Exact Single-Scale Outer Solution of the Abrikosov Vortex in the Extreme Type-II Limit

We determine the exact outer structure of the Abrikosov vortex in the extreme type-II limit, which occurs when the Ginzburg-Landau parameter $\kappa$ diverges. In this limit, Ginzburg-Landau theory simplifies, outside a shrinking core, to a closed nonlinear theory for the superfluid velocity subject to an algebraic density constraint. The resulting solution is asymptotically exact everywhere outside the vanishing vortex core, demonstrating that both magnetic field and superconducting density vary on the length scale of the London penetration depth. This establishes that the conventional two-length-scale picture of the vortex does not hold in the $\kappa \gg 1$ limit.