Vortex Analysis of the Periodic Ginzburg-Landau Model

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Vortex analysis of the periodic Ginzburg–Landau model

Hassen Aydi

, Etienne Sandier

aInstitut supérieur d’informatique de Mahdia, route de Réjiche Km 4, BP 53, Mahdia 5121, Tunisie bUniversité Paris-Est, LAMA – CNRS UMR 8050, 61, Avenue du Général de Gaulle, 94010 Créteil, France

cInstitut Universitaire de France, France

Received 8 January 2008; received in revised form 19 September 2008; accepted 19 September 2008 Available online 29 October 2008

Abstract

We study the vortices of energy minimizers in the London limit for the Ginzburg–Landau model with periodic boundary condi- tions. For applied fields well below the second critical field we are able to describe the location and number of vortices. Many of the results presented appeared in [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], others are new.

©2008 Elsevier Masson SAS. All rights reserved.

Résumé

Nous étudions les tourbillons des minimiseurs de l’énergie de Ginzburg–Landau en supraconductivité dans la limite de London, et pour des conditions aux limites périodiques. Lorsque le champ magnétique appliqué est petit devant le second champ critique, nous décrivons le nombre et la localisation de ces tourbillons. Certains de ces résultats étaient présents dans [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], d’autres sont nouveaux.

©2008 Elsevier Masson SAS. All rights reserved.

Keywords:Superconductivity; Ginzburg–Landau equations; Vortices; Periodic boundary conditions

Mots-clés :Supraconductivité ; Équations de Ginzburg–Landau ; Tourbillons ; Conditions aux limites périodiques

1. Introduction

Periodic solutions of the Ginzburg–Landau equations of superconductivity with vortices arranged in a lattice were first introduced in the famous work of A. Abrikosov [1], based on an analysis of the linearized equations about the normal solution, where the order parameter is 0. Since then many contributions to the study of this type of solutions have appeared both from physicists and mathematicians, establishing rigorously the existence of Abrikosov type so- lutions ([20], more recently [5] established the existence of many other families of periodic solutions) or investigating the energy or the minimality of these solutions [18,17] or their numeric analysis [12]. We may refer to the review paper [10] for a broad overview of the subject from a physicist’s point of view.

✩ This research was supported by the ANR, project ANR-05-JCJC-0213.

* Corresponding author at: Université Paris-Est, LAMA – CNRS UMR 8050, 61, Avenue du Général de Gaulle, 94010 Créteil, France.

E-mail address:sandier@univ-paris12.fr (E. Sandier).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.09.004

There is a convenient variational setting for the periodic Ginzburg–Landau equations, we may refer to [20,13] for a mathematically oriented presentation of this setting, the existence of minimizers is proved in [20], the regularity of solutions is established [13] together with other properties of minimizers or critical points of the Ginzburg–Landau functional. Let us also cite the recent works [6] and [14], without going into further detail.

Opposite to the case of solutions close to 0, the so-called London limit or London approximation deals with so- lutions where the order parameter has modulus close to 1, except in small areas (the vortex cores). This limit was investigated by A. Abrikosov from the beginning, see also [10] for the use of this approximation in numerous situa- tions, or the classical textbook [25]. The mathematical justification of this approach following the methods introduced in [8] may be found in [22], where it is in addition applied to describe the vortices of minimizers of the Ginzburg–

Landau energy in a fixed domain Ω (in units of the penetration depth), when the Ginzburg–Landau parameterκ is large and the applied field is small compared to theR²-upper critical field, i.e. in the parameter region where mini- mizers may indeed be analyzed using the London approximation.

In [7], the first author carried out for the case of periodic boundary conditions the analysis carried out in [22] in the case of natural boundary conditions in a bounded domain. It was established among other things that in the periodic case, the regime of the lower critical field isH_c1(κ)=logκ/2 to leading order asκ→ +∞, see below for a precise statement. Note that in the units used in [7], and in most regimes where vortices are present, there is a divergent number of vortices asκ→ ∞in each periodicity cell. The periodicity cell is large compared to the intervortex distance, and thus the periodicity constraint is not a strong one.

The present paper contains both results present in [7] and new results which complete them, in order to give a fairly unified description of the vortices of minimizers of the Ginzburg–Landau functional in the limitκ→ ∞, for applied fields well below the upper critical field.

2. Statement of the results

Notation. Throughout,K will denote a parallelogram with area 1 generated by two vectors(u, v). We will denote byLthe group of translations generated by(u, v).

The parameter of our asymptotic analysis will be the inverse of the Ginzburg–Landau parameterκ, denoted byε.

The applied fieldh_exis a positive function ofε∈R^∗₊, and we define Δ_ex=h_ex−1

2|logε|.

As noted in the introduction, ¹₂|logε|is the regime of the so-called lower critical fieldH_c1(ε), to be defined below, this motivates the notationΔ_ex.

Definition 1.We defineH_per¹ to be the set of(u, A)inH_loc¹ (R²)such that for any integersk, ∈Zthe configuration (u(· +ku+v), A( · +ku+v)) is gauge-equivalent to(u, A).

In a more geometrical language,uis a section of a complex line bundle over the torusR²/L, andAis a connection.

Then curlAis the curvature of the connection and 1

2π

K

curlA

is an integer, the first chern class of the line bundle.

Given(u, A)inH_per¹ , and an for eachε >0 an applied magnetic fieldhex(ε), we define for anyε >0 G_ε(u, A)=1

2

K

|∇u−iAu|²+1

2(curlA−h_ex)²+ 1 2ε²

1− |u|²2 .

Then the problem of minimizingGε(u, A)overH_per¹ is well posed.

Proposition 2.1.The minimum ofG_ε(u, A)overH_per¹ is achieved.

We refer to [20] for the proof.

A useful fact is

Proposition 2.2.Given any(u, A)∈H_per¹ , then 1

2π

K

curlA

is an integer. Moreover, if(u₁, A₁)minimizes the Ginzburg–Landau energy with parametersε >0andh_ex=h₁, and if(u2, A2)minimizes the Ginzburg–Landau energy with parametersε >0andhex=h2> h1, thenn2n1, where we have set fori=1,2

n_i= 1 2π

K

curlA_i.

Proof. The fact that

KcurlA∈2πZwas already mentioned.

For the second statement, denoteG1(resp.G2) the Ginzburg–Landau functional with parameters(ε, h1)(resp.

(ε, h2)), thenG1(u1, A1)G1(u2, A2)andG2(u1, A1)G2(u2, A2), hence (G1−G2)(u1, A1)(G1−G2)(u2, A2),

which translates as (h₂−h₁)

K

curlA₁+1 2

K

h²₂−h²₁

(h₂−h₁)

K

curlA₂+1 2

K

h²₂−h²₁ ,

hence the result. 2

Remark 2.1.As a consequence of the above proposition, for eachε >0 there is a well-defined valueH_c1(ε)which we call the first critical field, and such that the minimizers of the Ginzburg–Landau functional with parameters(ε, hex) satisfy n=0 if hex< Hc₁, and n=0 if hex> Hc₁. Note that in the former case, the minimizers are necessarily gauge-equivalent to the constant superconducting solutionu=1,A=0, see below.

Theorem 1.Denote by(u_ε, A_ε)any minimizer ofG_ε and leth_ε=curlA_ε. We define the integern_εby 2π nε=

K

h_ε.

Then the following behaviour ofh_ε,n_εholds, according to the regime considered for the applied fieldh_ex. (1) If1 Δ_ex 1/ε², then, asε→0,

h_ε

2π nε →1 inW^1,pfor anyp <2, andn_ε≈Δ_ex

2π . (2.1)

(2) IfΔexis bounded independently ofεthen so ishεW^1,p, for anyp <2. In particularnεis bounded independently ofε. If{ε}is a subsequence such that{h_ε}εconverges toh_∗andΔ_exconverges to a valueΔ^∗_ex, thenn_ε→n_∗∈N along the same subsequence, thus in particularn_ε=n_∗for small enoughε, and there aren_∗distinct points{a_i}i

inKsuch that

−Δh_∗+h_∗=2π

n_∗

i=1

δa_i. (2.2)

Moreover, denote byP the finite families of points inKand forp=(p1, . . . , p_k)∈Plet W (p)=lim

ρ→0

π nlogρ+1 2

K\ iB(p_i,ρ)

|∇hp|²+h²_p

+n

γ−2π Δ^∗_ex

, (2.3)

where h_p is the unique K-periodic solution of −Δh_p+h_p=2π_k

i=1δ_pi. Then (a₁, . . . , a_n∗)minimizes W overP.

The numberγin(2.3)was introduced in[8], we define it as in[22], Proposition3.11, as γ= lim

R→+∞−πlogR+1 2

B(0,R)

|∇u₀|²+(1− |u0|²)²

2 , (2.4)

whereu₀is the unique solution of−Δu₀=u₀(1−|u₀|²)inR²of the formu₀(r, θ )=f (r)e^iθ, withf:R+→R+. (3) There exists a possibly negativeΔ₁∈Rsuch that ifΔ_ex< Δ₁andεis small enough, thenn_∗=0. In this case

(u_ε, A_ε)is gauge-equivalent to the Meissner solution(1,0).

Remark 2.2.The first item in the above theorem was proved initially in [7].

The main difference between the periodic case and the case of a bounded domain is that in the former the distribu- tion of vortices is always uniform, and hence may be described by a unique number, namely the number of vortices, at least if it tends to+∞asε→0. This is a big simplification over the case of a bounded domain, together with the fact that here the number of vortices is given by the integral ofh.

Remark 2.3.The fact that the minimum ofW onP is achieved is a consequence of the above theorem, it could also be derived directly from (2.3). From Proposition 2.2, ifΔ₂> Δ₁∈Rand ifp1(resp.p2) minimizesWforΔ_ex=Δ₁ (resp.Δ_ex=Δ₂), then the number of points inp2is larger than the number of points inp1. This shows that there are critical values ofΔ_exfor which the number of points for a minimizer experiences a jump, and it would be interesting to show that it jumps by one unit only. This would show the existence of an increasing sequence of critical values (Δ_k)_k∈N∗for which the number of vortices jumps fromk−1 tok.

Note that at a jump there exists minimizers with different numbers of vortices, hence the minimizer of W need not be unique. Even ifΔexis not a critical value, i.e. if minimizers ofW all have the same number of points, if this number is 2 for instance then the symmetry of the periodicity is broken and there are several minimizers as well.

Remark 2.4.Regarding the finer structure of vortices, they are expected in general to arrange themselves in periodic lattices, and the hexagonal lattice is supposedly optimal. Several rigorous mathematical results in this direction have been proved (see for instance [3,2,23]). In the periodic setting, the strongest version of this conjecture should be true: If the lattice generated by(u, v) is the hexagonal lattice and ifn_ε=k²for some integerk, then the minimizing configuration should be periodic w.r.t. the vectors(u/k, v/k). A limiting form of this conjecture would be that – still in the case of a hexagonal lattice – in the case of a bounded number of vortices, and assumingn_∗=k², thenh_∗ is periodic w.r.t. the vectors(u/k, v/k).

Remark 2.5.Note that for a minimizer(u_ε, A_ε)ofG_εand arbitrary positive values of the parametersε,h_ex, we have 1

2

K

(h_ε−h_ex)²G_ε(u_ε, A_ε)G_ε(1,0)=1 2h²_ex, whereh_ε=curlA_ε. Therefore the following holds

0nε, Gε(uε, Aε)1

2h²_ex. (2.5)

Moreover, ifnε=0, thenhε=0 as well. Then(uε, Aε)is gauge equivalent to a configuration(u_ε,0)andGε(u_ε,0) Gε(1,0). But, since(1,0)is clearly the unique minimizer ofGε among configurations(u, A)such thatA=0, we deduce thatu_ε=1 and that(uε, Aε)is gauge-equivalent to the Meissner solution.

The paper is organized as follows: In Section 3 we construct a test configuration which will be useful in the proof of case (1) of the theorem. In Section 4, matching the upper bound of the previous section with appropriate lower bounds, we prove case (1) of the theorem, with anL²instead ofW^1,pconvergence, we also prove anL²bound forh_ε in case (2), and we prove case (3) completely. In Section 5 we discuss the improvement fromL²toW^1,pconvergence without going into the details, since similar arguments appear in [22] in the context of natural, instead of periodic,

boundary conditions. Finally in the last section we finish the proof of case (2), i.e. identify the limith_∗as a minimizer ofW. This involves arguments from [8,9] adapted to the periodic setting, that are in part sketched.

3. Upper bound

The theorem is proved by energy comparison with an appropriate test configuration(v_ε, B_ε), which will be peri- odic. It is defined below, rather quickly since this construction appears elsewhere (see for instance [21,7] or [4]). In the case whereΔ_exC, which corresponds to cases (2) and (3) of the theorem, the upper bound in (2.5) will suffice, hence we assume from now on that

1 Δ_ex 1/ε².

Then, we definem_ε as the integer such that√m_ε is the integer part of Δex

2π .

SinceΔextends to+∞asε→0 we have m_ε≈Δex

2π.

We then divideKintom_εidentical parallelograms generated by the vectorsu_ε= u/√

m_ε andv_ε= v/√

m_ε. Denote byK_εone of these parallelograms, bya_εits center, and byf_εthe solution inK_ε, with periodic boundary conditions, of the equation

−Δf_ε=2π δa_ε− 2π

|K_ε|.

Note that such a solution exists since the integral of the right-hand side overKεis 0. It is then naturally extended by periodicity to a functionfεdefined in all ofR²and satisfying

−Δf_ε=2π

k,∈Z

δ_aε+kuε+vε− 2π

|K_ε|. Then, we letB_ε:R²→R²be such that

curlB_ε= 2π

|K_ε|

and writev_ε=ρ_εe^iϕε, whereρ_ε is periodic with respect to the vectors (u_ε,v_ε) and is defined inK_ε by ρ_ε(x)= min(|x−a_ε|/ε,1), and whereϕ_εis such that−∇^⊥f_ε= ∇ϕ_ε−B_ε. This latter equation can be solved in the sense that the curl of(B_ε− ∇^⊥f_ε)is

2π

k,∈Z

δ_aε+kuε+vε,

hence there exists a functionϕ_ε defined modulo 2π except at the pointsa_ε+ku_ε+v_ε which satisfies the identity.

Then lettingvε=ρεe^iϕε is legitimate, since precisely from the definition ofρε we haveρε(aε+kuε+vε)=0 for any integersk, .

To estimate the energy, note that the integrand ofGε(vε, Bε)is precisely 1

2

ρ_ε²|∇f_ε|²+ |∇ρ_ε|²+ 1 2ε²

1−ρ_ε²2

+ 2π

|Kε|−h_ex ²

. (3.1)

Its integral overK_εis easily estimated (see [21,7] or [4]). Putting aside the last term, the contribution from the region whereρ_ε=1, which isB(a_ε, ε)is bounded by a constant independent ofεfrom scaling arguments. The integrand outsideB(a_ε, ε)reduces to ¹₂|∇f_ε|²which, with the price of an error bounded independently ofε, may be replaced by ¹₂|∇log|x−a_ε||², whose integral overK_εisπlog√m¹_εε +O(1)asε→0. Then, integrating (3.1) overK_εyields

πlog 1

√m_εε+|K_ε| 2

2π

|K_ε|−h_ex

²+O(1) asε→0.

Since there arem_ε=1/|K_ε|squares inK, we deduce Gε(vε, Bε)=π mεlog 1

√m_εε+1

2|2π mε−hex|²+O(mε).

After expanding and replacingh_ex=Δ_ex+ |logε|/2 we find G_ε(v_ε, B_ε)=2π²m²_ε−2π mεΔ_ex+1

2h²_ex+o m²_ε

.

Finally, using the fact thatmε≈Δex/2πand tends to+∞asε→0 we get G_ε(v_ε, B_ε)=1

2

h²_ex−Δ²_ex +o

Δ²_ex

. (3.2)

4. Lower bound, identification of the limits

High fields. The case where Δex≈hex, or equivalently|logε| hex ε⁻² is particularly simple. In fact in this case one could even get more detailed information (see [21,22]). Indeed the existence of a test configuration satisfying (3.2) implies thatG_ε(u_ε, A_ε)is smaller than the right-hand side of (3.2), which is itself o(h²_ex). it follows in particular that h_ε−h_ex²_L2(K) is o(h²_ex)and therefore, dividing byh²_ex, thath_ε/ h_ex→1 inL²(K), hence also inL¹(K). In particular

nε≈h_ex 2π ≈Δ_ex

2π,

and of courseh_ε/(2π nε)→1 inL²(K). In fact in this case we even have strong convergence ofh_ε/(2π nε)inH¹. Indeed, any critical point ofG_εsatisfies the so-called second Ginzburg–Landau equation

−∇^⊥h_ε=j_ε, wherej_ε=(iu_ε,∇u_ε−iA_εu_ε)=ρ_ε²(∇ϕ_ε−A_ε), (4.1) where the last equality is an alternative expression forjεwhenuε=ρεe^iϕεis not zero. A consequence of (4.1) is that pointwise|∇hε||∇uε−iAεuε|. Therefore

1 2

K

|∇h_ε|²+ |h−h_ex|²

G_ε(u_ε, A_ε),

and we are able to deduce as above, since the right-hand side is o(h²_ex), thath_ε/ h_ex→1 inH¹.

Low fields.We now deal with the rest of the cases in the theorem, namely case (1) withh_ex=O(|logε|)and cases (2) and (3). We thus assume the estimateh_exC|logε|.

First, we recall the following construction which can either be adapted from [15] or [22], Theorem 4.1 to this periodic setting. We first define for anyr∈(0,1)thefree energyof(u, A)∈H_per¹ as

F_r,ε(u, A)=1 2

K

|∇u−iAu|²+r²h²+ 1 2ε²

1− |u|²2

, (4.2)

with the notationh=curlA. In particular we have the following relation betweenF_r,εandG_ε Gε(u, A)=Fr,ε(u, A)+1−r²

2

K

h²−hex

K

h+1

2h²_ex. (4.3)

Proposition 4.1.AssumeG_ε(u_ε, A_ε)C|logε|²for anyε >0, where(u_ε, A_ε)∈H_per¹ . Then there existsε₀>0and r₀>0such that, for anyε < ε₀and any1> rε^1/3, the following holds.

There exists a collection of disjoint closed ballsB= {B_i}i∈I which is periodic with respect toKand finite in any compact subset ofR². Moreover, denoting byB0a minimal subset ofBsuch thatB= _τ∈LτB0,

(1) Denoting byr(B)the sum of the radii of balls belonging to a collectionB, we haver(B0)=r.

(2) Abusing notation,{||u_ε| −1|1/2} ⊂B. (3) Writingd_B=deg(u, ∂B),

Fr,ε(uε, Aε, K∩B)π D

log r Dε−C

, (4.4)

whereD=

B∈B0|d_B|is assumed to be nonzero andCis a universal constant.

(4) Finally,

DCFε(uε, Aε, K)

|logε| , (4.5)

whereCis a universal constant.

Finally, ifr₁> r₂andB1,B2are the corresponding families of balls, then every ball inB2is included in one of the balls ofB1.

The above result is adapted from [22], Theorem 4.1, applied withα=2/3. The proof of the equivalent of Theo- rem 4.1 in the periodic setting poses no difficulty, the way to do it is to consideru_ε as a section of a complex line bundle over the torusT =R²/L, whereLis the group of translations generated by(u, v), and A_ε as a connection over this bundle. This is described somewhat in [4], Proposition 4.7. Note that we must impose a boundr < r₀which is here to ensure that any ball of radiusr < r₀onT is indeed a topological ball, it suffices to taker₀smaller than the injectivity radius ofT.

We also note the following

Lemma 4.1.For any(u, A)∈H_per¹ , if{B_i}i∈I is a finite collection of disjoint closed balls inR²such that|u|>0in K\ iB_i and such that _iB_i does not intersect∂K, then

i

deg

u/|u|, ∂B_i

=

K

curlA.

Proof. This is standard. By periodicity, there exists two functionsf, g:R²→Rsuch that u(x+ u)=u(x)e^{if (x)}, A(x+ u)=A(x)+ ∇f (x),

and the same relations hold, replacingubyvandf byg.

The sum D of the degrees is the degree of u/|u| restricted to∂K. Letting ϕ be the phase of u, we thus have, denoting byτ the positively oriented unit vector tangent to∂K,

D=

∂K

∂_τϕ= 1 0

1 u

d ds

ϕ(su) −ϕ(su+ v) +

1 0

1 v

d ds

ϕ(u+sv) −ϕ(sv) ds.

Then sinceϕ(x+ u)−ϕ(x)=f (x)andϕ(x+ v)−ϕ(x)=g(x)it follows that

D= 1 0

1 u

d

dsg(su) + 1 v

d dsf (sv),

and then, sinceA(x+ u)−A(x)= ∇f (x)andA(x+ v)−A(x)= ∇g(x), D=

∂K

A·τ =

K

curlA. 2

Let us now consider for anyε >0 the minimizer (u_ε, A_ε)of G_ε, and let h_ε=curlA_ε. For any r∈(0,1)we may construct using Proposition 4.1 a collection of ballsB of total radiusr. Moreover ifr is small enough, then a translationτ can be chosen so that∂(τ K)does not intersectB. We may then choose as the minimal subsetB0of

Proposition 4.1 the collectionBrconsisting of the balls inBwhich are included inτ K. The previous lemma combined with Proposition 4.1 then implies that

F_r,ε(u_ε, A_ε,Br)π d_ε

log r d_εε−C

, 2π dε

τ K

h_ε=

K

h_ε=n_ε,

whered_ε=

B∈Br|d_B|, and the equality between the integrals overτ K andK follows from the periodicity of the configurations. Note that sinceFr,ε(uε, Aε)Ch²_exandhexC|logε|, the a priori bound (4.5) reads

d_εC|logε|.

we deduce from the above a lower bound forG_ε: Gε(uε, Aε)π|logε|nε−2π hexnε+1

2h²_ex+1−r² 2

K

h²_ε+Rε, (4.6)

where

R_ε=π|logε|(d_ε−n_ε)+π d_ε

log r dε −C

. (4.7)

We now choose a fixed radiusr <1/4.Sinced_εC|logε|, it is easy to check, that ifd_ε>2nεandεis small enough, thenR_ε0 while ifn_εd_ε2nε, then clearly

R_ε−Cn_ε(logn_ε−C_r), (4.8)

whereC_r depends onr. Returning to the lower bound forG_ε, we writeh_ex=Δ_ex+ |logε|/2 and obtain G_ε(u_ε, A_ε)−2π Δexn_ε+1

2h²_ex+1−r² 2

K

h²_ε+R_ε. (4.9)

Now we distinguish two cases.

Δ_exC. This corresponds to cases (2) and (3) of the theorem. Combining (4.9) with the upper bound (2.5) and the estimate (4.8) gives

1−r² 2

K

h²_ε−Cnε(lognε−Cr)2π ΔexnεCnε (4.10)

but the integral of h²_ε overK is bounded below by 4π²n²_ε using Cauchy–Schwarz, and thus the above inequality implies that for somec, C >0 independent ofε,

cn²_ε−Cnε(lognε+1)2π ΔexnεCnε.

Thusn_εis bounded independently ofε, and going back to (4.10)h_εis bounded inL²(K)independently ofε.

Moreover, sincen_ε0, the above inequality implies that ifΔ_exis smaller than a certain, possibly negative value Δ₁∈Rwhich could be expressed in terms of the constantsc, Cappearing in the inequality, thenn_ε=0 ifεis small enough. Then the inequality

1 2

K

(h_ε−h_ex)²G_ε(u_ε, A_ε)G_ε(1,0)=1 2h²_ex

implies thath_ε=0 and|u_ε| =1, thus(u_ε, A_ε)is the Meissner solution, proving case (3) of the theorem.

1 Δ_exC|logε|. In this case as in the previous one we first combine (4.9) with the upper bound (2.5) to obtain 1−r²

2

K

h²_ε−Cn_ε(logn_ε−C_r)2π Δexn_ε,

which allows to conclude thatn_ε/Δ_exremains bounded asε→0, and then thath_ε/Δ_exis bounded inL²(K)inde- pendently ofε. It remains to identify the limit of bothn_ε/Δ_exandh_ε/Δ_ex. More precisely, from any sequence{ε_n}n

tending to zero we may extract a subsequence such that the limits exist, let us call themn_∗andh_∗. If we compute h_∗andn_∗independently of the particular subsequence, then we will have proved the convergence, and identified the limits. Since we have the relation

2π n_∗=

K

h_∗, (4.11)

it suffices, to finish the proof of the theorem, to prove thath_∗=1.

First we may combine the more precise upper bound (3.2) with (4.9) and (4.8) to obtain 1−r²

2

K

h²_ε−Δex

K

hε−Cnε(lognε−Cr)−1

2Δ²_ex+o Δ²_ex

.

Then dividing the above byΔ²_exand lettingε→0 we find 1−r²

2

K

h²_∗−

K

h_∗−1 2.

The left-hand side is bounded below by the minimum of the function−x+x²/2, i.e. by−1/2. It follows thath_∗=1.

5. Improved convergence

To improve the convergence ofh_ε/Δex(resp.h_ε) in case (1) (resp. case (2)) of the theorem, we invoke the classical Jacobian estimate of [16], together with a small-large ball type argument. Note that in the casehex |logε|, we have already proved theH¹convergence, hence we assume below thathex< C|logε|.

We recall the London equation satisfied by critical points of the functionalGε:

−Δhε+hε=με, whereμε=curljε+hε, (5.1)

where the superconducting currentjε is defined in (4.1).

The proof for the improved convergence is then the following. We construct for anyε >0 using Proposition 4.1 a collection of vortex balls with total radiusr_ε=ε¹³, which we denoteB, and we let

ν_ε=2π

B∈B

d_Bδ_aB, (5.2)

wherea_Bdenotes the center ofB.

It then follows from the Jacobian estimate (see for instance [22], Theorem 6.2) that for anyβ∈(0,1), we have

εlim→0(μ_ε−ν_ε)=0, in the dual ofC^0,β. Now we letd_ε=

B∈B|d_B|, and we claim that

d_ε=O(nε), asεtends to 0. (5.3)

Assume a moment that this is true (sincen_ε=

B∈Bd_B, it amounts to proving that the balls have mostly positive degrees), then writing

μ_ε=ν_ε+(μ_ε−ν_ε),

we find that in the case 1 Δ_exC|logε|and sincen_ε=O(Δex), the sequence{μ_ε/Δ_ex}ε is bounded in the dual ofC^0,β for anyβ∈(0,1). By compact embedding ofW^1,q intoC^0,β for anyq >2 and well chosenβ, we deduce that for anyp <2 we may extract a convergent subsequence inW^−1,p. Using (5.1), this gives the convergence of {h_ε/Δ_ex}ε inW^1,p. The same argument holds, without normalizing byΔ_ex in caseΔ_exis bounded independently ofε.

It remains to prove (5.3). This is done by inspecting carefully (4.6) and (4.7), lettingr=ε¹³ there, and comparing with the upper bound (2.5). These yield after some simplifications

−2π nεΔex+π d_ε−nε

|logε| −d_ε

3 |logε| −d_εlogd_ε−Cd_ε

0.

Since we have assumedΔ_exC|logε|, the above may be written π2

3d_ε|logε| −Cn_ε|logε| +o

d_ε|logε|

0,

which implies thatd_ε=O(nε)asε→0, and concludes the proof of theW^1,pconvergence.

6. The case of a finite number of vortices

We now finish the proof of case (2) of the theorem, i.e. identifyh_∗as a minimizer of some renormalized energy, in the terminology of [8].

The proof of this fact relies first on the construction of test configurations with prescribed vortices, a rather straight- forward task. The second part is to compute a precise expansion of the energy of a minimizer(u_ε, A_ε)in terms of the vortex locations. This expansion can be found in [8] for the model without magnetic field and Dirichlet bound- ary condition, or in the case with magnetic field in [9], the definition of γ there is different from (2.4), but a result of Mironescu [19] shows the two are equivalent. The vortex locations in these references is to be understood as the limiting locations asε→0. A similar expansion may be found in [24] for configurations which are not necessarily minimizers of the energy. Another option is to give an expansion of the energy in terms of the actual vortex locations for fixedε. This is the approach used in [11] in the case without magnetic field or in [22] for the case with magnetic field. The latter approach gives information for eachε >0, but is less elementary than the former, that we adopt.

The proof we sketch for the convenience of the reader is borrowed from [22,9] and [8].

Upper bound.Givenp=(a₁, . . . , a_n)∈P, the test configuration is constructed as follows. First we lethbe the solution of−Δh+h=2π

iδ_ai inKwith periodic boundary conditions. We still denote byhthe extension ofhto R²by periodicity (note thathdoes not depend onε). Letting

A= {ai+ku+v|k, ∈Z, 1in}, we have−Δh+h=2π

a∈Aδ_ainR².

ThenA is chosen such that curlA=h andϕ is defined modulo 2π inR²\Aand such that∇ϕ−A= ∇^⊥h.

Such aϕexists precisely because curl(A+ ∇^⊥h)= −Δh+his equal to 2π

a∈Aδ_a.Finally we define, for a fixed, arbitrarily chosenR >0,

ρ_ε(x)=

1 ifx /∈ a∈AB(a, Rε),

f (|x−a|/ε)

f (R) ifx∈B(a, Rε),

wheref (r)= |u₀|(r), andu₀is the radial vortex which appears in (2.4), the definition ofγ.

Then we let v_ε =ρ_εe^iϕ. It is clear that (v_ε, A) is K-periodic since the gauge-invariant quantities h=curlA, ρ_ε= |v_ε|are and since∇ϕ−A= ∇^⊥h. It remains to evaluate the energy of(v_ε, A)overK. First we note that, since the integral ofhoverKis 2π n,

Gε(vε, A)=h²_ex

2 −2π nεhex+G_ε(vε, A), (6.1)

where

G_ε(v_ε, A)=1 2

K

|∇v_ε−iAv_ε|²+h²+ 1 2ε²

1− |v_ε|²2

. (6.2)

We evaluateG_ε(v_ε, A). LetB_i=B(a_i, Rε)andK_R=K\ _iB_i. OnK_Rwe have|v_ε| =1 and∇ϕ−A= ∇^⊥hfrom which it follows that the energy density there reduces to(|∇h|²+ |h|²)/2. Therefore, from the definition (2.3), we may write

εlim→0

G_ε(vε, A, KR)+π nlog(Rε)

=W (p)−n

γ−2π Δ^∗_ex .

On the other hand, from [22], Chapter 10, we have

R→+∞lim lim

ε→0

G_ε

v_ε, A,

i

B_i

−π nlogR−nγ

=0.

It follows that

εlim→0

G_ε(v_ε, A)+π nlogε

=W (p)+2π nΔ^∗_ex and therefore

εlim→0

Gε(vε, A)−h²_ex 2

=W (p, n). (6.3)

Lower bound.Now we assume that(u_ε, A_ε)is a minimizer ofG_εand we try to compute a matching lower bound for (6.3).

First, from (5.1), we know thath_εconverges inW^1,pfor anyp <2 to the solution of

−Δh_∗+h_∗=μ_∗,

whereμ_∗is the common limit of{με}εand{νε}εasε→0. Then (5.2), (5.3) and the fact thatnε→n_∗imply thatμ_∗ is of the form 2π_k

i=1d_iδ_ai,whered_i∈Zand

id_i=n_∗. Note that the pointsa_i need not (yet) be distinct.

From the lower and upper bounds (4.6) and (2.5), we draw some further consequences, in view of (4.7). Indeed, in the lower bound (4.6), we have not used all the terms in the integrand ofG_ε. More precisely, we have only taken into account outside the ballsBr the term(h_ε−h_ex)²/2. It follows that a by-product of the upper bound (2.5) is that, as ε→0,

1 2

K\Br

|∇uε−iAεuε|²+ 1 2ε²

1− |uε|²2

C.

Since this is true for anyr (the constantC then depending onr), we obtain in particular, lettingρε= |uε|and since

|∇u_ε−iA_εu_ε||∇ρ_ε|, that

|∇ρ_ε|²+ 1 2ε²

1−ρ_ε²2

is bounded inL¹_loc(K\ {a₁, . . . , a_n}).

Letr₀be half the minimal distance between two pointsa_i,a_j, and fixr∈(0, r0). It follows from the above and from the strong convergence ofh_εtoh_∗inW^1,p, using a mean-value argument that for anyε >0, there existsr/2< r_ε< r such that for every 1inand lettingγ_i,ε=∂B(a_i, ε),

γ_i,ε

|∇ρ_ε|²+ 1 2ε²

1−ρ_ε²2

=O(1), h_ε−h_∗W^1,p(γ_i,ε)=o(1), (6.4)

asε→0.

We bound from belowG_ε(uε, Aε, Bi,ε), whereBi,ε=B(ai, rε). To this aim we assume that we are in the Coulomb gauge. Then (see [9]),{A_ε}εis bounded inW^2,p(K)for anyp <2, hence inL^∞. Moreover, since

|∇u_ε−iA_εu_ε|²|∇u_ε|²−2

A_ε·j_ε+ |A_ε|²|u_ε|² , we deduce thatG_ε(u_ε, A_ε, B_i,ε)is bounded below by

1 2

Bi,ε

|∇u_ε|²+(1− |u_ε|²)² 2ε²

−C

A_ε²_L2(B_i,ε)+ A_εL⁴(B_i,ε)j_εL⁴_(Bi,ε) ,

wherej_ε=(iu_ε,∇u_ε−iA_εu_ε). But from (4.1) we havej_ε→ −∇^⊥h_∗inL^p for anyp <2. It follows easily (using also theW^2,pbound forAε), that

G_ε(u_ε, A_ε, B_i,ε)1 2

Bi,ε

|∇u_ε|²+ 1 2ε²

1− |u_ε|²2

−δ_r,ε,