🔬 mesoscale physics

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

This paper demonstrates that in non-Hermitian magnetic skyrmions, topological protection splits into two distinct behaviors: a homotopy-protected charge derived from the right state remains stable, while a biorthogonal charge loses quantization and breaks down at exceptional points, revealing a real-space manifestation of non-Hermitian degeneracy.

Original authors: Kejun Liu

Published 2026-06-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kejun Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: When "Perfect" Protection Breaks

Imagine a magnetic skyrmion as a tiny, swirling tornado of magnetism. In the normal world (what physicists call the "Hermitian" world), this tornado is incredibly stable. It has a topological charge, which is like a permanent tattoo or a unique ID number (an integer like 1, 2, or 3). Because of the laws of geometry, you cannot smooth this tornado out or change its ID number without tearing it apart. It is "topologically protected."

This paper asks a simple question: What happens to this protection if we introduce "gain and loss" into the system?

In the real world, systems often lose energy (friction) or gain energy (pumps). In physics, we model this with "non-Hermitian" math. The authors found that when you add this gain and loss, the skyrmion's protection doesn't just get weaker; it splits in two.

The Two Faces of the Skyrmion

The paper reveals that a non-Hermitian skyrmion actually carries two different charges that look the same when everything is perfect, but behave very differently when things get messy.

1. The "Right-Hand" Charge (The Stubborn Tourist)

Think of this charge as a tourist who is strictly following a map.

How it works: This charge only looks at the "right" side of the math. Even when you add gain and loss, this tourist stays on a perfect sphere (the map).
The Result: Because they stay on the sphere, their ID number (the charge) never changes. It is still perfectly protected. No matter how much you shake the system, this specific number remains an integer.
The Analogy: Imagine a hiker walking on a circular trail. Even if the wind blows (gain/loss), as long as they stay on the trail, they can't magically teleport off the path. Their position is "homotopy-protected."

2. The "Biorthogonal" Charge (The Drifting Ghost)

This is the "real" charge. It looks at both the "left" and "right" sides of the math together, which represents the actual physical reality of the system.

How it works: This charge is like a ghost that can float off the map. As soon as you turn on the gain and loss, this charge stops being a whole number. It becomes a complex number (with a real part and an imaginary part).
The Result: It loses its "integer" identity. It drifts away from 1, 2, or 3. It is no longer quantized.

The Breaking Point: The "Exceptional Ring"

The most dramatic part of the paper happens at a specific moment called an Exceptional Point (EP).

In normal physics, things usually break at a single point. But here, the authors found that the skyrmion breaks along a ring around its equator (its middle).

The Metaphor: Imagine the skyrmion is a spinning top. The "Exceptional Ring" is a belt around the middle of the top.
What happens there: When the strength of the gain/loss matches the strength of the skyrmion itself, the "Ghost Charge" (the biorthogonal one) goes crazy.
- The math describing the magnetic field at this ring diverges (it goes to infinity).
- The "phase rigidity" (how tightly the magnetic spins are locked together) collapses to zero.
- The charge loses its definition entirely. It's as if the ID number on the ghost's tattoo smears out and becomes unreadable.

The Two Manifestations of the Same Problem

The paper draws a fascinating parallel between two very different things:

A Response Function: In some systems, if you push them too hard, the mathematical function that describes their reaction loses its smoothness (analyticity) and breaks.
A Skyrmion: In this system, the topological charge loses its integer nature.

The authors argue these are two sides of the same coin. Both are caused by the same "Exceptional Point" mechanism. One breaks the rules of calculus (analyticity), and the other breaks the rules of topology (quantization).

Summary of the Findings

Before the Break: The skyrmion has two charges that are identical and equal to an integer (e.g., +1).
During the Break (at the Ring): The "Right-Hand" charge stays safe and remains +1. The "Ghost" (Biorthogonal) charge starts to drift, gains an imaginary part, and eventually crashes at the ring where the math explodes.
After the Break: The Ghost charge jumps to a new, negative value and never recovers its integer status. The "Right-Hand" charge remains stable but slowly changes over time.

Why This Matters (According to the Paper)

The paper concludes that "Topological Protection" is no longer a single, simple statement in non-Hermitian systems. It depends on which "charge" you are looking at.

If you look at the mathematical "right" state, the skyrmion is still safe.
If you look at the physical reality (the biorthogonal state), the skyrmion's protection collapses at the Exceptional Ring.

This suggests that in open systems (like those with lasers or active materials), you might be able to create or destroy these magnetic tornadoes at a specific threshold of gain and loss, a process that would be impossible in a perfectly closed, Hermitian system.

Note: The paper focuses entirely on the theoretical and mathematical description of this phenomenon. It suggests that these effects could be observed in specific experimental setups like magnonic systems or optical microcavities, but it does not claim any immediate clinical or commercial applications.

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

Problem Statement
Topological protection in condensed matter physics relies on integer invariants (such as the winding number of a magnetic skyrmion) that remain robust against smooth local perturbations. In Hermitian systems, these charges are defined by the global structure of a single field or wavefunction. However, in non-Hermitian systems—characterized by gain, loss, or $PT$-symmetric anisotropy—the standard framework faces a fundamental challenge: the system possesses distinct left and right eigenstates, and spectral degeneracies known as exceptional points (EPs) can occur where eigenvalues and eigenvectors coalesce. While the breakdown of analyticity in causal response functions at EPs is well-established, it remains unclear whether real-space topological charges, specifically the skyrmion winding number, can similarly break down or lose quantization at an EP.

Methodology
The authors investigate a non-Hermitian magnetic skyrmion described within the $CP^{N-1}$ field theory framework (explicitly detailed for $N=2$ , i.e., $CP^1$ ). The study distinguishes between two definitions of topological charge:

Right-State Charge ( $Q_R$ ): Constructed solely from the right eigenstate (normalized spinor), analogous to the standard Hermitian definition.
Biorthogonal Charge ( $Q_B$ ): Constructed from the biorthogonal pair of left and right eigenstates, representing the physical observable in non-Hermitian systems.

The authors analyze the dynamics of a skyrmion under a $PT$-symmetric deformation, modeled by a generator $H = H_0 + i\gamma \sigma_z$ , where $\gamma$ represents balanced gain and loss.

Analytical Approach: They derive the projective evolution equation for the right Bloch vector and demonstrate its equivalence to a Gilbert-type relaxation flow. They also analytically solve for the local eigenstates of a hedgehog skyrmion texture to identify the locus of exceptional points.
Numerical Approach: The authors perform a numerical sweep of the non-Hermiticity parameter $\gamma$ relative to the texture scale $h$ . They compute $Q_R$ and $Q_B$ on a discretized grid, tracking the phase rigidity (the overlap between left and right eigenvectors) to detect the collapse of the biorthogonal frame.

Key Contributions and Results

Dichotomy of Topological Charges: The paper establishes that a non-Hermitian skyrmion carries two distinct charges that coincide only in the Hermitian limit ( $\gamma = 0$ ).
- $Q_R$ is Homotopy-Protected: The right-state charge remains an integer invariant under smooth time evolution. The authors prove that the $PT$ deformation induces a flow on the right Bloch vector that is mathematically identical to Gilbert damping toward an effective field. Since the vector remains on the unit sphere $S^2$ and the map remains smooth, the degree (winding number) cannot change.
- $Q_B$ Loses Quantization: The biorthogonal charge is complex-valued. As soon as gain/loss is introduced ( $\gamma \neq 0$ ), $Q_B$ acquires an imaginary part and drifts away from the integer value. Crucially, at the exceptional point, $Q_B$ ceases to be a well-defined quantized invariant.
Spatially Resolved Exceptional Ring: Unlike momentum-space EPs which are often isolated points, the EP in this real-space texture forms a continuous ring on the skyrmion's equator (where the out-of-plane magnetization vanishes).
- At the critical condition $\gamma = h$ , the local generator's discriminant vanishes on this ring.
- Here, the biorthogonal Bloch vector diverges, and the phase rigidity (the overlap $\langle z_L | z_R \rangle$ ) collapses to zero.
- Numerical results show that as $\gamma$ crosses $h$ , the real part of $Q_B$ jumps to a negative branch and then drifts continuously, never recovering a stable integer, while $Q_R$ remains smooth and close to unity.
Mechanism of Breakdown: The breakdown of the biorthogonal charge is directly linked to the singularity of the underlying Bloch field at the EP. The paper demonstrates that the loss of quantization is not a discretization artifact but a genuine consequence of the non-compact nature of the complex quadric manifold ( $\Sigma$ ) on which the biorthogonal vector lives, combined with the singularity at the EP.

Significance and Claims
The paper claims to resolve the question of whether topological protection survives in non-Hermitian dynamics by showing that the answer depends on the definition of the charge:

Protection of the Right State: The "standard" topological protection (based on the right state) survives non-Hermitian deformations because the dynamics are homotopically equivalent to Gilbert damping.
Fragility of the Physical Charge: The physically relevant biorthogonal charge is fragile. It loses its quantization precisely at the exceptional point where the left and right eigenstates coalesce.

The authors draw a parallel between this result and recent findings in confining gauge theories: just as a $PT$-symmetric deformation can break the analyticity of a causal response function at an EP, it can break the quantization of a real-space topological charge at the same type of degeneracy. Both phenomena are presented as manifestations of the same non-Hermitian degeneracy mechanism.

The paper concludes by identifying the equatorial exceptional ring as a spatially resolved spectral feature that could, in principle, be observed via local probes (e.g., spin-polarized STM or NV centers) detecting a collapse in phase rigidity. It frames the detection of this charge breakdown as a testable direction for future experiments in magnonic and photonic systems, without claiming that such breakdown has already been observed or that it implies specific new applications beyond the theoretical framework presented.