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arxiv: 2509.03247 · v3 · pith:6L3IOK5Lnew · submitted 2025-09-03 · ❄️ cond-mat.supr-con

Inherent momentum-dependent gap structure of altermagnetic superconductors

Christian L. H. Rasmussen , Jannik Gondolf , Mats Barkman , Merc\`e Roig , Daniel F. Agterberg , Andreas Kreisel , Brian M. Andersen This is my paper

Pith reviewed 2026-05-21 22:44 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords altermagnetssuperconductivitygap nodessublattice structurespin splittingtriplet pairingBrillouin zone
0
0 comments X

The pith

Altermagnetic sublattice structure forces nodes in the superconducting gap on Brillouin zone edges

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

The paper studies how altermagnets, which break time-reversal symmetry through compensated collinear magnetic order and produce spin-split bands, can become superconducting at low temperatures. It demonstrates that the two-sublattice structure built into these materials constrains the gap function, requiring nodes on the Brillouin zone boundaries whenever the pairing comes from momentum-independent attractions. This constraint disappears with longer-range interactions, which then permit both spin-singlet and equal-spin triplet pairing, the latter becoming preferred when the magnetic splitting is much larger than the gap. A reader would care because the result identifies a built-in mechanism that sets the symmetry and nodal structure of the superconducting state in this new class of materials.

Core claim

Using microscopic models that retain the altermagnetic sublattice degrees of freedom, the sublattice structure imposes nodes in the gap on the Brillouin zone edges for superconductors stabilized by momentum-independent bare attraction channels. In contrast, superconductivity generated by extended range interactions allows pairing on the Brillouin zone edges and stabilizes both spin-singlet and equal-spin-pairing triplet states. Equal-spin-pairing triplet superconductivity is generically favored in the limit of large altermagnetic spin splitting compared to the gap scale and features nonunitary properties arising from the altermagnetic order.

What carries the argument

The altermagnetic sublattice structure on a compensated Néel state with collinear moments, which enforces momentum-dependent constraints on the superconducting gap through the normal-state band structure.

If this is right

  • Momentum-independent pairing produces nodes on Brillouin zone edges.
  • Extended interactions lift the nodes and allow both singlet and triplet channels.
  • Triplet equal-spin pairing is favored when altermagnetic splitting greatly exceeds the gap.
  • The triplet state acquires nonunitary character from the underlying altermagnetic order.

Where Pith is reading between the lines

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

  • The enforced nodes should produce power-law temperature dependence in low-energy observables such as specific heat.
  • Materials with tunable interaction range could switch between nodal and nodeless regimes.
  • The same sublattice mechanism may constrain pairing in other compensated magnetic systems.

Load-bearing premise

The altermagnetic order is modeled as a compensated Néel state with collinear moments, and the superconducting instability is treated within a mean-field or weak-coupling framework on top of the normal-state bands.

What would settle it

Angle-resolved photoemission or tunneling measurements that find a nodeless gap on the Brillouin zone edges in an altermagnetic superconductor driven by short-range attraction would contradict the predicted nodes.

Figures

Figures reproduced from arXiv: 2509.03247 by Andreas Kreisel, Brian M. Andersen, Christian L. H. Rasmussen, Daniel F. Agterberg, Jannik Gondolf, Mats Barkman, Merc\`e Roig.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Sublattice structure of the 2D lattice model with orange sites defining the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. In the upper panel, Fourier coefficients [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The superconducting order parameter stabilized [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The superconducting order parameter stabilized by NNN bare attraction. Each horizontal row displays the momentum [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Altermagnetic metals break time-reversal symmetry and feature spin-split Fermi surfaces generated by compensated N\'eel-ordered collinear magnetic moments. Being metallic, such altermagnets may undergo a further instability at low temperatures to a superconducting state, and it is an interesting open question what the salient features are of such altermagnetic superconductors. We address this question on the basis of realistic microscopic models that capture the altermagnetic sublattice degrees of freedom. We find that the sublattice structure can strongly affect the superconducting gap structure in altermagnetic superconductors. In particular, it imposes nodes in the gap on the Brillouin zone edges for superconductors stabilized by momentum-independent bare attraction channels. We contrast this to the case of superconductivity generated by extended range interactions where pairing is allowed on the Brillouin zone edges and both spin-singlet and equal-spin-pairing triplet states can be stabilized. Equal-spin-pairing triplet superconductivity is generically favored in the limit of large altermagnetic spin splitting of the bands compared to the superconducting gap scale, and features characteristic nonunitary properties arising from the altermagnetic order.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that the sublattice structure in altermagnetic superconductors, modeled via compensated Néel order with collinear moments, imposes nodes in the superconducting gap on Brillouin zone edges for pairing driven by momentum-independent bare attractions. For extended-range interactions, pairing on zone edges is allowed and both singlet and equal-spin triplet states can be stabilized, with triplet states generically favored (and nonunitary) when altermagnetic band splitting greatly exceeds the gap scale. The results follow from analytic solution of the gap equation on normal-state bands taken from the altermagnetic Hamiltonian.

Significance. If the central derivation holds, the work supplies a concrete, model-derived mechanism by which altermagnetic sublattice degrees of freedom enforce specific momentum-dependent gap features, offering a clear distinction from conventional superconductors and falsifiable predictions for nodes and pairing symmetry. The analytic treatment within the stated microscopic model and weak-coupling framework is a strength, as is the explicit contrast between local and extended interactions.

major comments (2)
  1. [microscopic model and gap equations] § on microscopic Hamiltonian and gap equation: the result that the gap vanishes on BZ edges for momentum-independent attraction follows from the fixed compensated Néel background and standard mean-field decoupling on the normal-state bands. This fixed-order assumption is load-bearing for the central claim; any self-consistent feedback in which the superconducting condensate renormalizes the effective exchange or interaction vertex could permit finite matrix elements on the zone edges, as the stress-test concern correctly identifies.
  2. [results on triplet pairing] Results section on large-splitting limit: the statement that equal-spin triplet pairing is generically favored when altermagnetic splitting ≫ gap scale is derived under the weak-coupling mean-field treatment; the manuscript should quantify the regime in which this remains valid when the two energy scales become comparable.
minor comments (2)
  1. [notation] The notation distinguishing the different bare attraction channels would be clearer if summarized in a short table or explicit equation reference.
  2. [figures] Figures displaying the gap structure on the Brillouin zone would benefit from explicit labels or shading marking the zone edges where nodes are claimed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [microscopic model and gap equations] § on microscopic Hamiltonian and gap equation: the result that the gap vanishes on BZ edges for momentum-independent attraction follows from the fixed compensated Néel background and standard mean-field decoupling on the normal-state bands. This fixed-order assumption is load-bearing for the central claim; any self-consistent feedback in which the superconducting condensate renormalizes the effective exchange or interaction vertex could permit finite matrix elements on the zone edges, as the stress-test concern correctly identifies.

    Authors: We agree that the fixed compensated Néel order is central to our analytic treatment. Within the weak-coupling framework of the manuscript, the altermagnetic order is the primary instability setting the normal-state bands, while superconductivity develops at a parametrically lower scale; back-action from the condensate on the exchange field is therefore higher-order and does not alter the leading gap structure. Nevertheless, to address the concern we will add a dedicated paragraph in the revised manuscript that explicitly states the regime of validity of the fixed-order approximation and notes that a fully self-consistent treatment of both orders lies beyond the present analytic scope. revision: yes

  2. Referee: [results on triplet pairing] Results section on large-splitting limit: the statement that equal-spin triplet pairing is generically favored when altermagnetic splitting ≫ gap scale is derived under the weak-coupling mean-field treatment; the manuscript should quantify the regime in which this remains valid when the two energy scales become comparable.

    Authors: We thank the referee for this suggestion. Our analytic preference for nonunitary equal-spin triplet pairing is obtained in the limit where the altermagnetic splitting greatly exceeds the gap. In the revised manuscript we will add a short quantitative discussion that estimates the range of validity by comparing the two scales inside the gap equation and indicates the corrections expected when they become comparable; we will also note that a numerical solution of the coupled equations would be needed outside this regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from microscopic Hamiltonian

full rationale

The paper constructs explicit microscopic models incorporating altermagnetic sublattice degrees of freedom and compensated Néel order, then solves the superconducting gap equation in the weak-coupling mean-field limit on the resulting normal-state bands. The reported nodes on Brillouin-zone edges for momentum-independent bare attractions follow directly from the symmetry properties encoded in the Hamiltonian and the structure of the gap equation; this does not reduce to a fitted parameter, a renamed input, or a self-citation chain. Normal-state bands are treated as given input from the altermagnetic model, and the superconducting instability is analyzed on top without mutual renormalization feedback entering the central claim. The derivation therefore remains independent of the target result and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard BCS-style mean-field treatment of superconductivity on top of a two-sublattice altermagnetic band structure. No new particles or forces are introduced. One free parameter is the relative strength of local versus extended interactions.

free parameters (1)
  • interaction range parameter
    The distinction between momentum-independent bare attraction and extended-range interactions is introduced to contrast the two regimes; its specific value is not fitted but chosen to illustrate the cases.
axioms (2)
  • domain assumption Compensated Néel-ordered collinear magnetic moments generate spin-split Fermi surfaces while preserving overall time-reversal properties in the normal state.
    This is the defining property of altermagnets used to construct the normal-state Hamiltonian.
  • standard math Superconducting instability can be treated within a weak-coupling gap equation on the normal-state bands.
    Standard mean-field approach for pairing instabilities.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Direction-selective triplet pairing and spin-edge locking in altermagnetic metals

    cond-mat.supr-con 2026-05 unverdicted novelty 6.0

    Altermagnetic spin splitting selects direction-selective triplet pairing in 2D d-wave metals and generates spin-locked Majorana edge states in both spin-conserving and Rashba-mixed regimes.

Reference graph

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    How- 11 ever, as we will demonstrate below, fort z ̸= 0 and Nz ̸= 0, features unique to altermagnetism will be im- printed on the triplet state

    The triplet state would be stabilized for this set of param- eters even in the absence of altermagnetic order. How- 11 ever, as we will demonstrate below, fort z ̸= 0 and Nz ̸= 0, features unique to altermagnetism will be im- printed on the triplet state

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