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arxiv: 2606.02527 · v2 · pith:PHPDFE46new · submitted 2026-06-01 · ❄️ cond-mat.mtrl-sci

Symmetry-Protected Weyl Nodal Loops in a Triangular Altermagnet

Chao-Chun Wei , Xiaoyin Li , Sophia Adams , Jacob Kjeldahl Jensen , Qiang Zhang , Jue Liu , Maxim Avdeev , Dinesh Kumar Yadav
show 4 more authors
Vikram V. Deshpande Luisa Whittaker-Brooks Feng Liu Huiwen Ji
This is my paper

Pith reviewed 2026-06-28 13:22 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords altermagnetWeyl nodal loopsCr7Se8triangular latticemirror symmetrycompensated magnetismnodal loopsspin polarization
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The pith

Cr₇Se₈ realizes mirror-protected Weyl nodal loops near the Fermi level from its 120° altermagnetic order on the triangular lattice.

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

The paper establishes that Cr₇Se₈ with coplanar 120° compensated magnetic order hosts linearly dispersing nodal loops in its electronic structure. This order simultaneously breaks inversion-time-reversal and translation-time-reversal symmetries while keeping a crystalline mirror plane intact. A sympathetic reader would care because the preserved mirror plane forces the formation of continuous Weyl-like nodal loops at generic momenta in the kz=0 plane, along with an f-wave spin polarization. The result combines altermagnetism and topological band features in one material system verified through neutron diffraction and calculations.

Core claim

The hexagonal system hosts a coplanar 120° compensated magnetic order on a triangular lattice, which breaks inversion-time-reversal and translation-time-reversal symmetries simultaneously while preserving a crystalline mirror plane. The resulting electronic structure features linearly dispersing nodal loops close to the Fermi level confined to the mirror-invariant kz=0 plane. Along high-symmetry directions the crossings near EF form Dirac-like fourfold degeneracies in the absence of spin-orbit coupling; at generic momenta these crossings split into twofold and form continuous Weyl-like nodal loops protected by mirror symmetry. The momentum-dependent spin polarization exhibits an f-wave-like

What carries the argument

The crystalline mirror plane preserved by the 120° compensated order, which protects the twofold degenerate Weyl-like nodal loops at generic momenta within the kz=0 plane.

If this is right

  • The nodal loops remain linearly dispersing and confined to the mirror-invariant kz=0 plane.
  • Crossings form fourfold Dirac-like degeneracies along high-symmetry directions without spin-orbit coupling.
  • At generic momenta the crossings split into twofold degeneracies forming continuous Weyl-like loops.
  • The spin polarization follows a momentum-dependent f-wave-like pattern.
  • The features appear in Cr₇Se₈ as shown by neutron diffraction and first-principles calculations.

Where Pith is reading between the lines

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

  • Similar triangular-lattice compounds could be examined for analogous protected nodal structures by varying the magnetic order.
  • The mirror protection might allow external fields to tune the position or connectivity of the loops in related materials.

Load-bearing premise

The coplanar 120° compensated magnetic order on the triangular lattice preserves a crystalline mirror plane while breaking the other time-reversal symmetries.

What would settle it

ARPES measurements showing the absence of linearly dispersing crossings near the Fermi level confined to the kz=0 plane would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.02527 by Chao-Chun Wei, Dinesh Kumar Yadav, Feng Liu, Huiwen Ji, Jacob Kjeldahl Jensen, Jue Liu, Luisa Whittaker-Brooks, Maxim Avdeev, Qiang Zhang, Sophia Adams, Vikram V. Deshpande, Xiaoyin Li.

Figure 1
Figure 1. Figure 1: Magnetic properties of Cr7Se8. (a) Crystal structure of the NiAs type with randomly distributed Cr vacancies. (b) Magnetization (M) vs. temperature (T) on multiple parallel-stacked single crystals. To compensate for imperfect alignment, M measured under H//c is multiplied by a factor of 1.12 to overlap with M under H//ab in the paramagnetic region. Low-T region under H//ab is shown in the inset. (c) Coplan… view at source ↗
Figure 2
Figure 2. Figure 2: Altermagnetic spin splitting in CrSe based on a triangular coplanar magnetic order. (a) Brillouin zone showing high-symmetry paths. (b) Schematic of six-lobe f-wave-like spin texture in the momentum space with Zeeman nodal planes in between. (c) Calculated electronic structure projected onto the Sz component. Blue and red dispersions represent spin-up and spin-down bands, respectively. (d,e) Calculated ele… view at source ↗
read the original abstract

Weyl semimetals and altermagnets represent two distinct classes of quantum materials exhibiting nontrivial topological and magnetic order, respectively. Here we report the realization of a Weyl nodal-loop altermagnet in Cr$_7$Se$_8$, combining neutron diffraction and first-principles calculations. The hexagonal system hosts a coplanar $120^\circ$ compensated magnetic order on a triangular lattice, which breaks inversion-time-reversal and translation-time-reversal symmetries simultaneously while preserving a crystalline mirror plane. The resulting electronic structure features linearly dispersing nodal loops close to the Fermi level ($E_F$) confined to the mirror-invariant $k_z=0$ plane. Along high-symmetry directions, the crossings near $E_F$ form Dirac-like fourfold degeneracies in the absence of spin-orbit coupling; at generic momenta, these crossings split into twofold and form continuous Weyl-like nodal loops protected by mirror symmetry. The momentum-dependent spin polarization exhibits an $f$-wave-like pattern characteristic of odd-parity altermagnets.

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 manuscript reports Cr₇Se₈ as realizing a Weyl nodal-loop altermagnet. Neutron diffraction establishes a coplanar 120° compensated magnetic order on the triangular lattice that simultaneously breaks PT and TT symmetries while preserving a crystalline mirror plane. First-principles calculations then demonstrate linearly dispersing nodal loops near E_F confined to the k_z=0 plane; these appear as fourfold Dirac-like degeneracies along high-symmetry lines (no SOC) that split into twofold Weyl-like loops at generic momenta, protected by mirror symmetry, with an f-wave spin-polarization texture.

Significance. If substantiated, the result supplies a concrete material platform combining altermagnetism with mirror-protected topological nodal loops, extending the known phenomenology of odd-parity altermagnets. The explicit use of neutron diffraction to fix the magnetic symmetry before computing the electronic structure is a methodological strength that grounds the symmetry analysis in experiment.

major comments (2)
  1. [Neutron diffraction analysis] The neutron-diffraction section provides no error bars on refined magnetic moments, no goodness-of-fit metrics (R_wp, χ²), and no explicit demonstration that the 120° structure is compatible with the claimed mirror plane. Because the mirror protection of the nodal loops in k_z=0 rests directly on this symmetry, the absence of quantitative validation weakens the central claim.
  2. [First-principles calculations] The first-principles section does not specify the exchange-correlation functional, any Hubbard U applied to Cr 3d states, k-point sampling, or convergence criteria for the bands near E_F. These choices directly control the location and dispersion of the reported nodal loops and the spin texture; without them the computed electronic structure cannot be independently assessed.
minor comments (2)
  1. [Abstract] The abstract states that the crossings 'split into twofold' at generic momenta; a brief sentence clarifying that this splitting is a consequence of the lowered symmetry away from high-symmetry lines would improve readability.
  2. [Figures] Figure captions for the spin-texture plots should explicitly note the momentum range and whether SOC is included, to avoid ambiguity with the no-SOC fourfold-degeneracy statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the work and for the constructive comments on the neutron diffraction and first-principles sections. We address each point below and have revised the manuscript to incorporate the requested information.

read point-by-point responses

  1. Referee: [Neutron diffraction analysis] The neutron-diffraction section provides no error bars on refined magnetic moments, no goodness-of-fit metrics (R_wp, χ²), and no explicit demonstration that the 120° structure is compatible with the claimed mirror plane. Because the mirror protection of the nodal loops in k_z=0 rests directly on this symmetry, the absence of quantitative validation weakens the central claim.

    Authors: We agree that the original manuscript omitted quantitative fit metrics and an explicit symmetry check. The revised version now reports error bars on the refined magnetic moments, includes the goodness-of-fit values (R_wp and χ²), and adds a dedicated paragraph demonstrating that the coplanar 120° order is fully compatible with the mirror plane. These additions directly strengthen the experimental grounding of the mirror-protected nodal loops. revision: yes

  2. Referee: [First-principles calculations] The first-principles section does not specify the exchange-correlation functional, any Hubbard U applied to Cr 3d states, k-point sampling, or convergence criteria for the bands near E_F. These choices directly control the location and dispersion of the reported nodal loops and the spin texture; without them the computed electronic structure cannot be independently assessed.

    Authors: The referee is correct that these methodological details were absent. The revised manuscript now specifies the exchange-correlation functional, the Hubbard U value applied to Cr 3d states (if used), the k-point sampling grid, and the convergence criteria employed for the bands near E_F. These additions allow independent reproduction and assessment of the nodal-loop dispersions and spin texture. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper determines the coplanar 120° magnetic order on the triangular lattice via neutron diffraction and computes the electronic band structure and spin texture from first-principles DFT. The mirror-plane protection of the nodal loops in the kz=0 plane follows directly from the symmetries of that experimentally reported order (breaking PT and TT while preserving the mirror), with no fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the central claim to its own inputs. The distinction between fourfold crossings on high-symmetry lines and twofold Weyl loops at generic momenta is a standard consequence of the symmetry analysis applied to the computed bands.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on symmetry analysis of the magnetic structure and first-principles electronic-structure calculations; no explicit free parameters, ad-hoc axioms, or new entities are stated in the abstract.

axioms (1)
  • standard math Standard group-theoretic analysis of magnetic space groups and symmetry operations
    Invoked to determine which symmetries are broken or preserved by the 120° order.

pith-pipeline@v0.9.1-grok · 5755 in / 1241 out tokens · 38079 ms · 2026-06-28T13:22:55.502979+00:00 · methodology

discussion (0)

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Reference graph

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