On the Bogolubov-de Gennes Equations

0.33 MB·

by  Li Chen

ON THE BOGOLUBOV-DE GENNES EQUATIONS
LI CHEN AND I.M. SIGAL
7
1
0
Abstract. We consider the Bogolubov-de Gennes equations giving an equivalent for-
2
mulationoftheBCStheoryofsuperconductivity. Weareinterestedinthecasewhenthe

n magnetic field is present. We (a) discuss their general features, (b) isolate key physical
a classes of solutions (normal, vortex and vortex lattice states) and (c) prove existence of
J
thenormal,vortexandvortexlatticestatesandstability/instabilityofthenormalstates

1 for large/small temperature or/and magnetic fields.
2

1. Introduction
]
h
p The Bogolubov-de Gennes equations describe the remarkable quantum phenomenon of
- superconductivity.1 They present an equivalent formulation of the BCS theory and are
h
t among the latest additions to the family of important effective equations of mathematical
a
physics. Together with the Hartree-Fock (-Bogolubov), Ginzburg-Landau and Landau-
m
Lifshitz equations, they are the quantum members of this illustrious family consisting of
[
such luminaries as the heat, Euler, Navier-Stokes and Boltzmann equations.

1
Therearestill many fundamentalquestions aboutthese equations which arecompletely
v
open, namely
0
8 • Derivation;
0
• Well-posedness;
6
0 • Existence and stability of stationary magnetic solutions.
1. By the magnetic solutions we mean (physically interesting) solutions with non-zero
0 magnetic fields. In this paper we address the third problem. The well-posedness (or
7
existence) theory will be addressed elsewhere (cf. [2]).
1
The key special solutions of Bogoliubov-de Gennes (BdG) equations are normal, super-
:
v
conducting and mixed or intermediate states. The latter appear only for non-vanishing
i
X magnetic fields. For type II superconductors, they consist of the vortices and (magnetic)
r vortexlattices. Inthispaper,weprovetheexistenceofthenormalstatesfornon-vanishing
a
magnetic fields and of the vortex lattices and investigate the stability of the former.
There is a considerable physics literature devoted to the BdG equations, but, despite
the role played by magnetic phenomena in superconductivity, it deals mainly with the
zero magnetic field case, with only few disjoint remarks about the case when the magnetic
fields are present, the main subject of this work.2
As for rigorous work, it also deals exclusively with the case of zero magnetic field.
The general (variational) set-up for the BdG equations is given in [3]. We use, like all
subsequent papers, this set-up. The next seminal works on the subject are [8], where the
Date: January 18, 2017.
1For some physics background,see books [5, 12] and thereview papers [4, 10].
2The Ginzburg-Landauequations give agood account of magnetic phenomenain superconductors but
only for temperatures sufficiently close to the critical one.
1
2 LICHENANDI.M.SIGAL
authors prove the existence of superconducting states (the existence of the normal states
under the assumptions of [8] is trivial), to which our work is closest, and [6], deriving the
(macroscopic) Ginzburg-Landau equations. For an excellent, recent review of the subject,
with extensive references and discussion see [9].
In the rest of this section we introduce the BdG equations, describe their properties
and the main issues and present the main results of this paper. In the remaining sections
we prove the these results, with technical derivations delegated to appendices. In the last
appendix, following [2], we discuss a formal, but natural, derivation of the BdG equations.
1.1. Bogoliubov-de Gennes equations. In the Bogoliubov-de Gennes approach states
of superconductors are described by the pair of operators γ and α, acting on the one-
particle state space and satisfying (after peeling off the spin variables)
0≤ γ = γ∗ ≤ 1 and α∗ = α, (1.1)
where γ := CγC, with C, the operation of complex conjugation. γ is a one-particle density
operator,ordiagonalcorrelationandαisatwo-particlecoherenceoperator,oroff-diagonal
correlation. γ(x,x) is interpreted as the one-particle density, so that Trγ = γ(x,x)dx is
the total number of particles.
R
The Bogoliubov-de Gennes equations form a system of self-consistent equations for γ
and α. It is convenient to organize the operators γ and α into the self-adjoint operator-
matrix
γ α
η := . (1.2)
α∗ 1−γ¯
(cid:18) (cid:19)
The relations (1.1) and the structure (1.2) of η are equivalent to the following relations
(cf. [3])
0 1
0≤ η = η∗ ≤ 1 and J∗ηJ = 1−η¯, J := . (1.3)
−1 0
(cid:18) (cid:19)
Since the BdG equations describe the phenomenon of superconductivity, they are natu-
rally coupled to the electromagnetic field. We describe the latter by the vector and scalar
potentials a and φ. Then the time-dependent BdG equations state (see e.g. [5, 4, 12])
i(∂ +iφ)η = [Λ(η ,a ),η ], (1.4)
t t t t t
with
Λ(η,a) = hγa v♯α ,
v♯α¯ −hγa
wheretheoperato(cid:0)rv♯isdefi(cid:1)nedthroughtheintegralkernelsas(v♯α)(x;y) := v(x,y)α(x;y),
v(x,y) is the pair potential, and
h = h +v∗γ−v♯γ, (1.5)
γa a
where (v∗γ)(x) = (v ∗d )(x) := v(x,y)d (y)dy, d (x) := γ(x,x). v∗γ and v♯γ are the
γ γ γ
direct and exchange self-interaction potentials, and h = −∆ . Eq (1.4) is coupled to the
a a
R
Maxwell equation (Amp`ere’s law)
∂ (∂ a+∇φ) = −curl∗curla+j(γ,a), (1.6)
t t
ON THE BOGOLUBOV-DE GENNES EQUATIONS 3
where j(γ,a)(x) := 1[−i∇ ,γ](x,x) is the superconducting current. (Above, A(x,y)
2 a
stands for the integral kernel of an operator A.) In what follows, we assume that
v(x,y) = v(y,x).
Remarks. 1) In general, h might contain also an external potentials V(x) and A(x),
a
due to the impurities, which, for simplicity of exposition, we do not consider.
2) For α = 0, Eq (1.4) becomes the time-dependent von Neumann-Hartree-Fock equa-
tion for γ (and a).
3) We may assume that the physical space is a finite box, Ω, in Rd and γ and α are
trace class and Hilbert-Schmidt operators, respectively, see Subsection 1.6 for the precise
formulation.
4) One can extend a formal derivation of (1.4) given in Appendix C to the coupled
system(1.4)-(1.6)bystartingwiththehamiltonian (C.4)coupledtothequantized electro-
magnetic filed.
Connection with the BCS theory. Eq (1.4)can bereformulated as an equation on
the Fock space involving an effective quadratic hamiltonian (see [4, 5, 9] and [2], for the
bosonicversion). ThesearetheeffectiveBCSequationsandtheeffectiveBCShamiltonian.
1.2. Symmetries and conservation laws. The equations (1.4) - (1.6) are invariant
under the gauge transformations,
Tgauge : (γ,α,a,φ) 7→ (eiχγe−iχ,eiχαeiχ,a+∇χ,φ+∂ φ), (1.7)
χ t
foranysufficientlyregularfunctionχ :Rd → R,and,iftheexternalpotentialsarezeroand
considering, for simplicity, the entire space Rd, then also under translation and rotation
transformations,
Ttrans : (γ,α,a) 7→ (U γU−1,U αU−1,U a), (1.8)
h h h h h h
Trot : (γ,α,a) 7→ (U γU−1,U αU−1,ρU a), (1.9)
ρ ρ ρ ρ ρ ρ
for any h ∈ Rd and ρ ∈ O(d). Here Utransl and Urot are the standard translation and
h ρ
rotation transforms Utransl : φ(x) 7→ φ(x+h) and Urot : φ(x) 7→ φ(ρ−1x). In terms of η,
h ρ
gauge
say the gauge transformation, T , could be written as
χ
eiχ 0
Tgauge : η → U ηU−1, where U = (1.10)
χ χ χ χ 0 e−iχ
(cid:18) (cid:19)
gauge gauge
(extended correspondingly to (η,a) by T (η,a) = (T (η),a + ∇χ)). Notice the
χ χ
difference in action of this transformation on the diagonal and off-diagonal elements of η.
The invariance under the gauge transformations can be proven by using the relation
Λ(T (η,a)) =T (Λ(η,a)), (1.11)
χ χ
gauge
shown by using the operator calculus and the fact that U is unitary.
χ
If the external fields are zero, as we assume in this paper, then the equations are
translationally invariant (when considered in R2). Because of the gauge invariance, it is
natural to consider the simplest, gauge (magnetically) translationally invariant solutions,
i.e. solutions invariant under the transformations
T : (η,a) → (Tgauge)−1Ttrans(η,a), (1.12)
bs χs s
4 LICHENANDI.M.SIGAL
for any s ∈ R2, where χ (x) := b(s∧x) (modulo ∇f). (For a gauge-free expression, we
s 2
give χ (x) := x ·a (s), where a (x) is the vector potential with the constant magnetic
s b b
field, curla = b.) We have
b
Lemma 1.1. The operators T defined in (1.12) are unitary and satisfy
bs
T T = Iˆ T , (1.13)
bs bt bst b(s+t)
Iˆ u:= I uI−1, I := ei2b(t∧s) 0 (1.14)
bst bst bst bst 0 e−i2b(t∧s) !
Proof. TheunitarityofT isobvious. Forthesecondstatement,letUmt := (Ugauge)−1Utrans.
bs bs χs s
Then T u = Umtu(Umt)−1 and UmtUmt = I Umt = Umt I where we used that
bs bs bs bs bt bst bs+t bs+t bst
g (x) := χ (x)+χ (x+t)−χ (x) = b(t∧s). Hence, the result follows. (cid:3)
st s t s+t 2
Particle-hole symmetry. The evolution under the equations (1.4) - (1.6) preserves the
relations in (1.3), i.e. if an initial condition has one of these properties, then so does the
solution. This follows from the relation
J∗ΛJ = −Λ. (1.15)
The second relation in (1.3) is called the particle-hole symmetry.
Conservation laws. Eqs (1.4) – (1.6) conserve the energy E(η,a,e) := E(η,a)+ |e|2,
where e is the electric field and E(η,a) is given by
R
1 1
E(η,a) = Tr (−∆ )γ + Tr (v∗d )γ − Tr (v♯γ)γ
a γ
2 2
+ 1T(cid:0)r α∗(v♯α(cid:1)) + d(cid:0)x|curla(x(cid:1))|2. (cid:0) (cid:1) (1.16)
2
Z
(cid:0) (cid:1)
Theenergy functional E(η,a) originates as E(η,a) := ϕ(H ), where ϕ is a quasi-free state
a
in question (see Appendix C) and H is the standard many-body given in (C.4), coupled
a
to the vector potential a.
Furthermore, the global gauge invariance implies the evolution conserves the number of
particles, N := Trγ.
Finally, an important role in our analysis is played by the reflections symmetry. Let
the reflection operator trefl be given by conjugation by the reflections, urefl. We say that
a state (η,a) is even (reflection symmetric) iff
treflγ = γ and urefla = −a. (1.17)
The reflections symmetry of the BdG equations implies that if an initial condition is
even then so is the solution every moment of time.
In what follows we always assume that solutions (η,a) are even.
Remark. We do not specify here the spaces and, consequently, the domain of integra-
tion, for γ, α and a. These are defined in Subsection 1.6.
ON THE BOGOLUBOV-DE GENNES EQUATIONS 5
Hamiltonian structure. The BdG equations (1.4) - (1.6) are hamiltonian, with the
hamiltonian
H(κ,κ∗,α,α∗,a,e) = E(κ∗κ,α,a)+ |e|2.
Z
given in terms the canonically conjugate variables κ,κ∗, α,α¯∗ and a,−e, where e is the
electric field, and the symplectic form (to be checked)
Im[Tr(κ∗κ′)+Tr(α∗α′)]+[e·a′−e′·a].
1.3. Stationary Bogoliubov-de Gennes equations. We consider stationary solutions
to (1.4) of the form
η := Tgaugeη = Ugaugeη(Ugauge)−1, (1.18)
t χ χ χ
with η and χ˙ ≡ µ independentof t, χ independentof x, and a independentof t and φ= 0.
We have
Proposition 1.2. (1.18), with η and χ˙ ≡ −µ independent of t, is a solution to (1.4) iff
η solves the equation
[Λ ,η] = 0, (1.19)
ηa
1 0
where Λ ≡Λ := Λ(η,a)−µS, with S := , and is given explicitly
ηa ηaµ 0 −1
(cid:18) (cid:19)
h v♯α
Λ := γaµ , (1.20)
ηa v♯α∗ −h¯
γaµ
(cid:18) (cid:19)
with h := h −µ.
γaµ γa
Proof. Plugging (1.18) into (1.4) and using (1.11) and that χ is independent of x, we see
that
µ[S,U ηU−1]=i∂ η = [Λ(U ηU−1,a),η ] (1.21)
χ χ t t χ χ t
=[Λ(U ηU−1,a+∇χ),U ηU−1] (1.22)
χ χ χ χ
=U [Λ(η,a),η](U )−1. (1.23)
χ χ
Since U and U are diagonal, so the commute. It follows then [Λ(η,a)−µS,η] = 0, which
χ
is just the statement of the proposition. (cid:3)
For any reasonable function f, solutions of the equation
1
η = f( Λ ), (1.24)
ηa
T
solve (1.19) and therefore give stationary solutions of (1.4). Under some conditions, the
converse is also true. (The parameter T > 0, the temperature, is introduced here for the
future references.)
The physical function f is selected by either a thermodynamic limit (Gibbs states) or
by a contact with a reservoir (or imposing the maximum entropy principle) It is given by
the Fermi-Dirac distribution
f(h)= (1+e2h/T)−1. (1.25)
6 LICHENANDI.M.SIGAL
Inverting the function f, one can rewrite (1.24) as Λ = Tf−1(η). Let f−1 =:g′. Then
ηa
the stationary Bogoliubov-de Gennes equations can be written as
Λ −Tg′(η) = 0, (1.26)
ηa
curl∗curla = j(η,a). (1.27)
Here Λ ≡ Λ(η,a) and T ≥ 0 (temperature) and, as follows from the equations g′ = f−1
ηa
and (1.25), the function g is given by
1
g(λ) = − (λlnλ+(1−λ)ln(1−λ)), (1.28)
2
so that
1 λ
g′(λ) = − ln . (1.29)
2 1−λ
Remarks. 1) One can express these equations in terms of eigenfunctions of the operator
Λ , which is the form appearing in physics literature (see [3, 2]).
ηa
2) For (1.24) to give η of the form (1.2), the function f(h) should satisfy the conditions
f(h¯)= f(h) and f(−h)= 1−f(h). (1.30)
For g(x) given in (1.28), the function f(h) satisfies these conditions as can be checked
from its explicit form (1.25). However, (1.30) is more general than (1.25). Indeed, the
first condition in (1.30) means merely that f is a real function, while the second condition
in (1.30) is satisfied by functions f(h):= (1+eg˜(h))−1, with g˜(h), any odd function.
From now on, we assume g(λ) either is given in (1.28) or satisfies the conditions (1.30),
with f(h) := (g′)−1(h), and
g′(1−x)= −g′(x). (1.31)
3) If we drop the direct and exchange self-interactions from h , then Λ becomes
γaµ ηa
independent of the diagonal part, γ, of η and the equation (1.24) implies that (1.26) has
always the solution
1
η = f( Λ ), where Λ := Λ . (1.32)
Ta T a a ηa η=0
(cid:12)
1.4. Free energy. The Bogoliubov-de Gennes equations arise as the Euler-Lagrange
(cid:12)
equations for the free energy functional
F (η,a) := E(η,a)−TS(η)−µN(η), (1.33)
T
where S(η) = −Tr(ηlnη) is the entropy (see Remark 2 after (1.35)), N(η) := Trγ is the
number of particles, and E(η,a) is the energy functional given in (1.16). With the spaces
defined in Subsection 1.6, we have
Theorem 1.3. (a) The free energy functional F is well defined on the space D1×(a +
T b
~h1 ).
δ,ρ
(b) F is continuously (Gaˆteaux or Fr´echet) differentiable at (η,a) ∈ D1 ×(a +~h1 ),
T b δ,ρ
s.t. ηlnη is trace class, with respect of perturbations (η′,a′), with η′ satisfying
J∗η′J = −η¯′, (η′)2 . [η(1−η)]2, (1.34)
where J is defined in (1.3).
ON THE BOGOLUBOV-DE GENNES EQUATIONS 7
(c) If 0 < η < 1, strictly, (η,a) is even in the sense of the definition (1.17) and if v is
even, then critical points of F satisfy the BdG equations.
T
(d) Minimizers of F over D1×(a +~h1 ) are its critical points.
T b δ,ρ
This theorem is proven in Section 2. For the translation invariant case, it is proven in
[8]. In general case, but with a = 0 (which is immaterial here), the fact that BdG is the
Euler-Lagrange equation of BCS used in [6], but it seems with no proof provided.
As a result of Theorem 1.3, we write the Bogoliubov-de Gennes equations as
F′(η,a) = 0. (1.35)
T
Remarks. 1) Usually in physics, Eq (1.35) appears in the minimization of E(η,a), while
keeping S(η) and N(η) fixed.
2) Due to the symmetry (1.3) of η, we see that
Tr(ηlnη) = Tr((1−η)ln(1−η)) (1.36)
which implies that S(η) = Trg(η), with g(λ) given in (1.28).
3) The map F′(η,a) can be thought of a gradient map.
T
4) Condition (1.34) on perturbations are designed to handle a delicate problem of non-
differentiability of s(λ) := λlnλ at λ = 0, while allowing for sufficiently rich set to derive
the BdG equations.
From now on, we consider only the cylindrical geometry, i.e. we assume the dimension
is 2.
1.5. Special solutions of Bogoliubov-de Gennes equations. In general, equations
(1.26)-(1.27) have the following special solutions:
(1) Normal state: (η,a), with α = 0.
(2) Superconducting state: (η,a), with α6= 0 and a = 0.
(3) Mixed state: (η,a), with α 6= 0 and a 6= 0.
Wethinkofv livingonamicroscopicscale(1),normalandsuperconductingstatesasliving
on a macroscopic scale (δ′−1, the scale of the sample) and mixed states, on a mesoscopic
one (δ−1), with the scales related as: micro ≪ meso ≪ macro or
1≪ δ−1 ≪ δ′−1.
In what follows, we take δ′ → 0, so we are left with 1 ≪ δ−1.
We discuss the above states in more detail.
Superconducting states. The existence of superconducting, translationally invariant
solutions is proven in [8] (see this paper and [9] for the references to earlier results).
Normal states. For b = 0, we can choose a = 0. In this case, if we drop the direct and
exchange self-interactions from h , then, as was mentioned above, the normal state is
γaµ
given by (1.32), with a = 0. This result can be extended to the situation when the direct
and exchange self-interactions are present ([7]).
These are normal translationally invariant states. For b 6= 0, the simplest normal states
are the magnetically translation (mt-) invariant ones. The existence of the mt-invariant
normal states for b 6= 0 is stated in
8 LICHENANDI.M.SIGAL
Theorem 1.4. Drop the exchange term v♯γ and let | v| be small. Then the BdG equa-
tions have a mt-invariant normal solution, which is unique in among even, in the sense
R
of the definition (1.17), solutions.
Moreover, this solution is of the form (η = η , a = a ), where (cf. (1.32))
T,b b
γ 0
η := Tb , (1.37)
Tb 0 1−γ¯
Tb
(cid:18) (cid:19)
with γ a solution to the equation
Tb
1
γ = g♯( h ),
T γ,ab
where g♯ := (g′)−1, and a (x) is the magnetic potential with the constant magnetic field b
b
(curla = b). (For g(x) = −(xlnx+(1−x)ln(1−x)), we have g♯(h) = e2h+1 −1 and
b
−1
therefore γTb solves the equation γ = e2hγ,ab/T +1 .) (cid:0) (cid:1)
(cid:16) (cid:17)
The fact that γ in (1.37) is diagonal should not come as a surprise as a corresponds
Tb b
to aconstant magnetic field bthroughoutthesampleanditcorrespondstoa normalstate.
It can be also seen from the following elementary statement
Proposition 1.5. If η is mt-invariant, then α = 0.
Proof. The mt-invariance implies that α = e−ibs·ab(t)α for all s,t ∈ R2, which yields that
α = 0. (cid:3)
Remark. Since F is mt-invariant, it is infinite on mt-invariant states. Hence we can
T
introduce the mt-invariant free energy density. As it turns out the latter reduces to the
free energy density in the translation invariant case.
We address the question of the energetic stability of the mt-invariant states. We con-
sider perturbations η′ satisfying the condition (1.34). Physically, the most important
perturbations are of the form η′ = φ(α), where φ(α) is the off-diagonal operator-matrix,
defined by
0 α
φ(α) := , (1.38)
α∗ 0
(cid:18) (cid:19)
and α is a Hilbert-Schmidt operator with appropriate smoothness (and equivariance con-
ditions, e.g. from the space I¯2,2,δ ≥ 0, of Subsection 1.6). Then η′ = φ(α) satisfies (1.34)
δ,ρ
iff α satisfies
αα∗ . [γ (1−γ )]2. (1.39)
Tb Tb
Let hL and hR stand for the operators acting on other operators by multiplication from
the left by the operator h and from right by the operator h¯ , respectively, and recall v♯ is
defined after (1.4). We define the operator
L := K +v♯, (1.40)
Tb Tb
hL +hR
K := Tb Tb . (1.41)
Tb tanh(hL /T)+tanh(hR /T)
Tb Tb
where h := h . Let hα,α′i := Tr(α∗α′). We have the following result proven in
Tb γTb,ab
Section 4, which generalizes that of [8] for a = 0:
ON THE BOGOLUBOV-DE GENNES EQUATIONS 9
Proposition 1.6. For perturbations η′ = φ(α) satisfying, (1.34), we have
F (η +ǫη′,a ) =F (η ,a )+ǫ2hα,L αi+O(ǫ3). (1.42)
T Tb b T Tb b Tb
Since h := h + a uniformly bounded perturbation, and since h ≥ 1b−µ, we see
Tb ab ab 2
that h ≥ 1b−const. Using this, it is not hard to show that σ (L ) ⊂ [2Tµ(b,T),∞),
Tb 2 ess Tb
where µ(b,T) is monotonically non-decreasing in b and T, µ(0,T) = 1 and µ(b,T) → ∞
as b → ∞. Hence, since v is independent of either T or b, we have
Proposition 1.7. For either T or b sufficiently large, L > 0 and consequently the
Tb
normal state (η ,a ) is energetically stable under general α−perturbations.
T,b b
On the other hand, for T = 0, L =v♯, where v♯ is defined after (1.4) and therefore we
Tb
can choose α satisfying (1.39) and hα,L αi < 0 given that v ≤ 0, v 6≡ 0 (by taking first
Tb
α′ satisfying (1.39) and then passing to α := χ♯α′, where χ is a cut-off function supported
in the set {v ≤ 0}). Though L is not continuous at T = 0, we can extend this result to
Tb
small T and b:
Theorem 1.8. Suppose that v ≤ 0, v 6≡ 0. For T and b sufficiently small, the operator
L has a negative eigenvalue and consequently the normal state (η ,a ) is energetically
Tb T,b b
unstable under general α−perturbations.
A proof of this proposition is given in Section 5.
Let T (b)/T′(b) be the largest/smallest temperature s.t. the normal solution is ener-
c c
getically unstable/stable under α−perturbations, for (T < T (b))/(T > T (b)′). Clearly,
c c
∞ ≥ T′(b) ≥ T (b) ≥ 0. Proposition 1.7 and Theorem 1.8 imply
c c
Corollary 1.9. Suppose that v ≤ 0, v 6≡ 0. Then T (b) > 0 for b sufficiently small and
c
T′(b) = 0 for b sufficiently large.
c
We conjecture that T (b) = T′(b).
c c
The next corollary provides a convenient criterion for the determination of T (b) and
c
T (b)′.
c
Corollary 1.10. AtT = T (b) and T = T′(b), zero is the lowest eigenvalue of the operator
c c
L .
Tb
A proof of energetically stability under general perturbations, for either T or b suffi-
cientlylarge,ismoresubtle. Forit,onehastousethefulllinearizedoperator,dF′(η ,a ),
T Tb b
which is discussed in Appendix B. Our expressions in Appendix B suggest that 0 is the
lowest eigenvalue of L iff 0 is the lowest eigenvalue of dF′(η ,a ) and, consequently,
Tb T Tb b
T (b) and T′(b) apply also to the general perturbations.
c c
The statement T = T (0) = T′(0) > 0 for a = 0 and therefore b = 0 and for a large
c c c
class of potentials is proven, by the variational techniques, in [8].
In conclusion of this paragraph, we mention that a simple computation shows
Proposition 1.11. The operator L commutes with the magnetic translations. The same
Tb
is true for dF′(η ,a ).
T Tb b
Remarks. 1) The question of when L has negative spectrum for a larger range of
Tb
T’s is delicate one. For T close to T , this depends, besides of the parameters T and b,
c
also on whether the superconductor is of Type I or II.
2) Since the components of magnetic translations (1.12) do not commute, the fiber
decomposition of L is somewhat subtle (see [1]).
Tb
10 LICHENANDI.M.SIGAL
Mixed states. We consider d = 2, which means effectively the cylinder geometry, and let
L = δ−1(Z+τZ), where τ ∈ C,Imτ > 0. Recall that the small parameter δ > 0 defines
δ
the ratio of the microscopic and mesoscopic scales. For the mixed states, there are the
following specific possibilities:
(a) Magnetic vortices: Trot(η,a) = Tgauge(η,a) for every ρ∈ O(2) and some functions
ρ gρ
g :O(2)×R2 → R;
ρ
(b) Vortex lattices: Ttrans(η,a) = Tgauge(η,a), for every s ∈ L and for somefunctions
s χs δ
χ :L ×R2 → R.
s δ
The maps g :O(2)×R2 → R and χ : L ×R2 → R satisfy the co-cycle conditions, e.g.
ρ s δ
χ (x)−χ (x+t)−χ (x) ∈ 2πZ, ∀s,t ∈ L , (1.43)
s+t s t δ
and are called the summands of automorphy (see [13] for a relevant discussion). (The
maps eig : O(2) × R2 → U(1) and eiχ : L × R2 → U(1), where g(x,ρ) ≡ g (x) and
δ ρ
χ(x,s) ≡ χ (x) are called the factors of automorphy.) One can take g to be independent
s ρ
of x.
We think of L as the (mesoscopic) vortex lattice of the solution.
δ
Magnetic flux quantization. Denote by Ω a fundamental cell of L. One has the
L
following results
(a) Magnetic vortices: 1 curla = degg ∈Z;
2π R2
(b) Vortex lattices: 1 curla = c (χ) ∈Z.
2π ΩLR 1
δ
Here degg is the degree (wRinding number) of the map eig : O(2) → U(1) (which is map
of a circle into itself, here we assume that g(ρ) ≡ g is independent of x) and c (χ) is the
ρ 1
first Chern number associated to the summand of automorphy χ :L ×R2 → R (see [13]).
δ
Existence of vortex lattice solutions. With the spaces defined in Subsection 1.6, we
have the following result on the existence of vortex lattices proven in Section 6:
Theorem 1.12. Drop the self-interaction terms 2v∗d and −v♯γ in (1.5), for simplicity.
γ
Assume that T ≥ 0 and kvk is small. Then
∞
(i) for a every Chern number c , there exists a (generalized) solution (η,a) ∈ D1×~h1
1 δ,b
of the BdG equations (1.26)–(1.27); in particular, it satisfies Ttrans(η,a) = Tˆgauge(η,a);
s χs
(ii) this solution minimizes the free energy F (with the corresponding terms dropped,
T
see (6.1)) for a given c ;
1
(iii) if the operator L , given in (1.40) and defined on I¯2,2, has a negative eigenvalue,
Tb
then (η,a) has α 6= 0, i.e. this solution is a vortex lattice. In particular, the latter holds
if T and b are sufficiently small, provided v ≤ 0,v 6≡ 0.
The statement (iii) follows from (i) and Proposition 1.6.
Remarks. 1)Theself-interaction terms2v∗γ andv♯γ in(1.5)areinessentialforphysics
and analysis and are dropped for simplicity. We expect that these terms can be readily
added back to the equation and will not drastically effect our proof.
2) The proof in Section 6 is a variational one and does not give much information
about the solutions. The most interesting unanswered question here, which is related to
the Ginzburg-Landau limit of the BdG equations, is the behaviour of the minimizer as
δ → 0. (By the magnetic flux quantization 1 curla = n ∈ Z, where, recall, Ω a
2π ΩL Lδ
δ
fundamental cell of L , and therefore curla,b = O(δ2).
δ R