Why is the magnetic flux not quantized in a standard Aharonov-Bohm (infinite) solenoid setup, whereas in a superconductor setting, flux is quantized?
Answer
Just adding to @Xcheckr's answer, which I think is the most correct: quantum fields are always single-valued. In a superconductor, it is energetically favorable to minimize the kinetic term $|D_A\psi|^2$, where $\psi$ is the superconducting order parameter. $D_A\psi=0$ implies that the phase of $\psi$ is determined through parallel transport by exponentiating $iq\int A$, and this together with the single-valuedness of $\psi$ enforces flux quantization.
In an AB effect setup by contrast, there is no energetic reason to set $D_A\psi=0$, and so the phase of $\psi(x)$ will not be determined by $\exp(iq\int A)$. This means that for some generic value of the flux, the magnitude $|\psi|$ will not be constant (e.g. it will pass through zero at some point), which is where the interference in the AB effect comes from. In a superconductor $|\psi|$ must be constant for energetic reasons, and this is why the flux is quantized in a SC.
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