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.

1 Introduction

In the 1950s, Vitaly Ginzburg and Lev Landau developed a powerful phenomenological theory to provide a mathematical description of superconductors near the critical temperature, see e.g. [4, 5, 16]. The theory revolves around the so-called Ginzburg–Landau equations, whose non-dimensional form is

$$\begin{aligned} \Big (\frac{1}{\kappa }\nabla -iA\Big )^2\Psi&=(|\Psi |^2-1)\Psi \,,\\ -{{\,\textrm{curl}\,}}{{\,\textrm{curl}\,}}A&=\frac{i}{2\kappa }(\bar{\Psi }\nabla \Psi -\Psi \nabla \bar{\Psi })+|\Psi |^2A\,. \end{aligned}$$

The unknowns are the vector field $A:\Omega \rightarrow {\mathbb {R}}^3$, which is the magnetic vector potential, and the complex-valued function $\Psi :\Omega \rightarrow {\mathbb {C}}$, which is the order parameter representing the superconducting electron pairs. Here $\Omega \subset {\mathbb {R}}^3$ is a bounded domain and $\kappa >0$ is the Ginzburg–Landau parameter, whose value determines whether the superconductor is of type I or II. The usual boundary conditions are

$$\begin{aligned} N\cdot (\nabla \Psi -i\Psi A)=0 \end{aligned}$$

(1.1)

on $\partial \Omega$.

A major problem in the study of high-temperature superconductors is the question of whether (and how) neighboring vortex lines in the glass or liquid phase may entangle around each other [3, 12,13,14], as was proposed by Nelson [11] in 1988. Despite more than three decades of theoretical, numerical and experimental studies, the question of whether vortices can form an entangled state has not yet been convincingly answered. We recall that vortices are defined as the nodal lines

$$\begin{aligned} Z_\Psi :=\{x\in \Omega : \Psi (x)=0\}, \end{aligned}$$

which are the defects or singularities of the phase of the order parameter $\Psi$.

Our objective in this paper is to construct solutions to the Ginzburg–Landau system that indeed exhibit entangled vortices with complicated topologies. The solutions we shall construct have small amplitude, so it is convenient to write the Ginzburg–Landau equations as

$$\begin{aligned} (\kappa ^{-2}\Delta +1)\Psi&=2i\kappa ^{-1}A\cdot \nabla \Psi +|A|^2\Psi +|\Psi |^2\Psi +i\kappa ^{-1}{{\,\textrm{div}\,}}A\Psi \,,\\ \Delta A&=\nabla {{\,\textrm{div}\,}}A +\frac{i}{2\kappa }(\bar{\Psi }\nabla \Psi -\Psi \nabla \bar{\Psi })+|\Psi |^2A\,, \end{aligned}$$

where the Laplacian $\Delta$ acts on the vector field A componentwise.

Given any smooth function $\chi :\Omega \rightarrow {\mathbb {R}}$, it is easy to check that the gauge transformation

$$\begin{aligned} (A,\Psi )\mapsto (A+\nabla \chi ,\; \Psi e^{i\chi }) \end{aligned}$$

takes solutions to solutions, so one can always pick a gauge where A is divergence-free. However, for the construction later, it will be convenient not to fix such a gauge.

For concreteness, throughout the paper we will consider the Ginzburg–Landau equations on the cylindrical domain

$$\begin{aligned} \Omega :=D_\rho \times (-\ell ,\ell ), \end{aligned}$$

as is usually done in applications. Here $2\ell >0$ is the height of the cylinder and $\rho$ is its radius; $D_\rho$ denotes the two-dimensional disk of radius $\rho$ centered at the origin.

Nelson’s hypothesis concerns the existence of entangled (i.e., braided) vortices in high-temperature superconductors (see Fig. 1), whose precise definition is the following:

Definition 1.1

A braid in the domain $\Omega =D_\rho \times (-\ell ,\ell )$ is a finite collection of pairwise disjoint smooth lines, diffeomorphic to $(-\ell ,\ell )$, contained in $\Omega$, whose endpoints are one at the bottom boundary $\partial _{\textrm{B}}\Omega :=D_\rho \times \{-\ell \}$ and the other at the top boundary $\partial _{\textrm{T}}\Omega :=D_\rho \times \{\ell \}$. These lines may be knotted and linked [2, Chapter 5.4].

Fig. 1

A braid consisting of three lines

The following theorem is the main result of this article. We show that there exist solutions $(A,\Psi )$ to the Ginzburg–Landau equations on the cylinder $\Omega$, satisfying the natural boundary conditions on the lateral boundary $\partial _{\textrm{L}}\Omega := \partial D_\rho \times (-\ell ,\ell )$, which exhibit a subset of isolated vortex lines isotopic to any prescribed braid L in $\Omega$. These vortex lines are structurally stable in the sense that any other complex-valued function that is close to $\Psi$ exhibits an isotopic subset of vortex lines.

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 $C^\infty$ solution $(A,\Psi )$ to the Ginzburg–Landau equations in $\Omega$ satisfying the boundary condition (1.1) on $\partial _{\textrm{L}}\Omega$, such that $\Phi (L)$ is a subset of isolated vortex lines of $\Psi$.

Here $\Phi$ is a smooth diffeomorphism of ${{\overline{\Omega }}}$, close to the identity in the sense that $\Vert \Phi -\textrm{id}\Vert _{C^r(\Omega )}<\varepsilon$. Furthermore, these vortex lines are structurally stable. More precisely, take an open set $V\subset \Omega$ which contains L and any $\varepsilon '>0$. Then there exists some $\delta >0$ such that any function $\Psi _1$ satisfying $\Vert \Psi -\Psi _1\Vert _{C^r(V)}<\delta$ has a subset of vortex lines given by $\Phi _1\circ \Phi (L)$, where $\Phi _1$ is a diffeomorphism of ${{\overline{\Omega }}}$ bounded as $\Vert \Phi _1-\textrm{id}\Vert _{C^r(\Omega )}<\varepsilon '$.

Several remarks are in order. First, the nodal set $Z_\Psi$ may contain other components, but they are at a positive distance from $\Phi (L)$ because $\Phi (L)$ is an isolated nodal set. Second, note that there is no Neumann condition at the bottom and top boundaries of $\Omega$, but this does not seem to be very important for the braid structure. Third, we emphasize that $(A,\Psi )$ is a small amplitude solution, so proving a similar result for large values of the order parameter function $\Psi$ (which forces the Ginzburg–Landau system to operate far from the linear regime) remains an interesting, and probably very hard, open problem.

The proof of Theorem 1.2, which consists of five steps, is presented in Sect. 2. The proof of the auxiliary Proposition 2.1, which is more technical, is relegated to Sect. 3.

2 Proof of the Theorem

The solution $(A,\Psi )$ we construct to the Ginzburg–Landau equations is a perturbation of a solution to the linearized equations, so the Ginzburg–Laudau system essentially operates in the monochromatic wave regime. The proof is divided in five steps. In the first one we construct a solution $\psi$ to the Helmholtz equation in $\Omega$ which exhibits a subset of structurally stable vortex lines that are isotopic to the braid L. This solution is promoted to a solution of the nonlinear Ginzburg–Landau equations by using a fixed point argument in steps 2 to 4. To achieve this, it is crucial to solve certain elliptic PDEs in a cylindrical domain with suitable boundary conditions. The proof is completed in step 5 by using the structural stability of the nodal lines of $\psi$. All along this section, the integer r (which appears in the statement of Theorem 1.2) is an arbitrary positive integer.

2.1 Step 1: A Monochromatic Wave with Prescribed Braided Vortex Lines

In this first step we consider the extended cylinder $\Omega ':= D_\rho \times (-2\ell ,2\ell )$ and a braid $L'\subset \Omega '$ that is an “extension” of the braid L, i.e., $L'\cap \Omega =L$. We prove that there is a monochromatic wave in $\Omega '$ with suitable boundary conditions whose nodal set contains a braid that is isotopic to $L'$, the isotopy being as close to the identity as desired. In the proof we use techniques that we developed to study the level sets of harmonic functions in Euclidean space [6] and to tackle a conjecture of Berry on knotted nodal lines for eigenfunctions of some Schrodinger operators [7].

The main new difficulty that arises is that we want to construct solutions in a bounded domain with Neumann boundary data, so the method of proof in [6, 7] does not work directly and new technicalities have to be introduced. For future reference, for any $\varepsilon >0$, we will say that two $C^r$ functions $f_1,f_2$ on $\Omega '$ are $\varepsilon$-close in the $C^r$-norm if $\Vert f_1-f_2\Vert _{C^r(\Omega ')}<C\varepsilon$ for some $\varepsilon$-independent constant C.

Proposition 2.1

For any $\varepsilon >0$, there exists a function $\psi \in C^\infty (\overline{\Omega '},\mathbb C)$ and a diffeomorphism $\Phi _0:\overline{\Omega '}\rightarrow \overline{\Omega '}$ such that:

• (i) $\psi$ satisfies the Helmholtz equation $(\kappa ^{-2}\Delta +1)\psi =0$ in $\Omega '$ and the Neumann boundary condition $N\cdot \nabla \psi =0$ on the lateral boundary $\partial _L\Omega ':=\partial D_\rho \times (-2\ell ,2\ell )$.

• (ii) $\Phi _0(L')$ is a subset of structurally stable vortex lines of $\psi$. More precisely, take an open set $V\subset \Omega '$ such that V contains the braid $L'$. Then there exists some $\delta >0$ such that any function $\psi _1\in C^r(V)$ satisfying

  $$\begin{aligned} \Vert \psi -\psi _1\Vert _{C^r(V)}<\delta \end{aligned}$$

  has a subset of vortex lines $S_1$ with $\Phi _1(S_1)= \Phi _0(L')$. Here $\Phi _1$ is a diffeomorphism of ${\overline{V}}$ which is $\delta$-close to the identity in the $C^r$-norm.

• (iii) $\Phi _0$ is $\varepsilon$-close to the identity in the $C^r$-norm on $\Omega '$.

The proof of this result is presented in Sect. 3.

2.2 Step 2: Setting up a Fixed Point Argument

Let us take a contractible axisymmetric domain $\Omega _1\subset {\mathbb {R}}^3$ with $C^\infty$ boundary such that $\Omega \subset \Omega _1\subset \Omega '$ (e.g., think of adding two caps to the domain $\Omega$). Note, in particular, that $\partial \Omega _1\supset \partial _{\textrm{L}}\Omega$. It is standard (see e.g. [17]) that, by slightly changing the boundary of this domain away from $\partial _{\textrm{L}}\Omega$ if necessary, we can safely assume that $\kappa ^2$ is not a Neumann eigenvalue of the Laplacian $-\Delta$ in the domain $\Omega _1$. The situation is sketched in Fig. 2.

Fig. 2

The cylinder $\Omega =D_\rho \times (-\ell ,\ell )$ is contained in a longer cylinder $\Omega '=D_\rho \times (-2\ell ,2\ell )$, and we consider a smooth domain $\Omega _1$ which contains the short cylinder yet is contained in the long one. The monochromatic wave $\psi$ satisfies a Neumann boundary condition on $\partial _L\Omega '$ and the boundary conditions on $\partial \Omega _1$ are given by the problems (2.3) and (2.4)

The linearization of the Ginzburg–Landau equation at the trivial solution $(A,\Psi ):=(0,0)$ is

$$\begin{aligned} (\kappa ^{-2}\Delta +1)\dot{\Psi }&=0\,,\\ \Delta \dot{A}-\nabla {{\,\textrm{div}\,}}\dot{A}&=0\,, \end{aligned}$$

where $(\dot{A},\dot{\Psi })$ is our unknown. Therefore, our argument will be constructed around a fixed solution

$$\begin{aligned} (\dot{A},\dot{\Psi }):=(a,\psi ) \end{aligned}$$

of this linear system, where a is any $C^\infty$ divergence-free vector field in ${\mathbb {R}}^3$ such that $\Delta a=0$ and $N\cdot a = 0$ on $\partial _{\textrm{L}}\Omega '$ (e.g., $a(x):=(0,0,1)$ or $a(x):=(x_2,-x_1,0)$), and where $\psi \in C^\infty (\overline{\Omega '},\mathbb C)$ is a monochromatic wave as in Proposition 2.1.

Specifically, in the next step we shall set an iteration of the form

$$\begin{aligned} \begin{aligned} (\kappa ^{-2}\Delta +1)\Psi _{k+1}&=H(\Psi _{k},A_{k}),\\ \Delta A_{k+1}&=F(\Psi _{k},A_{k}), \end{aligned} \end{aligned}$$

(2.1)

where $(A_0,\Psi _0)$ is constructed using $(a,\psi )$ and where

$$\begin{aligned} H(\Psi ,A )&:=2i\kappa ^{-1}A \cdot \nabla \Psi +|A |^2\Psi +|\Psi |^2\Psi +i\kappa ^{-1}({{\,\textrm{div}\,}}A) \, \Psi \,,\\ F(\Psi ,A )&:=\frac{i}{2\kappa }(\bar{\Psi }\nabla \Psi -\Psi \nabla \bar{\Psi })+|\Psi |^2A \,. \end{aligned}$$

For future reference, note that the Hölder norm of these functions can obviously be estimated using Young’s inequality for products as

$$\begin{aligned}{} & {} \Vert H(\Psi ,A )\Vert _{C^{r-2,\alpha }}+ \Vert F(\Psi ,A )\Vert _{C^{r-2,\alpha }}\nonumber \\{} & {} \quad \le C\big (\Vert \Psi \Vert _{C^{r-1,\alpha }}^2+ \Vert \Psi \Vert _{C^{r-1,\alpha }}^3+ \Vert A\Vert _{C^{r-1,\alpha }}^2+\Vert A\Vert _{C^{r-1,\alpha }}^3\big ), \end{aligned}$$

(2.2)

where the constant C depends on $r,\alpha$ and $\kappa ^{-1}$.

2.3 Step 3: Reduction to a Modified Boundary Problem

For $\eta >0$ small enough and each $k\ge 0$, we consider the following functions defined on $\Omega _1$:

$$\begin{aligned} \Psi _{k}&:=\frac{\eta }{2C_1} \psi +\widetilde{\Psi }_{k}\,,\\ A_{k}&:=\frac{\eta }{2C_2} a+{\widetilde{A}}_{k}\,. \end{aligned}$$

Here $({\widetilde{A}}_k,\widetilde{\Psi }_k)$ will be solutions to the system (2.1), with $({\widetilde{A}}_0,\widetilde{\Psi }_0)=(0,0)$, and the constants are chosen as $C_1:=\Vert \psi \Vert _{C^{r,\alpha }(\Omega _1)}$ and $C_2:=\Vert a\Vert _{C^{r,\alpha }(\Omega _1)}$.

As we shall see in Lemma 2.2 below, key ingredient in the iteration is the somewhat non-standard choice of boundary conditions. Specifically, $\widetilde{\Psi }_{k+1}$ is the solution to the boundary value problem for the Helmholtz operator

$$\begin{aligned} (\kappa ^{-2}\Delta +1)\widetilde{\Psi }_{k+1}=H(\Psi _{k},A_{k}) \quad \text { in } \Omega _1, \end{aligned}$$

(2.3)

with Neumann condition

$$\begin{aligned} N\cdot \nabla \widetilde{\Psi }_{k+1}=i\Psi _kA_k\cdot N\quad \text { on }\partial \Omega _1. \end{aligned}$$

By the Fredholm alternative, this solution exists and is unique because we assumed $\kappa ^2$ is not a Neumann eigenvalue of $-\Delta$ in $\Omega _1$. Moreover, since the couple $(a,\psi )$ is $C^\infty$, the function $\widetilde{\Psi }_{k+1}$ is $C^\infty$ as well by elliptic regularity.

Likewise, ${\widetilde{A}}_{k+1}$ is the solution to the boundary problem:

$$\begin{aligned} \Delta {\widetilde{A}}_{k+1}=F(\Psi _{k},A_{k}) \quad \text { in } \Omega _1 \end{aligned}$$

(2.4)

with relative boundary conditions:

$$\begin{aligned} {\widetilde{A}}_{k+1}\times N=0\quad \text { and }\quad {{\,\textrm{div}\,}}{\widetilde{A}}_{k+1}=0\quad \text { on } \partial \Omega _1. \end{aligned}$$

Since $\Omega _1$ is a contractible domain (and hence the space of harmonic forms satisfying the relative boundary conditions is trivial), it is well known that ${\widetilde{A}}_{k+1}$ exists and is unique [15, Lemma 3.5.6], and, as before, it is $C^\infty$ by elliptic regularity.

It is clear that

• 1. The couple $(A_{k+1},\Psi _{k+1})$ solves the equations of the iteration in $\Omega _1$.

• 2. $N\cdot (\nabla \Psi _{k+1}-i\Psi _kA_k)=0$ on $\partial _L\Omega$ for all $k\ge 0$.

• 3. ${{\,\textrm{div}\,}}A_{k+1}=0$ on $\partial \Omega _1$ for all $k\ge 0$.

Assuming that the sequence $(A_k,\Psi _k)$ converges to a couple $(A,\Psi )$ in the $C^{r,\alpha }(\Omega _1)$ norm, it follows from the definition of the iteration that $(A,\Psi )$ solves the equation

$$\begin{aligned} \begin{aligned} (\kappa ^{-2}\Delta +1)\Psi&=H(\Psi ,A),\\ \Delta A&=F(\Psi ,A), \end{aligned} \end{aligned}$$

(2.5)

in $\Omega _1$, with boundary conditions $N\cdot (\nabla \Psi -i\Psi A)=0$ on $\partial _L\Omega$ and ${{\,\textrm{div}\,}}A=0$ on $\partial \Omega _1$. Again, since the system of PDEs (2.5) is elliptic, by elliptic regularity, the $C^{r,\alpha }(\Omega _1)$ solution $(A,\Psi )$ is actually $C^\infty$.

The following lemma shows that, in fact, the vector field A is divergence-free in $\Omega _1$. To prove this we use a property of the vector field $F(\Psi ,A)$ when $(A,\Psi )$ is a solution to Eqs. (2.5), and the boundary condition ${{\,\textrm{div}\,}}A=0$ on $\partial \Omega _1$. This crucially uses our choice of relative boundary conditions to solve Eq. (2.4).

Lemma 2.2

The solution $(A,\Psi )$ satisfies ${{\,\textrm{div}\,}}A=0$ in $\Omega _1$, so in particular it solves the Ginzburg–Landau equations in $\Omega _1$.

Proof

An easy computation shows that the couple $(A,\Psi )$ satisfies ${{\,\textrm{div}\,}}F(A,\Psi )=0$. Indeed,

$$\begin{aligned}&{{\,\textrm{div}\,}}F(A,\Psi )=\frac{i}{2\kappa }(\bar{\Psi }\Delta \Psi -\Psi \Delta \bar{\Psi })+A\cdot \nabla |\Psi |^2+|\Psi |^2{{\,\textrm{div}\,}}A\\&=\frac{i}{2\kappa }(2i\kappa \bar{\Psi }A\cdot \nabla \Psi +2i\kappa \Psi A\cdot \nabla \bar{\Psi }+2i\kappa |\Psi |^2{{\,\textrm{div}\,}}A)+A\cdot \nabla |\Psi |^2+|\Psi |^2{{\,\textrm{div}\,}}A\\&=0\,, \end{aligned}$$

where we have used the first equation that satisfies the solution $(A,\Psi )$ to pass to the second line. Accordingly, taking ${{\,\textrm{div}\,}}$ in the second equation that satisfies the couple $(A,\Psi )$, we get

$$\begin{aligned} \Delta ({{\,\textrm{div}\,}}A)=0 \text { in } \Omega _1,\qquad {{\,\textrm{div}\,}}A=0 \text { on } \partial \Omega _1, \end{aligned}$$

and hence ${{\,\textrm{div}\,}}A=0$ on $\Omega _1$ as claimed. Finally, notice that this implies that the couple $(A,\Psi )$ satisfies the Ginzburg–Landau equations in the domain $\Omega _1$. $\square$

2.4 Step 4: Convergence of the Scheme

Armed with these estimates, one can now show the convergence of our iteration scheme by a standard induction argument. In particular, we show below that

$$\begin{aligned} \Psi _{k+1} = \frac{\eta }{2C_1}\psi + O(\eta ^2)\,,\\ A_{k+1}=\frac{\eta }{2C_2}a+ O(\eta ^2)\,, \end{aligned}$$

where we use the notation $q= O(\eta ^2)$ for terms bounded as $\Vert q\Vert _{C^{r,\alpha }(\Omega _1)}\le C\eta ^2$.

To prove the convergence of the iteration, let us make the induction hypothesis that

$$\begin{aligned} \Vert A_{k'}\Vert _{C^{r,\alpha }(\Omega _1)}<\eta ,\qquad \Vert \Psi _{k'}\Vert _{C^{r,\alpha }(\Omega _1)}<\eta \end{aligned}$$

for all $k'\le k$, where $\eta$ is a small enough constant. We shall next show that the same bounds hold for $A_{k+1}$ and $\Psi _{k+1}$.

Indeed, by definition

$$\begin{aligned} \Vert A_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le \frac{\eta }{2}+\Vert {\widetilde{A}}_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}, \end{aligned}$$

where ${\widetilde{A}}_{k+1}$ is the unique solution to the Hodge boundary problem (2.4). Standard Schauder estimates yield [15, Lemma 3.4.7]

$$\begin{aligned} \Vert {\widetilde{A}}_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le C\Vert F(\Psi _k,A_k)\Vert _{C^{r-2,\alpha }(\Omega _1)}, \end{aligned}$$

where $C>0$ is a constant that depends on $r,\alpha ,\Omega _1$ but not on k. By the estimate (2.2), we then get

$$\begin{aligned} \Vert {\widetilde{A}}_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}&\le C\big (\Vert \Psi _k\Vert _{C^{r-1,\alpha }}^2+ \Vert \Psi _k\Vert _{C^{r-1,\alpha }}^3+ \Vert A_k\Vert _{C^{r-1,\alpha }}^2+\Vert A_k\Vert _{C^{r-1,\alpha }}^3\big )\\&\le C\eta ^2\,. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert A_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le \frac{\eta }{2}+C\eta ^2<\eta , \end{aligned}$$

provided that $\eta$ is small enough.

Analogously, we can estimate

$$\begin{aligned} \Vert \Psi _{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le \frac{\eta }{2}+\Vert \widetilde{\Psi }_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}, \end{aligned}$$

where $\widetilde{\Psi }_{k+1}$ is the unique solution to the boundary problem (2.3). Again, by the Fredholm alternative, since $\kappa ^2$ is not a Neumann eigenvalue of $-\Delta$ in $\Omega _1$, there exists a unique solution to this boundary problem, which by standard Schauder estimates is bounded as [10, Section 10.5]

$$\begin{aligned} \Vert \widetilde{\Psi }_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le C\big (\Vert H(\Psi _k,A_k)\Vert _{C^{r-2,\alpha }(\Omega _1)}+\Vert \Psi _kA_k\Vert _{C^{r-1,\alpha }(\Omega _1)}\big ), \end{aligned}$$

with C a constant that depends on $r,\alpha ,\Omega _1$ but not on k. Together with (2.2), this results in

$$\begin{aligned} \Vert \widetilde{\Psi }_{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}&\le C\big (\Vert \Psi _k\Vert _{C^{r-1,\alpha }}^2+ \Vert \Psi _k\Vert _{C^{r-1,\alpha }}^3+ \Vert A_k\Vert _{C^{r-1,\alpha }}^2+\Vert A_k\Vert _{C^{r-1,\alpha }}^3\big )\\&\le C\eta ^2\,. \end{aligned}$$

Accordingly,

$$\begin{aligned} \Vert \Psi _{k+1}\Vert _{C^{r,\alpha }(\Omega _1)}\le \frac{\eta }{2}+C\eta ^2<\eta , \end{aligned}$$

provided that $\eta$ is small enough.

Since $({\widetilde{A}}_0,\widetilde{\Psi }_0)=(0,0)$, we then infer that

$$\begin{aligned} \Vert \Psi _{k}\Vert _{C^{r,\alpha }(\Omega _1)}<\eta ,\qquad \Vert A_{k}\Vert _{C^{r,\alpha }(\Omega _1)}<\eta \end{aligned}$$

(2.6)

for all $k\ge 0$, and

$$\begin{aligned} \Vert \widetilde{\Psi }_{k}\Vert _{C^{r,\alpha }(\Omega _1)}<C\eta ^2,\qquad \Vert {\widetilde{A}}_{k}\Vert _{C^{r,\alpha }(\Omega _1)}<C\eta ^2 \end{aligned}$$

(2.7)

for all $k\ge 0$ and some k-independent constant $C>0$ that does not depend on $\eta$.

Finally, to show that the sequence $(A,\Psi )$ is Cauchy, we notice that

$$\begin{aligned} A_{k+1}-A_k={\widetilde{A}}_{k+1}-{\widetilde{A}}_k,\qquad \Psi _{k+1}-\Psi _k=\widetilde{\Psi }_{k+1}-\widetilde{\Psi }_k, \end{aligned}$$

and these differences satisfy the boundary problems

$$\begin{aligned} \Delta ({\widetilde{A}}_{k+1}-{\widetilde{A}}_k)=F(\Psi _{k},A_{k})-F(\Psi _{k-1},A_{k-1}) \text { in } \Omega _1 \end{aligned}$$

with relative boundary conditions on $\partial \Omega _1$, and

$$\begin{aligned} (\kappa ^{-2}\Delta +1)(\widetilde{\Psi }_{k+1}-\widetilde{\Psi }_k)=H(\Psi _k,A_k)-H(\Psi _{k-1},A_{k-1}) \text { in } \Omega _1 \end{aligned}$$

with Neumann boundary condition

$$\begin{aligned} N\cdot \nabla (\widetilde{\Psi }_{k+1}-\widetilde{\Psi }_k)=iN\cdot (\Psi _kA_k-\Psi _{k-1}A_{k-1}) \quad \text { on } \quad \partial \Omega _{1}. \end{aligned}$$

For any $A_{k'},\Psi _{k'}$ bounded as $\Vert A_{k'}\Vert _{C^{r-1,\alpha }(\Omega _1)}+\Vert \Psi _{k'}\Vert _{C^{r-1,\alpha }(\Omega _1)}<2\eta$ for $k'=k,k-1$, thanks to the mean value theorem, one can argue as in the case of (2.2) to conclude that

$$\begin{aligned}{} & {} \Vert F(\Psi _{k},A_{k})-F(\Psi _{k-1},A_{k-1})\Vert _{C^{r-2,\alpha }(\Omega _1)} + \Vert H(\Psi _{k},A_{k})-H(\Psi _{k-1},A_{k-1})\Vert _{C^{r-2,\alpha }(\Omega _1)}\\{} & {} \qquad +\Vert \Psi _kA_k-\Psi _{k-1}A_{k-1}\Vert _{C^{r-1,\alpha }(\Omega _1)}\\{} & {} \quad \le C\eta \Big (\Vert A_{k}-A_{k-1}\Vert _{C^{r-1,\alpha }(\Omega _1)}+\Vert \Psi _{k}-\Psi _{k-1}\Vert _{C^{r-1,\alpha }(\Omega _1)}\Big ), \end{aligned}$$

where the constant is independent of $\eta <1$ and k. Therefore, using the bound (2.6) and Schauder estimates as before, we conclude that

$$\begin{aligned}{} & {} \Vert A_{k+1}-A_k\Vert _{C^{r,\alpha }(\Omega _1)}+\Vert \Psi _{k+1}-\Psi _k\Vert _{C^{r,\alpha }(\Omega _1)}\\{} & {} \quad \le C\eta \Big (\Vert A_{k}-A_{k-1}\Vert _{C^{r,\alpha }(\Omega _1)}+\Vert \Psi _{k}-\Psi _{k-1}\Vert _{C^{r,\alpha }(\Omega _1)}\Big ), \end{aligned}$$

which yields convergence in the $C^{r,\alpha }$-norm provided that $\eta$ is chosen small enough so that $C\eta <\frac{1}{4}$. This completes the proof that the sequence $(A_k,\Psi _k)$ converges to a couple $(A,\Psi )$ in $C^{r,\alpha }(\Omega _1)$.

2.5 Step 5: Completion of the Proof

In the previous steps we have constructed a $C^\infty$ solution $(A,\Psi )$ to the Ginzburg–Landau equations in $\Omega _1\supset \Omega$ that satisfies the boundary condition

$$\begin{aligned} N\cdot (\nabla \Psi -i\Psi A)=0 \end{aligned}$$

on $\partial _L\Omega$. Moreover, using the bound (2.7), we conclude that it satisfies the estimate

$$\begin{aligned} \Vert 2C_1\eta ^{-1}\Psi -\psi \Vert _{C^{r,\alpha }(\Omega _1 )}+\Vert 2C_2\eta ^{-1}A-a\Vert _{C^{r,\alpha }(\Omega _1)}\le C\eta . \end{aligned}$$

By construction, cf. Proposition 2.1, since $\Omega _1\subset \Omega '$, the function $\psi |_{\Omega _1}$ has a subset of structurally stable vortex lines that is isotopic to $L'\cap \Omega _1$, the diffeomorphism being $\varepsilon$-close to the identity in the $C^r$-norm. In particular, $\psi |_{\Omega }$ has a subset of structurally stable vortex lines isotopic to $L=L'\cap \Omega$. This fact and the previous estimate imply that $2C_1\eta ^{-1}\Psi |_{\Omega }$, and hence $\Psi |_{\Omega }$, has a subset of vortex lines isotopic to L. Specifically, there is a smooth diffeomorphism $\Phi :\overline{\Omega }\rightarrow \overline{\Omega }$ which is $(\varepsilon +\eta )$-close to the identity in the $C^r$-norm, and such that $\Phi (L)$ is a subset of vortex lines of $\Psi |_{\Omega }$. Since $\varepsilon +\eta$ can be taken as small as desired, this completes the proof of the theorem.

Remark 2.3

The solution A is a perturbation of the vector field a in $\Omega$. Since a is fairly explicit, this provides a good understanding of the orbits of A in the perturbative regime, but this does not seem to be relevant for applications.

3 Proof of Proposition 2.1

It is convenient to take a larger cylindrical domain $\widetilde{\Omega }=D_\rho \times (-\widetilde{\ell },\widetilde{\ell })$, with $\widetilde{\ell }\in (2\ell ,3\ell )$, and a braid ${\widetilde{L}}\subset \widetilde{\Omega }$ which is an “extension” of the braid $L'$, i.e.,

$$\begin{aligned} {\widetilde{L}}\cap \Omega '=L'. \end{aligned}$$

An easy application of Whitney’s approximation theorem ensures that, by perturbing the braid ${\widetilde{L}}$ slightly if necessary (see e.g. [9, Section 2.5]), we can assume that it is a real analytic submanifold of $\widetilde{\Omega }$. Let us denote by $\{{\widetilde{L}}_k\}_{k=1}^n$ the connected components of ${\widetilde{L}}$. Each component ${\widetilde{L}}_k$ is an analytic open curve without self-intersections, and we claim that we can write the curve ${\widetilde{L}}_k$ as the transverse intersection of two surfaces $\Sigma _k^1$ and $\Sigma _k^2$.

Indeed, each curve ${\widetilde{L}}_k$ is contractible, so it has trivial normal bundle. Then there exists an analytic submersion $\Theta _k:W_k\rightarrow {\mathbb {R}}^2$, where $W_k\subset \widetilde{\Omega }$ is a tubular neighborhood of ${\widetilde{L}}_k$ and $\Theta _k^{-1}(0)={\widetilde{L}}_k$. We can then take the analytic surfaces

$$\begin{aligned} \Sigma ^1_k:=\Theta _k^{-1}((-1,1)\times \{0\})\subset W_k, \qquad \Sigma ^2_k:=\Theta _k^{-1}(\{0\}\times (-1,1))\subset W_k. \end{aligned}$$

Since $\Theta _k$ is a submersion, these surfaces intersect transversally at

$$\begin{aligned} {\widetilde{L}}_k=\Sigma ^1_k\cap \Sigma ^2_k. \end{aligned}$$

Now that we have expressed the component ${\widetilde{L}}_k$ as the intersection of two real analytic surfaces $\Sigma ^1_{k}$ and $\Sigma ^2_k$, we can consider the following Cauchy problems, with $m=1,2$:

$$\begin{aligned} \kappa ^{-2}\Delta u_k^m+u_k^m=0,\qquad u_k^m|_{\Sigma ^m_k}=0,\qquad \partial _N u_k^m|_{\Sigma ^m_k}=1. \end{aligned}$$

Here $\partial _N$ denotes a normal derivative at the corresponding surface. The Cauchy–Kovalevskaya theorem then grants the existence of solutions $u_k^m$ to this Cauchy problem in the closure of small neighborhoods $U^m_{k}\subset \widetilde{\Omega }$ of each surface $\Sigma ^m_{k}$. We can safely assume that the tubular neighborhoods $U^1_k\cap U^2_k$ are small enough so that the neighborhoods corresponding to distinct components are disjoint. Now we take the union of these pairwise disjoint tubular neighborhoods,

$$\begin{aligned} U:=\bigcup _{k}(U^1_k\cap U^2_k)\subset \widetilde{\Omega }, \end{aligned}$$

and define a complex-valued function $\varphi$ on the set U as

$$\begin{aligned} \varphi |_{U^1_{k}\cap U^2_k}:=u_k^1+iu_k^2. \end{aligned}$$

The following properties of $\varphi$ are clear from the construction:

• (i) $\varphi$ satisfies the equation

  $$\begin{aligned} \kappa ^{-2}\Delta \varphi +\varphi =0 \end{aligned}$$

  in the tubular neighborhood U of the braid ${\widetilde{L}}$.

• (ii) U can be taken small enough so that the nodal set of $\varphi$ is precisely ${\widetilde{L}}$, i.e., ${\widetilde{L}}=\varphi ^{-1}(0)$.

• (iii) The intersection of the zero sets of the real and imaginary parts of $\varphi$ on ${\widetilde{L}}$ is transverse, i.e.,

  $$\begin{aligned} {\text {rank}}(\nabla {{\,\textrm{Re}\,}}\varphi (x),\nabla {{\,\textrm{Im}\,}}\varphi (x))=2 \end{aligned}$$

  (3.1)

  for all $x\in {\widetilde{L}}$.

Denote by S a compact subset of U whose interior contains $\overline{L'}$ and $\widetilde{\Omega }\backslash S$ is connected. Our next goal is to construct a solution of the Helmholtz equation in $\Omega '$ that approximates the local solution $\varphi$ in the set $S\cap \Omega '$. To this end, let us take a smooth function $\chi :{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ equal to 1 in a neighborhood of S and identically zero outside $U\cap \Big (D_\rho \times (-\widetilde{\ell }+\delta _0,\widetilde{\ell }-\delta _0)\Big )$, with $\delta _0>0$ a small but fixed constant so that $\widetilde{\ell }-\delta _0>2\ell$ and $S\subset D_\rho \times (-\widetilde{\ell }+\delta _0,\widetilde{\ell }-\delta _0)$. We then define a smooth extension $\varphi _0$ of the function $\varphi$ to $\widetilde{\Omega }$ by setting

$$\begin{aligned} \varphi _0:=\chi \varphi . \end{aligned}$$

By construction, this function is compactly supported in $\widetilde{\Omega }$ and $\varphi _0=\varphi$ in a neighborhood of S.

Let us now write the function $\varphi _0$ as an integral involving the Neumann Green’s function of the domain $\widetilde{\Omega }$. We first observe that, by the monotonicity principle, for any fixed $\kappa >0$ there is a constant $\widetilde{\ell }\in (2\ell ,3\ell )$ such that $\kappa ^2$ is not a Neumann eigenvalue of $-\Delta$ in $\widetilde{\Omega }$. In what follows we fix such a constant $\widetilde{\ell }$ (which depends on $\kappa$). The next lemma on the existence of a Neumann Green’s function is elementary, but we provide a proof for the sake of completeness:

Lemma 3.1

Let ${{\,\textrm{diag}\,}}$ denote the diagonal of $\widetilde{\Omega }\times \widetilde{\Omega }$. For some $\widetilde{\ell }\in (2\ell ,3\ell )$, there exists a distribution $G\in C^\infty (\widetilde{\Omega }\times \widetilde{\Omega })\backslash {{\,\textrm{diag}\,}}$ with the following properties:

• (i) G is symmetric: $G(x,y)=G(y,x)$.

• (ii) $(\kappa ^{-2}\Delta _x +1) G( x,y)=\delta (x-y)$.

• (iii) $N(x)\cdot \nabla _x G(x,\cdot )=0$ for all $x\in \partial \widetilde{\Omega }$.

• (iv) $|G(x,y)|\le \frac{C}{|x-y|}$ for some constant $C>0$ and any $(x,y)\in (\widetilde{\Omega }\times \widetilde{\Omega })\backslash {{\,\textrm{diag}\,}}$

Proof

Since $\kappa ^2$ is not a Neumann eigenvalue of $-\Delta$ in $\widetilde{\Omega }$, it follows from Fredholm’s alternative that there exists a unique Green’s function $G\in C^\infty (\widetilde{\Omega }\times \widetilde{\Omega })\backslash {{\,\textrm{diag}\,}}$ satisfying

$$\begin{aligned} (\kappa ^{-2}\Delta _x +1) G( x,y)=\delta (x-y) \end{aligned}$$

in $\widetilde{\Omega }$ and the Neumann condition $N(x)\cdot \nabla _x G(x,\cdot )=0$ for all $x\in \partial \widetilde{\Omega }$. We claim that G is symmetric. Indeed, using Green’s second identity it follows that

$$\begin{aligned} G(z,y)-G(y,z)&=\int _{\partial \widetilde{\Omega }}\big (G(x,y)\nabla _x G(x,z)\cdot N(x)-G(x,z)\nabla _x G(x,y)\cdot N(x)\big )d\sigma (x)\\&=0\,, \end{aligned}$$

where we have used that G satisfies the Neumann condition on $\partial \widetilde{\Omega }$. Finally, the estimate in the item (iv) follows from a general property of singularities of solutions to second order elliptics PDEs, cf. [8]. This completes the proof of the lemma. $\square$

Setting $\rho := \kappa ^{-2}\Delta \varphi _0+\varphi _0$, it follows from the fact that $\kappa ^2$ is not a Neumann eigenvalue of $-\Delta$ in $\widetilde{\Omega }$ that

$$\begin{aligned} \varphi _0(x)=\int _{\widetilde{\Omega }} G(x,y)\, \rho (y)\, dy. \end{aligned}$$

(3.2)

The complex-valued function $\rho$ vanishes in S because $\kappa ^{-2}\Delta \varphi +\varphi =0$ and $\varphi =\varphi _0$ in S. In fact, the compact support of $\rho$ is contained in the open set $\widetilde{\Omega }\backslash S$, and its distance to the caps $D_\rho \times \{-\widetilde{\ell }\}$ and $D_\rho \times \{\widetilde{\ell }\}$ is at least $\delta _0>0$. Next, observe that an easy continuity argument ensures that one can approximate the integral (3.2) uniformly in the compact set S by a finite Riemann sum of the form

$$\begin{aligned} \varphi _1(x):=\sum _{j=-J}^J \rho _j\, G(x,x_j). \end{aligned}$$

(3.3)

Specifically, for any $\delta >0$ there is a large integer J, complex numbers $\rho _j$ and points $x_j\in \text {supp } \rho \subset \widetilde{\Omega }\backslash S$ such that the finite sum (3.3) satisfies

$$\begin{aligned} \Vert \varphi _1-\varphi _0\Vert _{C^0(S)}<\delta . \end{aligned}$$

(3.4)

In the following lemma we show how to “sweep” the singularities of the function $\varphi _1$ in order to approximate it in the set S by another function $\varphi _2$ whose singularities are contained in $D_\rho \times (\widetilde{\ell }-\frac{\delta _0}{2},\widetilde{\ell })$. The proof is based on a duality argument and the Hahn–Banach theorem, and is an adaptation to our setting of the proof of [7, Lemma 4.1].

Lemma 3.2

For any $\delta >0$, there is a finite set of points $\{z_j\}_{j=-J'}^{J'}$ in $D_\rho \times (\widetilde{\ell }-\frac{\delta _0}{2},\widetilde{\ell })\subset \widetilde{\Omega }$ and complex numbers $c_j$ such that the finite linear combination

$$\begin{aligned} \psi (x):=\sum _{j=-J'}^{J'} c_j\, G(x,z_j) \end{aligned}$$

(3.5)

approximates the function $\varphi _1$ uniformly in S:

$$\begin{aligned} \Vert \psi -\varphi _1\Vert _{C^0(S)}<\delta . \end{aligned}$$

(3.6)

Proof

Consider the space ${{{\mathcal {U}}}}$ of all complex-valued functions on $\widetilde{\Omega }$ that are finite linear combinations of the form (3.5), where the points $z_j$ range over $D_\rho \times (\widetilde{\ell }-\frac{\delta _0}{2},\widetilde{\ell })$ and where the constants $c_j$ may take arbitrary complex values. Restricting these functions to the set S, ${{{\mathcal {U}}}}$ can be regarded as a subspace of the Banach space $C^0(S)$ of continuous complex-valued functions on S.

By the Riesz–Markov theorem, the dual of $C^0(S)$ is the space ${{{\mathcal {M}}}}(S)$ of the finite complex-valued Borel measures on $\widetilde{\Omega }$ whose support is contained in the set S. Let us take any measure $\mu \in {{{\mathcal {M}}}}(S)$ such that $\int _{\widetilde{\Omega }} f\, d\mu =0$ for all $f\in {{{\mathcal {U}}}}$. Using that the Green’s function G is symmetric, cf. item (i) in Lemma 3.1, we now define a complex-valued function F as

$$\begin{aligned} F(x):=\int _{\widetilde{\Omega }} G({\widetilde{x}}, x)\,d\mu ({\widetilde{x}}), \end{aligned}$$

which is in $L^1(\widetilde{\Omega })$ by the estimate (iv) in Lemma 3.1. Furthermore, it is clear that F satisfies the equation

$$\begin{aligned} \kappa ^{-2}\Delta F+F=\mu \end{aligned}$$

distributionally in $\widetilde{\Omega }$: for any $\phi \in C^2_c(\widetilde{\Omega })$, one has

$$\begin{aligned} \int _{\widetilde{\Omega }} F\, (\kappa ^{-2}\Delta +1)\phi \, dx =\int _{\widetilde{\Omega }} \phi \, d\mu . \end{aligned}$$

Notice that F is identically zero on $D_\rho \times (\widetilde{\ell }-\frac{\delta _0}{2},\widetilde{\ell })$ by the definition of the measure $\mu$ and that F satisfies the elliptic equation

$$\begin{aligned} \kappa ^{-2}\Delta F+F=0 \end{aligned}$$

in $\widetilde{\Omega }\backslash S$, so F is analytic in this set. Hence, since $\widetilde{\Omega }\backslash S$ is connected and contains the set $D_\rho \times (\widetilde{\ell }-\frac{\delta _0}{2},\widetilde{\ell })$, by analyticity the function F must vanish on the complement of S. It then follows that the measure $\mu$ also annihilates any complex-valued function of the form $G(x,x_j)$ because, as the points $x_j$ do not belong to S,

$$\begin{aligned} 0=F(x_j)=\int _{\widetilde{\Omega }} G(x,x_j)\,d\mu (x). \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{{\mathbb {R}}^3}\varphi _1\,d\mu = \sum _{j=-J}^J \rho _j \int _{\widetilde{\Omega }} G(x,x_j)\,d\mu (x)=0, \end{aligned}$$

which implies that $\varphi _1$ can be uniformly approximated on S by elements of the subspace ${{{\mathcal {U}}}}$ as a consequence of the Hahn–Banach theorem. This completes the proof of the lemma. $\square$

To complete the proof of Proposition 2.1, we observe that

$$\begin{aligned} \kappa ^{-2}\Delta \psi +\psi =0 \end{aligned}$$

(3.7)

in the set $D_\rho \times (-\widetilde{\ell }+\delta _0,\widetilde{\ell }-\delta _0)$, which contains S and $\Omega '$. Moreover, it satisfies the Neumann boundary condition $N\cdot \nabla \psi =0$ on the lateral boundary $\partial D_\rho \times (-\widetilde{\ell }+\delta _0,\widetilde{\ell }-\delta _0)\supset \partial _L\Omega '$, and the estimate

$$\begin{aligned} \Vert \psi -\varphi \Vert _{C^0(S)}<2\delta , \end{aligned}$$

where we have used the bounds (3.4) and (3.6) and that $\varphi _0=\varphi$ on a neighborhood of S. In fact, since $\varphi$ also satisfies the Helmholtz equation in a neighborhood of the compact set S, standard elliptic estimates imply that the uniform estimate can be promoted to the $C^r$ bound

$$\begin{aligned} \Vert \psi -\varphi \Vert _{C^r(S)}<C_r\delta . \end{aligned}$$

(3.8)

Finally, taking into account that the braid ${\widetilde{L}}$ is exactly the nodal set of $\varphi$ and satisfies the transversality condition (3.1), and using the estimate (3.8), the fact that $\overline{L'}$ is contained in S and Thom’s isotopy theorem [1, Theorem 20.2], we conclude that there is a diffeomorphism $\Phi _0$ of $\overline{\Omega '}$ such that $\Phi _0(L')$ is a union of components of the zero set $\psi ^{-1}(0)\cap \Omega '$. Moreover, the diffeomorphism $\Phi _0$ is $C^r$-close to the identity. The structural stability of the link $\Phi _0(L')$ for the function $\psi |_{\Omega '}$ also follows from Thom’s isotopy theorem and the fact that $\psi |_{\Omega '}$ satisfies the transversality condition

$$\begin{aligned} {\text {rank}}(\nabla {{\,\textrm{Re}\,}}\psi (x),\nabla {{\,\textrm{Im}\,}}\psi (x))=2 \end{aligned}$$

for all $x\in \Phi _0(L')$. This last equation is a consequence of the $C^r$-estimate (3.8), the fact that the function $\varphi$ satisfies the transversality estimate (3.1) by definition, and the fact that transversality is an open property under $C^1$-small perturbations. This completes the proof of the proposition.