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Entanglement of vortices in the Ginzburg--Landau equations for superconductors

Alberto Enciso, Daniel Peralta-Salas

Abstract

In 1988, Nelson proposed that neighboring vortex lines in high-temperature superconductors may become entangled with each other. In this article we construct solutions to the Ginzburg--Landau equations which indeed have this property, as they exhibit entangled vortex lines of arbitrary topological complexity.

Entanglement of vortices in the Ginzburg--Landau equations for superconductors

Abstract

In 1988, Nelson proposed that neighboring vortex lines in high-temperature superconductors may become entangled with each other. In this article we construct solutions to the Ginzburg--Landau equations which indeed have this property, as they exhibit entangled vortex lines of arbitrary topological complexity.
Paper Structure (3 sections, 5 theorems, 63 equations)

This paper contains 3 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem
  3. Proof of Proposition \ref{['L.psi']}

Key Result

Theorem 1.2

Let $L$ be a braid in the cylinder $\Omega$ and fix any positive integer $r$ and any $\varepsilon>0$. Then there exists a solution $(A,\Psi)$ to the Ginzburg--Landau equations in $\Omega$ satisfying the boundary condition E.BC on $\partial_{\mathrm{L}}\Omega$, such that $\Phi(L)$ is a subset of isol

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof