymmetric bulk g-wave altermagnets also realize a distinctive quantum-geometric magnetization response. In CrSb, inversion symmetry forbids the linear electric-field-induced spin magnetization, and $\Gamma$17 symmetry suppresses the equilibrium contribution, leaving the magnetic-field-induced linear response as the only allowed linear quantum-geometric magnetization. The response is controlled entirely by the spin-rotation quantum metric,

$\Gamma$18

and representative centrosymmetric altermagnets are predicted to exhibit a giant linear spin magnetization of order $\Gamma$19 at magnetic fields of $\Gamma$20, with CrSb serving as the g-wave prototype (Chakraborti et al., 30 Apr 2026).

Nonequilibrium probes expose additional bulk specificity. Real-time TDDFT for g-wave CrSb shows that laser-induced ultrafast demagnetization is strongly governed by incidence direction. Under normal incidence along $\Gamma$21, the two Cr sublattices demagnetize symmetrically, preserving the net-zero magnetization, whereas off-normal incidence drives asymmetric demagnetization and a transient ferrimagnetic-like state through anisotropic optical intersite spin transfer. The authors relate this direction dependence to the characteristic nodal structure of the bulk g-wave electronic states (Zhou et al., 18 Sep 2025). More generally, the dynamic coupling of a d- or g-wave altermagnetic order parameter to strain produces hybridized paramagnon-polarons; in 3D the dominant effect, even at finite temperature, is suppression of quantum fluctuations and an extended ordered regime (Steward et al., 2023).

6. Surfaces, thin films, and interpretation of bulk measurements

Bulk g-wave order does not project trivially to surfaces. In CrSb, a symmetry analysis of surfaces and thin films shows that if the surface coincides with a symmetry plane of the bulk altermagnetic order, the resulting two-dimensional Brillouin zone can become spin-degenerate. For the $\Gamma$22 orientation, however, the effective nonrelativistic surface term is

$\Gamma$23

which is d-wave-like in the surface Brillouin zone even though the bulk is g-wave. The same work emphasizes that $\Gamma$24 surfaces can appear ferromagnet-like, $\Gamma$25 surfaces can be nonrelativistically spin-degenerate, and only selected orientations retain a direct signature of the bulk g-wave order (Sorn et al., 17 Jun 2026).

A related slab study formulates this as functionalization of a bulk g-wave altermagnet by surfaces. In a $\Gamma$26 slab, surface-induced d-wave altermagnetism produces a finite linear spin-splitter effect absent in the bulk. The two surfaces contribute with the same sign, so their spin-splitter signals add rather than cancel, and the spin-splitter angle reaches up to $\Gamma$27 degrees. The same symmetry reduction also permits a surface-induced weak ferromagnetism, offering a route to control altermagnetic domains by an external magnetic field (Sorn, 6 Jul 2026).

These results matter directly for experimental interpretation. Surface-sensitive probes such as ARPES, SARPES, and STM may observe a d-wave-like, spin-degenerate, or even ferromagnetic-like surface texture in a material whose bulk classification is g-wave. Conversely, the absence of translational symmetry does not preclude a bulk phase: the Ammann–Beenker octagonal tiling supports a thermodynamically defined g-wave altermagnetic phase with nonzero order parameter over a broad range of filling and coupling, despite having no Brillouin zone and no unit cell (Chen et al., 24 Jul 2025). A recurrent misconception is therefore that “bulk g-wave” can be read directly from any single surface spectrum; the present literature shows instead that the bulk classification is controlled by the symmetry and nodal topology of the three-dimensional or thermodynamic phase, while surfaces and dimensional reduction can project that order into lower-symmetry forms.