Superconductor Ginzburg-Landau model

[vidi]

Homework Statement

Why is it that for a superconductor $\alpha<0,\beta>0$ in the Ginzburg-Landau model wit free energy formulation
$$E=\int d^3x\,\, \frac{\vec B^2}{2\mu}+\frac{1}{2m}|(-i\hbar\nabla-e\vec A)\psi|^2+\alpha|\psi|^2+\beta|\psi|^4$$

Homework Equations

$$E=\int d^3x\,\, \frac{\vec B^2}{2\mu}+\frac{1}{2m}|(-i\hbar\nabla-e\vec A)\psi|^2+\alpha|\psi|^2+\beta|\psi|^4$$

The Attempt at a Solution

I have no idea how to start! perhaps it has to do with some critical temperature?

 

[Hypersphere]

$\beta >0$, otherwise the free energy could be minimized by $|\psi |\rightarrow \infty$. This may not be obviously unphysical, but the Ginzburg-Landau model does after all aim to model the phase transition around $|\psi |=0$.

$\alpha <0$, then. The parameter $\alpha$ is indeed related to the critical temperature, but you don't need to look at the temperature dependence here. Just consider the simpler case with $\vec{A}=0$ and $\nabla \psi \approx 0$ and minimize the free energy. This will give you the sign. Of course, you'll probably have to extend the arguments a bit, to account for the inhomogeneous case and non-zero vector potentials, but that's the basic idea anyway.

 

[vidi]

Thanks, Hypersphere.