Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 8 / d / Solution

The scalar potential is $V=-{\mu }^2|\phi {|}^2/2+\lambda |\phi {|}^4/4$. Minimizing it with respect to $|\phi {|}^2$ givesChoose a gauge with constant condensate phase and take the homogeneous-amplitude London limit, $\phi =v$. Then c reduces toThe London screening with an unnormalized complex condensate has a factor of two from the complex-field kinetic term as printed; no $1/\sqrt{2}$ has been included in the definition of the real positive $v$. Equivalently, the gauge mass is $m_A^2=2e^2{v}^2$.

For a planar surface at $x=0$ and conductor in $x>0$, take a tangential field $B_z(x)$. The solution bounded at large depth isThe exponentially growing solution is excluded. The screening current $j_y=-{\partial }_x{B}_z=B_z(0)e^{-x/{\lambda }_L}/{\lambda }_L$ is nonzero near the surface, demonstrating the Meissner effect through a persistent surface current. The constant-amplitude approximation describes London screening away from any region where the condensate itself is strongly suppressed.

In the stipulated static gauge, $\mathbf{E}=-{\partial }_t\mathbf{A}-\nabla A_0=0$. Where this equilibrium screening current is nonzero, the assumed local Ohm's law $\mathbf{E}=\rho \mathbf{j}$ therefore demands $\rho =0$. The simultaneous equality $\mathbf{E}=\mathbf{j}=0$ deep in the bulk would by itself place no constraint on $\rho$; it is not a valid standalone proof of superconductivity.

The lossless condensate also supplies the dynamical check. With fixed amplitude and phase in temporal gauge, differentiating the London relation gives ${\partial }_t\mathbf{j}=2e^2{v}^2\mathbf{E}$. A steady field accelerates the supercurrent rather than producing a stationary dissipative current. Thus the ideal condensate has zero electrical resistivity, or an infinite electrical conductivity, consistent with the static screening solution. This is the superconducting component of the toy model; normal carriers and dissipation are not included. The nonzero field expectation is understood in a gauge-fixed description of the Higgs phase, not as a gauge-invariant local observable.