Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 36C / Solution

In the Landau-Ginzburg theory functional, $B^2/(2{\mu }_0)$ is the magnetic field energy density; the covariant-gradient term is the carrier kinetic energy including phase gradients and electromagnetic coupling. The terms $\alpha |\psi {|}^2+\beta |\psi {|}^4$ give the local condensation energy. With the normalization in the question, $n=|\psi {|}^2$ is the carrier density.

For a uniform field-free equilibrium, minimize $\alpha n+\beta n^2$ over $n\ge 0$. Stability for large $n$ requires $\beta >0$, and a nonzero minimum requires $\alpha <0$. ThereforeThe factor two follows from the question's quartic coefficient $\beta$, rather than a convention using $\beta /2$.

Let $D_A=-i\hslash \nabla -qA$. Under the gauge transformation,Its squared magnitude is unchanged, $|\psi |$ is unchanged, and $B=\nabla \times A$ is unchanged. Thus the full energy has gauge invariance.

For uniform $n$, write $\psi =\sqrt{n}{e}^{i\varphi }$. The kinetic density is $n(\hslash \nabla \varphi -qA{)}^2/(2m)$. Varying $A$ with compactly supported variations and integrating the magnetic term by parts givesAt a minimum its coefficient vanishes, giving the London equationTaking another curl, using $\nabla \cdot B=0$ and a smooth phase so $\nabla \times \nabla \varphi =0$, yieldsFor a planar surface this has the decaying solution $B(x)=B(0)e^{-x/{\lambda }_L}$ inside the material; the growing solution is excluded by boundedness in the bulk. Hence magnetic fields penetrate only a surface layer of thickness ${\lambda }_L$ and vanish deep in the macroscopic bulk, the Meissner effect. This conclusion uses the uniform-density, smooth-phase setting; a vortex has a singular phase and a nonuniform core and is outside those assumptions.