= Solution
The scalar potential is $V=-\mu^2|\varphi|^2/2+\lambda|\varphi|^4/4$. Minimizing it with respect to $|\varphi|^2$ gives
$$
\boxed{v^2=\frac{\mu^2}{\lambda},\qquad v=\frac{\sqrt{\mu^2}}{\sqrt\lambda}.}
$$
Choose a gauge with constant condensate phase and take the homogeneous-amplitude London limit, $\varphi=v$. Then <8/c> reduces to
$$
\boxed{\mathbf j=-2e^2v^2\mathbf A,\qquad
\nabla^2\mathbf B=2e^2v^2\mathbf B.}
$$
The <London screening with an unnormalized complex condensate> has a factor of two from the complex-field kinetic term as printed; no $1/\sqrt2$ has been included in the definition of the real positive $v$. Equivalently, the gauge mass is $m_A^2=2e^2v^2$.
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 is
$$
\boxed{B_z(x)=B_z(0)e^{-x/\lambda_L},\qquad
\lambda_L=\frac1{\sqrt2|e|v}=\frac{\sqrt\lambda}{\sqrt2|e|\sqrt{\mu^2}}.}
$$
The exponentially growing solution is excluded. The screening current $j_y=-\partial_xB_z=B_z(0)e^{-x/\lambda_L}/\lambda_L$ 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 \b[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 $\partial_t\mathbf j=2e^2v^2\mathbf E$. 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.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2015-london-field-and-screening-current.png]
{title=Exponential magnetic screening and the persistent surface current}
{description=Original London-limit half-space solution. The magnetic field and screening-current magnitude share the decay length lambda L; their boundary values set the plotted normalization.}
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