Non-integer flux quanta for a spherical superconductor

A thin film superconductor shaped into a spherical shell at whose center lies the end of long thin solenoid in which there is an integer flux $N\Phi_0$ has been previously extensively studied numerically as a model of a two-dimensional superconductor. The emergent flux from the solenoid produces a radial B-field at the superconducting shell and $N$ vortices in the superconducting film. We study here the effects of including a second solenoid (carrying a flux $f$) which is inserted inside the first solenoid but passing right across the sphere. This Aharonov-Bohm (AB) flux does not have to be quantized to make the order parameter single valued. The Ginzburg-Landau (GL) free energy is minimized at fixed $N$ as a function of $f$ and it is found that the minimum is usually achieved when the AB flux $f$ is half a flux quantum, but depending on $N$ the minimum may be at $f=0$ or values which are not obvious rational fractions.