What is the relation between vortex and quantized magnetic flux in superconductor? A vortex is a topological defect of the order parameter. As far as I am concerned, I think a vortex is a phase singularity point and a vortex always has a quantized flux. And we know that the magnetic flux is also quantized in superconductor. So what is the relation between them? I read from a book that the quantization of a magnetic flux is not due to topological, but energetic considerations. What does that mean? Is it saying that in type II superconductors, the magnetic flux is quantized so that the energy of the system is minimized and then this phenomenon just has a topological property?
 A: Think of the superconducting order parameter as given by the minimum of the Ginzburg-Landau functional: 
$$\mathscr{F}=\mathscr{F}_{N}+\int d\boldsymbol{x}\left[\dfrac{1}{2m}\left|\left(-\mathbf{i}\hslash\nabla-2e\mathbf{A}\right)\Delta\right|^{2}+\alpha\left|\Delta\right|^{2}+\beta\left|\Delta\right|^{4}-\dfrac{\left(\nabla\times\mathbf{A}\right)^{2}}{2\mu}\right]$$
with $\mathscr{F}_{N}$ the free-energy of the normal metal, $\Delta$ the order-parameter, $\mathbf{A}$ the magnetic (vector) potential, and $\alpha<0$ and $\beta>0$ two real parameters. $m$ and $e$ are the (effective) mass and charge, and $\mu$ the magnetic permeability of the material.
Minimize this functional ($\delta\mathscr{F}/\delta\Delta^{\ast}=0$) using the ansatz $\Delta\left(\boldsymbol{x}\right)=\left|\Delta_{0}\right|e^{\mathbf{i}\varphi\left(\boldsymbol{x}\right)}$, with only the phase being space-dependent. It makes the argument simpler to follow at first. When $\mathbf{A}\neq\mathbf{0}$, one has $\varphi = \left(2e/\hslash\right)\int\mathbf{A}\cdot d\boldsymbol{x}$. Thus a magnetic flux line and a phase singularity are indeed the same. Note the double meaning of the word phase : here it is in the sense of the argument of the exponential. When we also insist for the minimum configuration of the order parameter $\Delta_{0}$ to be position dependent, it turns out that at the vortex core, $\Delta_{0}\rightarrow 0$, so a vortex core is also a phase singularity, in the sense that the superconducting phase disappears (the order parameter disappears).
One says this is a topological property because the phase of the order-parameter is non-local : it is an integral which maps the value of the field $\mathbf{A}$ at any position to a flux at position $\boldsymbol{x}$. So you need (somehow) to define it for the entire material. For instance, even for a single flux line inside the superconductor, the test-boundary $B$ you choose to calculate $\varphi = \left(2e/\hslash\right)\int_{B}\mathbf{A}\cdot d\boldsymbol{x}$ can be as large as you want : the phase (call it a phase-difference if you wish) will always be the same as long as the $B$ encloses a phase singularity. The flux inside a vortex is quantised by construction: it is the phase(-difference) $\varphi$ which does not change the order-parameter if you shift it by multiples of $2\pi$.
The vortex is clearly an energy-argument construction, since we minimised the Ginzburg-Landau functional. 
So in short: you want to find the minimum of the energy of a superconductor under magnetic field. The superconductor has this fancy phase singularity when turning around a region with a non-vanishing vector potential. Moreover you can turn around this flux line at any distance you want, the phase singularity after one turn will be the same : one says this is a topological effect.
