For fermions on an Abrikosov-Nielsen-Olesen vortex, the vacuum fermion number equals [sgn(m+e sqrt(2) v)+sgn(m-e sqrt(2) v)] n/4, and disk edge states carry charge e/2.
Charged Vortex in Superconductor
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We find the charged spinless vortices in the effective field theory of a Schr\"{o}dinger type complex scalar field of Cooper pair, a U(1) gauge field of electromagnetism, and a gapless neutral scalar field of acoustic phonon. We show that regular static vortex solutions are obtained only for the nonzero critical cubic Yukawa type coupling between neutral and complex scalar fields. Since the Coulombic electric field is exactly cancelled by the phonon, the obtained charged vortices have finite energy. When the quartic self-interaction coupling of complex scalar field has the critical value, the BPS (Bogomolny-Prasad-Sommerfield) bound is saturated for multiple charged vortices of arbitrary separations and hence the borderline of type I and II superconductors is achieved in nonperturbative regime.
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Vortex Fractional Fermion Number through Heat Kernel methods and Edge States
For fermions on an Abrikosov-Nielsen-Olesen vortex, the vacuum fermion number equals [sgn(m+e sqrt(2) v)+sgn(m-e sqrt(2) v)] n/4, and disk edge states carry charge e/2.