Altermagnetic Weyl Semimetals

• Altermagnetic Weyl semimetals are topological materials in magnetically compensated systems where momentum-dependent exchange splitting drives robust Weyl topology.
• They exhibit large exchange-driven spin splitting that produces isolated bulk Weyl nodes and extended spin-polarized Fermi arcs, as demonstrated in CrSb.
• Theoretical studies and materials design reveal tunable node configurations and pure spin nonlinear optical responses, advancing applications in topological spintronics.

Altermagnetic Weyl semimetals are topological semimetals realized in altermagnets, a class of magnetically compensated ordered materials that combine vanishing net magnetization with momentum-dependent exchange spin splitting. In these systems, broken time-reversal symmetry does not arise from a uniform ferromagnetic moment, but from crystal-symmetry-governed spin-lattice structure, so Weyl topology can be generated by exchange on an energy scale that may greatly exceed relativistic spin-orbit coupling (SOC). The resulting landscape includes experimentally established bulk Weyl nodes and spin-polarized Fermi arcs in CrSb, symmetry-enforced same-spin Weyl nodes carrying an additional magnetic quantum number, tunable two-dimensional Dirac-Weyl analogs, and closely related mirror-protected nodal-loop and node-network phases (Li et al., 2024, Lu et al., 2024, Antonenko et al., 2024, Qu et al., 2024).

1. Altermagnetic symmetry and the route to Weyl topology

Altermagnets occupy a symmetry class distinct from both ferromagnets and conventional collinear antiferromagnets. In a ferromagnet, broken time-reversal symmetry and nonzero net magnetization generically produce exchange-split bands throughout momentum space. In a conventional collinear antiferromagnet, opposite magnetic sublattices are commonly related by a symmetry such as combined parity-time symmetry $PT$ or a translation-plus-spin operation, and the band structure remains doubly degenerate. In an altermagnet, by contrast, the crystal and magnetic sublattices are related by proper or improper rotations, mirrors, screws, or more general spin-lattice operations rather than by translation or inversion alone; the system remains magnetically compensated, but global spin degeneracy is not enforced, so momentum-dependent spin splitting appears even without SOC (Li et al., 2024, Lu et al., 2024).

This nonrelativistic spin splitting is nodal and symmetry structured. It changes sign under crystal operations, vanishes on symmetry-enforced submanifolds of the Brillouin zone, and can be large away from those loci. The spin-momentum locking discussed across the literature is therefore not the ferromagnetic Zeeman pattern but a $d$-, $g$-, $i$-, or $f$-wave-like alternation tied to the magnetic crystal class (Yoshida et al., 6 Sep 2025, Wei et al., 1 Jun 2026). In the spin-group formulation for collinear altermagnets without SOC, the internal spin symmetry is written as $\mathrm{SO}(2)\rtimes \mathbb Z_2$, and this setting is central for the classification of same-spin Weyl nodes and spin-resolved nonlinear responses (Yoshida et al., 6 Sep 2025).

The connection to Weyl physics follows from the same symmetry breaking. Once $PT$ or analogous degeneracy-enforcing operations are absent, crossings between nondegenerate bands can form isolated Weyl points or, in related settings, twofold Weyl-like nodal loops. Several works emphasize that altermagnetism is not merely a Zeeman perturbation added to a topological band structure: the momentum dependence of the exchange field can selectively split Dirac points, reshape band-inversion surfaces, or stabilize crossings between same-spin bands of opposite sublattice character (Liu et al., 24 Jan 2026, Antonenko et al., 2024, Wan et al., 30 Dec 2025).

2. Weyl-node taxonomy and altermagnetic topological invariants

The topological charge of a Weyl node remains its chirality $\chi=\pm1$, conventionally written as

$$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$

with $\boldsymbol{\Omega}(\mathbf{k})$ the Berry curvature on a small enclosing surface $d$0 (Li et al., 2024). What changes in altermagnetic Weyl semimetals is the coexistence of chirality with a spin or magnetic label that need not be trivial.

The clearest formulation appears in CrSb, where two classes of Weyl points were identified. Opposite-spin Weyl points are formed by crossing states of opposite spin and carry $d$1. Same-spin Weyl points are intrinsic to the altermagnetic structure, are formed by crossing states of the same spin, and in the altermagnetic limit carry $d$2 in units of $d$3. With weak SOC, $d$4 is no longer exactly quantized but remains close to $d$5. Each same-spin node is therefore characterized by the pair $d$6, which the CrSb work describes as a chirality and a magnetic quantum number carried by the same bulk singularity (Li et al., 2024).

To summarize the relation between chirality and spin across symmetry-related node sets, CrSb introduced the locking invariant

$d$7

This quantity measures chirality-spin locking. In CrSb, symmetry enforces $d$8, but the same work states that other magnetic space groups could support nonzero integer $d$9 (Li et al., 2024). A later symmetry classification generalized this perspective by identifying 34 spin space groups with symmetry-enforced Weyl points compatible with quantized pure spin circular photogalvanic effect, making same-spin Weyl topology a searchable symmetry class rather than a material-by-material accident (Yoshida et al., 6 Sep 2025).

A recurring terminological issue is dimensionality. Several papers use “Weyl points” for isolated twofold crossings in two-dimensional systems. Those works explicitly note that such crossings are not conventional three-dimensional Weyl nodes with Berry-flux monopole physics in full 3D momentum space; their boundary manifestation is a one-dimensional Fermi-line edge state or a chiral edge mode rather than a two-dimensional surface Fermi arc (Liu et al., 24 Jan 2026, Tan et al., 2024, Meng-Han et al., 2024). The term “altermagnetic Weyl semimetal” is therefore used most strictly for three-dimensional phases such as CrSb, CoNb$g$0Se$g$1, or the SOC-induced Weyl phases of $g$2 and $g$3 (Li et al., 2024, Sah et al., 24 Oct 2025, Qu et al., 2024).

3. CrSb as the prototype topological Weyl altermagnet

CrSb is the central experimental prototype. It crystallizes in the NiAs-type hexagonal structure with ordinary space group $g$4 and, in the altermagnetic state, magnetic space group $g$5. Its opposite-spin Cr sublattices are related by a sixfold screw rotation, and inversion connects atoms with the same magnetization. In the absence of SOC, there are four mirror planes on which the altermagnetic bands are spin degenerate; away from those planes, broken $g$6, zero net moment, preserved crystalline symmetries, and nonsymmorphic screw symmetry allow large exchange-driven spin splitting and Weyl physics (Li et al., 2024).

The combined symmetry analysis, first-principles calculations, ARPES, and spin-resolved ARPES establish CrSb as a topological Weyl altermagnet. The measured momentum-dependent spin splitting reaches about $g$7, far larger than the weak SOC scale. In the energy window from $g$8 to $g$9 around the Fermi level, calculations identify 13 groups of Weyl points: 10 groups of opposite-spin Weyl points within $i$0 of the Fermi level and 3 groups of same-spin Weyl points. The experimentally emphasized same-spin pair lies on $i$1 at $i$2 and $i$3, with opposite chirality and large projected momentum separation on the (001) surface, which produces unusually extended Fermi arcs (Li et al., 2024).

The experimental data resolve both the bulk altermagnetic splitting and the surface topology. Broad photon-energy scans from 40 to 120 eV establish the $i$4 periodicity, and bulk Fermi surfaces away from $i$5 and $i$6 form the characteristic hexagram of two intersecting equilateral triangles associated with opposite-spin Fermi sheets. Band dispersions along $i$7 confirm symmetry-enforced spin degeneracy on the $i$8 and $i$9 planes and strong splitting in between. Spin-resolved ARPES at 102 eV shows opposite $f$0-spin signatures on the two branches of the split band. Surface calculations for both Sb-terminated and Cr-terminated (001) surfaces show long, strongly spin-polarized Fermi arcs connecting projected Weyl nodes of opposite chirality, and the same-spin arcs connect nodes of identical spin projection but opposite chirality (Li et al., 2024).

A complementary ARPES study on the naturally cleaved (100) side surface reached a closely related conclusion. Using soft X-ray ARPES for bulk $f$1 mapping and VUV ARPES for high-resolution surface spectroscopy, it observed band spin splitting up to $f$2 near the Fermi level at generic momenta, identified surface states absent in bulk calculations and soft X-ray spectra, and assigned them as surface Fermi arcs on the (100) surface. The associated topology analysis found 12 Weyl points with chirality $f$3 in the planes $f$4, while also noting that overlapping projected Weyl nodes on the (100) surface make point-by-point endpoint identification experimentally difficult. That work additionally emphasized that CrSb is a room-temperature altermagnetic candidate with $f$5 (Lu et al., 2024).

Taken together, the CrSb studies established the main physical claim of the field: altermagnetic exchange symmetry can generate and stabilize Weyl topology on an exchange scale rather than an SOC scale, and the resulting surface arcs inherit strong spin selectivity (Li et al., 2024, Lu et al., 2024).

4. Two-dimensional analogs and minimal model constructions

A substantial theoretical literature develops altermagnetic Weyl physics in minimal models and in strictly two-dimensional settings. In a 2D Young–Kane nonsymmorphic Dirac semimetal, a $f$6-wave altermagnetic exchange field

$f$7

selectively splits the Dirac points at $f$8 and $f$9 while leaving the $\mathrm{SO}(2)\rtimes \mathbb Z_2$0-point Dirac node untouched, because the form factor vanishes at $\mathrm{SO}(2)\rtimes \mathbb Z_2$1. Near $\mathrm{SO}(2)\rtimes \mathbb Z_2$2, the Weyl nodes occur at $\mathrm{SO}(2)\rtimes \mathbb Z_2$3, so the pair separation scales as $\mathrm{SO}(2)\rtimes \mathbb Z_2$4, and the separation direction is always perpendicular to the in-plane altermagnetic axis. Rotating $\mathrm{SO}(2)\rtimes \mathbb Z_2$5 therefore rotates the node separation axis and reconfigures the Fermi-line edge-state connectivity (Liu et al., 24 Jan 2026).

The same model distinguishes in-plane and out-of-plane order. In-plane altermagnetism produces a 2D Dirac-Weyl semimetal composed of a surviving fourfold Dirac point at $\mathrm{SO}(2)\rtimes \mathbb Z_2$6 and momentum-separated twofold Weyl nodes at $\mathrm{SO}(2)\rtimes \mathbb Z_2$7. Out-of-plane altermagnetism gaps the $\mathrm{SO}(2)\rtimes \mathbb Z_2$8 sectors while preserving the $\mathrm{SO}(2)\rtimes \mathbb Z_2$9-point Dirac node and yields a gapless phase with chiral edge modes and quantized line polarizations $PT$0 and $PT$1 equal to $PT$2 (Liu et al., 24 Jan 2026). A related honeycomb-lattice altermagnet similarly realizes Dirac crossings and a 2D Weyl phase without SOC, with Berry-curvature singularities concentrated at Weyl points, chiral edge states after mass gapping, and Chern numbers up to $PT$3 (Meng-Han et al., 2024).

Another 2D formulation is the bipolarized Weyl semimetal. In a four-band square-lattice model with altermagnetic spin-group symmetry,

$PT$4

the condition $PT$5 generates two spin-up-polarized and two spin-down-polarized Weyl points. First-principles calculations on monolayer $PT$6 and $PT$7 ($PT$8) identify ideal 2D type-I bipolarized Weyl semimetals without SOC. With SOC, $PT$9, $\chi=\pm1$0, and, according to the main-body discussion, $\chi=\pm1$1 realize a quantum crystal valley Hall phase, while $\chi=\pm1$2 remains a Weyl semimetal with only a single pair of Weyl points controlled by the Néel-vector direction. The same paper notes an internal inconsistency between its abstract and main text on whether $\chi=\pm1$3 or $\chi=\pm1$4 is the exceptional SOC-stable single-pair Weyl semimetal; the main-body discussion and conclusion point to $\chi=\pm1$5 (Tan et al., 2024).

A different minimal route uses an altermagnetic mass rather than an altermagnetic exchange splitting added to an existing Dirac problem. Replacing the isotropic Wilson mass by

$\chi=\pm1$6

or, on the lattice, $\chi=\pm1$7, drives a direct $\chi=\pm1$8 transition between opposite nontrivial Chern phases. In a 3D extension,

$\chi=\pm1$9

the $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$0-dependent mass generates Weyl nodes and, on the same surface, coexisting counter-propagating helical Fermi arcs associated with opposite slice-Chern sectors. For $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$1, $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$2, $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$3, the Weyl nodes lie at $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$4 and $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$5, and the slice Chern number jumps from $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$6 for $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$7 to $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$8 for $$\chi=\frac{1}{2\pi}\oint_S \boldsymbol{\Omega}(\mathbf{k})\cdot d\mathbf S,$$9 (Wan et al., 30 Dec 2025).

5. Nodal-loop and node-network extensions

Not all topological altermagnets are point-node Weyl semimetals in the strict sense. A major branch of the subject concerns Weyl nodal loops and node networks generated by altermagnetic symmetry. In a rutile-type two-sublattice model with out-of-plane staggered moments, same-spin opposite-sublattice crossings survive SOC on the $\boldsymbol{\Omega}(\mathbf{k})$0 plane because $\boldsymbol{\Omega}(\mathbf{k})$1 remains a good symmetry and the crossing bands have opposite mirror eigenvalues. The resulting phase is a mirror-protected Weyl nodal-loop semimetal, distinct from the opposite-spin altermagnetic Zeeman line crossings that are gapped by SOC. The same paper further showed that the analogous two-dimensional crossings are gapped by SOC and produce mirror Chern bands, while the 3D system hosts four Weyl nodal loops on the $\boldsymbol{\Omega}(\mathbf{k})$2 plane that can undergo a Lifshitz transition into two larger loops as SOC is varied (Antonenko et al., 2024).

An experimentally backed noncollinear extension appears in $\boldsymbol{\Omega}(\mathbf{k})$3. Neutron diffraction identifies a coplanar $\boldsymbol{\Omega}(\mathbf{k})$4 compensated magnetic structure with propagation vector $\boldsymbol{\Omega}(\mathbf{k})$5 and magnetic space group $\boldsymbol{\Omega}(\mathbf{k})$6. First-principles calculations then show linearly dispersing crossings near $\boldsymbol{\Omega}(\mathbf{k})$7 that form continuous nodal loops confined to the mirror-invariant $\boldsymbol{\Omega}(\mathbf{k})$8 plane. Along high-symmetry directions the crossings remain Dirac-like and fourfold, while at generic momenta they split into twofold Weyl-like crossings protected by the horizontal mirror plane. The momentum-dependent spin polarization is dominated by $\boldsymbol{\Omega}(\mathbf{k})$9, satisfies $d$00 with $d$01, and exhibits an $d$02-wave-like six-lobe pattern characteristic of odd-parity altermagnetism (Wei et al., 1 Jun 2026).

A more elaborate nonrelativistic electronic topology is realized in $d$03 and $d$04. In the absence of SOC, these systems are predicted to be altermagnetic node-network semimetals protected by spin space-group symmetry. Mirror symmetry $d$05 protects Weyl nodal rings on the $d$06 and $d$07 planes, while the spin symmetry $d$08 protects generic Weyl nodal networks between them. Nonsymmorphic spin symmetries protect 16 Dirac points on high-symmetry lines. Once SOC is included, the nodal rings and nodal networks are mostly gapped and $d$09 becomes a type-I Weyl semimetal with 84 Weyl points, 12 of them symmetry independent; Fermi arcs were calculated on both (001) and (100) surfaces (Qu et al., 2024).

These nodal-loop and node-network phases broaden the meaning of Weyl altermagnetism. The common thread is not isolated point monopoles alone, but the fact that compensated magnetic order and spin-lattice symmetry organize topological twofold crossings—points, loops, or networks—within a spin-split yet net-zero-moment background (Antonenko et al., 2024, Wei et al., 1 Jun 2026, Qu et al., 2024).

6. Classification, responses, materials design, and points of caution

The most systematic symmetry program to date concerns nonlinear optical response. A recent classification of collinear altermagnets without SOC showed that the spin-current analog of the quantized circular photogalvanic effect is forbidden in antiferromagnets by the mirror-plus-$d$10 constraints needed there for pure spin current, but is intrinsically allowed in altermagnets. The work classified all 27 non-centrosymmetric altermagnetic spin point groups and found 10 for which

$d$11

so the quantized response is purely spin. It further identified 34 spin space groups with symmetry-enforced Weyl points compatible with this response, constructed a tight-binding model for $d$12, and obtained

$d$13

$d$14 was proposed as a first-principles candidate, although the calculated stoichiometric material is insulating, so doping would be required to bring the Weyl crossings to the Fermi level (Yoshida et al., 6 Sep 2025).

Materials design has also expanded beyond CrSb. In magnetically intercalated transition-metal dichalcogenides $d$15, the A-type antiferromagnetic members $d$16, $d$17, $d$18, and $d$19 were identified as altermagnets. The most developed case, $d$20, hosts 12 pairs of Weyl points within $d$21 of $d$22, including a representative pair at $d$23, with chiral charges $d$24, (001) Fermi arcs, and projected double-arc features caused by overlap of same-chirality projections. That work also proposed a structural selection rule: the ratio $d$25 of interlayer to intralayer spacing correlates with the magnetic ground state, with $d$26 tending to favor A-type AFM and hence altermagnetism (Sah et al., 24 Oct 2025).

Several points require careful interpretation. First, SOC does not play the same role across the field. In CrSb, same-spin Weyl points are described as “purely altermagnetic” in origin and only weakly perturbed by SOC; in rutile models, by contrast, SOC is what distinguishes 2D mirror Chern bands from 3D Weyl nodal loops; in $d$27, SOC converts a spin-symmetry-protected node-network phase into a Weyl semimetal (Li et al., 2024, Antonenko et al., 2024, Qu et al., 2024). Second, some experimental evidence is more direct than others: in CrSb on the (100) surface, the bulk-topological interpretation is strongly calculation-backed because overlapping projected Weyl nodes obscure arc endpoints in ARPES (Lu et al., 2024). Third, “Weyl” in 2D papers is a controlled but nonstandard extension of 3D language, and those works themselves warn that their nodes are isolated 2D crossings rather than 3D Berry-flux monopoles (Liu et al., 24 Jan 2026, Meng-Han et al., 2024).

The broader significance is nevertheless consistent across the literature. Altermagnetic Weyl systems define a topological magnetic class in which robust bulk and boundary topology can be generated by exchange symmetry in magnetically compensated matter, enabling spin-polarized Fermi arcs, tunable node geometry, pure spin nonlinear optics, and interface-relevant topological spin selectivity without stray ferromagnetic fields (Li et al., 2024, Yoshida et al., 6 Sep 2025).