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 |DAψ|2, where ψ is the superconducting order parameter. DAψ=0 implies that the phase of ψ is determined through parallel transport by exponentiating iq∫A, and this together with the single-valuedness of ψ enforces flux quantization.
In an AB effect setup by contrast, there is no energetic reason to set DAψ=0, and so the phase of ψ(x) will not be determined by exp(iq∫A). This means that for some generic value of the flux, the magnitude |ψ| 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 |ψ| must be constant for energetic reasons, and this is why the flux is quantized in a SC.
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