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arxiv: 2510.25959 · v4 · submitted 2025-10-29 · ✦ hep-th · cond-mat.supr-con· hep-lat· math-ph· math.MP

Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions

Gabriel Meyniel , Fei Zhou This is my paper

Pith reviewed 2026-05-18 02:56 UTC · model grok-4.3

classification ✦ hep-th cond-mat.supr-conhep-latmath-phmath.MP
keywords Weyl fermionslattice modelstopological superconductorstime-reversal symmetrytopological quantum critical pointsnodal pointschiral fermionssuperconductors
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The pith

Lattice models with single Weyl fermions in three dimensions all belong to one equivalence class in the infrared, isomorphic to topological superconductor critical points or their duals.

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

This paper develops a generic method to build three-dimensional lattice models that host exactly one Weyl cone in the low-energy limit by breaking charge symmetry in topological states. It examines three specific constructions based on symmetry-protected topological phases: pushing to a topological quantum critical point with a minimal change in winding number, applying time-reversal breaking fields to peel off degrees of freedom, and a combination of both. The central finding is that despite different starting points, all these models flow in the infrared to either a time-reversal symmetric critical point in a class-DIII topological superconductor or its time-reversal breaking dual phase of superconducting nodal points. A sympathetic reader would care because this unifies various attempts to realize chiral fermions on lattices while respecting no-go theorems, and it connects lattice models directly to physical systems like gapless superconductors and superfluids.

Core claim

In the infrared limit, all the lattice models with single Weyl fermions studied here are isomorphic to either a tQCP in a DIII class topological superconductor with a protecting T-symmetry, or its dual, a T-symmetry breaking superconducting nodal point phase, and therefore form an equivalent class. The proposal relies on spontaneous charge U(1) symmetry breaking to evade the usual no-go theorem of a single Weyl cone in a 3d lattice. For a generic T-symmetric tQCP the conserved-charge operators span a six-dimensional linear space while for a T-symmetry breaking gapless state charge operators typically span a two-dimensional linear space instead.

What carries the argument

Spontaneous breaking of U(1) charge symmetry applied to fermionic topological symmetry-protected states, which generates single Weyl cones or equivalent real-fermion nodal points whose infrared theories are isomorphic to DIII-class topological quantum critical points or their duals.

If this is right

  • For generic T-symmetric tQCPs the conserved-charge operators span a six-dimensional linear space.
  • For T-symmetry breaking gapless states the charge operators span a two-dimensional linear space.
  • All the studied single-Weyl-fermion lattice models form one equivalence class in the infrared.
  • Three-dimensional lattice chiral fermion models connect to gapless real fermions that appear in superfluids or superconductors.

Where Pith is reading between the lines

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

  • Numerical simulations of the proposed models should reproduce identical low-energy spectra and correlation functions as the DIII tQCP or its dual.
  • The charge-operator dimension difference offers a diagnostic to distinguish the T-symmetric and T-breaking members of the class.
  • The constructions may extend to other symmetry classes or dimensions by applying analogous minimal topological changes or symmetry-breaking fields.

Load-bearing premise

The infrared effective theory after symmetry breaking contains precisely one Weyl cone or its real-fermion equivalent without extra gapless modes or instabilities.

What would settle it

A numerical diagonalization or renormalization-group analysis of one constructed lattice model that finds additional Weyl cones, extra gapless excitations, or instabilities differing from the expected DIII tQCP or nodal phase would disprove the isomorphism.

Figures

Figures reproduced from arXiv: 2510.25959 by Fei Zhou, Gabriel Meyniel.

Figure 1
Figure 1. Figure 1: FIG. 1. Top row from the left to right: shown on the left is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A T-symmetric tQCP in SPTs when the mass pa [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Phase diagram for the number of Weyl cones in model [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Nodal phase) Spectrum for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Exemple of definition of the orientation number in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Charges (in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Charges (in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. If [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

In this article, we put forward a practical but generic approach towards constructing a large family of $(3+1)$ dimension lattice models which can naturally lead to a single Weyl cone in the infrared (IR) limit. Our proposal relies on spontaneous charge $U(1)$ symmetry breaking to evade the usual no-go theorem of a single Weyl cone in a 3d lattice. We have explored three concrete paths in this approach, all involving fermionic topological symmetry protected states (SPTs). Path a) is to push a gapped SPT in a 3d lattice with time-reversal symmetry (or $T$-symmetry) to a gapless topological quantum critical point (tQCP) which involves a minimum change of topologies,i.e. $\delta N_w=2$ where $\delta N_w$ is the change of winding numbers across the tQCP. Path b) is to peal off excessive degrees of freedom in the gapped SPT via applying $T$-symmetry breaking fields which naturally result in a pair of gapless nodal points of real fermions. Path c) is a hybrid of a) and b) where tQCPs, with $\delta N_w \geq 2$, are further subject to time-reversal-symmetry breaking actions. In the infrared limit, all the lattice models with single Weyl fermions studied here are isomorphic to either a tQCP in a DIII class topological superconductor with a protecting $T$-symmetry, or its dual, a $T$-symmetry breaking superconducting nodal point phase, and therefore form an equivalent class. For a generic $T$-symmetric tQCP along Path a), the conserved-charge operators span a six-dimensional linear space while for a $T$-symmetry breaking gapless state along Path b), c), charge operators typically span a two-dimensional linear space instead. Finally, we pinpoint connections between three spatial dimensional lattice chiral fermion models and gapless real fermions that can naturally appear in superfluids or superconductors studied previously.

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

1 major / 1 minor

Summary. The paper proposes a generic approach to construct (3+1)D lattice models realizing a single Weyl cone in the IR limit via spontaneous charge U(1) symmetry breaking to evade no-go theorems. It outlines three paths from gapped 3d DIII-class SPTs: (a) driving to a tQCP with minimal topological change δN_w=2, (b) applying T-symmetry breaking fields to obtain a pair of gapless real-fermion nodal points, and (c) a hybrid of the two. The central claim is that all such models form an equivalence class isomorphic in the IR to either a T-symmetric tQCP in a DIII topological superconductor or its T-breaking dual (a superconducting nodal phase), with conserved-charge operators spanning a six-dimensional space in the former case and a two-dimensional space in the latter. Connections to gapless real fermions in superfluids and superconductors are noted.

Significance. If the constructions are shown to isolate precisely one Weyl cone (or real-fermion nodal pair) without extra gapless modes or instabilities, the work would provide a unifying framework for emergent single-Weyl lattice models, linking high-energy chiral fermions to condensed-matter gapless phases in topological superconductors and superfluids. The explicit distinction between six-dimensional and two-dimensional charge-operator spaces for T-symmetric versus T-breaking cases offers a concrete organizing principle. The approach of using minimal δN_w=2 changes and symmetry breaking to evade no-go theorems is a strength worth developing.

major comments (1)
  1. Abstract (central claim paragraph): The assertion that the models are isomorphic to a DIII tQCP or its T-breaking dual, with the IR containing precisely one Weyl cone without extra gapless modes, is load-bearing for the equivalence-class conclusion. This rests on the assumption that δN_w=2 plus the chosen symmetry-breaking fields (charge-U(1) breaking and T-breaking) isolate the target mode; however, no explicit lattice Hamiltonians, dispersion calculations, or full Brillouin-zone spectra are supplied to verify the absence of additional crossings or instabilities once the fields are applied.
minor comments (1)
  1. The abstract introduces δN_w without a brief definition or reference to the underlying topological invariant (e.g., winding number); a short clarification in the introduction would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and constructive feedback. We address the major comment below and will strengthen the manuscript with additional explicit details.

read point-by-point responses

  1. Referee: Abstract (central claim paragraph): The assertion that the models are isomorphic to a DIII tQCP or its T-breaking dual, with the IR containing precisely one Weyl cone without extra gapless modes, is load-bearing for the equivalence-class conclusion. This rests on the assumption that δN_w=2 plus the chosen symmetry-breaking fields (charge-U(1) breaking and T-breaking) isolate the target mode; however, no explicit lattice Hamiltonians, dispersion calculations, or full Brillouin-zone spectra are supplied to verify the absence of additional crossings or instabilities once the fields are applied.

    Authors: We agree that the central claim would be strengthened by explicit verification. The manuscript develops a general construction based on minimal topological changes (δN_w=2) and controlled symmetry breaking to evade no-go theorems, with the equivalence class defined via IR effective theories and conserved charge operators (six-dimensional for T-symmetric tQCPs, two-dimensional for T-breaking cases). However, to directly address the concern, the revised manuscript will include concrete lattice Hamiltonians realizing each of the three paths, together with explicit low-energy dispersion relations and full Brillouin-zone spectra confirming that only the target single Weyl cone (or real-fermion nodal pair) remains gapless, with no additional crossings or instabilities introduced by the symmetry-breaking fields. These additions will make the isolation of the modes fully explicit while preserving the symmetry-based arguments. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no load-bearing reductions to inputs or self-citations

full rationale

The paper constructs explicit lattice models via three paths starting from gapped 3d SPTs in DIII class, driving them to tQCPs with δN_w=2 or applying T-breaking fields, then argues IR equivalence to known phases. No equations reduce the final isomorphism claim to a fitted parameter or prior self-citation by definition; the equivalence is presented as a consequence of the explicit constructions and symmetry analysis rather than a renaming or tautology. The isolation of single Weyl cones is an assumption about the spectrum after symmetry breaking, but this is an external physical claim, not a circular redefinition of the input models. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on standard domain assumptions from topological band theory and the Nielsen-Ninomiya no-go theorem; no new free parameters or invented particles are introduced.

axioms (2)
  • domain assumption Nielsen-Ninomiya no-go theorem prohibits a single Weyl cone on a lattice without symmetry breaking
    Invoked to motivate the need for spontaneous U(1) breaking and SPT starting points.
  • domain assumption Topological quantum critical points change winding number by δN_w = 2
    Used to define the minimal topology change that produces gapless single-Weyl states.

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Forward citations

Cited by 2 Pith papers

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    Topological quantum critical points between chiral topological orders are captured by a non-compact conformal manifold whose topological angle satisfies Θ_cft^{-1} = ½(Θ₁^{-1} + Θ₂^{-1}).

  2. Minimal-doubling and single-Weyl Hamiltonians

    hep-lat 2025-12 unverdicted novelty 6.0

    Minimal-doubling lattice fermion Hamiltonians yield single-Weyl phases when supplemented by a species-splitting mass term, but one-parameter symmetry-preserving deformations introduce additional Weyl nodes above a cri...

Reference graph

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