Vortex lines interaction in the three-dimensional magnetic Ginzburg--Landau model

We complete our study of the three dimensional Ginzburg--Landau functional with magnetic field, in the asymptotic regime of a small inverse Ginzburg--Landau parameter $\varepsilon$, and near the first critical field $H_{c_1}$ for which the first vortex filaments appear in energy minimizers. Under a nondegeneracy condition, we show a next order asymptotic expansion of $H_{c_1}$ as $\varepsilon \to 0$, and exhibit a sequence of transitions, with vortex lines appearing one by one as the intensity of the applied magnetic field is increased: passing $H_{c_1}$ there is one vortex, then increasing $H_{c_1}$ by an increment of order $\log |\log\varepsilon|$ a second vortex line appears, etc. These vortex lines accumulate near a special curve $\Gamma_0$, solution to an isoflux problem. We derive a next order energy that the vortex lines must minimize in the asymptotic limit, after a suitable horizontal blow-up around $\Gamma_0$. This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near $\Gamma_0$ compete. This elucidates the shape of vortex lines in superconductors.