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

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.