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

Carlos Román, Etienne Sandier, Sylvia Serfaty

TL;DR

The paper analyzes the three-dimensional Ginzburg–Landau functional with gauge field in the London limit ε→0 near the first vortex onset field H_c1. It derives a next-order expansion for H_c1, shows vortices appear one by one and concentrate near a special isoflux curve Γ_0, and introduces a renormalized energy W_N governing the interaction of multiple vortex filaments via a horizontal blow-up. The main contribution is a sharp Γ-convergence framework that combines 3D lower bounds (via Rom’s method and slicing) with a novel Biot–Savart-based upper bound construction, yielding precise energy expansions and vortex configurations that minimize W_N. This work clarifies the shape and arrangement of vortex lines in 3D superconductors, revealing curvature effects and boundary interactions that cause lines to bend near Γ_0 and to organize into nearly parallel, curved filaments dictated by the geometry of the isoflux problem.

Abstract

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 $Γ_0$, 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 $Γ_0$. This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near $Γ_0$ compete. This elucidates the shape of vortex lines in superconductors.

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

TL;DR

The paper analyzes the three-dimensional Ginzburg–Landau functional with gauge field in the London limit ε→0 near the first vortex onset field H_c1. It derives a next-order expansion for H_c1, shows vortices appear one by one and concentrate near a special isoflux curve Γ_0, and introduces a renormalized energy W_N governing the interaction of multiple vortex filaments via a horizontal blow-up. The main contribution is a sharp Γ-convergence framework that combines 3D lower bounds (via Rom’s method and slicing) with a novel Biot–Savart-based upper bound construction, yielding precise energy expansions and vortex configurations that minimize W_N. This work clarifies the shape and arrangement of vortex lines in 3D superconductors, revealing curvature effects and boundary interactions that cause lines to bend near Γ_0 and to organize into nearly parallel, curved filaments dictated by the geometry of the isoflux problem.

Abstract

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 , and near the first critical field for which the first vortex filaments appear in energy minimizers. Under a nondegeneracy condition, we show a next order asymptotic expansion of as , and exhibit a sequence of transitions, with vortex lines appearing one by one as the intensity of the applied magnetic field is increased: passing there is one vortex, then increasing by an increment of order a second vortex line appears, etc. These vortex lines accumulate near a special curve , 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 . This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near compete. This elucidates the shape of vortex lines in superconductors.
Paper Structure (36 sections, 26 theorems, 317 equations, 3 figures)

This paper contains 36 sections, 26 theorems, 317 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Energy splitting
  4. The isoflux problem and nondegeneracy assumption
  5. Main theorem
  6. Proof approach
  7. Plan of the paper
  8. Preliminaries
  9. Vector fields, forms, currents and notation
  10. Reference magnetic field and splitting formula
  11. Biot-Savart vector fields and a new constant
  12. ε-level lower bounds
  13. Bounded vorticity
  14. Tube coordinates, energy rewriting and horizontal rescaling
  15. Tube coordinates
  16. ...and 21 more sections

Key Result

Theorem 1.1

Assume that the smooth simple curve $\Gamma_0$ is a unique nondegenerate maximizer (in the sense of Definition qdeu) of the ratio $\mathop{\mathrm{R}}\nolimits$. There exists $c_\varepsilon\to 0$ as $\varepsilon\to 0$ such that the following holds. Assume that with $N\ge 1$ independent of $\varepsilon$. Let $(\mathbf{u},\mathbf{A})$ be a minimizer (depending on $\varepsilon$) of $GL_\varepsilon$

Figures (3)

  • Figure 1: Vortex filament with boundary closure.
  • Figure 2: 2D and 3D views of 4 vortex lines.
  • Figure 3: Piecewise graph: $z'(t) = 1$ on $[0,z_2]$ and $[z_2+h,{L_0}+h]$, $z'(t) = -1$ on $[z_2,z_2+h]$.

Theorems & Definitions (62)

  • Definition 1.1: Isoflux problem
  • Definition 1.2: Strong nondegeneracy
  • Definition 1.3
  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Proposition A
  • Definition 2.1
  • Proposition 2.1
  • proof
  • ...and 52 more