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arxiv: 2605.11587 · v1 · submitted 2026-05-12 · ❄️ cond-mat.supr-con · cond-mat.mes-hall

Recognition: 2 theorem links

· Lean Theorem

Euler Topology in Superconducting Honeycomb Lattices

Rasoul Ghadimi , Chiranjit Mondal , Bohm-Jung Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:37 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hall
keywords Euler topologysuperconducting honeycomb latticesspace-time inversion symmetryhelical domain-wall modesnon-Abelian braidingDirac nodesvalley-Euler superconductorf-wave pairing
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The pith

Superconducting pairings on honeycomb lattices give rise to Euler superconductors with protected helical modes.

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

This paper examines band topology in honeycomb lattices that have space-time inversion symmetry and undergo superconductivity. It finds that s-wave spin-singlet pairings produce valley-Euler superconductors while f-wave spin-triplet pairings produce Euler superconductors. In both cases the Euler topology creates helical modes at domain walls that mirror symmetry protects. Adding anisotropic hopping to the f-wave state further causes Dirac nodes to braid non-Abelianly in momentum space. The findings present superconducting order as a route to Euler topology in Dirac materials.

Core claim

In IST-symmetric superconducting honeycomb lattices, s-wave spin-singlet (SWSS) pairings give rise to valley-Euler superconductors and f-wave spin-triplet (FWST) pairings give rise to Euler superconductors. The Euler topology in both pairing states leads to mirror-symmetry-protected helical domain-wall modes. In the FWST state, this topology induces non-Abelian braiding of Dirac nodes in momentum space when anisotropic hopping is introduced. Superconducting electronic instabilities thus provide a natural route to realizing nontrivial Euler band topology in Dirac materials.

What carries the argument

The Euler topology of the bands enabled by space-time inversion symmetry, which classifies the superconducting states and enforces helical domain-wall modes plus Dirac-node braiding.

If this is right

  • Mirror symmetry protects helical domain-wall modes in both the valley-Euler and Euler superconducting states.
  • Anisotropic hopping in the f-wave spin-triplet state induces non-Abelian braiding of Dirac nodes.
  • Superconducting instabilities offer a route to Euler topology in systems with Dirac bands.
  • The distinction between valley-Euler and full Euler superconductivity arises from the choice of pairing symmetry.

Where Pith is reading between the lines

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

  • The same pairing-induced mechanism could appear in other two-dimensional Dirac lattices that preserve space-time inversion symmetry.
  • Anisotropy provides an experimental tuning parameter to control the braiding of Dirac nodes in fabricated samples.
  • The helical modes could be probed through local spectroscopy or transport at engineered domain walls in real materials.

Load-bearing premise

The superconducting honeycomb lattices must possess space-time inversion symmetry for their bands to host nontrivial Euler topology.

What would settle it

Numerical or experimental absence of mirror-protected helical modes at domain walls, or a zero Euler invariant in the calculated band structure for the predicted pairings, would disprove the claims.

Figures

Figures reproduced from arXiv: 2605.11587 by Bohm-Jung Yang, Chiranjit Mondal, Rasoul Ghadimi.

Figure 1
Figure 1. Figure 1: FIG. 1. Euler superconductor (ES) and valley-Euler superconduc [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Domain-wall setup, where the sign of the pairing poten [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Normal-state energy dispersion of the honeycomb lattice [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Electronic bands in systems with space-time inversion (IST) symmetry can host nontrivial Euler topology. Here, we investigate the band topology of IST-symmetric superconducting honeycomb lattices and demonstrate that s-wave spin-singlet (SWSS) and f-wave spin-triplet (FWST) superconducting pairings give rise to valley-Euler and Euler superconductors, respectively. We find that Euler topology in both pairing states gives rise to mirror-symmetry-protected helical domain-wall modes. Furthermore, we show that Euler topology in the FWST state induces non-Abelian braiding of Dirac nodes in momentum space when anisotropic hopping is introduced. Our work establishes superconducting electronic instabilities as a natural route to realizing nontrivial Euler band topology in Dirac materials.

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 IST-symmetric superconducting honeycomb lattices with s-wave spin-singlet (SWSS) pairing realize valley-Euler superconductors while f-wave spin-triplet (FWST) pairing realizes Euler superconductors. Euler topology in both cases produces mirror-symmetry-protected helical domain-wall modes, and anisotropic hopping in the FWST state further induces non-Abelian braiding of Dirac nodes in momentum space. The work positions superconducting instabilities as a route to nontrivial Euler topology in Dirac materials.

Significance. If the IST symmetry is preserved by the chosen pairings and the Euler class is correctly computed, the results would identify a concrete mechanism for realizing Euler topology and its protected modes in superconductors, with potential relevance to helical edge states and non-Abelian statistics. The approach relies on standard symmetry classification rather than ad-hoc parameters.

major comments (2)
  1. [Abstract and BdG Hamiltonian section] The central claims require that both the SWSS and FWST order parameters preserve IST symmetry in the BdG Hamiltonian so that the bands remain real and admit a well-defined Euler class. The abstract asserts IST symmetry of the lattices but supplies no explicit commutation relation, matrix representation, or check that the pairing terms commute with the IST operator; without this verification the valley-Euler/Euler classification and the subsequent domain-wall and braiding results do not follow.
  2. [Domain-wall modes subsection] The statement that Euler topology produces mirror-symmetry-protected helical domain-wall modes needs an explicit symmetry argument or invariant calculation (e.g., a mirror-resolved Euler number or Pfaffian invariant along the domain wall). If this is only asserted from bulk topology without a domain-wall calculation, the claim is not yet load-bearing.
minor comments (2)
  1. [Introduction] Define 'valley-Euler' versus 'Euler' topology at first use and clarify whether the former is a standard term or a new designation for the valley-contrasting Euler class.
  2. [Figures] Ensure all figures showing band structures or topological invariants are labeled with the specific pairing (SWSS or FWST) and include the IST operator action if relevant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We have carefully considered the major comments and revised the manuscript accordingly to provide the requested explicit verifications and calculations. Our point-by-point responses are as follows.

read point-by-point responses

  1. Referee: [Abstract and BdG Hamiltonian section] The central claims require that both the SWSS and FWST order parameters preserve IST symmetry in the BdG Hamiltonian so that the bands remain real and admit a well-defined Euler class. The abstract asserts IST symmetry of the lattices but supplies no explicit commutation relation, matrix representation, or check that the pairing terms commute with the IST operator; without this verification the valley-Euler/Euler classification and the subsequent domain-wall and braiding results do not follow.

    Authors: We agree with the referee that an explicit verification of IST symmetry preservation is essential for the validity of the Euler classification. In the revised manuscript, we have added a dedicated subsection in the BdG Hamiltonian section detailing the IST operator in matrix form for the honeycomb lattice. We explicitly demonstrate that [IST, H] = 0 for the normal state and show that both the SWSS and FWST pairing terms commute with the IST operator, preserving the reality of the bands and allowing the Euler class to be defined. The commutation relations are now provided explicitly, along with the basis choice that makes the Hamiltonian real. revision: yes

  2. Referee: [Domain-wall modes subsection] The statement that Euler topology produces mirror-symmetry-protected helical domain-wall modes needs an explicit symmetry argument or invariant calculation (e.g., a mirror-resolved Euler number or Pfaffian invariant along the domain wall). If this is only asserted from bulk topology without a domain-wall calculation, the claim is not yet load-bearing.

    Authors: We thank the referee for pointing this out. While the bulk-boundary correspondence for Euler topology suggests the presence of protected modes, we acknowledge the need for a more explicit argument. In the revision, we have included a symmetry analysis showing that the mirror symmetry commutes with the Euler class in a way that protects helical modes at the domain wall. Furthermore, we have added a calculation of the domain-wall spectrum using a ribbon geometry, confirming the existence of helical modes crossing the gap, protected by the mirror symmetry. We also discuss the mirror-resolved Euler invariant to make the protection rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard Euler classification to IST-symmetric BdG Hamiltonians

full rationale

The paper assumes IST symmetry for the superconducting honeycomb lattices and classifies SWSS and FWST pairings as inducing valley-Euler or Euler topology via standard band-structure analysis in the BdG framework. No step reduces a claimed prediction or topological invariant to a fitted parameter defined by the result itself, nor does any load-bearing premise collapse to a self-citation chain or self-definitional loop. The helical modes and braiding follow from the topology once the symmetry is granted; the symmetry preservation is an input assumption rather than an output derived from the Euler class. The work therefore remains self-contained against external benchmarks of symmetry-protected topology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claims rest on the presence of space-time inversion symmetry and on standard assumptions of band theory in superconductors; no explicit free parameters or invented entities are stated.

axioms (1)
  • domain assumption The system possesses space-time inversion (IST) symmetry.
    Invoked as the condition that allows nontrivial Euler topology in the bands.

pith-pipeline@v0.9.0 · 5420 in / 1222 out tokens · 33880 ms · 2026-05-13T01:37:28.026180+00:00 · methodology

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Reference graph

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