Altermagnetic Weyl Semimetal

• Altermagnetic Weyl semimetals are quantum materials that feature symmetry-induced, momentum-dependent spin splitting with zero net magnetization.
• They exhibit robust topological characteristics, including Weyl nodes with nontrivial Chern numbers and Fermi arc surface states detectable via ARPES.
• The interplay of altermagnetism and topology leads to distinctive transport and optical effects, such as quantized spin circular photogalvanic and Hall responses.

Altermagnetic Weyl semimetals (AMWSMs) are a class of quantum materials that incorporate the symmetry-driven, zero-net-moment collinear magnetism of altermagnets with band structures hosting topologically protected Weyl fermions. Altermagnets are collinear magnets that, unlike conventional antiferromagnets, possess large, momentum-dependent spin splitting without net magnetization, due to nonsymmorphic or spin-lattice combined symmetries that break parity-time ($\mathcal{PT}$) invariance. The absence of $\mathcal{PT}$ symmetry enables substantial spin splitting throughout the Brillouin zone, even in the absence of strong spin-orbit coupling (SOC), and generically opens avenues for the realization of Weyl nodes—band touchings characterized by nontrivial Chern numbers and associated Fermi-arc surface states. The interplay between altermagnetic symmetry, nontrivial band topology, and tunable spin textures endows AMWSMs with a rich suite of transport and optical phenomena, and positions them as prime platforms for spintronic and topological device concepts.

1. Altermagnetism: Symmetry, Spin Splitting, and Microscopic Origin

Altermagnetism represents a distinct symmetry class of collinear magnetic order, separate from ferromagnetism (uniform magnetization, broken $\mathcal{T}$, nonzero net $M$) and simple antiferromagnetism (compensated moments, $\mathcal{PT}$ preserved, Kramers degeneracy at every $k$). In altermagnets, compensated sublattice moments are related not by pure translation or $\mathcal{PT}$ but by a point-group operation (rotation, mirror, glide, screw) often intertwined with spin rotation or fractional translation (Lu et al., 2024, Li et al., 2024). The upshot is that while the net magnetization vanishes, $\mathcal{PT}$ symmetry is broken, enabling strong, momentum-dependent spin splitting—typically with high-order angular harmonics (e.g., $d$-wave, $g$-wave, or $\mathcal{PT}$0-wave form factors), and a vanishing splitting at symmetry-protected nodal planes.

This “altermagnetic” spin splitting can be described at low energy by Hamiltonians of the form:

$\mathcal{PT}$1

where $\mathcal{PT}$2 transforms with even ($\mathcal{PT}$3- or $\mathcal{PT}$4-wave) or odd ($\mathcal{PT}$5-wave) parity, and $\mathcal{PT}$6 introduces weaker relativistic splitting. In hexagonal systems like CrSb, the minimal exchange term is $\mathcal{PT}$7 with $\mathcal{PT}$8, vanishing along mirror-protected lines (Li et al., 2024).

Unlike in antiferromagnets, momentum-dependent spin splitting is maximal away from these high-symmetry points/planes, and can reach values up to $\mathcal{PT}$9 eV (e.g., CrSb), overwhelmingly larger than typical SOC energy scales (Li et al., 2024, Lu et al., 2024).

2. Band Topology: Emergence of Weyl Nodes and Nodal Features

The core topological feature of AMWSMs is the presence of symmetry-protected Weyl nodes—twofold degeneracies with linear dispersion and nontrivial chirality $\mathcal{T}$0—arising generically when $\mathcal{T}$1 is broken but nontrivial space (especially crystalline and spin) symmetries are preserved (Sah et al., 24 Oct 2025, Nag et al., 2023, Yoshida et al., 6 Sep 2025). The types of nodal features in AMWSMs depend on the dimension, crystal symmetry, and presence/absence of SOC:

• Weyl Points: Discrete monopoles of Berry curvature in 3D hosts such as transition metal dichalcogenides (TMDs) (Sah et al., 24 Oct 2025), GdAlSi (Nag et al., 2023), and CrSb (Li et al., 2024, Lu et al., 2024). They are characterized by linear crossings, Chern number $\mathcal{T}$2, and are robust against weak SOC if protected by non-symmorphic or mirror symmetries.
• Nodal Lines and Loops: In mirror- or glide-protected cases, bands may cross along 1D lines (nodal rings or nodal loops), as realized in Nb$\mathcal{T}$3FeB$\mathcal{T}$4, Ta$\mathcal{T}$5FeB$\mathcal{T}$6 (Qu et al., 2024), and Cr$\mathcal{T}$7Se$\mathcal{T}$8 (where mirror symmetry $\mathcal{T}$9 pins the loop to $M$0) (Wei et al., 1 Jun 2026). Upon inclusion of SOC, rings generally gap out except at points (residual Weyl nodes).
• Dirac Points and Node Networks: Fourfold crossings arising from nonsymmorphic or magnetic symmetry (such as along high-symmetry lines) (Qu et al., 2024), decomposed into Weyl fermions under SOC.

The low-energy theory near a generic Weyl point has the form:

$M$1

with $M$2 the velocity matrix, $M$3. Each node acts as a Berry flux source, with

$M$4

and

$M$5

In 2D models (Fe$M$6WTe$M$7, Fe$M$8MoZ$M$9, or artificial Rashba–altermagnet superlattices), pointlike and line nodes emerge due to the $\mathcal{PT}$0-wave or $\mathcal{PT}$1-wave exchange terms, leading to “bipolarized” Weyl cones of opposite spin along perpendicular axes (Tan et al., 2024, Wan et al., 30 Dec 2025).

3. Model Hamiltonians, Symmetry Classification, and Topological Invariants

Altermagnetic symmetry strongly constrains both the form of the effective Hamiltonian and the allowed topological features. Symmetry analysis using spin space groups (SSG) and spin point groups (SPG) classifies those magnetic space groups where symmetry-enforced Weyl points—and quantized pure spin responses—are generic (Yoshida et al., 6 Sep 2025). Key mechanisms include:

• Even-parity altermagnetic mass terms (e.g., $\mathcal{PT}$2- or $\mathcal{PT}$3-wave): Even in $\mathcal{PT}$4, flips sign under certain rotation/mirror operations, enables Chern number inversion without passing through a trivial phase (Wan et al., 30 Dec 2025).
• SSG and SPG structure: 34 SSGs were identified that generically enforce Weyl points via two-coset decompositions and pure-spin CPGE-allowed space groups (Yoshida et al., 6 Sep 2025). Table S9 of (Yoshida et al., 6 Sep 2025) details SSGs, $\mathcal{PT}$5-points, and chirality $\mathcal{PT}$6.
• Topological invariants: The quantized Berry curvature and associated Chern numbers remain the key invariants:

$\mathcal{PT}$7

For nodal loops, the Berry phase accumulated along a closed path $\mathcal{PT}$8 encircling the loop is $\mathcal{PT}$9.

4. Materials Realizations and Electronic Structure

Multiple families of experimentally confirmed and theoretically predicted AMWSMs now exist:

Material Family	Notable Features	Reference
CrSb	Room-temperature, giant $k$0 eV spin splitting, Fermi arcs on (100)	(Li et al., 2024, Lu et al., 2024)
Magnetically intercalated TMDs XY$k$1Z$k$2 (CoNb$k$3Se$k$4, FeNb$k$5Se$k$6)	A-type AFM, g-wave PMDS, kagome flat bands, Weyl nodes at $k$7 eV	(Sah et al., 24 Oct 2025)
GdAlSi	Collinear AFM, $k$8-wave splitting, 32 Weyl nodes, Fermi arcs, device concepts	(Nag et al., 2023)
Fe$k$9WTe$\mathcal{PT}$0, Fe$\mathcal{PT}$1MoZ$\mathcal{PT}$2	2D, bipolarized Weyl points, strain-tuned QCVH effect/Chern transitions	(Tan et al., 2024)
Nb$\mathcal{PT}$3FeB$\mathcal{PT}$4, Ta$\mathcal{PT}$5FeB$\mathcal{PT}$6	Nodal rings, node networks, 84 Weyl nodes w/ SOC, large AHE, optical transitions	(Qu et al., 2024)
Cr$\mathcal{PT}$7Se$\mathcal{PT}$8	$\mathcal{PT}$9 triangular order, $\mathcal{PT}$0-wave texture, symmetry-protected nodal loops	(Wei et al., 1 Jun 2026)

CoNb$\mathcal{PT}$1Se$\mathcal{PT}$2 and FeNb$\mathcal{PT}$3Se$\mathcal{PT}$4 in particular offer robust altermagnetic splitting, Weyl nodes near the Fermi level, and proximate kagome bands, forming an ideal platform for combined ARPES, magnetotransport, and scanning-probe investigation of coupled altermagnetic and topological phenomena (Sah et al., 24 Oct 2025).

5. Surface States, Fermi Arcs, and Bulk-Boundary Correspondence

A defining feature of Weyl semimetals—including the altermagnetic subclass—is the presence of open Fermi-arc surface states, connecting surface-projected Weyl nodes of opposite chirality (Sah et al., 24 Oct 2025, Li et al., 2024, Lu et al., 2024, Nag et al., 2023). Calculation and measurement details include:

• On the (001) surface of magnetically intercalated TMDs, Fermi arcs connect projected Weyl points; double arcs occur when nodes of equal chirality overlap (Sah et al., 24 Oct 2025).
• In CrSb, ARPES directly resolves Fermi arcs on (100), confirmed by slab DFT calculations, whose momentum connectivity aligns with bulk Weyl node projections (Lu et al., 2024, Li et al., 2024).
• GdAlSi features distinct “fish-tail” Fermi arc features and diamond-like central pockets on (001), again matching ab initio tight-binding and ARPES spectra (Nag et al., 2023).
• Multilayer Rashba–altermagnet engineered systems theoretically enable coexisting helical Fermi arcs of opposite chirality on the same surface, unlike conventional magnetic WSMs (Wan et al., 30 Dec 2025).

Robustness of these Fermi arcs—owing to the large energy scale of altermagnetic splitting versus SOC—enables experimental detection even in the presence of moderate disorder and at elevated temperature.

6. Macroscopic Phenomena: Transport, Optical, and Topological Responses

AMWSMs host a number of unconventional macroscopic responses, uniquely tied to their symmetry and topology:

• Quantized Spin Circular Photogalvanic Effect (CPGE): Pure spin-current analogs of the quantized CPGE become allowed—quantized by $\mathcal{PT}$5—in SSGs that support symmetry-enforced Weyl points, in stark contrast with conventional AFMs, where CPGE is forbidden by symmetry (Yoshida et al., 6 Sep 2025).
• Quantum Crystal Valley Hall Effect (QCVH): In 2D AMWSMs such as Fe$\mathcal{PT}$6WTe$\mathcal{PT}$7, the valley Hall conductivity is $\mathcal{PT}$8 per valley, tunable by the Néel vector and affected by SOC and strain, with predicted plateaus in nonlocal resistance (Tan et al., 2024).
• Anomalous Hall and Nernst Effects: Large intrinsic AHE ($\mathcal{PT}$9 S/cm) predicted for node-network altermagnetic semimetals; anisotropic responses tied to the $d$0/$d$1/$d$2-wave symmetry of spin splitting and the distribution of Berry curvature (Qu et al., 2024, Wei et al., 1 Jun 2026).
• Magneto-optical and dichroic phenomena: Strong valley-selective (and spin-selective) optical transitions are expected due to broken time-reversal and valley-contrasting Berry curvature (Tan et al., 2024).

7. Experimental Signatures and Device Proposals

The unique combination of symmetry, spin, and topology in AMWSMs enables a diverse array of device and experimental paradigms:

• Direct ARPES observation of spin splitting—up to 1 eV (CrSb), Fermi arcs, and Weyl node connectivity (Lu et al., 2024, Li et al., 2024, Sah et al., 24 Oct 2025).
• Spin-resolved ARPES—red/blue (up/down) Weyl cone polarization as function of $d$3 (Tan et al., 2024).
• Nonlocal and quantized Hall transport—plateaued resistance and chiral edge state dynamics under strain (Tan et al., 2024).
• Scanning tunneling and magneto-optical Kerr microscopy—surface localized LDOS, van-Hove singularities, and large optical anisotropy (Wan et al., 30 Dec 2025, Qu et al., 2024).
• Topotronic device proposals—spin-twister valves and spin-junction transistors exploiting symmetry-enforced altermagnetic splitting and Weyl arc-mediated transport (Nag et al., 2023).

• "Altermagnetism, Kagome Flat Band, and Weyl Fermion States in Magnetically Intercalated Transition Metal Dichalcogenides" (Sah et al., 24 Oct 2025)
• "Observation of surface Fermi arcs in altermagnetic Weyl semimetal CrSb" (Lu et al., 2024)
• "Topological Weyl Altermagnetism in CrSb" (Li et al., 2024)
• "GdAlSi: An antiferromagnetic topological Weyl semimetal with non-relativistic spin splitting" (Nag et al., 2023)
• "Symmetry-Protected Weyl Nodal Loops in a Triangular Altermagnet" (Wei et al., 1 Jun 2026)
• "Bipolarized Weyl semimetals and quantum crystal valley Hall effect in two-dimensional altermagnetic materials" (Tan et al., 2024)
• "Helical Fermi Arc in Altermagnetic Weyl Semimetal" (Wan et al., 30 Dec 2025)
• "Quantization of spin circular photogalvanic effect in altermagnetic Weyl semimetals" (Yoshida et al., 6 Sep 2025)
• "Altermagnetic Weyl node-network semimetals protected by spin symmetry" (Qu et al., 2024)
• "Tunable two-dimensional Dirac-Weyl semimetal phase induced by altermagnetism" (Liu et al., 24 Jan 2026)

---

Altermagnetic Weyl semimetals represent a symmetry- and topology-enriched frontier in correlated quantum materials. They realize robust, high-temperature spin splitting at zero net magnetization, and host Weyl nodes, Fermi arcs, quantum-geometry-driven transport, and quantized spin-optical effects. The interplay between unconventional magnetic order and band topology in these systems provides a versatile foundation for manipulating spin, charge, and topological degrees of freedom in next-generation electronic and spintronic devices.