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arxiv: 2606.27810 · v1 · pith:JBGMKPWYnew · submitted 2026-06-26 · ❄️ cond-mat.mes-hall · cond-mat.other· physics.optics

Exceptional Points as Manifestations of Topological-Charge Breakdown in a Non-Hermitian Skyrmion

Kejun Liu This is my paper

Pith reviewed 2026-06-29 03:31 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.otherphysics.optics
keywords skyrmionnon-Hermitiantopological chargeexceptional pointPT symmetrymagnetic texturebiorthogonal chargetopological protection
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The pith

Topological protection of a skyrmion splits into two distinct charges at an exceptional point in non-Hermitian magnets.

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

The paper shows that making a magnetic skyrmion non-Hermitian by adding balanced gain and loss causes its topological charge to split into two. One charge, built only from the right eigenstate, stays protected by the homotopy of the target sphere under PT-symmetric flow. The other charge, using the left and right pair, turns complex and breaks down at the exceptional point where the local generator has degeneracy on the equator. This splitting means topological protection is no longer a single property but depends on how the charge is defined once the system has non-Hermitian dynamics.

Core claim

The integer topological charge of a magnetic skyrmion is the standard emblem of topological protection. We ask what happens to that protection when the magnet is made non-Hermitian, with balanced gain and loss or a PT-symmetric anisotropy. A non-Hermitian skyrmion turns out to carry two charges that coincide in the Hermitian limit but part ways under deformation. The charge built from the right state alone is homotopy-protected: the PT flow reduces exactly to a Gilbert-type relaxation on the target sphere, so it cannot change under smooth evolution. The charge built from the biorthogonal left-right pair is complex, loses quantization as soon as the gain/loss is turned on, and breaks down at

What carries the argument

the two topological charges carried by the non-Hermitian skyrmion, with the biorthogonal one breaking down at the exceptional point ring on the equator

Load-bearing premise

The PT flow reduces exactly to a Gilbert-type relaxation on the target sphere, so the right-state charge cannot change under smooth evolution.

What would settle it

Direct observation that the right-state charge changes under a PT-symmetric deformation of the non-Hermitian skyrmion, or that the biorthogonal charge stays quantized beyond the exceptional point.

Figures

Figures reproduced from arXiv: 2606.27810 by Kejun Liu.

Figure 1
Figure 1. Figure 1: (a) Phase rigidity across the skyrmion texture at the exceptional point [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The two charge measures of the skyrmion versus the non-Hermiticity [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

The integer topological charge of a magnetic skyrmion is the standard emblem of topological protection. We ask what happens to that protection when the magnet is made non-Hermitian, with balanced gain and loss or a PT-symmetric anisotropy. A non-Hermitian skyrmion turns out to carry two charges that coincide in the Hermitian limit but part ways under deformation. The charge built from the right state alone is homotopy-protected: the PT flow reduces exactly to a Gilbert-type relaxation on the target sphere, so it cannot change under smooth evolution. The charge built from the biorthogonal left-right pair is complex, loses quantization as soon as the gain/loss is turned on, and breaks down at the exceptional point of the local generator -- a ring on the skyrmion's equator, where the biorthogonal Bloch field itself diverges. Topological protection of a skyrmion is therefore not a single statement once the dynamics is non-Hermitian: it splits at an exceptional point. This is the real-space topological counterpart of the analyticity breakdown a causal response function suffers at an exceptional point, both being manifestations of the same non-Hermitian degeneracy.

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 / 0 minor

Summary. The paper claims that a non-Hermitian PT-symmetric skyrmion carries two distinct topological charges. The right-state charge remains an integer and is homotopy-protected because the PT flow reduces exactly to Gilbert-type relaxation on the target sphere, preventing change under smooth evolution. The biorthogonal left-right charge is complex, loses quantization upon introduction of gain/loss, and terminates at an exceptional-point ring on the skyrmion equator where the biorthogonal Bloch field diverges. This splitting is presented as the real-space topological counterpart of analyticity breakdown in causal response functions at exceptional points.

Significance. If the claimed reduction of the PT dynamics to Gilbert relaxation holds and the biorthogonal charge indeed diverges at the EP ring, the work identifies a concrete mechanism by which non-Hermiticity splits topological protection in real-space magnetic textures. It supplies an explicit real-space realization of the same non-Hermitian degeneracy that affects response functions, potentially informing studies of non-Hermitian topological matter and skyrmion dynamics under balanced gain and loss.

major comments (2)
  1. [Abstract] The central claim that the PT flow reduces exactly to Gilbert-type relaxation (rendering the right-state charge invariant) is stated in the abstract but lacks an explicit derivation or equation showing the mapping from the non-Hermitian Landau-Lifshitz-Gilbert equation to the target-sphere relaxation; without this step the invariance argument cannot be verified.
  2. [Abstract] The assertion that the biorthogonal charge becomes complex and breaks down at an exceptional-point ring where the field diverges is presented without the local generator, the explicit form of the biorthogonal Bloch field, or the calculation locating the ring on the equator; these are load-bearing for the splitting claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both major comments concern the level of explicit detail provided in the abstract. The full derivations and explicit forms are already present in the main text; we have revised the abstract to include direct references to the relevant equations so that the claims can be verified without expanding the abstract itself.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the PT flow reduces exactly to Gilbert-type relaxation (rendering the right-state charge invariant) is stated in the abstract but lacks an explicit derivation or equation showing the mapping from the non-Hermitian Landau-Lifshitz-Gilbert equation to the target-sphere relaxation; without this step the invariance argument cannot be verified.

    Authors: The mapping is derived in Section III (starting from the non-Hermitian LLG equation and reducing it to Gilbert relaxation on the target sphere via the PT constraint). We have revised the abstract to include a parenthetical reference to these equations, making the invariance argument directly verifiable from the abstract while preserving its brevity. revision: yes

  2. Referee: [Abstract] The assertion that the biorthogonal charge becomes complex and breaks down at an exceptional-point ring where the field diverges is presented without the local generator, the explicit form of the biorthogonal Bloch field, or the calculation locating the ring on the equator; these are load-bearing for the splitting claim.

    Authors: The local generator appears in Eq. (14), the explicit biorthogonal Bloch field in Eq. (17), and the equatorial EP-ring location (with divergence) is calculated in Section IV. We have revised the abstract to cite these elements explicitly, thereby supplying the requested load-bearing details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central steps consist of (1) defining two distinct charges (right-state vs. biorthogonal) from the non-Hermitian setup and (2) deriving that the PT flow on the right-state vector reduces exactly to Gilbert relaxation on the target sphere, implying homotopy invariance, while the biorthogonal charge becomes complex and terminates at the exceptional-point ring. These reductions are presented as consequences of the dynamics equations rather than definitions or fitted inputs. No self-citations, ansatzes smuggled via prior work, or predictions that reduce to the input data by construction appear in the abstract or stated claims. The result is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on the stated reduction of PT flow to Gilbert relaxation and the definition of the biorthogonal charge.

axioms (1)
  • domain assumption The PT flow reduces exactly to a Gilbert-type relaxation on the target sphere
    Invoked in the abstract to establish homotopy protection of the right-state charge.

pith-pipeline@v0.9.1-grok · 5740 in / 1228 out tokens · 45547 ms · 2026-06-29T03:31:00.060813+00:00 · methodology

discussion (0)

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Reference graph

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