The Goldstone--Wilczek Charge on the Rim

A solenoid flux $\Phi=2\pi\alpha$ threaded through the center of the Kaplan--Sen lattice disk binds a ground-state charge $Q_{\rm core}=\alpha-\mathrm{round}(\alpha)$, and the edge carries exactly the opposite charge. The edge charge is the Goldstone--Wilczek charge of the edge ladder, $Q_{\rm edge}=-\tfrac12\eta$, with $\eta$ its spectral asymmetry. At $\Phi=\pi$ one unit of charge crosses the disk. We ask how the rim operator, the Schur complement of the disk Hamiltonian on its boundary sites, registers this transfer. Its count of negative eigenvalues never changes over the flux cycle. Instead, the unit of charge enters the rim through a simple pole: the extremal eigenvalue of the rim operator passes through infinity in the same flux interval as the charge jump, with $\Delta Q_{\rm edge}=\tfrac12\Delta\,\mathrm{sign}\,\lambda_{\rm ext}$. The residue factorizes into the coupling of the inner edge mode to the rim and the inverse slope of that mode's level, and grows linearly with the radius. All statements are checked numerically on disks of radius $R=11$, $19$ and $34$.