2Vortices
III Classical and Quantum Solitons
2.3 Abelian Higgs/Gauged Ginzburg–Landau vortices
We now consider a theory where the complex scalar field Φ is coupled to a
magnetic field. This is a U(1) gauge theory, with gauge potential given by a
smooth real 1-form
A = A
1
dx
1
+ A
2
dx
2
.
The coupling between Φ and
A
is given by minimal coupling: this is enacted by
introduction of the covariant derivative
DΦ = D
A
Φ = dΦ − iAΦ =
2
X
j=1
D
j
Φ dx
j
.
To proceed, it is convenient to have a list of identities involving the covariant
derivative.
Proposition.
If
f
is a smooth real-valued function, and Φ and Ψ are smooth
complex scalar fields, then
D(fΦ) = (df) Φ + f DΦ,
d(Φ, Ψ) = (DΦ, Ψ) + (Φ, DΨ).
(Here (
·
) is the real inner product defined above.) In coordinates, these take
the form
D
j
(fΦ) = (∂
j
f) Φ + f D
j
Φ
∂
j
(Φ, Ψ) = (D
j
Φ, Ψ) + (Φ, D
j
Ψ).
The proofs just involve writing all terms out. The first rule is a version of
the Leibniz rule, while the second, called unitarity, is analogous to the fact that
if V, W are smooth vector fields on a Riemannian manifold, then
∂
k
g(V, W ) = g(∇
k
V, W ) + g(V, ∇
k
W )
for the Levi-Civita connection ∇ associated to a Riemannian metric g.
The curvature term is given by the magnetic field.
Definition
(Magnetic field/curvature)
.
The magnetic field, or curvature is given
by
B = ∂
1
A
2
− ∂
2
A
1
.
We can alternatively think of it as the 2-form
F = dA = B dx
1
∧ dx
2
.
The formulation in terms of differential forms is convenient for computations,
because we don’t have to constrain ourselves to working in Cartesian coordinates
— for example, polar coordinates may be more convenient.
Proposition. If Φ is a smooth scalar field, then
(D
1
D
2
− D
2
D
1
)Φ = −iBΦ.
The proof is again a direct computation. Alternatively, we can express this
without coordinates. We can extend D to act on
p
-forms by letting
A
act as
A∧
.
Then this result says
Proposition.
DDΦ = −iF Φ.
Proof.
DDΦ = (d − iA)(dΦ − iAΦ)
= d
2
Φ − id(AΦ) − iA dΦ − A ∧ A Φ
= −id(AΦ) − iA dΦ
= −idA Φ + iA dΦ − iA dΦ
= −i(dA) Φ
= −iF Φ.
The point of introducing the covariant derivative is that we can turn the global
U(1) invariance into a local one. Previously, we had a global U(1) symmetry,
where our field is unchanged when we replace Φ
7→
Φ
e
iχ
for some constant
χ ∈ R
.
With the covariant derivative, we can promote this to a gauge symmetry.
Consider the simultaneous gauge transformations
Φ(x) 7→ e
iχ(x)
Φ(x)
A(x) 7→ A(x) + dχ(x).
Then the covariant derivative of Φ transforms as
(d − iA)Φ 7→ (d − i(A + dχ))(Φe
iχ
)
= (dΦ + iΦdχ − i(A + dχ)Φ)e
iχ
= e
iχ
(d − iA)Φ.
Similarly, the magnetic field is also invariant under gauge transformations.
As a consequence, we can write down energy functionals that are invariant
under these gauge transformations. In particular, we have (using the real inner
product defined above)
(DΦ, DΦ) 7→ (e
iχ
DΦ, e
iχ
DΦ) = (DΦ, DΦ).
So we can now write down the gauged Ginzburg–Landau energy
V
λ
(A, Φ) =
1
2
Z
R
2
B
2
+ |DΦ|
2
+
λ
4
(1 − |Φ|
2
)
2
d
2
x.
This is then manifestly gauge invariant.
As before, the equations of motion are given by the Euler–Lagrange equations.
Varying Φ, we obtain
−(D
2
1
+ D
2
2
)Φ −
λ
2
(1 − |Φ|
2
)Φ = 0.
This is just like the previous vortex equation in the ungauged case, but since we
have the covariant derivative, this is now coupled to the gauge potential
A
. The
equations of motion satisfied by A are
∂
2
B = (iΦ, D
1
Φ)
−∂
1
B = (iΦ, D
2
Φ).
These are similar to one of Maxwell’s equation — the one relating the curl of
the magnetic field to the current.
It is again an exercise to derive these. We refer to the complete system as
the gauged Ginzburg–Landau, or Abelian Higgs equations. In deriving them, it
is helpful to use the previous identities such as
∂
j
(Φ, Ψ) = (D
j
Φ, Ψ) + (Φ, D
j
Ψ).
So we get the integration by parts formula
Z
R
2
(D
j
Φ, Ψ) d
2
x = −
Z
(Φ, D
j
Ψ) d
2
x
under suitable boundary conditions.
Lemma.
Assume Φ is a smooth solution of the gauged Ginzburg–Landau
equation in some domain. Then at any interior maximum point
x
∗
of
|
Φ
|
, we
have |Φ(x
∗
)| ≤ 1.
Proof. Consider the function
W (x) = 1 − |Φ(x)|
2
.
Then we want to show that
W ≥
0 when
W
is minimized. We note that if
W
is
at a minimum, then the Hessian matrix must have non-negative eigenvalues. So,
taking the trace, we must have ∆
W
(
x
∗
)
≥
0. Now we can compute ∆
W
directly.
We have
∂
j
W = −2(Φ, D
j
Φ)
∆W = ∂
j
∂
j
W
= −2(Φ, D
j
D
j
Φ) − 2(D
j
Φ, D
j
Φ)
= λ|Φ|
2
W − 2|∇Φ|
2
.
Thus, we can rearrange this to say
2|∇Φ|
2
+ ∆W = λ|Φ|
2
W.
But clearly 2
|∇
Φ
|
2
≥
0 everywhere, and we showed that ∆
W
(
x
∗
)
≥
0. So we
must have W (x
∗
) ≥ 0.
As before, this suggests we interpret
|
Φ
|
as an order parameter. This model
was first used to describe superconductors. The matter can either be in a
“normal” phase or a superconducting phase, and
|
Φ
|
measures how much we are
in the superconducting phase.
Thus, in our model, far away from the vortices, we have
|
Φ
| ≈
1, and so we
are mostly in the superconducting phase. The vortices represent a breakdown of
the superconductivity. At the core of the vortices, we have |Φ| = 0, and we are
left with completely normal matter. Usually, this happens when we have a strong
magnetic field. In general, a magnetic field cannot penetrate the superconductor
(the “Meissner effect”), but if it is strong enough, it will cause such breakdown
in the superconductivity along vortex “tubes”.
Radial vortices
Similar to the ungauged case, for
λ >
0, there exist vortex solutions of the form
Φ = f
N
(r)e
iNθ
A = Nα
N
(r) dθ.
The boundary conditions are f
N
, α
N
→ 1 as r → ∞ and f
N
, α
N
→ 0 as r → 0.
Let’s say a few words about why these are sensible boundary conditions from
the point of view of energy. We want
λ
Z
R
2
(1 − |Φ|
2
)
2
< ∞,
and this is possible only for
f
N
→
1 as
r → ∞
. What is less obvious is that we
also need α
N
→ 1. We note that we have
D
θ
Φ =
∂Φ
∂θ
− iA
θ
Φ = (iNf
N
− iNα
N
f
N
)e
iNθ
.
We want this to approach 0 as r → ∞. Since f
N
→ 1, we also need α
N
→ 1.
The boundary conditions at 0 can be justified as before, so that the functions
are regular at 0.
Topological charge and magnetic flux
Let’s calculate the topological charge. We have (assuming sufficiently rapid
approach to the asymptotic values as r → ∞)
Q =
1
π
Z
R
2
j
0
(Φ) d
2
x
= lim
R→∞
1
2π
I
|x|=R
(iΦ, dΦ)
= lim
R→∞
1
2π
I
(if
N
e
iNθ
, iNf
N
e
iNθ
) dθ
=
1
2π
· N lim
R→∞
Z
2π
0
f
2
N
dθ
= N.
Previously, we understood
N
as the “winding number”, and it measures how
“twisted” our field was. However, we shall see shortly that there is an alternative
interpretation of this
N
. Previously, in the sine-Gordon theory, we could think of
N
as the number of kinks present. Similarly, here we can think of this
N
-vortex
as a superposition of
N
vortices at the origin. In the case of
λ
= 1, we will
see that there are static solutions involving multiple vortices placed at different
points in space.
We can compute the magnetic field and total flux as well. It is convenient to
use the d
A
definition, as we are not working in Cartesian coordinates. We have
dA = Nα
0
N
(r) dr ∧ dθ =
1
r
Nα
0
N
dx
1
∧ dx
2
.
Thus it follows that
B = ∂
1
A
2
− ∂
2
A
1
=
N
r
α
0
N
.
Working slightly more generally, we assume given a smooth finite energy con-
figuration
A,
Φ, and suppose in addition that
|
Φ
|
2
→
1 and
r|A|
is bounded as
r → ∞, and also that D
θ
Φ = o(r
−1
). Then we find that
(dΦ)
θ
= iA
θ
Φ + o(r
−1
).
Then if we integrate around a circular contour, since only the angular part
contributes, we obtain
I
|x|=R
(iΦ, dΦ) =
I
|x|=R
(iΦ, iA
θ
Φ) dθ + o(1) =
I
|x|=R
A + o(1).
Note that here we are explicitly viewing
A
as a differential form so that we can
integrate it. We can then note that
|x|
=
R
is the boundary of the disk
|x| ≤ R
.
So we can apply Stokes’ theorem and obtain
I
|x|=R
(iΦ, dΦ) =
Z
|x|≤R
dA =
Z
|x|≤R
B d
2
x.
Now we let R → ∞ to obtain
lim
R→∞
Z
x≤R
B d
2
x = lim
R→∞
I
|x|=R
A = lim
R→∞
I
|x|=R
(iΦ, dΦ) = 2πQ.
Physically, what this tells us then is that there is a relation between the
topological winding number and the magnetic flux. This is a common property
of topological gauge theories. In mathematics, this is already well known —
it is the fact that we can compute characteristic classes of vector bundles by
integrating the curvature, as discovered by Chern.
Behaviour of vortices as r → 0
We saw earlier that for reasons of regularity, it was necessary that
f
N
, α
N
→
0
as r → 0.
But actually, we must have
f
N
∼ r
N
as
r →
0. This has as a consequence
that Φ
∼
(
re
iθ
)
N
=
z
N
. So the local appearance of the vortex is the zero of an
analytic function with multiplicity N .
To see this, we need to compute that the Euler–Lagrange equations are
−f
00
N
(r) −
1
r
f
0
N
(r) +
(N − α
N
)
2
r
2
f
N
=
λ
2
f
N
(1 − f
2
N
).
With the boundary condition that
f
N
and
α
N
vanish at
r
= 0, we can approxi-
mate this locally as
−f
00
N
−
1
r
f
0
N
+
N
2
r
2
f
N
≈ 0
since we have a
1
r
2
on the left hand side. The approximating equation is
homogeneous, and the solutions are just
f
N
= r
±N
.
So for regularity, we want the one that → 0, so f
N
∼ r
N
as r → 0.