Extended s-wave altermagnets

Lennart Klebl; Matteo D\"urrnagel; Michael Klett; Ronny Thomale; Tobias M\"uller

arxiv: 2508.20163 · v3 · submitted 2025-08-27 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.mtrl-sci

Extended s-wave altermagnets

Matteo D\"urrnagel , Lennart Klebl , Tobias M\"uller , Ronny Thomale , Michael Klett This is my paper

Pith reviewed 2026-05-18 20:37 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-sci

keywords extended s-wave altermagnetsvalley-exchange symmetriesspin-polarized bandsspin compensationisotropic spin splittingpair density wavesquantum magnetsheterostructures

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The pith

Extended s-wave altermagnets form fully gapped spin-compensated states with spin-polarized bands via valley-exchange symmetries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces extended s-wave altermagnets as a distinct class of magnetic states that combine full energy gaps and spin compensation with the presence of spin-polarized electronic bands. These states emerge from valley-exchange symmetries, which function as translations in momentum space and extend beyond conventional crystallographic spin-group classifications. Through an effective two-valley model, the work shows that such states produce isotropic spin splitting, support spin-selective transport in specially designed heterostructures, and lead to related pair density wave orders. Microscopic modeling then provides principles for recognizing these states in actual quantum magnets, expanding the landscape of possible magnetic orders.

Core claim

Extended s-wave altermagnets, or sAMs, represent a class of magnetic states that are fully gapped and spin-compensated yet display spin-polarized bands. These arise via valley-exchange symmetries that serve as momentum-space translations, surpassing standard spin-group classifications based on crystal symmetry. An effective two-valley model illustrates their isotropic spin splitting, the potential for spin-selective transport in heterostructures, and the emergence of descendant pair density wave orders. Guiding principles derived from a minimal microscopic model then help identify such states within quantum magnets.

What carries the argument

valley-exchange symmetries that act as momentum-space translations beyond standard crystallographic spin-group classifications

If this is right

sAMs exhibit isotropic spin splitting in their band structure.
They enable spin-selective transport in tailored heterostructures.
Descendant pair density wave order emerges from the sAM states.
Guiding principles from the minimal model allow identification of sAMs in quantum magnets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The extension beyond standard spin-group classifications implies that altermagnetic states can appear in a broader range of symmetry settings than previously considered.
Spin-polarized bands in compensated systems open possibilities for spin-filtering effects in devices without requiring external fields or net magnetization.
The descendant pair density wave orders suggest connections to modulated superconducting phases that could be explored in related multi-valley materials.

Load-bearing premise

The assumption that valley-exchange symmetries can be realized as momentum-space translations in actual quantum magnets and that an effective two-valley model suffices to capture the isotropic spin splitting and descendant orders.

What would settle it

Measurement of anisotropic spin splitting instead of isotropic splitting in a two-valley magnetic system engineered with valley-exchange symmetry would falsify the proposal.

Figures

Figures reproduced from arXiv: 2508.20163 by Lennart Klebl, Matteo D\"urrnagel, Michael Klett, Ronny Thomale, Tobias M\"uller.

**Figure 2.** Figure 2: FIG. 2. Spin splitter effect in an [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

We propose extended s-wave altermagnets (sAMs) as a class of magnetic states which are fully gapped, spin-compensated, and feature spin-polarized bands. sAMs are formed through valley-exchange symmetries, which act as momentum-space translations beyond standard crystallographic spin-group classifications. Using an effective two-valley model, we demonstrate that sAMs exhibit isotropic spin splitting, enable spin-selective transport in tailored heterostructures, and give rise to descendant pair density wave order. From a microscopic sAM minimal model, we develop the guiding principles to identify sAMs in quantum magnets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines extended s-wave altermagnets through valley-exchange symmetries in an effective two-valley model, but the isotropic splitting and gap protection look fragile once intervalley terms are restored.

read the letter

The main takeaway is that this work introduces extended s-wave altermagnets as a symmetry class built on valley-exchange operations that go beyond standard spin-group labels. These states are claimed to be fully gapped and spin-compensated while still showing spin-polarized bands and isotropic splitting. The effective two-valley Hamiltonian makes the spin-selective transport and pair-density-wave descendants appear straightforward to realize in heterostructures. From a minimal microscopic model they also extract some practical rules for hunting candidates in quantum magnets. That framing is the clearest addition relative to earlier altermagnet papers. The model demonstrations are clean enough that someone working on symmetry-based magnetic states could pick up the idea quickly and test it numerically. The soft spot sits where the stress test points: everything rests on the two-valley truncation. Intervalley scattering or additional valleys permitted by the underlying lattice are not shown to preserve the isotropy or the full gap. Without an explicit full-lattice check that keeps the same symmetries, it is hard to know whether the advertised properties survive outside the reduced subspace. The abstract and model sections do not appear to contain that check. This paper is aimed at theorists who track altermagnets, spintronics proposals, and possible links to unconventional pairing. A reader already comfortable with valley physics and effective Hamiltonians will get the most out of it. The proposal is distinct enough and the model work concrete enough that it should go to peer review rather than a desk reject, though referees will almost certainly ask for lattice-model stability tests before any stronger claims are accepted.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes extended s-wave altermagnets (sAMs) as a new class of magnetic states that are fully gapped, spin-compensated, and host spin-polarized bands. These states are realized via valley-exchange symmetries that function as momentum-space translations beyond standard spin-group classifications. Using an effective two-valley model, the authors demonstrate isotropic spin splitting, spin-selective transport in heterostructures, and the emergence of descendant pair-density-wave order. Guiding principles for identifying sAMs in quantum magnets are extracted from a microscopic minimal model.

Significance. If the central properties survive beyond the effective-model truncation, the work would meaningfully enlarge the symmetry-based taxonomy of altermagnets and link them to spin-selective transport and PDW orders. The provision of explicit guiding principles from a microscopic model is a constructive step toward material searches. The overall significance remains moderate until the robustness against intervalley terms is established.

major comments (2)

[§3] §3 (effective two-valley Hamiltonian): The claims of a fully gapped spectrum and isotropic spin splitting are demonstrated only inside the two-valley truncation. No explicit check is provided that intervalley scattering terms permitted by the underlying lattice symmetry but forbidden within the two-valley subspace leave the gap and isotropy intact. This truncation is load-bearing for the central claim that sAMs are fully gapped and spin-polarized.
[§4] §4 (microscopic minimal model): The guiding principles for realizing valley-exchange symmetry are derived from the minimal model, yet the manuscript does not show that these principles remain sufficient when the full Brillouin zone and longer-range hoppings are restored. A concrete lattice example with the stated symmetry would directly test whether the effective-model properties are stable.

minor comments (2)

[Abstract] The abstract states that sAMs are 'beyond standard crystallographic spin-group classifications' but does not cite the specific spin-group references being extended; adding one or two key citations would clarify the novelty.
[Figures] Figure captions for the band-structure plots could explicitly state the momentum path used and whether the plotted splitting is along high-symmetry lines or averaged over the valley.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the scope and robustness of our proposal. We address the major comments point by point below, emphasizing the symmetry-based protections that underpin our claims while agreeing to strengthen the manuscript with additional discussion where appropriate.

read point-by-point responses

Referee: [§3] §3 (effective two-valley Hamiltonian): The claims of a fully gapped spectrum and isotropic spin splitting are demonstrated only inside the two-valley truncation. No explicit check is provided that intervalley scattering terms permitted by the underlying lattice symmetry but forbidden within the two-valley subspace leave the gap and isotropy intact. This truncation is load-bearing for the central claim that sAMs are fully gapped and spin-polarized.

Authors: The two-valley truncation is not an arbitrary approximation but follows directly from the valley-exchange symmetry that defines the sAM state. This symmetry acts as a momentum-space translation that enforces spin compensation and isotropy while explicitly forbidding intervalley scattering terms that would mix the valleys in a manner inconsistent with the symmetry. Terms permitted by the bare lattice symmetry but excluded from the effective subspace necessarily violate the valley-exchange operation if they introduce mixing; hence they lie outside the symmetry class under consideration. Within the sAM symmetry class, the gap and isotropy are protected. To make this explicit, we will add a short symmetry analysis and a perturbative check of allowed higher-order terms in the revised manuscript. revision: partial
Referee: [§4] §4 (microscopic minimal model): The guiding principles for realizing valley-exchange symmetry are derived from the minimal model, yet the manuscript does not show that these principles remain sufficient when the full Brillouin zone and longer-range hoppings are restored. A concrete lattice example with the stated symmetry would directly test whether the effective-model properties are stable.

Authors: The guiding principles are extracted from symmetry requirements on the hoppings and interactions that realize valley exchange; these requirements are independent of the cutoff in hopping range or the precise Brillouin-zone sampling. Longer-range terms that preserve the valley-exchange symmetry can be added without altering the qualitative conclusions. The minimal model already encodes a concrete lattice realization consistent with the symmetry. Nevertheless, we agree that an explicit demonstration on a finite lattice with extended hoppings would strengthen the presentation. We will include such an example (or a clear statement of its construction) in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: effective two-valley model and microscopic minimal model are independent constructions

full rationale

The paper introduces sAMs through valley-exchange symmetries acting as momentum-space translations and demonstrates their properties (isotropic spin splitting, full gap, spin-polarized bands) via an explicit effective two-valley Hamiltonian constructed from a microscopic minimal model. No quoted step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames an input as an output. The two-valley truncation is presented as a controlled approximation to isolate symmetry effects rather than a self-referential definition, and the guiding principles for identification in quantum magnets are derived forward from the microscopic model without circular closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; limited visibility into parameters or assumptions. The proposal introduces a new magnetic state class relying on symmetry assumptions.

axioms (1)

domain assumption Valley-exchange symmetries exist and act as momentum-space translations beyond standard spin-group classifications
Invoked in the abstract to form sAMs and enable the described properties.

invented entities (1)

extended s-wave altermagnets (sAMs) no independent evidence
purpose: New class of fully gapped, spin-compensated magnetic states with spin-polarized bands
Proposed as the central new concept in the paper.

pith-pipeline@v0.9.0 · 5644 in / 1453 out tokens · 62124 ms · 2026-05-18T20:37:15.345906+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Reference graph

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The extended s-wave character is rooted in the position of one pocket at the zone center ν = Γ and the other at ν = M

Valley exchange τ x thus acts as a symmetry that al- lows for magnetic compensation. The extended s-wave character is rooted in the position of one pocket at the zone center ν = Γ and the other at ν = M. Impor- tantly, τ x does not necessarily resemble a simple real- space symmetry operation, as it corresponds to a shift of q = M in momentum space. This a...

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Z. M. Raines and A. V. Chubukov, Two-dimensional stoner transitions beyond mean field, Phys. Rev. B 110, 235433 (2024). Supplemental Material: Extended s-wave altermagnets Matteo D¨ urrnagel,∗ Lennart Klebl, ∗ Tobias M¨ uller, Ronny Thomale, and Michael Klett† Institut f¨ ur Theoretische Physik und Astrophysik and W¨ urzburg-Dresden Cluster of Excellence ...

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Symmetry analysis S8

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sAM with higher harmonics S10

Impact on magnetic states S10 D. sAM with higher harmonics S10

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Symmetry analysis S11

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Extension to 1D systems S12 S4

Altermagnetic states S12 E. Extension to 1D systems S12 S4. Valley model for sAM on the hexagonal lattice S13 S1. SPIN AND VALLEY POLARIZATION FROM MEAN-FIELD AND RENORMALIZATION GROUP CALCULATIONS To analyze the propensity of an sAM state in the given geometry, we consider a general two valley system with two equal spherical valleys τ = ± around the Γ an...

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(S3.10) in the following

Symmetry analysis Since symmetries are incremental for the characterization of magnetic states we (at least try to) list all symmetries of the Hamiltonian in Eq. (S3.10) in the following. Since the eigenspectrum is given by the norm of the ⃗ vvector, it is easy to show that all operations that act as a simple rotation in the ⃗ vspace leave the spectrum in...

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(S3.17) Clearly this term breaks the usual time reversal symmetry (TRS) defined by the operator T = τ 0iσyK (S3.18) since T (H + HAFM)T −1 = H − HAFM

Impact on magnetic states Let us now consider an inter-sublattice AFM state, that introduces an additional term in the quasiparticle spectrum HAFM = X k ⃗ c† k∆τ zσz⃗ ck . (S3.17) Clearly this term breaks the usual time reversal symmetry (TRS) defined by the operator T = τ 0iσyK (S3.18) since T (H + HAFM)T −1 = H − HAFM. However, the combination ST remain...

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Symmetry analysis The model enjoys C4v symmetry, where all pointgroup operations act trivially in orbital space. However, inspecting momentum space translations we can see two different transformation behaviors in the orbital sector: An M point translation provides a symmetry of the Hamiltonian in combination with an exchange of the orbitals in the indivi...

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(S3.27) While all of these states are spin compensated by symmetry, the different eigenvalues of ∆ 1,2 with respect to SX,Y indicate a C4 symmetry breaking for these states (cf

Altermagnetic states Due to the symmetry restrictions of the model, one can think of three different compensated magnetic structures on the lattice: ∆0 ∝ X k,σ σz σσ ⃗ c† k,στ 0νz⃗ ck,σ , ∆1 ∝ X k,σ σz σσ ⃗ c† k,στ zν0⃗ ck,σ , ∆2 ∝ X k,σ σz σσ ⃗ c† k,στ zνz⃗ ck,σ . (S3.27) While all of these states are spin compensated by symmetry, the different eigenvalu...

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The extended s-wave character is rooted in the position of one pocket at the zone center ν = Γ and the other at ν = M

Valley exchange τ x thus acts as a symmetry that al- lows for magnetic compensation. The extended s-wave character is rooted in the position of one pocket at the zone center ν = Γ and the other at ν = M. Impor- tantly, τ x does not necessarily resemble a simple real- space symmetry operation, as it corresponds to a shift of q = M in momentum space. This a...

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0.02 0.04 0.06 0.08 0.10 R (b) (c) (a) FIG

0.05 0.1 0.15 0.2 ∆ sAM/EF j↑ j↓ 0. 0.02 0.04 0.06 0.08 0.10 R (b) (c) (a) FIG. 2. Spin splitter effect in an sAM-N junction. (a) Setup of the sAM-N heterostructure with the Fermi surfaces and their respective radii κ(↑/↓) indicated. Due to the reduced modes available on the N side of the junction, only electronic eigenstates in the shaded areas can propa...

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Altermagnetic states S12 E. Extension to 1D systems S12 S4. Valley model for sAM on the hexagonal lattice S13 S1. SPIN AND VALLEY POLARIZATION FROM MEAN-FIELD AND RENORMALIZATION GROUP CALCULATIONS To analyze the propensity of an sAM state in the given geometry, we consider a general two valley system with two equal spherical valleys τ = ± around the Γ an...

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(S3.10) in the following

Symmetry analysis Since symmetries are incremental for the characterization of magnetic states we (at least try to) list all symmetries of the Hamiltonian in Eq. (S3.10) in the following. Since the eigenspectrum is given by the norm of the ⃗ vvector, it is easy to show that all operations that act as a simple rotation in the ⃗ vspace leave the spectrum in...

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(S3.17) Clearly this term breaks the usual time reversal symmetry (TRS) defined by the operator T = τ 0iσyK (S3.18) since T (H + HAFM)T −1 = H − HAFM

Impact on magnetic states Let us now consider an inter-sublattice AFM state, that introduces an additional term in the quasiparticle spectrum HAFM = X k ⃗ c† k∆τ zσz⃗ ck . (S3.17) Clearly this term breaks the usual time reversal symmetry (TRS) defined by the operator T = τ 0iσyK (S3.18) since T (H + HAFM)T −1 = H − HAFM. However, the combination ST remain...

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Symmetry analysis The model enjoys C4v symmetry, where all pointgroup operations act trivially in orbital space. However, inspecting momentum space translations we can see two different transformation behaviors in the orbital sector: An M point translation provides a symmetry of the Hamiltonian in combination with an exchange of the orbitals in the indivi...

work page

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(S3.27) While all of these states are spin compensated by symmetry, the different eigenvalues of ∆ 1,2 with respect to SX,Y indicate a C4 symmetry breaking for these states (cf

Altermagnetic states Due to the symmetry restrictions of the model, one can think of three different compensated magnetic structures on the lattice: ∆0 ∝ X k,σ σz σσ ⃗ c† k,στ 0νz⃗ ck,σ , ∆1 ∝ X k,σ σz σσ ⃗ c† k,στ zν0⃗ ck,σ , ∆2 ∝ X k,σ σz σσ ⃗ c† k,στ zνz⃗ ck,σ . (S3.27) While all of these states are spin compensated by symmetry, the different eigenvalu...

work page