Nodeless Altermagnets: Node-Free Spin Splitting

• Nodeless altermagnets are collinear, compensated magnetic systems that exhibit persistent spin splitting without symmetry-enforced nodal zeros.
• Their unique symmetry settings enable fully spin-polarized Fermi surfaces, supporting topological transport and triplet superconducting phenomena.
• Microscopic models show that controlled SOC and symmetry reduction allow nodeless altermagnets to achieve large anomalous Hall effects with minimal stray magnetization.

Searching arXiv for papers on nodeless altermagnets and closely related altermagnetic symmetry/transport work. Nodeless altermagnets are collinear, compensated magnetic systems with zero net magnetization whose electronic structure remains spin-split over the relevant Fermi surface without symmetry-enforced zeros of the spin splitting. In the broader altermagnetic framework, the defining contrast is with nodal altermagnets, where the momentum-dependent splitting changes sign and vanishes along specific lines, planes, or points in reciprocal space. Current work places nodeless altermagnets at the intersection of nonrelativistic spin splitting, spin-space-group symmetry, valley-selective band topology, superconducting proximity, and multipolar magnetism. The topic is shaped by several partially distinct usages of “nodeless”: fully spin-split Fermi surfaces with no altermagnetic nodal-line intersections (Li et al., 17 Sep 2025); SOC-gapped descendants of altermagnetic Weyl phases that are nodeless at the Fermi energy (Chen et al., 6 Mar 2026); and, in a different symmetry setting, Kramers-degenerate antiferromagnetic phases derived from altermagnetic insulators and lacking spin-split nodes altogether (Cao et al., 2024). Taken together, these works establish nodelessness not as a single universal mechanism, but as a family of symmetry-resolved regimes within altermagnetic materials science.

1. Definition and symmetry setting

Altermagnets are collinear antiferromagnets with zero net magnetization yet spin-polarized bands, with momentum-space spin polarization that is alternating rather than uniform (Chen et al., 6 Mar 2026). In the quasicrystal formulation, the basic observable can be written either as an energy splitting,

$$E_{\uparrow}(\mathbf{k}) \neq E_{\downarrow}(\mathbf{k}),$$

or operationally through spin-resolved spectral weights $A_\uparrow(\mathbf{k})$ and $A_\downarrow(\mathbf{k})$, whose difference encodes the altermagnetic texture (Chen et al., 24 Jul 2025). The central distinction between nodal and nodeless behavior is whether this difference necessarily vanishes on symmetry-related subsets of the Fermi surface.

In symmetry terms, nodal altermagnetism typically follows from operations that relate opposite-spin sectors while forcing the spin splitting to transform antisymmetrically under rotation. In quasicrystalline $g$- and $i$-wave altermagnets, preserved composite symmetries $C_8T$ and $C_{12}T$ imply

$$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$

which forces zeros along symmetry-determined directions (Chen et al., 24 Jul 2025). This makes those phases intrinsically nodal. A nodeless altermagnet, in that logic, requires either a different order-parameter representation or a symmetry setting in which such antisymmetry is absent (Chen et al., 24 Jul 2025).

A complementary symmetry classification uses oriented spin space groups and the sublattice-interchanging operations

$\{-1\|(2l)\},$

which connect opposite-spin sublattices by a $2l$-fold spatial rotation plus spin flip (Wang et al., 7 Jul 2026). In that framework, fourfold-connected altermagnets generate the canonical $A_\uparrow(\mathbf{k})$0-wave spin texture and thus generic nodal structures, whereas reducing or breaking those symmetries is the route toward nodeless spin splitting (Wang et al., 7 Jul 2026). This suggests that nodelessness is best understood not as the absence of altermagnetism, but as the absence of symmetry-enforced sign-changing form factors on the relevant Fermi surfaces.

A third, broader symmetry perspective comes from Landau theory for collinear $A_\uparrow(\mathbf{k})$1 altermagnets. There, ideal zero-SOC altermagnets are tied to nontrivial crystal irreps and multipolar form factors of rank $A_\uparrow(\mathbf{k})$2, which generically produce $A_\uparrow(\mathbf{k})$3-, $A_\uparrow(\mathbf{k})$4-, or higher-wave nodal structures (Schiff et al., 2024). This suggests that strict nodelessness is difficult in the ideal zero-SOC classification, and is more naturally realized either effectively on the Fermi surface, through SOC-assisted lifting of nodes, or in reduced-symmetry material-specific situations (Schiff et al., 2024).

2. Nodal versus nodeless spin splitting

The most explicit definition of nodeless altermagnets appears in the Josephson-junction study of collinear altermagnets with maximal spin-valley polarization. There, nodeless altermagnets are those whose spin-split Fermi surfaces do not intersect the altermagnetic nodal lines, so low-energy states remain fully spin-polarized at each valley (Li et al., 17 Sep 2025). The spin-valley polarization is defined by

$A_\uparrow(\mathbf{k})$5

with

$A_\uparrow(\mathbf{k})$6

Maximal spin-valley locking corresponds to

$A_\uparrow(\mathbf{k})$7

meaning one valley is fully spin-up and the symmetry-related valley fully spin-down, while net magnetization remains zero (Li et al., 17 Sep 2025). In this usage, “nodeless” refers specifically to the occupied Fermi pockets avoiding the nodal lines of the altermagnetic form factor.

A related but distinct realization appears in the gate-tunable topological transport framework. Without SOC, the model is an altermagnetic Weyl semimetal with valley-localized, spin-resolved Weyl points. With SOC, the valleys are fully gapped, and the resulting altermagnetic quantum spin Hall valley Hall and quantum anomalous valley Hall phases are nodeless at the Fermi energy, despite inheriting their structure from an underlying alternating spin texture (Chen et al., 6 Mar 2026). The low-energy valley Hamiltonian is

$A_\uparrow(\mathbf{k})$8

where the SOC term gaps the Weyl points and removes residual valley nodes at the chemical potential (Chen et al., 6 Mar 2026). Here “nodeless” denotes a fully gapped bulk, not a trivialized spin texture.

A third usage comes from spin-driven multiferroics derived from altermagnetic insulators. In that case, the nodeless phase is a Kramers-degenerate antiferromagnetic state generated by changing the spin arrangement so that spin-up and spin-down sublattices are related by

$A_\uparrow(\mathbf{k})$9

with $A_\downarrow(\mathbf{k})$0 a half-unit-cell translation, restoring spin degeneracy throughout reciprocal space (Cao et al., 2024). This is nodeless in the sense of having no spin-split nodes because there is no spin splitting at all. That usage differs conceptually from the fully spin-split nodeless metallic altermagnets emphasized elsewhere, but it has become part of the same discourse (Cao et al., 2024).

These distinct definitions can be summarized succinctly.

Usage of “nodeless”	Band-structure meaning	Representative paper
Fermi-surface nodeless	Spin-split Fermi pockets avoid nodal lines	(Li et al., 17 Sep 2025)
SOC-gapped nodeless	Bulk valleys or Weyl points are fully gapped	(Chen et al., 6 Mar 2026)
Kramers-degenerate nodeless	Spin splitting is absent everywhere	(Cao et al., 2024)

This multiplicity of meanings explains why the literature sometimes appears internally inconsistent. The common thread is the elimination of low-energy symmetry-enforced zeros relevant to transport or order-parameter diagnostics, but the microscopic content differs.

3. Microscopic models and representative realizations

A widely used minimal model for metallic nodeless altermagnets is the square-lattice Hamiltonian

$A_\downarrow(\mathbf{k})$1

with

$A_\downarrow(\mathbf{k})$2

The $A_\downarrow(\mathbf{k})$3-wave-like altermagnetic term vanishes along $A_\downarrow(\mathbf{k})$4, but for suitable chemical potential the Fermi pockets around $A_\downarrow(\mathbf{k})$5 and $A_\downarrow(\mathbf{k})$6 lie away from those nodal lines and are fully spin-polarized (Li et al., 17 Sep 2025). This is the canonical metallic nodeless-AM platform now used for superconducting transport.

A different minimal description appears in the altermagnet/topological-insulator framework, where altermagnetism is generated by sublattice-inequivalent hoppings together with an antiferromagnetic exchange field,

$A_\downarrow(\mathbf{k})$7

and SOC gaps the valley Weyl points (Chen et al., 6 Mar 2026). The resulting helical and chiral phases are characterized by the composite spin-valley Chern number

$A_\downarrow(\mathbf{k})$8

taking values $A_\downarrow(\mathbf{k})$9 in the helical phase and $g$0 in the valley-selected chiral phase (Chen et al., 6 Mar 2026). In this context, monolayer V$g$1STeO and VO-family materials are identified as realistic nodeless platforms once SOC is included (Chen et al., 6 Mar 2026).

The superconducting memory study uses a bond-dependent altermagnetic term equivalent in momentum space to a cosine $g$2-wave spin splitting,

$g$3

implemented through spin-dependent nearest-neighbor hoppings on a square lattice (Giil et al., 2023). Although that model is nodal in the full Brillouin zone, the paper shows that interface geometry can render the transport channel effectively nodeless, because quasiparticles crossing a straight interface sample a fixed-sign projection of the spin splitting (Giil et al., 2023). This suggests that device-level nodelessness can emerge from directional selection even when the bulk form factor is sign-changing.

By contrast, quasicrystalline altermagnets provide explicit examples where nodelessness is prohibited within the studied order parameters. Octagonal and dodecagonal quasicrystals host $g$4- and $g$5-wave altermagnetism with spectral differences obeying antisymmetry under $g$6 or $g$7, yielding zeros at

$g$8

or the corresponding twelvefold analogue (Chen et al., 24 Jul 2025). These are symmetry-enforced nodal phases and serve as a counterpoint that sharpens the definition of the nodeless case.

4. Spin-orbit coupling, quasi-symmetry, and Hall response

The relation between nodeless spin splitting and weak ferromagnetism is clarified by the quasi-symmetry analysis of SOC-enabled altermagnets. A generic minimal Hamiltonian is written as

$g$9

supplemented by

$i$0

Here $i$1 is the altermagnetic crystal-field term transforming in the same irrep as the Néel order, while $i$2 encodes SOC (Roig et al., 2024). In the realistic minimal models discussed, the bands are fully spin-split in the energy range of interest, i.e. effectively nodeless (Roig et al., 2024).

The central result is the emergence of a uniaxial spin space-group quasi-symmetry when one SOC component dominates. This quasi-symmetry constrains the order in SOC at which a ferromagnetic spin moment $i$3 can appear, even when the anomalous Hall effect is symmetry-allowed and large (Roig et al., 2024). The Berry curvature is computed from

$i$4

and the paper shows that the anomalous Hall conductivity is generically linear in the SOC component parallel to $i$5, whereas the net spin moment can be quadratic or cubic in SOC depending on point group and altermagnetic irrep (Roig et al., 2024).

This resolves a central puzzle: why some nodeless altermagnets have large AHE but almost no spin ferromagnetism. In RuO$i$6, DFT finds AHE of order $i$7 S/cm with moment $i$8; in MnTe the AHE is $i$9 S/cm with moment $C_8T$0; by contrast, FeSb$C_8T$1 supports both sizable AHE and a larger moment of about $C_8T$2 (Roig et al., 2024). The quasi-symmetry explanation is that SOC-linear $C_8T$3–$C_8T$4 couplings are forbidden in the former cases but allowed in the orthorhombic case (Roig et al., 2024).

The magnetic anisotropy energy is likewise derived microscopically in the form

$C_8T$5

with coefficients controlled by SOC-resolved band structure (Roig et al., 2024). For nodeless altermagnets, this gives a direct route to selecting Néel-vector orientation without relying on large net magnetization.

5. Superconducting proximity and spin-polarized Josephson transport

The most distinctive functional consequence of nodeless altermagnets so far is the emergence of magnetization-free spin-polarized Josephson transport. In a planar SC/AM/SC junction, the total Hamiltonian can be written as

$C_8T$6

where the superconducting leads are conventional $C_8T$7-wave, the AM region uses the $C_8T$8-wave-like altermagnetic term above, and a thin interfacial region carries Rashba SOC $C_8T$9 from local inversion-symmetry breaking (Li et al., 17 Sep 2025). The interfacial SOC converts singlet pairs into equal-spin triplets.

Inside the nodeless AM, the anomalous Green function is decomposed as

$C_{12}T$0

For a $C_{12}T$1-aligned junction, the singlet component $C_{12}T$2 decays very rapidly, whereas $C_{12}T$3 and $C_{12}T$4 are long-ranged and valley selective: one is carried by the $C_{12}T$5 valley, the other by the $C_{12}T$6 valley (Li et al., 17 Sep 2025). Because the valleys are oppositely spin-polarized but symmetry-related, the junction supports a fully triplet Josephson current with zero net magnetization.

The supercurrent is computed as

$C_{12}T$7

and decomposes into singlet, triplet, and mixed parts,

$C_{12}T$8

In the nodeless $C_{12}T$9 case, $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$0 and $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$1, leaving an almost purely triplet Josephson response (Li et al., 17 Sep 2025). This establishes a mechanism fundamentally different from ferromagnet-based equal-spin transport, because there is no net magnetization to generate the long-range triplets.

The same platform also supports a robust $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$2–$$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$3 transition controlled by the interface Rashba couplings $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$4 and $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$5. The sign of the critical current follows

$$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$6

so opposite signs of the local inversion breaking at the two interfaces produce a $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$7-junction, while equal signs produce a $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$8-junction (Li et al., 17 Sep 2025). This transition does not require fine tuning of the AM thickness or exchange scale.

Junction orientation is decisive. In a $$\Delta \mathcal{A}(C_n \mathbf{p}) = - \Delta \mathcal{A}(\mathbf{p}),$$9-aligned junction, transverse momentum no longer preserves valley separation, so inter-valley singlet pairing is allowed and the system crosses over from a pure-triplet to a mixed singlet-triplet regime (Li et al., 17 Sep 2025). This provides a direct geometrical control knob for selecting whether a nodeless altermagnet behaves as a triplet-only or mixed superconducting medium.

6. Topological, multiferroic, and device perspectives

Nodelessness in altermagnets is not limited to Josephson physics. In the topological setting, SOC-gapped altermagnets support helical spin-valley-momentum-locked edge states with

$\{-1\|(2l)\},$0

and, upon applying a sublattice-staggered potential $\{-1\|(2l)\},$1, a chiral phase with

$\{-1\|(2l)\},$2

(Chen et al., 6 Mar 2026). Because the bulk is fully gapped, the edge transport is quantized and robust against nonmagnetic and long-range magnetic disorder; the chiral phase is robust against all disorder types considered (Chen et al., 6 Mar 2026). This identifies SOC-gapped nodeless altermagnets as programmable topological devices rather than merely unusual metals.

A more speculative but symmetry-grounded direction concerns nonlinear Hall transport. In pure $\{-1\|(2l)\},$3-wave altermagnets with $\{-1\|(2l)\},$4 symmetry, both linear AHE and intrinsic Berry-curvature-dipole-driven second-order Hall responses are forbidden. However, a dc electric field can induce a Berry curvature dipole through the Berry connection polarizability and quantum metric,

$\{-1\|(2l)\},$5

leading to a field-induced second-order Hall current (Mukherjee et al., 16 Oct 2025). The paper argues that mixed-symmetry or Rashba-coupled altermagnets can become effectively nodeless on the Fermi surface, enhancing this nonlinear response (Mukherjee et al., 16 Oct 2025). This suggests that nodelessness may be diagnosable not only by spectroscopy but by quantum-geometric transport.

In multiferroics derived from altermagnetic insulators, the nodeless phase is Kramers-degenerate rather than spin-split, but it is still technologically relevant. The exchange-striction mechanism yields a Landau-Ginzburg functional

$\{-1\|(2l)\},$6

and supports polarization values up to $\{-1\|(2l)\},$7 with magnetoelectric coupling one to two orders of magnitude above SOC-driven multiferroics in the reported materials (Cao et al., 2024). This usage broadens the concept of nodelessness to include altermagnetic-derived states where removing spin splitting opens other functionalities.

From the device perspective, several papers point toward a common implication: nodeless or nearly nodeless altermagnets are especially attractive because they combine strong spin selectivity with negligible stray fields. In spin-space-group language, they can support large AHE with minimal net spin moment (Wang et al., 7 Jul 2026, Roig et al., 2024); in superconducting hybrids, they enable triplet-only supercurrents without ferromagnetism (Li et al., 17 Sep 2025); in topological monolayers, they permit electrically switchable helical–chiral conversion (Chen et al., 6 Mar 2026).

7. Open problems and conceptual boundaries

Several points remain unsettled. First, strict nodelessness in the ideal zero-SOC Landau classification appears highly constrained because nontrivial irreps generically imply sign-changing momentum-space form factors (Schiff et al., 2024). This suggests that many experimentally relevant “nodeless” altermagnets are effective rather than absolute: the nodal lines exist in principle, but do not intersect the Fermi surface or are lifted by SOC and lower-symmetry perturbations (Li et al., 17 Sep 2025, Wang et al., 7 Jul 2026).

Second, the literature uses “nodeless” in at least three non-equivalent senses: fully spin-split metallic Fermi surfaces, SOC-gapped topological descendants of nodal altermagnets, and Kramers-degenerate spin-split-node-free antiferromagnets (Li et al., 17 Sep 2025, Chen et al., 6 Mar 2026, Cao et al., 2024). This terminological spread is manageable but requires care. A plausible implication is that future classification schemes will need separate labels for “fully spin-split nodeless,” “gapped altermagnetic,” and “Kramers-degenerate derived” phases.

Third, many widely studied altermagnets remain nodal by symmetry. Quasicrystalline $\{-1\|(2l)\},$8- and $\{-1\|(2l)\},$9-wave altermagnets are explicitly nodal (Chen et al., 24 Jul 2025); perovskite altermagnets retain glide- or mirror-protected nodal planes (Naka et al., 2024); MnF$2l$0 and MnTe exhibit nodal magnon chirality patterns that polarized neutrons can directly detect (McClarty et al., 2024). These cases are not peripheral exceptions but major branches of the field. Nodeless altermagnets should therefore be viewed as a special subclass rather than the default realization of altermagnetism.

Finally, experimental identification of nodelessness demands probes sensitive to the full momentum dependence of spin splitting. The current toolkit includes spin-resolved spectral functions and conductance anisotropy (Chen et al., 24 Jul 2025), polarized-neutron mapping of magnon chirality (McClarty et al., 2024), Hall response versus Néel orientation and SOC scaling (Wang et al., 7 Jul 2026, Roig et al., 2024), and superconducting transport that isolates triplet channels (Li et al., 17 Sep 2025). A plausible implication is that combining these methods on the same material platform will be necessary to distinguish truly nodeless phases from merely weakly nodal ones.

Nodeless altermagnets therefore occupy a precise but evolving place in contemporary condensed-matter theory. They are best understood as altermagnetic states in which the low-energy electronic or collective spectrum avoids symmetry-enforced zeros that would otherwise fragment spin splitting. Whether realized through Fermi-surface placement, SOC-induced gapping, reduced symmetry, or derived Kramers-degenerate phases, they extend the original altermagnetic idea from “compensated yet spin-split” to “compensated, spin-selective, and low-energy node-free,” with direct implications for topological transport, Hall physics, superconducting spintronics, and multiferroic control (Li et al., 17 Sep 2025, Chen et al., 6 Mar 2026, Roig et al., 2024).