2026/06/26 by Kejun Liu
#cond-mat.mes-hall #cond-mat.other #physics.optics
The integer topological charge of a skyrmion is the standard example of topological protection. We ask what happens to that protection when the local generator is made non-Hermitian by polarization-selective gain or loss. The texture charge of a smoothly evolving state remains homotopy-protected under the usual fixed-boundary condition: after projective normalization, the gain/loss contribution has the Gilbert relaxation form on the target sphere. The spectral charges of the local generator behave differently. Its biorthogonal eigenframe charge is quantized while the texture avoids exceptional points. At linewidth matching, the skyrmion equator becomes a real-space exceptional ring on which the biorthogonal Bloch field diverges, and the eigenframe class jumps from +1 to 0 in a parameter-driven spectral transition. The closed contour relies on constant splitting along the equator, or an equivalent design constraint; a partial or amplitude-anisotropic realization instead leaves isolated real-space exceptional points. We use an operational Jones parametrization of an isolated polarization doublet and give a Stokes-tomography reconstruction of the left-right frame. The result identifies a spatially resolved exceptional contour that mediates an integer transition of a real-space eigenframe charge.