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

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.


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.[4,16,5].The theory revolves around the so-called Ginzburg-Landau equations, whose non-dimensional form is The unknowns are the vector field A : Ω → R 3 , which is the magnetic vector potential, and the complex-valued function Ψ : Ω → C, which is the order parameter representing the superconducting electron pairs.Here Ω ⊂ R 3 is a bounded domain and κ > 0 is the Ginzburg-Landau parameter, whose value determines whether the superconductor is of type I or II.The usual boundary conditions are 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 Z Ψ := {x ∈ Ω : Ψ(x) = 0} , which are the defects or singularities of the phase of the order parameter Ψ.
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 where the Laplacian ∆ acts on the vector field A componentwise.Given any smooth function χ : Ω → R, it is easy to check that the gauge transformation (A, Ψ) → (A + ∇χ, Ψe iχ ) takes solutions to solutions, so one can always pick a gauge where A is divergencefree.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 as is usually done in applications.Here ℓ > 0 is the height of the cylinder and ρ is its radius; D ρ denotes the two-dimensional disk of radius ρ centered at the origin.
Nelson's hypothesis concerns the existence of entangled (i.e., braided) vortices in high-temperature superconductors, whose precise definition is the following: The following theorem is the main result of this article.We show that there exist solutions (A, Ψ) to the Ginzburg-Landau equations on the cylinder Ω, satisfying the natural boundary conditions on the lateral boundary ∂ L Ω := ∂D ρ × (−ℓ, ℓ), which exhibit a subset of isolated vortex lines isotopic to any prescribed braid L in Ω.These vortex lines are structurally stable in the sense that any other complexvalued function that is close to Ψ exhibits an isotopic subset of vortex lines.Theorem 1.2.Let L be a braid in the cylinder Ω and fix any positive integer r and any ε > 0. Then there exists a solution (A, Ψ) to the Ginzburg-Landau equations in Ω satisfying the boundary condition Here Φ is a smooth diffeomorphism of Ω, close to the identity in the sense that Φ − id C r (Ω) < ε.Furthermore, these vortex lines are structurally stable.More precisely, take an open set V ⊂ Ω which contains L and any ε ′ > 0. Then there exists some δ > 0 such that any function Several remarks are in order.First, the nodal set Z Ψ may contain other components, but they are at a positive distance from Φ(L) because Φ(L) is an isolated nodal set.Second, note that there is no Neumann condition at the bottom and top boundaries of Ω, but this does not seem to be very important for the braid structure.Third, we emphasize that (A, Ψ) is a small amplitude solution, so proving a similar result for large values of the order parameter function Ψ (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 Section 2. The proof of the auxiliary Proposition 2.1, which is more technical, is relegated to Section 3.

Proof of the theorem
The solution (A, Ψ) we construct to the Ginzburg-Landau equations is a perturbation of a solution to the linearized equations, so the Ginzburg-Laudan system essentially operates in the monochromatic wave regime.The proof is divided in five steps.In the first one we construct a solution ψ to the Helmholtz equation in Ω 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 ψ.All along this section, the integer r (which appears in the statement of Theorem 1.2) is an arbitrary positive integer.
Step 1: A monochromatic wave with prescribed braided vortex lines.In this first step we consider the extended cylinder Ω ′ := D ρ × (−2ℓ, 2ℓ) and a braid L ′ ⊂ Ω ′ that is an "extension" of the braid L, i.e., L ′ ∩ Ω = L.We prove that there is a monochromatic wave in Ω ′ 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 ε > 0, we will say that two take an open set V ⊂ Ω ′ such that V contains the braid L ′ .Then there exists some δ > 0 such that any function The proof of this result is presented in Section 3.
Step 2: Setting up a fixed point argument.Let us take a contractible axisymmetric domain Ω 1 ⊂ R 3 with C ∞ boundary such that Ω ⊂ Ω 1 ⊂ Ω ′ (e.g., think of adding two caps to the domain Ω).Note, in particular, that ∂Ω 1 ⊃ ∂ L Ω.It is standard (see e.g.[17]) that, by slightly changing the boundary of this domain away from ∂ L Ω if necessary, we can safely assume that κ 2 is not a Neumann eigenvalue of the Laplacian −∆ in the domain Ω 1 .
Specifically, in the next step we shall set an iteration of the form where (A 0 , Ψ 0 ) is constructed using (a, ψ) and where For future reference, note that the Hölder norm of these functions can obviously be estimated using Young's inequality for products as , where the constant C depends on r, α and κ −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, Ψ k+1 is the solution to the boundary value problem for the Helmholtz operator with Neumann condition By the Fredholm alternative, this solution exists and is unique because we as- Likewise, A k+1 is the solution to the boundary problem: with relative boundary conditions: Since Ω 1 is a contractible domain (and hence the space of harmonic forms satisfying the relative boundary conditions is trivial), it is well known that A k+1 exists and is unique [15, Lemma 3.5.6].
It is clear that Assuming that the sequence (A k , Ψ k ) converges to a couple (A, Ψ) in the C r,α (Ω 1 ) norm, it follows from the definition of the iteration that (A, Ψ) solves the equation in Ω 1 , with boundary conditions N • (∇Ψ − iΨA) = 0 on ∂ L Ω and div A = 0 on ∂Ω 1 .By elliptic regularity, (A, Ψ) is C ∞ .
The following lemma shows that, in fact, the vector field A is divergence-free in Ω 1 .To prove this we use a property of the vector field F (Ψ, A) when (A, Ψ) is a solution to Equations (2.5), and the boundary condition div A = 0 on ∂Ω 1 .This crucially uses our choice of relative boundary conditions to solve Equation (2.4).
Proof.An easy computation shows that the couple (A, Ψ) satisfies div where we have used the first equation that satisfies the solution (A, Ψ) to pass to the second line.Accordingly, taking div in the second equation that satisfies the couple (A, Ψ), we get and hence div A = 0 on Ω 1 as claimed.Finally, notice that this implies that the couple (A, Ψ) satisfies the Ginzburg-Landau equations in the domain Ω 1 .
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 where we use the notation q = O(η 2 ) for terms bounded as q C r,α (Ω1) Cη 2 .
To prove the convergence of the iteration, let us make the induction hypothesis that for all k ′ k, where η is a small enough constant.We shall next show that the same bounds hold for A k+1 and Ψ k+1 .
Indeed, by definition where A k+1 is the unique solution to the Hodge boundary problem (2.4).Standard Schauder estimates yield [15, Lemma 3.5.6] where C > 0 is a constant that depends on r, α, Ω 1 but not on k.By the estimate (2.2), we then get Therefore, provided that η is small enough.
Analogously, we can estimate where Ψ k+1 is the unique solution to the boundary problem (2.3).Again, by the Fredholm alternative, since κ 2 is not a Neumann eigenvalue of −∆ in Ω 1 , there exists a unique solution to this boundary problem, which by standard Schauder estimates is bounded as [10, Section 10.5] with C a constant that depends on r, α, Ω 1 but not on k.Together with (2.2), this results in Accordingly, provided that η is small enough.
Since ( A 0 , Ψ 0 ) = (0, 0), we then infer that for all k 0, and for all k 0 and some k-independent constant C > 0 that does not depend on η.
Finally, to show that the sequence (A, Ψ) is Cauchy, we notice that and these differences satisfy the boundary problems with relative boundary conditions on ∂Ω 1 , and with Neumann boundary condition For any A k , Ψ k bounded as A k C r−1 (Ω1) + Ψ k C r−1 (Ω1) < 2η, thanks to the mean value theorem, one can argue as in the case of (2.2) to conclude that where the constant is independent of η < 1 and k.Therefore, using the bound (2.6) and Schauder estimates as before, we conclude that which yields convergence in the C r,α -norm provided that η is chosen small enough so that Cη < 1  4 .This completes the proof that the sequence (A k , Ψ k ) converges to a couple (A, Ψ) in C r,α (Ω 1 ).
Step 5: Completion of the proof.In the previous steps we have constructed a C ∞ solution (A, Ψ) to the Ginzburg-Landau equations in Ω 1 ⊃ Ω that satisfies the boundary condition N • (∇Ψ − iΨA) = 0 on ∂ L Ω.Moreover, using the bound (2.7), we conclude that it satisfies the estimate By construction, cf.Proposition 2.1, since Ω 1 ⊂ Ω ′ , the function ψ| Ω1 has a subset of structurally stable vortex lines that is isotopic to L ′ ∩ Ω 1 , the diffeomorphism being ε-close to the identity in the C r -norm.In particular, ψ| Ω has a subset of structurally stable vortex lines isotopic to L = L ′ ∩ Ω.This fact and the previous estimate imply that 2C 1 η −1 Ψ| Ω , and hence Ψ| Ω , has a subset of vortex lines isotopic to L. Specifically, there is a smooth diffeomorphism Φ : Ω → Ω which is (ε+η)-close to the identity in the C r -norm, and such that Φ(L) is a subset of vortex lines of Ψ| Ω .Since ε + η 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 Ω.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.

Proof of Proposition 2.1
It is convenient to take a larger cylindrical domain Ω = D ρ × (− ℓ, ℓ), with ℓ ∈ (2ℓ, 3ℓ), and a braid L ⊂ Ω which is an "extension" of the braid L ′ , i.e., An easy application of Whitney's approximation theorem ensures that, by perturbing the braid L slightly if necessary (see e.g.[9, Section 2.5]), we can assume that it is a real analytic submanifold of Ω.Let us denote by { L k } n k=1 the connected components of L. Each component L k is an analytic open curve without self-intersections, and we claim that we can write the curve L k as the transverse intersection of two surfaces Σ 1 k and Σ 2 k .Indeed, each curve L k is contractible, so it has trivial normal bundle.Then there exists an analytic submersion Θ k : We can then take the analytic surfaces Now that we have expressed the component L k as the intersection of two real analytic surfaces Σ 1 k and Σ 2 k , we can consider the following Cauchy problems, with m = 1, 2: Here ∂ N denotes a normal derivative at the corresponding surface.The Cauchy-Kovalevskaya theorem then grants the existence of solutions u m k to this Cauchy problem in the closure of small neighborhoods U m k ⊂ Ω of each surface Σ m k .We can safely assume that the tubular neighborhoods U 1 k ∩ 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, and define a complex-valued function ϕ on the set U as The following properties of ϕ are clear from the construction: (i) ϕ satisfies the equation in the tubular neighborhood U of the braid L. (ii) U can be taken small enough so that the nodal set of ϕ is precisely L, i.e., L = ϕ −1 (0).(iii) The intersection of the zero sets of the real and imaginary parts of ϕ on L is transverse, i.e., Denote by S a compact subset of U whose interior contains L ′ and Ω\S is connected.Our next goal is to construct a solution of the Helmholtz equation in Ω ′ that approximates the local solution ϕ in the set S ∩ Ω ′ .To this end, let us take a smooth function χ : R 3 → R equal to 1 in a neighborhood of S and identically zero outside U ∩ D ρ × (− ℓ + δ 0 , ℓ − δ 0 ) , with δ 0 > 0 a small but fixed constant so that ℓ − δ 0 > 2ℓ and S ⊂ D ρ × (− ℓ + δ 0 , ℓ − δ 0 ).We then define a smooth extension ϕ 0 of the function ϕ to Ω by setting By construction, this function is compactly supported in Ω and ϕ 0 = ϕ in a neighborhood of S.
Let us now write the function ϕ 0 as an integral involving the Neumann Green's function of the domain Ω.We first observe that, by the monotonicity principle, for any fixed κ > 0 there is a constant ℓ ∈ (2ℓ, 3ℓ) such that κ 2 is not a Neumann eigenvalue of −∆ in Ω.In what follows we fix such a constant ℓ (which depends on κ).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 diag denote the diagonal of Ω × Ω.For some ℓ ∈ (2ℓ, 3ℓ), there exists a distribution G ∈ C ∞ ( Ω × Ω)\ diag with the following properties: in Ω and the Neumann condition N (x) • ∇ x G(x, •) = 0 for all x ∈ ∂ Ω.We claim that G is symmetric.Indeed, using Green's second identity it follows that where we have used that G satisfies the Neumann condition on ∂ Ω.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.
Setting ρ := κ −2 ∆ϕ 0 + ϕ 0 , it follows from the fact that κ 2 is not a Neumann eigenvalue of −∆ in Ω that Specifically, for any δ > 0 there is a large integer J, complex numbers ρ j and points x j ∈ supp ρ ⊂ Ω\S such that the finite sum (3.3) satisfies In the following lemma we show how to "sweep" the singularities of the function ϕ 1 in order to approximate it in the set S by another function ϕ 2 whose singularities are contained in D ρ × ( ℓ − δ0 2 , ℓ).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 δ > 0, there is a finite set of points {z j } J ′ j=−J ′ in D ρ × ( ℓ − δ0 2 , ℓ) ⊂ Ω and complex numbers c j such that the finite linear combination approximates the function ϕ 1 uniformly in S: Proof.Consider the space U of all complex-valued functions on Ω that are finite linear combinations of the form (3.5), where the points z j range over D ρ × ( ℓ − δ0 2 , ℓ) and where the constants c j may take arbitrary complex values.Restricting these functions to the set S, 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 M(S) of the finite complex-valued Borel measures on Ω whose support is contained in the set S. Let us take any measure µ ∈ M(S) such that Ω f dµ = 0 for all f ∈ 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 Notice that F is identically zero on D ρ × ( ℓ − δ0 2 , ℓ) by the definition of the measure µ and that F satisfies the elliptic equation ∆F + F = 0 in Ω\S, so F is analytic in this set.Hence, since Ω\S is connected and contains the set D ρ × ( ℓ − δ0 2 , ℓ), by analyticity the function F must vanish on the complement of S. It then follows that the measure µ also annihilates any complex-valued function of the form G(x, x j ) because, as the points x j do not belong to S, 0 = F (x j ) = Ω G(x, x j ) dµ(x) .
(iv) |G(x, y)| C |x−y| for some constant C > 0 and any (x, y) ∈ ( Ω × Ω)\ diag Proof.Since κ 2 is not a Neumann eigenvalue of −∆ in Ω, it follows from Fredholm's alternative that there exists a unique Green's function G y) ρ(y) dy .The compact support of the complex-valued function ρ is contained in the open set Ω\S, and its distance to the caps D ρ × {− ℓ} and D ρ × { ℓ} is at least δ 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 (3.3) ϕ 1 (x) := J j=−J ρ j G(x, x j ) .