Quantification of the circulation in superfluids : why does the phase jump by $2 \pi$? In superfluidity we have the current density that is $j=\frac{h}{m} \vec{\nabla}(\phi)$
Where the wavefunction describing the system is : $\psi(\vec r)=\sqrt{n_s}e^{i \phi(\vec{r})}$
We then introduce the quantity :
$$ K = \oint \vec{v_s} . d\vec{r}$$ (we integrate the current over a closed loop in the superfluid phase).
And we say : The phase can only change by $2 \pi$ values after the loop as it is the same quantum state.
So we have 
$$K=\frac{h}{m} \oint \vec{\nabla} (\phi) d^3 \vec{r}= \frac{h}{m} (2 \pi n)$$
We thus have quantification of K.
My question : As $\psi_0$ is a global wavefunction, I would say that the phase has no physical meaning at all. So why do we say that it only can jump by $2 \pi$ values after a loop ? I would say that only the modulus of the wavefunction matters (so the phase can be anything). Indeed for me when we take care of the phase it is when we have a sum of wavefunctions (and the relative phases between them matters). But here we study a global wavefunction so we would'nt have such summations.
 A: If you do not believe in the effect of phase difference, then you reject the existence of interference phenomena ... 
What you do around a vortex in a superfluid is to take one part of the wave-function to propagate clockwise and an other part to propagate counter-clockwise. The reason you have to do both way is that all waves should be taken in a superposition due to the path integral postulate of quantum mechanics, and that the wave-function is not defined at the center of the vortex. 
The way to reconcile the propagation of matter (the superfluid current $j$) around the vortex and the ill-definition of the wave-function at the center of the vortex is to define the phase difference of the two paths around the vortex as 
$$j\sim\oint\nabla\varphi\cdot dl=2\pi$$
In many situations one finds $0$ and not $2\pi$. When it is so, there is no current, so no circulation of particle, and in fact no vortex, see the forthcoming argument. When the phase difference is $4\pi$, there are twice the number of particles moving as when the phase difference is $2\pi$ ; the current is doubled. What the principle of phase difference tells you is that without vortex, there is no difference between the clockwise and the counter-clockwise paths, you can deform the wave function to a straight line (going back and forth) and the phase difference is zero. So, without vortex you have no circulation of particle.
A: The complex field $\psi(\vec{r})$ studied in superconductivity is defined globally (although it's phase is not, because $\psi$ might have zeros. I made some comments on that on a different question, see  Vortex in superfluid? ).
However, it is not a wave-function of the system. In mean-field theory it arises as parametrizing the free energy. Recall that in Ginzburg-Landau theory we make an ansatz of the free energy as a functional of $\psi$ and then minimize this functional instead of solving the partition function exactly.
