On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

0.37 MB·

by  Li Chen

On Abrikosov Lattice Solutions of the Ginzburg-Landau
Equations
7
1 Li Chen, Panayotis Smyrnelis, Israel Michael Sigal
∗ † ‡
0
2
January 11, 2017

n
a
J

1
2 Abstract

] WeproveexistenceofAbrikosovvortexlatticesolutionsoftheGinzburg-Landau
h
equations of superconductivity, with multiple magnetic flux quanta per a funda-
p
- mental cell. We also revisit the existence proof for the Abrikosov vortex lattices,
h streamliningsomeargumentsandprovidingsomeessentialdetailsmissinginearlier
t
a proofs for a single magnetic flux quantum per a fundamental cell.
m
Keywords: magneticvortices,superconductivity,Ginzburg-Landauequations,Abrikosov
[
vortex lattices, bifurcations.

1
v
6 1 Introduction
7
0
1.1 The Ginzburg-Landau equations. The Ginzburg-Landau model of supercon-
6
0 ductivity describes a superconductor contained in Ω Rn, n = 2 or 3, in terms of a
1. complex order parameter Ψ : Ω C, and a magnet⊂ic potential A : Ω Rn1. The
→ →
0 Ginzburg-Landau theory specifies that the difference between the superconducting and
7
normal free energies2 in a state (Ψ,A) is
1
:
v κ2
i E (Ψ,A) := Ψ 2+ curlA2+ (1 Ψ 2)2, (1.1)
X Ω |∇A | | | 2 −| |
ZΩ
r
a
where is the covariant derivative defined as iA and κ is a positive constant that
A
∇ ∇−
depends on the material properties of the superconductor and is called the Ginzburg-
Landau parameter. In the case n = 2, curlA := ∂A2 ∂A1 is a scalar-valued function.
∂x1 − ∂x2
∗Dept of Mathematics, Univ of Toronto, Toronto, Canada; [email protected]
†Centro de Modelamiento Matema´atico (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile;
[email protected]
‡Dept of Mathematics, Univ of Toronto, Toronto, Canada; [email protected]
1TheGinzburg-Landautheoryisreviewedineverybookonsuperconductivityandmostofthebooks
onsolidstateorcondensedmatterphysics. Forreviewsofrigorousresultsseethepapers[11,12,20,29]
and the books [28, 17, 21, 27]
2In the problem we consider here it is appropriate to deal with Helmholtz free energy at a fixed
average magnetic field b:= 1 R curlA, where |Ω| is thearea or volumeof Ω.
|Ω| Ω
1
AbrikosovLattices, January 11, 2017 2
It follows from the Sobolev inequalities that for bounded open sets Ω, the energy E is
Ω
well-defined and C as a functional on the Sobolev space H1.
∞
The critical points of this functional must satisfy the well-known Ginzburg-Landau
equations inside Ω:
∆ Ψ = κ2(Ψ 2 1)Ψ, (1.2a)
A
| | −
curl curlA= Im(Ψ¯ Ψ). (1.2b)
∗ A
∇
Here ∆ = , and curl are the adjoints of and curl. Explicitly, F =
A −∇A∗∇A ∇A∗ ∗ ∇A ∇A∗
divF +iA F, and curl F = curlF for n = 3 and curl f = ( ∂f , ∂f ) for n = 2.
− · ∗ ∗ ∂x2 −∂x1
The key physical quantities for the Ginzburg-Landau theory are
the density of superconducting pairs of electrons, n := Ψ 2;
s
• | |
the magnetic field, B := curlA;
•
and the current density, J := Im(Ψ¯ Ψ).
A
• ∇
All superconductors are divided into two classes with different properties: Type I
superconductors, which have κ < κ and exhibit first-order phase transitions from the
c
non-superconducting state to the superconducting state, and Type II superconductors,
which have κ > κ and exhibit second-order phase transitions and the formation of
c
vortex lattices. Existence of these vortex lattice solutions is the subject of the present
paper.
1.2 Abrikosov lattices. In 1957, Abrikosov [1] discovered solutions of (1.2) in n = 2
whosephysicalcharacteristics n , B,andJ are(non-constantand)periodicwithrespect
s
to a two-dimensional lattice, while independent of the third dimension, and which have
asinglefluxperlattice cell3. Wecallsuchsolutionsthe( )Abrikosov vortex lattices or
L−
the ( )Abrikosov lattice solutions or an abbreviation of thereof. (In physics literature
L−
theyarevariouslycalledmixedstates, Abrikosov mixedstates, Abrikosov vortexstates.)
Due to an error of calculation Abrikosov concluded that the lattice which gives the
minimum average energy per lattice cell4 is the square lattice. Abrikosov’s error was
corrected by Kleiner, Roth, and Autler [22], who showed that it is in fact the triangular
lattice which minimizes the energy.
Since their discovery, Abrikosov lattice solutions have been studied in numerous
experimental and theoretical works. Of more mathematical studies, we mention the
articles of Eilenberger [16], Lasher [23], Chapman [10] and Ovchinnikov [26].
The rigorous investigation of Abrikosov solutions began soon after their discovery.
Odeh [25] sketched a proof of existence for various lattices using variational and bifur-
cation techniques. Barany, Golubitsky, and Turski [8] applied equivariant bifurcation
theory and filled in a number of details, and Tak´a˘c [30] has adapted these results to
3Such solutions correspond to cylindrical geometry.
4SinceforlatticesolutionstheenergyoverR2 (thetotalenergy)isinfinite,oneconsiderstheaverage
energy perlattice cell, i.e. energy perlattice cell divided by thearea of the cell.
AbrikosovLattices, January 11, 2017 3
study the zeros of the bifurcating solutions. Further details and new results, in both,
variational and bifurcation, approaches, were provided by [15, 14]. In particular, [15]
proved partialresultsontherelation between thebifurcationparameter andtheaverage
magneticfieldb(leftopenbypreviousworks)andontherelation betweentheGinzburg-
Landau energy and the Abrikosov function, and [14] (see also [15]) found boundaries
between superconducting, normal and mixed phases.
Among related results, a relation of the Ginzburg-Landau minimization problem,
for a fixed, finite domain and external magnetic field, in the regime of κ , to the
→ ∞
Abrikosov lattice variational problem was obtained in [3, 6].
The above investigation was completed and extended in [31, 32]. To formulate the
results of these papers, we introduce some notation and definitions. We define the
following function on lattices R2:
L ⊂
1 1
κ ( ) := 1 , (1.3)
c
L s2 − β( )
(cid:18) L (cid:19)
where β( ) is the Abrikosov parameter, see e.g. [31, 32]. For a lattice R2, we
L L ⊂
denote by Ω and Ω the basic lattice cell and its area, respectively. The following
L L
| |
results were proven in [31, 32]:
Theorem 1.1. For every lattice satisfying
L
1 b/κ2 1 and (κ κ ( ))(κ2 b) 0, where b := 2πn, (1.4)
− ≪ − c L − ≥ ΩL
| |
(cid:12) (cid:12)
with n(cid:12)= 1, the(cid:12)following holds
(I) The equations (1.2) have an Abrikosov lattice solution in a neighbourhood of
L−
the branch of normal solutions.
(II) The above solution is unique, up to symmetry, in a neighbourhood of the normal
branch.
(III) For (κ κ ( ))(κ2 b) = 0, the solution above is real analytic in b in a neigh-
c
− L − 6
bourhood of κ2.
Due to the flux quantization (see below), the quantity b := 2π , entering the theo-
ΩL
rem, is the average magnetic flux per lattice cell, b := 1 cu|rlA| . We note that due
ΩL ΩL
to the reflection symmetry of the problem we can assu|me| that b 0.
R ≥
All the rigorous results proven so far deal with Abrikosov lattices with one quantum
of magnetic flux per lattice cell. Partial results for higher magnetic fluxes were proven
in [10, 5].
1.3 Result. In this paper, we prove existence of Abrikosov vortex lattice solutions of
the Ginzburg-Landau equations, with multiple magnetic flux quanta per a fundamental
cell, for certain lattices and for certain flux quanta numbers.
AbrikosovLattices, January 11, 2017 4
We also revisit the existence proof for the Abrikosov vortex lattices, streamlining
some arguments and providing some essential details missing in earlier proofs for a
single magnetic flux quantum per a fundamental cell.
As in the previous works, we consider only bulk superconductors filling all R3, with
no variation along one direction, so that the problem is reduced to one on R2.
To formulate our results, we need some definitions. Motivated by the idea that most
stable (i.e. most physical) solutions are also most symmetric, we look for solutions
which are most symmetric among vortex lattice solution for a given lattice and given
the number of the flux quanta per fundamental cells. Following [13], we denote
1. G( ) to be the group of symmetries of the lattice ,
L L
2. T( ) to be the subgroup of G(L) consisting of lattice translations, and
L
3. H( ) := G( ) O(2) G( )/T( ), the maximal non-translation subgroup.
L L ∩ ≈ L L
Note that the non-SO(2) part of H( ) := G( ) O(2) comes from reflections and
L L ∩
since all reflections in G( ) O(2) can be obtained as products of rotations and one
L ∩
fixed reflection, which we take to be z z¯, it suffices for us to consider the conjugation
7→
action. Since the conjugation is not holomorphic, we show in Section A that there is no
solutions having this symmetry. This implies that the maximal point symmetry group
of the GL equations is
SH( ) := G( ) SO(2)(= H( ) SO(2)).
L L ∩ L ∩
Hence, welook forsolutions amongfunctionswhosephysicalpropertiesareinvariant
under action of SH( ).
L
Definition 1.2 (Maximal symmetry). We say vortex lattice solution on R2 is max-
imally symmetric iff all related physical quantities (i.e. n := Ψ 2, B := curlA,
s
| |
J := Im(Ψ¯ Ψ)) are invariant under the action of the group SH( ), where is the
A
∇ L L
underlying lattice of the solution.
Furthermore, we are interested in vortex lattice solutions with the following natural
property
Definition 1.3 ( irreducibility). We say that a solution is irreducible iff there
L− L−
are no finer lattice for which it is a vortex lattice solution.
Our main result is the following
Theorem 1.4. Assume either n is one of 2,4,6,8,10 and is a hexagonal lattice or
L
n = 3 and is arbitrary. Then the GLEs have an irreducible, maximally symmetric
L L−
solution brach (λ(s),Ψ(s),A(s)), s R small. This branch is, after rescaling (4.1), of
∈
the form (8.2).
AbrikosovLattices, January 11, 2017 5
As was mentioned above, we revisit the existence proof of [31, 32] streamlining some
arguments and providing some essential details either missing or only briefly mentioned
([31, 32]) in earlier proofs of the existence of Abrikosov vortex lattices.
After introducing general properties of (1.2) in Sections 2-4, we prove an abstract
conditionalresultinSections 5-8, fromwhichwederiveTheorem1.1 inSection 9(giving
a streamlined proof of this result) and Theorem 1.4, in Section 10.
Acknowledgements It is a pleasure to thank Max Lein for useful discussions. The
first and third authors’ research is supported in part by NSERC Grant No. NA7901.
During the work on the paper, they enjoyed the support of the NCCR SwissMAP. The
secondauthor (P.S.)was partially supportedbyFondoBasal CMM-ChileandFondecyt
postdoctoral grant 3160055
2 Properties of the Ginzburg-Landau equations
2.1 Symmetries. The Ginzburg-Landau equations exhibit a number of symmetries,
that is, transformations which map solutions to solutions:
The gauge symmetry,
(Ψ(x),A(x)) (eiη(x)Ψ(x),A(x)+ η(x)), η C2(R2,R); (2.1)
7→ ∇ ∀ ∈
The translation symmetry,
(Ψ(x),A(x)) (Ψ(x+t),A(x+t)), t R2; (2.2)
7→ ∀ ∈
The rotation and reflection symmetry,
(Ψ(x),A(x)) (Ψ(R 1x),RA(R 1x)), R O(2). (2.3)
− −
7→ ∀ ∈
An important role in our analysis is played by the reflections symmetry. Let the
reflection operator Trefl be given as
Trefl : (Ψ(x),A(x)) (Ψ( x), A( x)). (2.4)
7→ − − −
We say that a state (Ψ,A) is even (reflection symmetric) iff
Trefl(Ψ,A) = (Ψ,A). (2.5)
Thereflections symmetryof theGLEequations implies thatwecan restricttheclass
of solutions to even ones. In what follows we always assume that solutions (Ψ,A) are
even.
2.2 Elementary solutions. There are two immediate solutions to the Ginzburg-
Landau equations that are homogeneous in Ψ. These are the perfect superconductor
solution where Ψ 1 and A 0, and the normal (or non-superconducting) solution
S S
≡ ≡
where Ψ = 0 and A is such that curlA =: b is constant. (We see that the perfect
N N N
superconductor is a solution only when the magnetic field is absent. On the other
hand, there is a normal solution, (Ψ = 0, A , curlA = constant), for any constant
N N N
magnetic field.)
AbrikosovLattices, January 11, 2017 6
3 Lattice equivariant states
3.1 Periodicity. Our focus in this paper is on states (Ψ,A) defined on all of R2, but
whose physical properties, the density of superconducting pairs of electrons, n := Ψ 2,
s
| |
the magnetic field, B := curlA, and the current density, J := Im(Ψ¯ Ψ), are doubly-
A
∇
periodic with respect to some lattice . We call such states lattice states.
L L−
One can show that a state (Ψ,A) H1 (R2;C) H1 (R2;R2) is a -lattice state if
∈ loc × loc L
and only if translation by an element of the lattice results in a gauge transformation of
the state, that is, for each t , there exists a function g H2 (R2;R) such that
∈ L t ∈ loc
Ψ(x+t) = eigt(x)Ψ(x) and A(x+t) = A(x)+ g (x), t , (3.1)
t
∇ ∀ ∈L
almosteverywhere. Statessatisfying(3.1)willbecalled( )equivariant(vortex)states.
L−
It is clear that the gauge, translation, and rotation symmetries of the Ginzburg-
Landau equations map lattice states to lattice states. In the case of the gauge and
translation symmetries, the lattice with respect to which the solution is periodic does
not change, whereas with the rotation symmetry, the lattice is rotated as well. It is a
simple calculation to verify that the magnetic flux per cell of solutions is also preserved
under the action of these symmetries.
Note that (Ψ,A) is definedby its restriction to a singlecell and can bereconstructed
from this restriction by lattice translations.
3.2 Flux quantization. The important property of lattice states is that the mag-
netic flux through a lattice cell is quantized,
curlA= 2πn (3.2)
ZΩL
for some integer n, with Ω any fundamental cell of the lattice. This implies that
L
2πn
Ω = , (3.3)
L
| | b
where b is the average magnetic flux per lattice cell, b := 1 curlA.
ΩL ΩL
Indeed, if Ψ > 0 on the boundary of the cell, we ca|n w| rite Ψ = Ψ eiθ and 0
R
| | | | ≤
θ < 2π. The periodicity of n and J ensure the periodicity of θ A and therefore by
s
∇ −
Green’s theorem, curlA = A= θ and this function is equal to 2πn since Ψ
Ω ∂Ω ∂Ω∇
is single-valued.
R H H
Equation (3.2) then imposes a condition on the area of a cell, namely, (3.3).
3.3 Lattice shape. We identify R2 with C, via the map (x ,x ) x +ix , and,
1 2 1 2
→
applying a rotation, if necessary, bring any lattice to the form
L
= r(Z+τZ), (3.4)
ω
L
where ω = (τ,r), r > 0, τ C, Imτ > 0, which we assume from now on. If satisfies
ω
∈ L
(3.3), then r = 2πn . Furthermore, we introduce the normalized lattice
bImτ
q
2π
τ := (Z+τZ) (3.5)
L Imτ
r
AbrikosovLattices, January 11, 2017 7
and let Ωτ stand for an elementary cell of the lattice τ. We note that Ωτ = 2π.
L | |
Since the action the modular group SL(2,Z) on C does not change the lattice, but
in general maps one basis, and therefore τ, Imτ > 0, into another (see Supplement I
of [29]), the modular invariance of γ(τ) means that it depends only on the lattice and
not its basis. Thus, it suffices to consider τ in the fundamental domain, H/SL(2,Z), of
SL(2,Z) acting on the Poincar´e half plane H := τ C : Imτ > 0 . The fundamental
{ ∈ }
domain H/SL(2,Z) is given explicitly as
1 1
H/SL(2,Z) = τ C : Imτ > 0, τ 1, < Reτ . (3.6)
{ ∈ | | ≥ −2 ≤ 2}
4 Fixing the gauge and rescaling
In this section we fix the gauge for solutions, (Ψ,A), of (1.2) and then rescale them to
eliminate the dependence of the size of the lattice on b. Our space will then depend
only on the number of quanta of flux and the shape of the lattice.
4.1 Fixing the gauge. The gauge symmetry allows one to fix solutions to be of
a desired form. Let Ab(x) = bJx bx , where x = Jx = ( x ,x ) and J is the
2 ≡ 2 ⊥ ⊥ − 2 1
symplectic matrix
0 1
J = − .
1 0
(cid:18) (cid:19)
We will use the following preposition, first used by [25] and proved in [30] (an alternate
proof is given in in Appendix A of [32]).
Proposition 4.1. Let (Ψ,A) be an -equivariant state, and let b be the average mag-
′ ′
L
netic flux per cell. Then there is a -equivariant state (Ψ,A), that is gauge-equivalent
L
to (Ψ,A), such that
′ ′
(i) Ψ(x+s) = ei(2bx·Js+cs)Ψ(x) and A(x+s)= A(x)+ 2bJs for all s ∈ L;
(ii) (A Ab) = 0, divA= 0.
Ω −
Here cRs satisfies the condition cs+t−cs −ct− 21bs∧t ∈ 2πZ.
4.2 Rescaling. Let τ C, Imτ > 0 and r = 2πn . We define the rescaled fields
∈ bImτ
(ψ,a) as q
2π n
(ψ(x),a(x)) := (r Ψ(r x),r A(r x)), r := r/ = . (4.1)
′ ′ ′ ′ ′
Imτ b
r r
Let and τ be the lattices defined in (3.4) and (3.5). We summarize the effects of
ω
L L
the rescaling above:
(A) Ψ and A solve the Ginzburg-Landau equations if and only if ψ and a solve
( ∆ λ)ψ = κ2 ψ 2ψ, λ = κ2n/b, (4.2a)
a
− − − | |
curl curla =Im(ψ¯ ψ). (4.2b)
∗ a
∇
AbrikosovLattices, January 11, 2017 8
(B) (Ψ,A) is a -equivariant state iff (ψ,a) is a τ-equivariant state. Moreover, if
ω
L L
(Ψ,A) is of the form described in Proposition 4.1, then (ψ,a) satisfies
n
ψ(x+t)= ein2x·Jt+ictψ(x), a(x+t) = a(x)+ Jt, t τ (4.3)
2 ∀ ∈ L
n
diva = 0, (a an)= 0, where an(x):= Jx, (4.4)
− 2
ZΩτ
and c , which satisfies the condition
t
1
c c c ns t 2πZ. (4.5)
s+t s t
− − − 2 ∧ ∈
(C) Ωr2τ EΩτ(Ψ,A) = Eλ(ψ,α), where a = an+α, with an(x):= n2Jx, λ = κ2r′2 = κ2nb
a|nd|
1 κ2 λ
(ψ,α) = ψ 2 + curla 2+ (ψ 2 )2 dx. (4.6)
Eλ Ωτ |∇a | | | 2 | | − κ2
| | ZΩτ (cid:18) (cid:19)
Our problem then is: for each n = 1,2,..., find (ψ,a), solve the rescaled Ginzburg-
Landau equations (4.2) and satisfying (4.3).
In what follows, the parameter τ is fixed and, to simplify the notation, we omit the
superindex τ at τ and Ωτ and write simply and Ω.
L L
5 The linear problem
In this section we consider the linearization of (4.2) satisfying (4.3) on the normal
solution (0,an), with, recall, an(x):= nJx. This leads to the linear problem:
2
∆anψ0 = λψ0, (5.1)
−
for ψ satisfying the gauge - periodic boundary condition (see (4.3))
0
ψ0(x+t) = ei(n2x·Jt+ct)ψ0(x), t . (5.2)
∀ ∈ L
Our goal is to prove the following
Proposition 5.1. The operator ∆an is self-adjoint on its natural domain and its
−
spectrum is given by
σ( ∆an) = (2m+1)n :m =0,1,2,... , (5.3)
− { }
with each eigenvalue is of the multiplicity n. Moreover,
Null( ∆an n)= ei2nx2(x1+ix2)Vn, (5.4)
− −
AbrikosovLattices, January 11, 2017 9
where V is spanned by functions of the form (below z = (x +ix )/ 2π )
n 1 2 Imτ
q
∞
θ(z,τ) := c ei2πmz, c = e inπzei2mπτc . (5.5)
m m+n − m
m=
X−∞
Such functions are determined entirely by the values of c ,...,c and therefore form
0 n 1
−
an n-dimensional vector space.
Proof. The self-adjointness of the operator ∆an is well-known. To find its spectrum,
−
we introduce the complexified covariant derivatives (harmonic oscillator annihilation
and creation operators), ∂¯an and ∂¯a∗n = −∂an, with
1
∂¯an := (∇an)1+i(∇an)2 = ∂x1 +i∂x2 + 2n(x1+ix2). (5.6)
One can verify that these operators satisfy the following relations:
[∂¯an,(∂¯an)∗]= 2curlan = 2n; (5.7)
∆an n= (∂¯an)∗∂¯an. (5.8)
− −
Asfortheharmonicoscillator(seeforexample[19]),thisgivesexplicitinformationabout
the spectrum of ∆an, namely (5.3), with each eigenvalue is of the same multiplicity.
−
Furthermore, the above properties imply
Null( ∆an n)= Null∂¯an. (5.9)
− −
We find Null∂¯an. A simple calculation gives the following operator equation
e−n2(ix1x2−x22)∂¯anen2(ix1x2−x22) = ∂x1 +i∂x2.
(The transformation on the l.h.s. is highly non-unique.) This immediately proves that
∂¯anψ = 0, (5.10)
if and only if θ = e−n2(ix1x2−x22)ψ satisfies (∂x1 +i∂x2)θ = 0. We now identify x ∈ R2
with z = x +ix C and see that this means that θ is analytic and
1 2
∈
ψ(x) = e−2πImnτ(|z|2−z2)θ(z,τ), z = (x1+ix2)/ 2π . (5.11)
Imτ
r
where we display the dependence of θ on τ. The quasiperiodicity of ψ transfers to θ as
follows
θ(z+1,τ) = θ(z,τ), θ(z+τ,τ) = e 2πinze inπτθ(z,τ).
− −
The first relation ensures that θ have a absolutely convergent Fourier expansion of
the form θ(z,τ) = ∞m= cme2πmiz. The second relation, on the other hand, leads to
relation for the coefficien−t∞s of the expansion: c = e inπzei2mπτc , which together
m+n − m
P
with the previous statement implies (5.5).
AbrikosovLattices, January 11, 2017 10
Next, we claim that the solution (5.11) satisfies
ψ(x) = ψ( x). (5.12)
−
By (5.11), it suffices to show that θ(z) = θ( z). We show this for n = 1. Denote the
−
corresponding θ by θ(z,τ). Iterating the recursive relation for the coefficients in (5.5),
we obtain the following standard representation for the theta function
θ(z,τ) = ∞ e2πi(12m2τ+mz). (5.13)
m=
X−∞
We observe that θ( z,τ) = θ(z,τ) and therefore ψ ( x) = ψ (x). Indeed, using the
0 0
− −
expression (5.13), we find, after changing m to m, we find
′
−
θ( z,τ) = ∞ e2πi(12m2τ−mz) = ∞ e2πi(12m′2τ+m′z) = θ(z,τ).
−
m= m′=
X−∞ X−∞
6 Setup of the bifurcation problem
In this section we reformulate the Ginzburg-Landau equations as a bifurcation problem.
We pick the fundamental cell with the center at the origin so that it is invariant under
the map x x.
→ −
We write a = an+α and substitute this into (4.2) to obtain
(Ln λ)ψ = h(ψ,α), Mα = J(ψ,α), (6.1)
− −
where h(ψ,α) := 2iα anψ+ α2ψ+κ2 ψ 2ψ and J(ψ,α) := Im(ψ¯ an+αψ) and
·∇ | | | | ∇
Ln := ∆an and M := curl∗curl. (6.2)
−
The pair (ψ,α) satisfies the conditions (4.3) - (4.4), with a = an +α, an(x) := nJx,
2
which we reproduce here
ψ(x+t) = ei(n2x·Jt+ct)ψ(x), (6.3)
α(x+t) = α(x) and divα = 0, α= 0, (6.4)
ZΩ
where t and c satisfies the condition (4.5). We take c = 0 on the basis vectors
t t
∈ L
t = 2π , 2π τ (see (3.5)). Then the relation (4.5) gives
Imτ Imτ
q q
2π
c = πnpq, for s = (p+qτ),p,q Z, (6.5)
s
Imτ ∈
r
which we assume in what follows.