2Vortices
III Classical and Quantum Solitons
2.2 Global U(1) Ginzburg–Landau vortices
We now put the theory into use. We are going to study Ginzburg–Landau vortices.
Our previous discussion involved a function taking values in the unit disk
D
.
We will not impose such a restriction on our vortices. However, we will later see
that any solution must take values in D.
The potential energy of the Ginzburg–Landau theory is given by
V (Φ) =
1
2
Z
R
2
(∇Φ, ∇Φ) +
λ
4
(1 −(Φ, Φ))
2
dx
1
dx
2
.
where λ > 0 is some constant.
Note that the inner product is invariant under phase rotation, i.e.
(e
iχ
a, e
iχ
b) = (a, b)
for χ ∈ R. So in particular, the potential satisfies
V (e
iχ
Φ) = V (Φ).
Thus, our theory has a global U(1) symmetry.
The Euler–Lagrange equation of this theory says
−∆Φ =
λ
2
(1 − |Φ|
2
)Φ.
This is the ungauged Ginzburg–Landau equation.
To justify the fact that our Φ takes values in
D
, we use the following lemma:
Lemma.
Assume Φ is a smooth solution of the ungauged 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
∇W = −2(Φ, ∇Φ)
∆W = ∇
2
W
= −2(Φ, ∆Φ) − 2(∇Φ, ∇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.
By itself, this doesn’t force
|
Φ
| ∈
[0
,
1], since we could imagine
|
Φ
|
having no
maximum. However, if we prescribe boundary conditions such that
|
Φ
|
= 1 on
the boundary, then this would indeed imply that
|
Φ
| ≤
1 everywhere. Often, we
can think of Φ as some “complex order parameter”, in which case the condition
|Φ| ≤ 1 is very natural.
The objects we are interested in are vortices.
Definition
(Ginzburg–Landau vortex)
.
A global Ginzburg–Landau vortex of
charge
N >
0 is a (smooth) solution of the ungauged Ginzburg–Landau equation
of the form
Φ = f
N
(r)e
iNθ
in polar coordinates (r, θ). Moreover, we require that f
N
(r) → 1 as r → ∞.
Note that for Φ to be a smooth solution, we must have
f
N
(0) = 0. In fact, a
bit more analysis shows that we must have
f
N
=
O
(
r
N
) as
r →
0. Solutions for
f
N
do exist, and they look roughly like this:
r
f
N
In the case of
N
= 1, we can visualize the field Φ as a vector field on
C
.
Then it looks like
This is known as a 2-dimensional hedgehog.
For general
N
, it might be more instructive to look at how the current looks
like. Recall that the current is defined by (
i
Φ
,
dΦ). We can write this more
explicitly as
(iΦ, dΦ) = (if
N
e
iNθ
, (df
N
)e
iNθ
+ if
N
N dθe
iNθ
)
= (if
N
, df
N
+ if
N
N dθ).
We note that
if
N
and d
f
N
are orthogonal, while
if
N
and
if
N
N
d
θ
are parallel.
So the final result is
(iΦ, dΦ) = f
2
N
N dθ.
So the current just looks this:
As
|x| → ∞
, we have
f
N
→
1. So the winding number is given as before,
and we can compute the winding number of this system to be
1
2π
lim
R→∞
I
(iΦ, dΦ) =
1
2π
lim
R→∞
I
f
2
N
N dθ = N.
The winding number of these systems is a discrete quantity, and can make the
vortex stable.
This theory looks good so far. However, it turns out this model has a problem
— the energy is infinite! We can expand out
V
(
f
N
e
iNθ
), and see it is a sum of a
few non-negative terms. We will focus on the
∂
∂θ
term. We obtain
V (f
N
e
iNθ
) ≥
Z
1
r
2
∂Φ
∂θ
2
r dr dθ
= N
2
Z
1
r
2
f
2
N
r dr dθ
= 2πN
2
Z
∞
0
f
2
N
r
dr.
Since f
N
→ 1 as r → ∞, we see that the integral diverges logarithmically.
This is undesirable physically. To understand heuristically why this occurs,
decompose dΦ into two components — a mode parallel to Φ and a mode
perpendicular to Φ. For a vortex solution these correspond to the radial and
angular modes respectively. We will argue that for fluctuations the parallel mode
is massive, while the perpendicular mode is massless. Now given that we just
saw that the energy divergence of the vortex arises from the angular part of the
energy, we see that it is the massless mode that leads to problems. We will see
below that in gauge theories, the Higgs mechanism serves to make all modes
massive, thus allowing for finite energy vortices.
We can see the difference between massless and massive modes very explicitly
in a different setting, corresponding to Yukawa mesons. Consider the equation
−∆u + M
2
u = f.
Working in three dimensions, the solution is given by
u(x) =
1
4π
Z
e
−M|x−y|
|x − y|
f(y) dy.
Thus, the Green’s function is
G(x) =
e
−M|x|
4π|x|
.
If the system is massless, i.e.
M
= 0, then this decays as
1
|x|
. However, if the
system is massive with
M >
0, then this decays exponentially as
|x| → ∞
. In
the nonlinear setting the exponential decay which is characteristic of massive
fundamental particles can help to ensure decay of the energy density at a rate
fast enough to ensure finite energy of the solution.
So how do we figure out the massive and massless modes? We do not have
a genuine decomposition of Φ itself into “parallel” and “perpendicular” modes,
because what is parallel and what is perpendicular depends on the local value of
Φ.
Thus, to make sense of this, we have to consider small fluctuations around a
fixed configuration Φ. We suppose Φ is a solution to the field equations. Then
δV
δΦ
= 0. Thus, for small variations Φ 7→ Φ + εϕ, we have
V (Φ + εϕ) = V (Φ) + ε
2
Z
|∇ϕ|
2
+ λ(ϕ, Φ)
2
−
2λ
4
(1 − |Φ|
2
)|ϕ|
2
dx + O(ε
3
).
Ultimately, we are interested in the asymptotic behaviour
|x| → ∞
, in which
case 1
− |
Φ
|
2
→
0. Moreover,
|
Φ
| →
1 implies (
ϕ,
Φ) becomes a projection along
the direction of Φ. Then the quadratic part of the potential energy density for
fluctuations becomes approximately
|∇ϕ|
2
+ λ|ϕ
parallel
|
2
for large
x
. Thus, for
λ >
0, the parallel mode is massive, with corresponding
“Yukawa” mass parameter
M
=
√
λ
, while the perpendicular mode is massless.
The presence of the massless mode is liable to produce a soliton with slow
algebraic decay at spatial infinity, and hence infinite total energy. This is all
slightly heuristic, but is a good way to think about the issues. When we study
vortices that are gauged, i.e. coupled to the electromagnetic field, we will see
that the Higgs mechanism renders all components massive, and this problem
does not arise.