Lowest Landau level approach in superconductivity for
the Abrikosov lattice close to Hc2
Amandine Aftalion∗ and Sylvia Serfaty†
February 12, 2007

Abstract
We study the Ginzburg-Landau energy for a superconductor submitted to an
applied magnetic field hex just below the “second critical field” Hc2 . When the
Ginzburg-Landau parameter ε is small, we show that the mean energy per unit
volume can be approximated by a reduced energy on a torus. Moreover, we expand
this reduced energy in terms of Hc2 − hex : when this quantity gets small, the problem
amounts to a minimization problem on a finite-dimensional space, equivalent to the“lowest Landau level” in other approaches. This connects the Ginzburg-Landau
energy to the “Abrikosov problem” of locating vortices optimally on a lattice.

1

Introduction

A superconducting material subject to an external magnetic field of intensity hex is described by its wave-function u, a complex-valued order parameter, and its potential-vector
A so that h := curl A is the induced magnetic field in the sample; |u|2 measures the local
density of superconducting electrons in the material. The response of the material varies
according to the value of the external field hex and the value of the Ginzburg-Landau parameter ε (inverse of the usual Ginzburg-Landau parameter κ), which is taken to be small
to model extreme type II superconductors. The state of the superconductor can be studied
through the minimization of the Ginzburg-Landau energyZ1(1 − |u|2 )2JΩ (u, A) =|∇A u|2 + |curl A − hex |2 +
.
(1.1)
2 Ω
2ε2
∗

Université Pierre et Marie Curie-Paris 6, CNRS-UMR 7598, Laboratoire Jacques-Louis Lions, 175 rue
du Chevaleret, Paris, F-75013, France. email: aftalion@ann.jussieu.fr†
Courant Institute of Mathematical Sciences, 251 Mercer st, New York NY 10012, USA. email:serfaty@cims.nyu.edu. Supported by NSF CAREER grant # DMS0239121 and a Sloan Foundation
fellowship

1Here ∇A = ∇ − iA is the covariant gradient and Ω is a two-dimensional bounded andsimply-connected domain, representing the cross-section of the material. We refer to thebook [SS1] for mathematical results that have been obtained, and to [SST, T] for physics
reference. See also [AD].
In this paper, we are interested in the regime where the intensity of the applied magnetic
field hex is of order 1/ε2 , thus we set
hex =

b
.
ε2

(1.2)

For fixed b, if ε is sufficiently small, it is known that A is such that curl A is very close
to hex (see e.g. Fournais-Helffer [FH2]). The behaviour of u, on the contrary, is strongly
dependent on b: when b is too large (b > b0 ' 1.695), the only critical point of the energy
is the normal solution u = 0, and A is such that curl A = hex (see [GP]). The behaviourof u is analyzed by considering the linearized problem around the normal solution u = 0,
curl A = hex . Two eigenvalues emerge: the value of b0 corresponds to an eigenvalue problem
for a half-space, studied by many authors, see e.g. [LP1, LP2, HM], while b = 1 is the
eigenvalue for the full space R2 . The value b = 1, that is hex = 1/ε2 is also called thesecond critical field Hc2 , while b0 /ε2 is the third critical field Hc3 .
For b ∈ (1, b0 ), the minimizer is such that |u| is very small, except on a thin boundary
layer close to the surface of the sample, referred to as surface superconductivity, see [P1,Al2, HP, FH1]. It is this boundary layer which is blown up and approximated by a halfspace problem. The formal computations of Abrikosov [Abr] indicate a bifurcation at b = 1.
As soon as b < 1, superconductivity is no longer present only on the surface of the sample
but also in the bulk. For b close to 1 (but smaller), it is expected that the modulus of theminimizer is small but nonzero and vanishes at isolated points, called vortices, located ona triangular lattice, referred to as the Abrikosov lattice. The rigorous proof of the optimal
location of the vortices is an open problem, related to the minimization of a reduced energy.
This is also called the Abrikosov problem. It is our aim in this paper to derive properties on
the minimal energy and relate it, for b close to 1, to the Abrikosov problem of minimizing
an energy on a torus.
The situation for vortices is quite different from the smaller field case hex  1/ε2 , orb → 0 (refer to [SS1] for this regime) where vortices are local perturbations of the wave
function, while in this paper,
R their size ε is comparable to the interdistance between them.
In particular the term in (1 − |u|2 )2 is no longer negligible.
Let us first be more precise about the space of minimization. The energy JΩ admits
the gauge invariance JΩ (u, A) = JΩ (ueiΦ , A + ∇Φ) for any smooth function Φ, and thephysically relevant quantities, such as the modulus of the wave function or the current are
gauge-invariant, see e.g. [SS1]. It is possible by a gauge change to restrict the minimizationof the energy to the space
H = {(u, A) s.t. u ∈ H 1 (Ω), A ∈ H 1 (Ω)}.
The problem of minimizing JΩ for b close to 1 has already been studied [Al1, SS2]. In2

[SS2], it is proved that the energy is uniformly distributed in the sample, leading to thedefinition of a function f (b):Theorem 1 ([SS2]). Let 0 ≤ b ≤ 1. There exists a continuous increasing function f from[0, 1] to [0, 41 ], such that, as ε → 0, for (u, A) any minimizer of JΩ in H, for all 1 ≥ Rε  ε,and all balls BRε of radius Rε included in Ω, we have
ε2 JBRε (u, A)
ε2 minH JBRε= lim
= f (b)ε→0ε→0|BRε ||BRε |
lim(1.3)

where |.| denotes the two-dimensional Lebesgue measure of a set. Moreover, there exists a
positive constant α such thatα(1 − b)2 ≤ 1 − 4f (b) ≤ (1 − b)2 .

(1.4)

Thus, in (1.3), the mean energy of the minimizer in each ball BRε is very close to
the minimal energy on the ball, provided that Rε is much larger than ε, that is there
should be a large number of vortices in the ball BRε . The value of this minimal energy
is independent of the ball, hence the equidistribution of the energy. The lower bound for1 − 4f (b) is obtained by constructing a test-function which is periodic with respect to somesquare lattice. Estimates on the convergenceof the average of the L2 and L4 norms of u,√establishing that they decrease like 1 − b as b → 1, thus showing an average decrease ofbulk-superconductivity, are also proved in [SS2].
In this paper, we want to better characterize f (b) in the limit when b tends to 1, andrelate it to a minimization problem on a torus. This relies on the study of the energy
on a reduced space, which is the first eigenspace for a magnetic operator, also called
lowest Landau level (LLL). This functional space is comprised of holomorphic functions
multiplied by a particular Gaussian. The lowest Landau level was used in [AB, ABN] toanalyze the vortex lattice in Bose-Einstein condensates. Our goal here is to show that asimilar framework applies to the Ginzburg-Landau energy. Because our problem is posedon a torus, the space is finite dimensional, and the dimension is related to the number ofzeroes, while in the case of BEC, the LLL is of infinite dimension. The results that weshow are related to those obtained by Almog in [Al1] for rectangles. We hope to provide
here a short and simple presentation of these results and to bridge between the works of[Al1] and [ABN].
In order to present our result, we rescale the functions u and A, around some origin inΩ, as follows:
 
εx
(1.5)
u→u √
b
 
εx
A → √εb A √(1.6)
band the energy JΩ turns into GΩ√b/ε where, for any domain D, we denoteZ1b(1 − |u|2 )2
GD (u, A) =|∇A u|2 + 2 |curl A − 1|2 +
.
2 D
ε
2b
3

(1.7)

Theorem 1 from [SS2] implies that if (u, A) is a minimizer of GΩ√b/ε , then for any Rε with
1/ε ≥ Rε  1, as ε tends to 0, we have
GBRε (u, A)
minH GBRε
f (b)
=
lim
=
lim
.
ε→0, Rε →∞ε→0, Rε →∞b|BRε ||BRε |(1.8)

We now want to reduce the Ginzburg-Landau energy to a periodic setting, in a sense that
we will make precise below. As pointed out in [SS2], in the relation (1.8), balls can be
replaced without loss of generality by squares, or other shapes like parallelograms. In what
follows, instead of considering balls, we will consider parallelograms, that is unit cells forperiodic problems: for R ∈ R∗+ , and τ = τ1 + iτ2 ∈ C\R, we set Kτ,R to denote the unit
parallelogram of the lattice
Lτ,R = R(Z ⊕ τ Z).
We will consider τ as fixed, and let R → ∞ in order to study large tori with fixed“shape”. Our main result is going to reduce the calculation of f (b), as b tends to 1, givenby (1.8), to a minimization problem for (u, A) where u lies in a finite-dimensional subspaceof “periodic” functions and A = A0 is given by
1
A0 = (−y, x) in Kτ,R .
2

(1.9)

For that purpose, we define the energy Gτ,R (u) = GKτ,R (u, A0 ), that isZ
1(1 − |u|2 )22
Gτ,R (u) =|∇A0 u| +.
2 Kτ,R2b(1.10)Let us point out that Gτ,R (u) no longer depends on ε, except through the fact that R
needs to be less than 1/ε. The next step is to set a framework for “periodic” boundary
conditions, in the sense that the gauge-invariant quantities are periodic, namely |u| and|∇A0 u| should be periodic, but not u itself. More precisely, u and ∇A0 u taken at the point(z + nR + mRτ ) for two integers n and m, have to be gauge equivalent to u and ∇A0 u[BGT]. This yields the definition of the following space for u (we will alternatively use
complex coordinates and coordinates in R2 ) :
iπN y

iπN

Eτ,R = {u ∈ H 1 (Kτ,R , C), s.t. u(z + R) = e Rτ2 u(z), u(z + Rτ ) = e Rτ2(τ1 y−τ2 x)u(z)}(1.11)
together with the quantization condition: |Kτ,R |/(2π) is an integer, that is there exists
N ∈ N∗ , such thatR2 τ2 = 2πN.
(1.12)
This procedure is performed on the full Ginzburg-Landau energy in [BGT, Du, Ay] andwe refer to [NV] and the references therein for the quantification of the torus. In fact, A0
itself can be extended by periodicity (see [BGT]), but we do not enter into the details here.
One may also check that u ∈ Eτ,R implies that the total degree of zeroes of u in the torus
is equal to N , also proportional to the flux of the magnetic field.
Our main result is the following.
4

Theorem 2. For every τ ∈ C \ R, let Kτ,R denote the unit parallelogram of the lattice
Lτ,R = R(Z ⊕ τ Z) and GKτ,R the Ginzburg-Landau energy (1.7) in this domain. Let f (b)be the function defined in Theorem 1. Then, for all b ∈ (0, 1), for all τ ∈ C \ R, we havef (b) =

minH GKτ,R
minEτ,R Gτ,R=lim
bε→0, R→∞, R<1/ε
R→∞, |Kτ,R |/(2π)∈N|Kτ,R |
|Kτ,R |
limb

(1.13)

where Gτ,R is defined in (1.10) and Eτ,R in (1.11). Moreover, we have the following expansion for f (b) as b approaches 1:
4f (b) − 1= lim min Fτ,R (v) ∀τ ∈ C \ R
R→∞ v∈Lτ,R
b→1 (1 − b)2
lim

where

1
Fτ,R (v) =|Kτ,R |

Z

|v|4 − 2|v|2(1.14)(1.15)

Kτ,R

and Lτ,R is a finite-dimensional subspace of Eτ,R defined by functions such that DA0 v :=(∂1 + i∂2 + 21 (x + iy))v = 0.
In (1.13), the first equality is a consequence of the definition of f (b) and follows from
(1.8), while the second equality is a reduction to an energy which no longer depends on ε.
The information provided by (1.13) allows us to reduce the minimization to a subspace ofEτ,R and thus obtain in (1.14), a more precise expansion of f (b) as b → 1 than the one of(1.4). Let us emphasize that the limit on the r.h.s of (1.14) is thus independent of τ .
The space Lτ,R can be viewed as the finite dimensional analogue of the “lowest Landau level” in [ABN]. It corresponds to the eigenspace for the smallest eigenvalue of the
Schrödinger operator with magnetic field −∇2A0 , as we will see later. The functions inthis space are explicit: they can be described using the Jacobi Theta function and arecompletely determined by their N zeroes on each lattice cell, N being defined by thequantization condition (1.12). The dimension of Lτ,R is also the integer N .
Let us give some ideas about the proof of our theorem. The first step consists in
reducing the minimization of GKτ,R over H, to a minimization for u ∈ Eτ,R , and A = A0 .
This reduction uses estimates on A for a minimizer and tricks introduced in [SS2]. This
proves (1.13). The error is unfortunately like O(R), which does not allow to carry out
estimates on how close the minimizers of these two problems are.
The second step consists in studying the minimization of Gτ,R when u ∈ Eτ,R : we willshow that if u ∈ Eτ,R , we have the crucial relation (see Corollary 3.1)Z1b
(1 − b)2u
Gτ,R (u)= +|DA0 u|2 +Fτ,R ( √),(1.16)
b|Kτ,R |
4 2|Kτ,R | Kτ,R
4
1−bwhere Fτ,R is defined in (1.15) and DA0 u is defined at the end of Theorem 2. When b tends
to 1, the last two terms are of different order and this forces, for an energy minimizer,
DA0 u to tend to 0. This leads us to analyze in details the space of functions u such that5

DA0 u = 0. It is the first eigenspace for −∇2A0 in Eτ,R , which we called Lτ,R . Thus, we may
project orthogonally a minimizer u of Gτ,R onto this first eigenspace Lτ,R and prove thatits projection uΠ is an almost minimizer of the energy Fτ,R restricted to Lτ,R . We also getthat u − uΠ is small in the following sense:√
Proposition 1.1. Let u be a minimizer of Gτ,R in Eτ,R and let v = u/ 1 − b. Let vΠ be
the L2 orthogonal projection of v onto Lτ,R and w = v − vΠ . Then for β < 1, there exists
a constant Cβ,τ such that, for R large,
kvkC 0,β (Kτ,R ) ≤ Cβ,τ R√
kwkC 0,β (Kτ,R ) ≤ Cβ,τ R 1 − b.
Moreover,√Fτ,R (vΠ ) = min Fτ,R + O(|Kτ,R | 1 − b).
Lτ,R(1.17)(1.18)(1.19)

If R is large, but b is sufficiently close to 1, the last term in the estimate is a small
error. The proof of this proposition relies on estimates of Lu and Pan [LP2], also provedin [AB], which are analogues of elliptic estimates applied to the operator −∇2A0 .
Let us finish with a discussion on related works in the literature.
Almog [Al1] studies a similar problem, but instead of considering general tori, he works
on rectangles only (that is τ = li for some l and the domain is Kli,R ), with some conditions
on their size related to the way b → 1. Here, we work on all tori shapes, we do nothave restrictions on the rate of convergence of b → 1. His proof is two-fold: on the one
hand, he proves that if u is a minimizer of GD , for an arbitrary domain D, then its L∞
norm is small in terms of b − 1. It is this estimate which allows him to truncate u on
the boundary of subrectangles to 0, so that the truncated function becomes an element of
Eli,R and the energy he considers is GKli,R . His constraints on the size of the rectangles
provide an estimate on how close the initial minimizer is to that on Eli,R . The core of his
results is an equivalent of Proposition 1.1, that is the projections onto the lowest Landaulevel, that relies on rather technical computations instead of elliptic estimates as we do.
But essentially the results are similar in nature. The price that we pay for our lack ofconstraints is that we do not have an estimate on the closeness of u to the minimizer ofFτ,R .
The minimization of Fτ,R over Lτ,R is another formulation of the Abrikosov problem:
it is expected that, as R tends to infinity, the zeroes of the minimizer are almost locatedon a triangular lattice whose unit cell has volume 2π. The main difficult open question
is to prove this statement. It would be already satisfactory (but difficult) to prove that if
τ = j = e2iπ/3 (that is the large torus is already triangular), then the minimum of Fj,R on
Lj,R is such that its zeroes are exactly on a triangular lattice of volume 2π. In other words,
we expect√ the modulus of the solution to be periodic with respect to the small lattice Lj,r0
with r02 3 = 4π, additionally to the initial one. F. Nier [Ni] was able to prove recently that
this solution with zeroes on a triangular lattice provides a local minimum of the energy
Fj,R . The other fact which was previously known is that if the zeroes of u are already6

assumed to be on a lattice of shape τ then the one minimizing Fτ,R is the triangular one,as proved in [NV, ABN].
The work in [ABN] deals with rotating Bose-Einstein condensates, described by the
Gross-Pitaevskii energy, when the rotational velocity Ω tends to the transverse trapping
velocity that is set to 1. This is the equivalent of the limit b → 1, with the role of theapplied magnetic field played by the rotational velocity. When Ω tends to 1, the minimizerof the Gross-Pitaevskii energy can be restricted to the lowest Landau level, that is the first
eigenspace for −∇2A0 in R2 (instead of a bounded domain) as done in [AB]. The proof relieson the projection onto the LLL and elliptic estimates, as we do here for Proposition 1.1.
The difference between our energy and the one for condensates is that the latter is posed in
the whole space R2 but a trapping potential makes the problem compact: the mean valueof |u|2 on several cells of vortices, instead of being almost constant, is close, on a large
scale, to an inverted parabola. In particular, there is no invariance of the mean energy
per unit volume in the domain, but on the contrary dependence on it. Nevertheless, atthe limit b = 1 or Ω = 1, the two limiting problems, for condensates and superconductors,
given by the minimization of Fτ,R should be the same, though the proof is still open in thecase of condensates [Aft, ABN].
The paper is organized as follows: first we prove (1.13) in Section 2, then we study theoperator DA0 , its eigenspace and its spectrum in Section 3, and finally, in Section 4, we
prove Proposition 1.1 and the rest of Theorem 2.
Open problems• The first main open problem is the one mentioned above, which consists in showing
that, as R tends infinity, the minimizer of Fτ,R in Lτ,R has zeroes that form an almost
triangular lattice, and an exact triangular lattice when the torus is already triangular,
i.e. τ = j.
• Does the fact that Fτ,R (vΠ ) ∼ min Fτ,R imply that vΠ is close in some sense to theset of minimizers of Fτ,R in Lτ,R ? In what norm is there convergence? In particular
do the zeroes of vΠ converge?
• Link more precisely the minimizers of GKτ,R to the the minimizers restricted to Eτ,R ,as done in [Al1] but with sharper estimates valid in a wider regime.
∞• This would probably require some√ better L estimates on minimizers u of GΩ : doestimates of the sort |u(x)| ≤ Cδ 1 − b when dist (x, ∂GΩ ) ≥ δ hold?

• Can anything else be said when b is fixed and does not tend to 1 and can f (b) thenbe better characterized? In particular, can we estimate the number of Landau levels
filled according to the distance of b to 1?

2

Reduction to a torus

In this section b is fixed in (0, 1). We are going to prove the following:7

Proposition 2.1. Let us recall that A0 is given by (1.9), i.e. a vector field such that
div A0 ≡ 0 and curl A0 ≡ 1. The function f (b) defined in Theorem 1 has the property(1.13).
This way, studying f (b) reduces to minimizing this simpler energy G on large tori (ofany shape). We first show intermediate lemmas. The first one recalls some useful a priori
estimates (see [FH2] Proposition 4.4):
Lemma 2.1. Let D be a bounded domain, (u, A) be a minimizer of GD , then |u| ≤ 1 andthere exists a constant C independent of ε and b such thatCεkcurl A − 1kC 1 (D) ≤ √ ,bCk∇A ukL∞ (D) ≤ √ .
bFollowing exactly the method of [SS2], Lemma 3.1, we can show the following result:Lemma 2.2. Let f (b) be given by Theorem 1, and (u, A) a minimizer of GΩ √b . Then for√εall R such that 1  R < 1/ε, and Kτ,R ⊂ Ω εb ,minH GKτ,R (u, A)
GKτ,R (u, A)f (b)=+ o(1) =+ o(1)|Kτ,R |
|Kτ,R |bas ε → 0,

so that (u, A) is an “almost minimizer” of GKτ,R . Moreover, as ε → 0,
Z1
|u|4 −→ 1 − 4f (b)|Kτ,R | Kτ,R
1 − 4f (b)1− o(1) ≤1−b|Kτ,R |

Z

|u|2 ≤

p
1 − 4f (b) + o(1).

(2.1)

(2.2)

Kτ,R

This amounts to writing the result of Theorem 1, but in parallelograms instead of balls.
The results corresponding to (2.1) and (2.2) are also proved in [SS2].
In the proof of Proposition 2.1, we can thus restrict, in a first step, to a minimizer ofGKτ,R . We next reduce to the case A = A0 .
Lemma 2.3. Let (v, B) be a minimizer of GKτ,R , then (v, B) is gauge-equivalent to some(u, A) such thatεRkA − A0 kL∞ (Kτ,R ) ≤ Cτ √bwhere Cτ is a constant depending only on τ .
Proof. Let us solve∆φ = −div B
in Kτ,R∂φ= A0 · ν − B · ν on ∂Kτ,R .
∂ν

8Then we take A = B + ∇φ. This way, A satisfies div A = 0 = div A0 and (A − A0 ) · ν = 0
on ∂Ω. Thus we may write A − A0 = ∇⊥ ξ with ξ = 0 on ∂Kτ,R . Moreover,∆ξ = curl A − curl A0 = curl B − 1CεIn view of Lemma 2.1, we have k∆ξkL∞ (Kτ,R ) ≤ √. By elliptic regularity, we deducebCτ Rεk∇ξkL∞ (Kτ,R ) ≤ √band the result follows.
Thus, by a gauge-transformation, we may reduce to such a (u, A) and are going toprove that its energy is close to GKτ,R (u, A0 ) = Gτ,R (u), which no longer depends on ε.
Lemma 2.4. We haveminH 1 Gτ,R.
R→∞
|Kτ,R |

f (b) = lim b
Proof. In view of Lemma 2.3, we haveZ
Z
2|∇A u| =(2.3)|∇A0 u|2 + O(ε2 R4 /b)

Kτ,R

Kτ,R

If follows that
1
GKτ,R (u, A) ≥2

Z|∇A u|2 +Kτ,R(1 − |u|2 )2= GKτ,R (u, A0 ) + O(ε2 R4 /b).
2b

Therefore, by minimality of (u, A), minH GKτ,R ≥ minH 1 GKτ,R (., A0 ) + O(ε2 R4 /b). Also
min GKτ,R ≤ min GKτ,R (., A0 ). We deduce that
Gτ,R + O(ε2 R4 /b)
min GKτ,R = min1HH

and that (u, A) is an almost minimizer of Gτ,R in that sense. In view of Lemma 2.2, wededuce thatminH 1 Gτ,R.
f (b) =limb
ε→0,1R1/ε|Kτ,R |
But since Gτ,R does not depend on ε, we may suppress the constraints on R and just replacethem with R → ∞.
Proof of Proposition 2.1. Let us consider χ a cut-off function which is identically equal to
1 in Kτ,R−2 , to 0 in Kτ,R \Kτ,R−1 and such that |∇χ| ≤ C. A similar proof to that of [SS2],
Lemma 3.1, allows to check that
Gτ,R (u) = GKτ,R (u, A0 ) = GKτ,R (χu, A0 ) + O(R) = Gτ,R (χu) + O(R).
9It follows that
minH01 (Kτ,R ) Gτ,R
minH 1 (Kτ,R ) Gτ,R∼
|Kτ,R |
|Kτ,R |

as R → ∞.

(2.4)

But H01 can be viewed as a subspace of Eτ,R (conditions (1.11) are satisfied) thus
minH01 (Kτ,R ) Gτ,R
minH 1 (Kτ,R ) Gτ,R
minEτ,R Gτ,R≥≥
|Kτ,R |
|Kτ,R |
|Kτ,R |
and from (2.4) and (2.3), it follows that Proposition 2.1 holds.

3

The operator DA0 and its first eigenspace

We now study the operator −∇2A0 over the space Eτ,R . For the sake of completeness, we
include the proof of results which are known in other contexts. We will need the Theta
function, naturally associated with the lattice Z ⊕ τ Z, defined by
+∞

1 X
2
(−1)n eiπτ (n+1/2) e(2n+1)πiv ,Θτ (v) =i n=−∞v ∈ C.

(3.1)

We refer the reader to [Cha] for details. The Theta function vanishes exactly on the latticeZ ⊕ τ Z and satisfiesΘτ (v) = −Θτ (−v),Θτ (v + 1) = −Θτ (v),Θτ (v + τ ) = −e−iπτ e−2πiv Θτ (v).

(3.2)

Proposition 3.1. The operator −∇2A0 is self-adjoint positive over the subspace Eτ,R . Its
lowest eigenvalue is equal to 1, and the associated eigenspace, called Lτ,R , has complex|Kτ,R |
dimension N = 2π
and is described by the functions22

u(z) = λe−|z| /4 ez /4−iπN z/R

N
Y
k=1

Θτ (

z − zk)R(3.3)

where λ is any complex number, zk are N points in Kτ,R , satisfying the constraintNX
k=1

zk = RN

1+τ
2

mod Lτ,R .

The second eigenvalue of −∇2A0 is greater than 3.
Let us point out that this space is the finite dimensional equivalent of the lowest Landaulevel as used e.g. in [GJ, ABN].

10

Corollary 3.1. Let u ∈ Eτ,R and let DA0 = ∂1 + i∂2 + 21 (x + iy), thenZ1(1 − |u|2 )2Gτ,R (u) = GKτ,R (u, A0 ) =|u|2 + |DA0 u|2 +2 Kτ,R2bZu
1|Kτ,R | 1 (1 − b)22+
Fτ,R ( √) (3.4).
=|DA0 u| +2 Kτ,Rb4
4
1−b
If additionally u ∈ Lτ,R , then DA0 u = 0 andZ
Z
1|Kτ,R |
1
(1 − |u|2 )22
Gτ,R (u) =
=
+
|u| +
|u|4 − 2(1 − b)|u|2 .
2 Kτ,R2b4b4b Kτ,Ru|Kτ,R |
2
1 + (1 − b) Fτ,R ( √) . (3.5)=4b1−bProof of Proposition 3.1. We introduce the operator
1z
DA0 = ∂1 + i∂2 + (x + iy) = 2∂z +
2
2

(3.6)

in complex coordinates notation, where ∂z = 12 (∂x + i∂y ). This operator corresponds tothe creation operator in second quantification and is naturally associated with −∇2A0 as
we will see below, cf. [GJ]. In [SS2], it was introduced for a general field A following theBogomolnyi trick for self-duality.
∗
The adjoint of DA0 with respect to the scalar product in L2 is DA= −2∂z + 12 z.
0Straightforward computations yield
1∗
DA0 DA= −4∂zz − z∂z + z∂z + |z|2 + 1,
04
1∗
DA0 DA0 = −4∂zz − z∂z + z∂z + |z|2 − 1,4hence we deduce the canonical relations:∗∗
DA0 DA+ DA
DA0 = −2∆ + 21 |z|2 − 2(iy∂y + ix∂y ) = −2∇2A0 ,00∗∗
DA0 DA− DA
DA0 = 2I.
0
0

(3.7)
(3.8)

∗
In particular, −∇2A0 = DA
DA0 + I. One may check that the periodicity conditions (1.11)
0ensure that ∇A0 u is really periodic with respect to Lτ,R . Thus the previous relation implies(after an integration by parts which yields no boundary term)
Z
Z
2|∇A0 u| =|DA0 u|2 + |u|2 .
(3.9)
Kτ,R

Kτ,R

Therefore, the lowest eigenvalue of −∇2A0 in Eτ,R , characterized via the Rayleigh quotientR|∇A0 u|2
Kτ,RRminu∈Eτ,R
|u|2
Kτ,R
11

is equal to 1 and the eigenspace Lτ,R identifies with the solutions of DA0 u = 0 in Eτ,R . In
2
particular, returning to the form (3.6), we see that it is made up of functions u = f (z)e−|z| /4
with f holomorphic. We can be more specific about the structure of f as we will see below.
⊥
2
Let us now consider u ∈ L⊥
τ,R = (KerDA0 ) (where the orthogonal is taken for the L∗scalar product on Eτ,R ). Clearly u belongs to the closure of the range of DA. We may0∗∗
thus assume u = DA0 v. In (3.9), we replace DA0 u by DA0 DA0 v and use (3.8) to find∗∗h−∇2A0 u, ui = h(I + DA
DA0 )u, ui = kuk2 + h2v + DA
DA0 v, DA0 ui
0
0
∗
∗
∗
2
∗
DA0 vk2 .
vi = 3kuk2 + kDA= kuk + 2hDA0 v, ui + hDA0 DA0 v, DA0 DA
00

We conclude that, for u ∈ L⊥τ,R ,h−∇2A0 u, ui ≥ 3kuk2 ,and this proves the last assertion of the proposition.
Let us now characterize better the eigenspace Lτ,R . We have seen that it is made up of2functions u = f (z)e−|z| /4 with f holomorphic, satisfying the periodicity condition (1.11),
which implies in particular that |u| is periodic. Since f is holomorphic, it has a finite
number of zeroes in Kτ,R . Let us call N0 this number and call zk the zeroes. We are going
to prove that (3.3) holds. This is a consequence of Hadamard’s
theorem. The
QNfactorizationz−zk
0
proof is similar to that in [ABN]. Since f and the function k=1 Θτ ( R ) have the samezeroes, their quotient is an analytic function which does not vanish in Kτ,R . Hence one canfind an analytic function φ such thatf (z) = eφ(z)

N0
Y

Θτ (

k=1
|z|2
− 4

The Lτ,R -periodicity of |u(z)| = ez − zk).
R|f (z)| implies the upper bound∀z ∈ C, eRe(φ(z))−

|z|2
4

N0
Y

Θτ (

k=1

z − zk) ≤ C1 .
R

Therefore, when the periodicity cell Q is chosen such that Lτ,R ∩ ∂Q = ∅, there exists a
constant C > 0 such that ∀z ∈ ∂Q + Lτ,R , Re (φ(z)) ≤ C |z|2 + ln(CC1 ) . Since Re (φ(z))
is a harmonic function, the maximum principle yields ∀z ∈ C, Re (φ(z)) ≤ C 0 (|z|2 + 1)
for some constant C 0 > 0. The Hadamard factorization theorem, see [Boa], then implies
that φ is a harmonic polynomial of degree 2, that is there exists (δ, η, β) ∈ C3 such that
δ+ηz+βz 2

f (z) = e

N0
Y
k=1

Θτ (

z − zk).
R

Conditions (1.11) and the properties on Theta (3.2) imply that
2

iπN y

2

eηR+βR −R /4 e2βRz−Rx/2 (−1)N0 = e Rτ2
2 2

2

2

(−1)N0 eηRτ +R τ /4−R |τ | /4+Rτ z/2−Rxτ1 /2−Ryτ2 −iπτ N0 −2iπN0 z/R+2iπ
12P

k zk /RiπN= e Rτ2(τ1 y−τ2 x)

The first equality yields
1
β= ,
4

R2 τ2 = 2πN,

η = −i(

πN
2kπ
+
)
R
R

(3.10)

for some integer k, while it follows from the second one that
N0 = N,andN
X

zk = RN

k=1

η = −iπN,
R

1+τ
mod Lτ,R .
2(3.11)(3.12)

This yields (3.3).
Proof of Corollary 3.1. Equality (3.4) is a direct consequence of (3.9) while (3.5) follows
from the definition of Lτ,R .

4

The behavior as b → 1

The aim of this section is to prove Proposition 1.1.

4.1

Upper bound for the energy

Lemma 4.1. We havemin Fτ,R ≤Lτ,R−1
2πγ(τ )

where γ(τ ) denotes the Abrikosov parameter of the lattice Lτ,ν defined by ν 2 = 2π/τ2 and
R
|v|4
Kτ,ν
γ(τ ) =
min
(4.1)

2 .
Lτ,ν ,ν 2 τ2 =2π R
2
|v|
Kτ,ν
The computation of γ(τ ) has been made in [ABN]: it is equal to the Abrikosov parameter and related to a series for Husimi functions for which Voros-Nonenmacher [NV] studied
the minimizer in terms of τ . It follows from these two papers that minτ γ is achieved when
τ = j, that is for the triangular lattice. An approximate value for 1/γ(j) is 0.86.
Proof. One specific choice of test function is to fix a lattice with a unit cell having a
volume 2π, that is Lτ,ν = ν(Z ⊕ τ Z) with ν 2 τ2 = 2π. Then as seen in Proposition 3.1, thecorresponding first eigenspace Lτ,ν is one-dimensional and spanned by uτ , which is multipleof Θτ ((z − (1 + τ )/2)/ν), as given by (3.3).

13

in Lτ,ν , we have v = λuτ and Fτ,ν (λuτ ) is minimal for λ2 =R Now if2 v isR any function
4( Kτ,ν |uτ | )/( Kτ,ν |uτ | ). This implies in particular that
2
2
|u
|
τ
K
1
R τ,ν
Fτ,ν (λuτ ) = −.
|Kτ,ν | Kτ,ν |uτ |4
R

Since γ(τ ) is invariant when the function v varies in Lτ,ν , we havemin Fτ,ν =
Lτ,ν

−1
.
2πγ(τ )

Now if R2 τ2 /2π is an integer, (1.12) is satisfied and a specific test function in Eτ,R
can be constructed by simply extending the previous λuτ by “periodicity” i.e. takingλuτ (z + ν(n + mτ )), for n, m integers. The periodicity of |uτ | implies that
Fτ,R (λuτ ) = Fτ,ν (λuτ ) = −1,2πγ(τ )

(4.2)

hence the result.
Remark 4.1. In order to try to minimize Fτ,R , we have taken test-configurations which
are periodic with respect to smaller lattices (of the same shape). However, we do not believethat the estimate above is optimal, unless τ = j. The natural conjecture is rather that
lim inf min Fτ,R = −R→∞ Lτ,R

1.
2πγ(j)

This corresponds to showing that in large tori, the optimal location of vortices is still on atriangular lattice, except for negligible boundary effects. Observe that the inequality
lim inf min Fτ,R ≤ −R→∞ Lτ,R

1
2πγ(j)is already not so easy to obtain.
We now easily deduce
Proposition 4.1. For any τ and any b ∈ (0, 1), we have




1
1
1
Gτ,R
2
2
min
≤
1 + (1 − b) min Fτ,R ≤
1 − (1 − b).
Eτ,R |Kτ,R |
Lτ,R
4b
4b
2πγ(τ )

(4.3)

Corollary 4.1. For any b ∈ (0, 1), we have
1 (1 − b)2
1 (1 − b)2 1
f (b) ≤ +inf lim inf min Fτ,R (v) ≤ −.
τ ∈C\R R→∞ v∈Lτ,R44442πγ(j)
14(4.4)

Proof of the proposition. It suffices√ to build a test-configuration: let us take v0 a minimizerof Fτ,R in Lτ,R , and take u0 = v0 1 − b. Since u0 ∈ Lτ,R , we use (3.5) and find|Kτ,R |
1 + (1 − b)2 Fτ,R (v0 )4b |Kτ,R |
2=
1 + (1 − b) min Fτ,R .
Lτ,R4b

Gτ,R (u0 ) =The second inequality follows from Lemma 4.1.
Proof of the corollary. In view of the characterisation of f by (1.13), we deducef (b) ≤

1 (1 − b)2+
lim inf min Fτ,R .
R→∞ Lτ,R
4
4

(4.5)

Since this is true for every τ , minimizing the right-hand side over τ and using Lemma 4.1
gives the result (since we know that γ is minimized for τ = j).
Remark 4.2. In [SS2] an upper bound for f of this sort was given by using configurations
periodic with respect to square lattices. We see here that this upper bound can be improvedby taking triangular lattices, but it is still open whether there is equality above, i.e. whether
triangular-periodic configurations are energetically optimal.

4.2

Lower bound as b → 1

Proposition 4.2. Let u be a minimizer of Gτ,R in Eτ,R . Then as R → ∞,1
1
(1 − b)2Gτ,R (u)≥+min Fτ,R + O(|Kτ,R |(1 − b) 2 ) .
Lτ,R|Kτ,R |
4b
4b

(4.6)

The proof consists in projecting the minimizer u onto the space Lτ,R and checking,using elliptic estimates, that u and its projection uΠ are close. Finally, we prove that
Fτ,R (uΠ ) is close to minLτ,R Fτ,R . Unfortunately, we are not able to prove that uΠ is close
to a minimizer of Fτ,R (see open problems). The proof is similar to that of [AB] for the
reduction of the minimization of the Gross-Pitaevskii energy to the lowest Landau level.
In [Al1], the projection onto the lowest Landau level is also used, but with different elliptic
estimates.
Proof. - Step 1: upper bounds. From Proposition 4.1 we have1
1 − (1 − b)2 /(2πγ(τ ))Gτ,R (u) ≤|Kτ,R |
4b
1
for any R such that R2 τ2 /(2π) ∈ N. Using (3.4), writing α = 2πγ(τ, and rearranging the)terms, we findZ1(|u|2 − (1 − b))2
(1 − α)(1 − b)22|DA0 u| +≤.
(4.7)|Kτ,R | Kτ,R
2b
2b

15

√
We let v = u/ 1 − b and deduce thatZ
1(1 − |v|2 )2 ≤ (1 − α)|Kτ,R | Kτ,R
1
|Kτ,R |Z

|DA0 v|2 ≤Kτ,R(1 − α)(1 − b)
.
2b

(4.8)

(4.9)

It follows that
Z

!1/2
|v|4

≤ k|v|2 − 1kL2 + k1kL2 ≤ 2|Kτ,R |1/2

(4.10)

Kτ,R

and also using the Cauchy-Schwarz inequality, thatZ
|v|2 ≤ 2|Kτ,R |.

(4.11)

Kτ,R- Step 2. The function v belongs to Eτ,R , we may project it orthogonally (for L2 (Kτ,R ))onto Lτ,R . Let vΠ denote its projection, and let us writev = vΠ + w(4.12)

Since w ∈ L⊥
τ,R , we have from Proposition 3.1 and (3.9) that
Z
Z
Z
2
2
3
|w| ≤
|∇A0 w| =|DA0 w|2 + |w|2
Kτ,R

Kτ,R

and thus

Z

Kτ,R

2

Z

|DA0 w|2

|w| ≤

2
Kτ,R

Kτ,R

But since vΠ ∈ Lτ,R and w ∈ L⊥
τ,R we have DA0 v = DA0 w and thus, in view of (4.9), we get
1|Kτ,R |

Z

|w|2 ≤

Kτ,R(1 − α)(1 − b).
4b(4.13)- Step 3: We use the estimates of Lu-Pan [LP2], or rather the adaptation proved in[AB]: for any (f, A), there exists a constant C such thatk∇A f kH 1 (Br ) ≤ C(k∆Ak∞ , kcurl Ak∞ ) k∇2A f kL2 (B2r ) + kf kL2 (B2r )(4.14)
where C is independent of r.
We deduce the following estimates16√Lemma 4.2. Let u be a minimizer of Gτ,R in Eτ,R , v = u/ 1 − b, and let vΠ its orthogonal
projection onto Lτ,R and w = v − vΠ . Then for β < 1 there exists a constant Cβ,τ suchthat, for R large enough,kvkC 0,β (Kτ,R ) ≤ Cβ,τ R√
kwkC 0,β (Kτ,R ) ≤ Cβ,τ R 1 − b.

(4.15)(4.16)

Proof. For R sufficiently large, R > 2 and Rτ2 > 2, so that for any x ∈ Kτ,R , B(x, 2) ⊂ MR ,defined by MR = Kτ,R (1 + mR + nRτ ), with m, n ∈ {−1, 0, 1}. We use the fact that sinceu is a minimizer of GKτ,R (., A0 ), v solves the Euler-Lagrange equationv(4.17)−∇2A0 v = (1 − (1 − b)|v|2 )bhence estimate (4.14) yields that for any x ∈ Kτ,R ,k∇A0 vkH 1 (B(x,1)) ≤ CkvkL2 (B(x,2)) + C(1 − b)kv|v|2 kL2 (B(x,2)) .

(4.18)

From (4.17)√ it is also easily shown (using the maximum principle) that |u| ≤ 1 hence|v| ≤ 1/ 1 − b. Moreover, since v ∈ Eτ,R , |v| is periodic, hence kvkL2 (MR ) ≤ 9kvkL2 (Kτ,R ) .
Using (4.11) we find kvkL2 (B(x,2)) ≤ CR. Similarly√(1 − b)kv|v|2 kL2 (B(x,2)) ≤ (1 − b)kvkL∞ k|v|2 kL2 (B(x,2) ≤ C 1 − bR
by (4.10). Inserting into (4.18), we conclude that k∇A0 vkH 1 (B1 (x)) ≤ CR, where C does
not depend on x in Kτ,R . The Sobolev embedding yields (4.15).
For the estimate on w, we use (4.17) and recall that −∇2A0 vΠ = vΠ , hence
1
1
1v−∇2A0 w = (1 − (1 − b)|v|2 ) − vΠ = w + vΠ (1 − b) − (1 − b)v|v|2 .
bbbb
Using (4.14), we find as above for all x ∈ Kτ,R ,√k∇A0 wkH 1 (B1 (x)) ≤ C(kwkL2 (Kτ,R ) + (1 − b)kvΠ kL2 (Kτ,R ) + 1 − bk|v|2 kL2 (Kτ,R )) .
√Estimates (4.11)-(4.10)-(4.13) yield k∇A0 wkH 1 (B1 (x)) ≤ CR 1 − b, where C does not depend on x in Kτ,R . The Sobolev embedding yields (4.16).
1√
1
2∞ ≤ C|Kτ,R | 2 .
It follows that kwkL∞ ≤
C|K
|
1
−
b
and
kv
k
τ,R
Π
L
R
- Step 4: We estimate Kτ,R |v|4 − 2|v|2 . By definition of vΠ and w, we have
Z
Z
Z
2
2
2
−2
|v| = −2|vΠ | + |w| ≥ −2|vΠ |2 − C|Kτ,R |(1 − b)
Kτ,R

Kτ,R

Kτ,R

where the last inequality comes from (4.13). Moreover,
Z
Z
Z

4
4
2
2
|v| ≥ |vΠ | + 4(vΠ · w)(|w| + |vΠ | ) ≥ |vΠ |4 − 2 kvΠ k2L∞ + kwk2L∞ kvΠ kL2 kwkL2
Z√≥ |vΠ |4 − C 1 − b|Kτ,R |2 .
17

We conclude that
Z

4

2

Z

|v| − 2|v| ≥Kτ,R√|vΠ |4 − 2|vΠ |2 − C|Kτ,R |2 1 − b.

Kτ,RInserting into (3.4), we findb1(1 − b)2(1 − b)2Gτ,R (u) − ≥Fτ,R (v) ≥Fτ,R (vΠ ) + O(|Kτ,R |(1 − b)5/2 ))|Kτ,R |
4
4
4

(4.19)and (4.6) follows.
We deduce immediately that if b tends to 1 and R → ∞ in such a way that0, then
Gτ,R (u)1 (1 − b)2b≥+min Fτ,R
lim
sup
√
Lτ,R
4
4
R→∞, 1−b|Kτ,R |→0 |Kτ,R |√1 − b|Kτ,R | →But by the characterisation (1.13), the left-hand side is equivalent as b → 1 to f (b)hence we deduce
4f (b) − 1
lim inf≥ lim sup min Fτ,R .
b→1(1 − b)2
R→∞ Lτ,R
Comparing with (4.5), we deduce that the liminf and limsup are equal and the second
assertion of Theorem 2 follows. The first assertion is also easily true.
Now if u minimizes Gτ,R , and vΠ is the projection on Lτ,R as above, comparing (4.3) to(4.19) we deduce also that√
Fτ,R (vΠ ) = min Fτ,R + O(|Kτ,R | 1 − b).
Lτ,R

With the content of Lemma 4.2, Proposition 1.1 is proved.

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20