Extended s-Wave Altermagnets (sAMs) Overview

• Extended s-wave altermagnets (sAMs) are magnetic states defined by valley-exchange symmetry that produces isotropic spin splitting and globally compensated spin polarization.
• They are modeled using a two-valley system where staggered spin polarization enables fully gapped electronic structures and supports unconventional superconducting instabilities such as pair-density waves.
• The unique valley-selective transport and spin filtering in sAMs pave the way for novel spintronic devices and deeper insights into multivalley magnetism.

Extended s-wave altermagnets, usually abbreviated sAMs, are a class of magnetic states that are fully gapped, spin-compensated, and feature spin-polarized bands. In the explicit formulation introduced in 2025, their defining symmetry is not a conventional crystallographic spin-group operation but a valley-exchange symmetry acting as a momentum-space translation between distinct Fermi pockets; this permits an $l=0$-like, isotropic spin splitting that alternates in valley space and therefore cancels in the total magnetization [2508.20163]. In this sense, sAMs generalize altermagnetism beyond the usual $d$-, $g$-, or $i$-wave crystallographic classifications by realizing spin compensation through valley structure while retaining spin-split single-particle bands. Related literature uses “extended $s$-wave” in adjacent but distinct senses—for example, to describe superconducting descendants on altermagnetic backgrounds or SOC-induced $s$-wave spin textures—and those distinctions are important for a precise taxonomy [2408.03999].

1. Definition and conceptual position

The central proposal identifies sAMs with the staggered spin polarization channel of a two-valley itinerant system. In that construction, the magnetic order parameter is
$$
\Delta^{\text{sAM}} \tau^z \sigma^z,
$$
where $\tau^i$ act in valley space and $\sigma^i$ in spin space. Each valley is spin-split, but the sign of the spin splitting reverses between valleys, so the state remains globally spin-compensated [2508.20163].

This places sAMs in a different conceptual slot from both conventional ferromagnets and the better-known crystallographic altermagnets. A ferromagnet corresponds to uniform spin polarization, $\tau^0\sigma^z$, and therefore a nonzero net magnetization. Standard altermagnets, by contrast, are usually classified by momentum-dependent even-parity waveforms such as $d$-, $g$-, or $i$-wave patterns associated with point-group relations between opposite-spin sublattices [2412.05377]. sAMs preserve the compensated character of altermagnetism, but their compensation is enforced by valley staggering rather than by the conventional crystallographic spin-group mechanisms emphasized in earlier classifications [2508.20163].

The analogy invoked by the original proposal is to extended $s$-wave superconductors of the iron-pnictide type: the order parameter is constant on each pocket, changes sign between pockets, and can therefore be fully gapped without being uniform over the full Brillouin zone. In the magnetic setting, this yields an $l=0$-like spin splitting that is isotropic on each Fermi surface but reverses between valleys, so the state is spin-polarized in momentum space yet compensated in the aggregate [2508.20163].

2. Valley-exchange symmetry and order-parameter structure

The minimal continuum setting is a two-valley electron gas with one pocket at $\Gamma$ and one at $M$. Using Pauli matrices $\tau^i$ for the valley degree of freedom and $\sigma^i$ for spin, the low-energy Hamiltonian is
$$
H_c = \int_{|\mathbf k|<k_c} d^2k~ \sum_{\nu,\sigma} \frac{\mathbf k^2}{2m}\, c^\dagger_{\mathbf k+\nu,\sigma}\, c^{}_{\mathbf k+\nu,\sigma},
$$
with $\nu=\Gamma,M$ [2508.20163].

Within this two-valley setting, three collinear magnetic channels were highlighted.

Channel	Order parameter	Interpretation
Spin polarization	$\tau^0\sigma^z$	Conventional ferromagnet
Staggered spin polarization	$\tau^z\sigma^z$	sAM order
Spin inter-valley coherence	$\tau^{x,y}\sigma^z$	Off-diagonal in valley space

For realistic interactions, specifically strong intra-valley repulsion $U$ and sizeable Hund’s coupling $J$, mean-field and one-loop RG analysis favor the staggered spin polarization channel, which is the sAM state [2508.20163].

The decisive symmetry is the valley-exchange symmetry. In the continuum model with sAM order,
$$
H = H_c + \Delta^{\text{sAM}} \tau^z \sigma^z,
$$
the system is invariant under valley exchange combined with spin flip,
$$
\big[H_c + \Delta^{\text{sAM}}\tau^z\sigma^z,\;\tau^x\sigma^x\big]=0.
$$
Here $\tau^x$ exchanges the $\Gamma$ and $M$ valleys, which in momentum space is a translation by $(\pi,\pi)$. The defining statement of the proposal is therefore that sAMs are formed through valley-exchange symmetries, which act as momentum-space translations beyond standard crystallographic spin-group classifications [2508.20163].

Because the bare dispersions of the two valleys are taken to be isotropic and identical, the resulting spin splitting is itself isotropic:
$$
E_{\nu,\sigma}(\mathbf k)=\frac{\mathbf k^2}{2m}+\nu\,\Delta^{\text{sAM}}\,\sigma.
$$
The spin-up and spin-down Fermi radii differ on each pocket, but the total spin polarization cancels when both valleys are included. This is the sense in which sAMs are simultaneously spin-polarized and spin-compensated [2508.20163].

3. Microscopic models and symmetry construction

To move beyond the continuum picture, the proposal introduced a bilayer lattice model in which the two layers play the role of the two valleys:
$$
\begin{multline}
H_l = -t_\parallel \sum_{\langle ij\rangle,\sigma}
\vec c^{\,\dagger}_{i,\sigma}\,\tau^z\, \vec c^{}_{j,\sigma}
- t^\prime_\perp \sum_{\langle!\langle ij\rangle!\rangle,\sigma}
\vec c^{\,\dagger}_{i,\sigma}\,\tau^x\, \vec c^{}_{j,\sigma} \
- t_\perp\sum_{i,\sigma} \vec c^{\,\dagger}_{i,\sigma}\,\tau^x\, \vec c^{}_{i,\sigma}.
\end{multline}
$$
In momentum space this becomes
$$
H_l=\sum_{\mathbf k,\sigma}\vec c^{\,\dagger}_{\mathbf k,\sigma}
\big[v_x(\mathbf k)\tau^x+v_z(\mathbf k)\tau^z\big]\vec c^{}_{\mathbf k,\sigma},
$$
with
$$
v_x(\mathbf k)=-t_\perp-4t_\perp^\prime\cos k_x\cos k_y,\qquad
v_z(\mathbf k)=-2t_\parallel(\cos k_x+\cos k_y).
$$
The model has $C_{4v}$ symmetry and a nontrivial symmetry $\mathcal S$ defined by
$$
\mathcal S\,\vec c^{\,\dagger}_{\mathbf k,\sigma}=\tau^x\,\vec c^{\,\dagger}_{\mathbf k+M,\sigma},
$$
which is the lattice analogue of valley exchange by momentum translation [2508.20163].

The magnetic sAM order is again
$$
H_{\rm sAM}=\Delta^{\text{sAM}}\tau^z\sigma^z.
$$
The full Hamiltonian obeys
$$
\big[H_l+\Delta^{\text{sAM}}\tau^z\sigma^z,\;\mathcal S\,\sigma^x\big]=0,
$$
so the combined operation “layer exchange at $\mathbf k\to\mathbf k+M$” plus spin flip enforces compensation while leaving the spin splitting intact [2508.20163].

A notable feature of the lattice theory is that the sAM gap transforms as the totally symmetric irrep $A_1$, but the valley-exchange symmetry further constrains which lattice harmonics can occur. In the simplest realization, the nodes are pinned to lines such as $\cos k_x+\cos k_y=0$; in alternative realizations, the allowed harmonic can instead be $\cos k_x\cos k_y$, while still remaining in the $A_1$ channel [2508.20163]. This means that “extended $s$-wave” in the magnetic context refers not merely to isotropy on a local pocket, but to a fully symmetric, valley-structured order whose harmonic content is selected jointly by point-group symmetry and momentum-translation symmetry.

The proposed identification strategy follows directly from this structure: one should look for multivalley band structures with valleys related by momentum translations and an internal orbital or layer degree of freedom that is exchanged under that translation. A staggered spin polarization in that internal space then realizes an sAM state [2508.20163].

4. Electronic structure, full gap, and spin-selective transport

The single-particle signature of an sAM is a set of spin-split but globally compensated Fermi surfaces. In the simplest two-valley picture, each pocket is spin-polarized, but the majority spin switches between $\Gamma$ and $M$, so the total spin-up and spin-down Fermi volumes are equal [2508.20163].

The “fully gapped” characterization is a statement about the absence of the low-energy nodal structure that typifies conventional $d$-, $f$-, or $g$-wave altermagnets. In sAMs, the spin splitting is isotropic on each pocket and the sign change occurs between pockets. The nodes of the extended-$s$-like magnetic form factor are therefore displaced away from the actual low-energy Fermi surfaces, much as in an $s^\pm$ superconductor [2508.20163]. This distinguishes sAMs sharply from nodal altermagnets, where symmetry forces vanishing spin splitting along lines or points on the Fermi surface.

This full gap has an immediate transport consequence in heterostructures. The proposal analyzed an sAM–normal-metal junction in which the normal side has only a single pocket. Because momentum parallel to the interface is conserved, only one of the two sAM valleys can transmit into the normal region; the other becomes evanescent. The resulting transmission probability for a propagating mode was written as
$$
T_\sigma(E,k_y)=
\frac{4v^\sigma_{\text{sAM}}v_{\text{N}}\,
\Theta(\kappa^\sigma_{\text{sAM}}-|k_y|)\Theta(\kappa_{\text{N}}-|k_y|)}
{4Z^2+(v^\sigma_{\text{sAM}}+v_{\text{N}})^2},
$$
with spin-resolved current
$$
J_\sigma=\frac{2e}{h}\int\frac{dk_y}{2\pi}\int dE\,[f(E-eV)-f(E)]\,T_\sigma(E,k_y).
$$
The associated spin conversion factor,
$$
R=\left|\frac{J_\uparrow-J_\downarrow}{J_\uparrow+J_\downarrow}\right|,
$$
can become large because the contact selects a single spin-polarized valley out of a globally compensated magnetic state [2508.20163].

The logic of this device concept is characteristic of sAMs: filtering is valley-selective rather than nodal-direction-selective. This is a major distinction from conventional altermagnetic spin filters, which rely on the angular structure of nodal $d$- or $g$-wave spin splitting [2508.20163].

5. Superconducting descendants and the pair-density-wave channel

The superconducting descendant emphasized for sAMs is not ordinary zero-momentum singlet pairing. Because each valley is strongly spin-polarized, conventional spin-singlet pairing at $\mathbf q=0$ is energetically disfavored. The proposal instead isolates two competing possibilities: intra-valley triplet pairing at zero momentum and inter-valley spin-singlet pairing at finite momentum [2508.20163].

The latter is the key instability. Pairing an electron from $\Gamma$ with one from $M$ naturally yields a finite center-of-mass momentum equal to the valley separation vector, giving a pair density wave:
$$
\Delta_{\rm PDW}(\mathbf q=M)\propto
\sum_{\mathbf k}\langle c_{\mathbf k,\Gamma,\uparrow}
c_{-\mathbf k,M,\downarrow}\rangle.
$$
Because the two valleys are symmetry-related by the same valley-exchange structure that defines the parent sAM, this PDW can be spin-singlet, finite-momentum, and fully gapped [2508.20163].

This PDW channel is conceptually distinct from the “extended $s$-wave” superconducting pairing discussed elsewhere in the altermagnet literature. In a square-lattice $d_{x^2-y^2}$ altermagnet, it was shown that uniform onsite $s$-wave spin-singlet pairing is not possible to achieve in altermagnets, whereas nearest-neighbor extended $s$-wave singlet pairing with
$$
\Delta_{\text{ext-}s}(\mathbf k)\propto \cos k_x+\cos k_y
$$
is symmetry-allowed [2408.03999]. In a separate study of a $g$-wave altermagnetic metal on a hexagonal lattice, weak altermagnetic fields stabilized non-chiral $s$-, extended $s$-, or $f$-wave superconducting states, while stronger altermagnetic splitting favored chiral $p$- or $d$-wave phases [2602.22736].

These superconducting results are directly related in spirit but not identical in meaning. The sAM of the magnetic proposal is a normal-state magnetic order based on valley-staggered spin polarization [2508.20163]. The extended-$s$ superconducting states in the pairing literature are pairing symmetries that live on top of an altermagnetic background [2408.03999, 2602.22736]. The common element is the repeated appearance of a fully symmetric but nontrivial momentum structure once compensation is enforced by altermagnetic kinematics rather than by simple ferromagnetic exchange.

6. Relation to broader altermagnetism, SOC, and adjacent usages

sAMs broaden rather than replace the standard altermagnetic taxonomy. Earlier reviews of altermagnets emphasized non-relativistic spin splitting with $d$-, $g$-, and $i$-wave waveforms and momentum-dependent sign changes generated by rotation-related opposite-spin sublattices [2412.05377]. The explicit sAM proposal shows that an $l=0$-like order can remain compensated if the compensation is transferred from crystallographic sublattice space to valley space [2508.20163]. This suggests a broader classification in which both crystallographic and emergent momentum-space symmetries must be considered.

A distinct but nearby line of work concerns SOC-induced spin-orbit magnetism in altermagnets. Using oriented spin Laue groups and SOC tensor expansions, it was shown that only altermagnets with opposite-spin sublattices connected by a fourfold rotation exhibit different perturbative orders for orbital and spin magnetization, with $\mathbf M_O$ first order and $\mathbf M_S$ second order in SOC, and that such systems can display a coaxial Hall effect [2607.06730]. That framework is not the same as the valley-exchange definition of sAMs, but it demonstrates that altermagnetic transport and magnetization can depend on symmetries lying beyond simple net-moment considerations.

Another nearby usage of “extended $s$-wave” appears in discussions of symmetry-allowed $s$-wave spin-density contributions in otherwise higher-wave altermagnets. In that setting, when an $s$-wave spin-density contribution is symmetry-allowed, a small magnetization and an anomalous Hall effect emerge; for “pure” altermagnets, where the $s$-wave component is symmetry-forbidden even in the presence of SOC, both the zero-field magnetization and the AHE vanish [2502.03517]. This is conceptually different from the valley-staggered sAM definition, because the latter preserves full spin compensation by construction [2508.20163].

Relativistic spin-momentum locking provides yet another adjacent perspective. In orthorhombic YVO$3$, the relativistic locking decomposes into **$s$-, $d{xy}$-, and $d_{xz}$-wave** channels for different spin components, while in hexagonal MnTe the dominant non-relativistic $g$-wave texture is lowered by SOC and Néel-vector symmetry breaking to a combination of $d_{xz}$-, $d_{yz}$-, and $s$-wave components [2510.23855]. These results do not define sAMs in the 2508 sense, but they show that SOC can generate robust $s$-wave spin textures inside compensated altermagnetic states.

A common misconception is therefore to treat every $s$-wave-like feature in altermagnets as the same phenomenon. The explicit sAM phase is a valley-staggered magnetic state with isotropic spin splitting and zero net magnetization [2508.20163]. Extended-$s$ superconducting gaps on altermagnetic backgrounds are pairing states, not magnetic orders [2408.03999, 2602.22736]. SOC-generated $s$-wave spin textures in materials such as YVO$_3$ or MnTe are relativistic components of broader spin-momentum-locking patterns, not necessarily valley-exchange sAMs [2510.23855]. Distinguishing these uses is essential for a consistent encyclopedia-level taxonomy.