Spectral Theory for Schrodingeroperator with magnetic field andanalysis of the third critical field

in superconductivity

Bernard HelfferMathematiques -

Universite Paris Sud- UMR CNRS 8628supported by the PROGRAMME HPRN-CT-2002-00277

and the ESF programme SPECT.(After S. Fournais and B. Helffer)

Λoυτρακι October 2005

– Typeset by FoilTEX –

Main goals

Using recent results by the authors on thespectral asymptotics of the Neumann Laplacianwith magnetic field, we give precise estimates onthe critical field, HC3, describing the appearanceof superconductivity in superconductors of type II.Furthermore, we prove that the local and globaldefinitions of this field coincide. Near HC3 onlya small part, near the boundary points wherethe curvature is maximal, of the sample carriessuperconductivity. We give precise estimates onthe size of this zone and decay estimates in both thenormal (to the boundary) and parallel variables.

Ginzburg-Landau functional

The Ginzburg-Landau functional is given by

Eκ,H[ψ, ~A] =∫Ω

|pκH ~Aψ|

2 − κ2|ψ|2 + κ2

2 |ψ|4

+κ2H2| curl ~A− 1|2dx ,

with (ψ, ~A) ∈W 1,2(Ω; C)×W 1,2(Ω; R2) and where

p ~A = (−i∇− ~A).

We fix the choice of gauge by imposing that

Div ~A = 0 in Ω , ~A · ν = 0 on ∂Ω .

Minimizers (ψ, ~A) of the functional satisfy theGinzburg-Landau equations,

p2κH ~A

ψ = κ2(1− |ψ|2)ψcurl 2 ~A = − i

2κH(ψ∇ψ − ψ∇ψ)− |ψ|2 ~A

in Ω ;

(1a)

(pκH ~Aψ) · ν = 0curl ~A− 1 = 0

on ∂Ω .

(1b)

Here curl (A1, A2) = ∂x1A2 − ∂x2A1,

curl 2 ~A = (∂x2( curl ~A),−∂x1( curl ~A)) .

Let ~F denote the vector potential generating theconstant exterior magnetic field

Div ~F = 0curl ~F = 1

in Ω , ~F · ν = 0 on ∂Ω .

The pair (0, ~F ) is called the Normal State.

A minimizer (ψ,A) for which ψ never vanishes willbe called SuperConducting State.

In the other cases, one will speak about Mixed State.

The general question is to determine the topologyof the sets of (κ,H) corresponding to minimizersbelonging to each of these three situations.

Existence of the third critical field HC3(κ)

It is known that, for given values of the parametersκ,H, the functional E has minimizers.

However, after some analysis of the functional, onefinds (see [GiPh]) that given κ there exists H(κ) such

that if H > H(κ) then (0, ~F ) is the only minimizerof Eκ,H (up to change of gauge).

Following Lu and Pan [LuPa1], we define

HC3(κ) = infH > 0 : (0, ~F ) minimizer of Eκ,H .

A central question in the mathematical treatment ofType II superconductors is to establish the asymptoticbehavior of HC3

(κ) for large κ.

We will also discuss the relevance of this definitionand describe how HC3

(κ) can be determined by thestudy of a linear problem.

Our first result is the following strengthening of aresult in [HePa].

Theorem ASuppose Ω is a bounded simply-connected domain inR2 with smooth boundary. Let kmax be the maximalcurvature of ∂Ω. Then

HC3(κ) =

κ

Θ0+C1

Θ320

kmax +O(κ−12) , (2)

where C1,Θ0 are universal constants.

RemarkThe constants Θ0, C1 are defined in terms of auxiliaryspectral problems.

Localization at the boundaryFrom the work of Helffer-Morame [HeMo2](improving Del Pino-Fellmer-Sternberg and Lu-Pan) (see also Helffer-Pan [HePa] for the non-linear case) we know that, when H is sufficientlyclose to HC3(κ), minimizers of the Ginzburg-Landau functional are exponentially localized to aregion near the boundary. This is called SurfaceSuperconductivity.

Note that this localization leads to the proof of :

||ψ||L2(Ω) ≤ Cκ−14||ψ||L4(Ω) , (3)

which is true for κ large enough.

Localization at the points of maximal curvature

The statement is that, when H is rather close to thethird critical field, the minimizers are also localizedin the tangential variable to a small zone around thepoints of maximal curvature.

This leads in particular to the better

||ψ||L2(Ω) ≤ Cκ−38||ψ||L4(Ω) , (4)

Discussion of critical fieldsActually, we should define more than one criticalfield, instead of just HC3

. We define an upper thirdcritical field, by

HC3(κ)= infH > 0 : ∀H ′ > H , (0, ~F )

unique minimizer of Eκ,H′ ,

Of course we have

HC3(κ) ≤ HC3(κ) .

Note that one can prove that the asymptotics givenbefore is valid for both fields.

The Schrodinger operator with magnetic field

Let, for B ∈ R+, the magnetic Neumann LaplacianH(B) be the self-adjoint operator (with Neumannboundary conditions) associated to the quadraticform

W 1,2(Ω) 3 u 7→∫

Ω

|(−i∇−B ~F )u|2 dx ,

We define λ1(B) as the lowest eigenvalue of H(B).

The local upper critical fields can now be defined :

Hloc

C3(κ) = infH > 0 : ∀H ′ > H,λ1(κH ′) ≥ κ2 ,

and

H locC3

(κ) = infH > 0 : λ1(κH) ≥ κ2 .

The coincidence between Hloc

C3(κ) and H loc

C3(κ) is

immediately related to lack of strict monotonicity ofλ1.

These critical fields appear when analyzing the (local)stability of the normal solution.

Comparison Theorem CLet Ω be a bounded simply-connected domain inR2 with smooth boundary and let κ > 0, then thefollowing general relations hold

HC3(κ) ≥ Hloc

C3(κ) ,

and

HC3(κ) ≥ H loc

C3(κ) .

EASY and GENERAL.

Next theorem is new and more delicate !

Theorem DLet Ω be a bounded simply-connected domain in R2

with smooth boundary. Then ∃ κ0 > 0 such that,for κ > κ0, we have

HC3(κ) = Hloc

C3(κ) ,

and

HC3(κ) = H loc

C3(κ) ,

So the monotonicity of λ1(B) for B largeimmediately give the coincidence of the four fields !!

The second identity is a remark of R. Frank (but theproof is essentially analogous to the first one due toFournais-Helffer)

This monotonicity has been shown in great generalityunder generic assumptions by Fournais-Helffer, whoget in addition a complete asymptotic expansion.

Around the proof of Theorem D

The crucial point leads in the following argument.

If for some H there is a non trivial minimizer (ψ,A)so

E(ψ, ~A) ≤ 0 .

then

0 < ∆ := κ2||ψ||22 −QκH ~A[ψ] = κ2||ψ||44 ,

where QκH ~A[ψ] is the energy of ψ.

The last equality is a consequence of the first G-Lequation.

Combining with (3), this gives

||ψ||2 ≤ Cκ−34∆

14 .

By comparison of the quadratic forms Q respectivelyassociated with ~A et ~F , we get, with ~a = ~A− ~F :

∆ ≤[κ2 − (1− ρ)λ1(κH ~F )

]‖ψ‖2

2 + ρ−1(κH)2∫

Ω

|~aψ|2 dx ,

(5)

for all 0 < ρ < 1.

Note that by the regularity of the system Curl-Div,combined with the Sobolev’s injection theorem, weget

‖~a‖4 ≤ C1‖~a‖W 1,2 ≤ C2‖ curl ~a‖2 .

Now ∆ is also controlling ‖ curl ~a‖22, so we get :

(κH)2‖~a‖24 ≤ C∆ .

Combining all these inequalities leads to :

0 < ∆ ≤≤

[κ2 − (1− ρ)λ1(κH ~F )

]‖ψ‖2

2 + ρ−1(κH)2‖~a‖24‖ψ‖2

4

≤[κ2 − λ1(κH ~F )

]‖ψ‖2

2

+Cρλ1(κH)∆12κ−

32 + Cρ−1∆

32κ−1 .

Chosing ρ =√

∆κ−34, and using the rough upper

bound λ1(κH ~F ) < Cκ2, we find

0 < ∆ ≤[κ2 − λ1(κH)

]‖ψ‖2

2 + C∆κ−14 .

This shows finally, for κ large enough independentlyof H sufficiently close to “any” third critical field(they have the same asymptoics)

0 < ∆ ≤ C[κ2 − λ1(κH)

]‖ψ‖2

2 ,

so in particular

κ2 − λ1(κH) > 0 .

Coming back to the definitions this leads to thestatement.

Perspectives

This is far to be the end of the story. Here are someadditional questions :

1. One can instead consider the more physicalfunctional :

Eκ,H[ψ, ~A] =∫Ω

|pκH ~Aψ|

2 − κ2|ψ|2 + κ2

2 |ψ|4

+κ2H2∫

R2 | curl ~A− 1|2dx ,

The difference is that the last integration is overR2 ! This is particularly important if Ω is notsimply connected !

2. What is going on in Dimension 3 ?Results by Pan, Helffer-Morame, Fournais-Helffer.

3. Note also that other conditions than Neumanncould be interesting.

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