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I have a question essentially the same with this one "Aharonov-Bohm Effect and Flux Quantization in superconductors" which is why we can say the flux is quantized in superconducting disk but not in AB effect.

I think David Bar Moshe's answer is really reasonable: In AB effect the wave function is the real wave function of the electron while in superconductor case the pair potential $\Delta(r)$ is actually a correlation function thus it must be single valued.

However, this argument seems cannot explain the Laughin's argument of IQHE. Since in his argument it is the wave function of the extended state(which is the real state of electron not macro correlation) around the loop forces the flux to be quantized.

Does anyone have any better explanation when we can have quantized flux and when cannot?

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Laughlin says nothing about quantization. In fact any flux is allowed inside the cylinder. What he need however, is that an equivalent of a full flux quantum has no effect on the physics in the system. – Fabian Jan 3 at 8:48
So why only full flux quanta can leave the Hamiltonian unchanged? – Knightq Jan 3 at 9:20
Because the charge is quantized in units of $e$. If you write down the Schrödinger equation for this problem, you will notice that you can gauge away full (and only full) flux quanta. – Fabian Jan 3 at 9:51
Sorry I still don't quite understand why only full flux quanta can give the original Hamiltonian, why it have to do with quantization of charge?could you tell me more in detail? – Knightq Jan 4 at 21:53

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