2Vortices
III Classical and Quantum Solitons
2.1 Topological background
Sine-Gordon kinks
We now review what we just did for sine-Gordon kinks, and then try to develop
some analogous ideas in higher dimension. The sine-Gordon equation is given by
∂
2
θ
∂t
2
−
∂
2
θ
∂x
2
+ sin θ = 0.
We want to choose boundary conditions so that the energy has a chance to
be finite. The first part is, of course, to figure out what the energy is. The
energy-momentum conservation equation given by Noether’s theorem is
∂
t
θ
2
t
+ θ
2
x
2
+ (1 − cos θ)
+ ∂
x
(−θ
t
θ
x
) = ∂
µ
P
µ
= 0.
The energy we will be considering is thus
E =
Z
R
P
0
dx =
Z
R
θ
2
t
+ θ
2
x
2
+ (1 − cos θ)
dx.
Thus, to obtain finite energy, we will want
θ
(
x
)
→
2
n
±
π
for some integers
n
±
as x → ±∞. What is the significance of this n
±
?
Example. Consider the basic kink
θ
K
(x) = 4 tan
−1
e
x
.
Picking the standard branch of tan
−1
, this kink looks like
x
φ
0
2π
This goes from θ = 0 to θ = 2π.
To better understand this, we can think of
θ
as an angular variable, i.e. we
identify
θ ∼ θ
+ 2
nπ
for any
n ∈ Z
. This is a sensible thing because the energy
density and the equation etc. are unchanged when we shift everything by 2
nπ
.
Thus, θ is not taking values in R, but in R/2πZ
∼
=
S
1
.
Thus, for each fixed t, our field θ is a map
θ : R → S
1
.
The number
Q
=
n
+
− n
−
equals the number of times
θ
covers the circle
S
1
on going from
x
=
−∞
to
x
= +
∞
. This is the winding number, which is
interpreted as the topological charge.
As we previously discussed, we can express this topological charge as the
integral of some current. We can write
Q =
1
2π
Z
θ(R)
dθ =
1
2π
Z
∞
−∞
dθ
dx
dx.
Note that this formula automatically takes into account the orientation. This is
the form that will lead to generalization in higher dimensions.
This function
dθ
dx
appearing in the integral has the interpretation as a topo-
logical charge density. Note that there is a topological conservation law
∂
µ
j
µ
=
∂j
0
∂t
+
∂j
1
∂x
= 0,
where
j
0
= θ
x
, j
1
= −θ
t
.
This conservation law is not a consequence of the field equations, but merely a
mathematical identity, namely the commutation of partial derivatives.
Two dimensions
For the sine-Gordon kink, the target space was a circle
S
1
. Now, we are concerned
with the unit disk
D = {(x
1
, x
2
) : |x|
2
< 1} ⊆ R
2
.
We will then consider fields
Φ : D → D.
In the case of a sine-Gordon kink, we still cared about moving solitons. However,
here we will mostly work with static solutions, and study fields at a fixed time.
Thus, there is no time variable appearing.
Using the canonical isomorphism
R
2
∼
=
C
, we can think of the target space
as the unit disk in the complex plane, and write the field as
Φ = Φ
1
+ iΦ
2
.
However, we will usually view the D in the domain as a real space instead.
We will impose some boundary conditions. We pick any function
χ
:
S
1
→ R
,
and consider
g = e
iχ
: S
1
→ S
1
= ∂D ⊆ D.
Here
g
is a genuine function, and has to be single-valued. So
χ
must be single-
valued modulo 2π. We then require
Φ
∂D
= g = e
iχ
.
In particular, Φ must send the boundary to the boundary.
Now the target space
D
has a canonical choice of measure dΦ
1
∧
dΦ
2
. Then
we can expect the new topological charge to be given by
Q =
1
π
Z
D
dΦ
1
∧ dΦ
2
=
1
π
Z
D
det
∂Φ
1
∂x
1
∂Φ
1
∂x
2
∂Φ
2
∂x
1
∂Φ
2
∂x
2
!
dx
1
∧ dx
2
.
Thus, the charge density is given by
j
0
=
1
2
ε
ab
ε
ij
∂Φ
a
∂x
i
∂Φ
b
∂x
j
.
Crucially, it turns out this charge density is a total derivative, i.e. we have
j
0
=
∂V
i
∂x
i
for some function
V
. It is not immediately obvious this is the case. However, we
can in fact pick
V
i
=
1
2
ε
ab
ε
ij
Φ
a
∂Φ
b
∂x
j
.
To see this actually works, we need to use the anti-symmetry of
ε
ij
and observe
that
ε
ij
∂
2
Φ
b
∂x
i
∂x
j
= 0.
Equivalently, using the language of differential forms, we view the charge density
j
0
as the 2-form
j
0
= dΦ
1
∧ dΦ
2
= d(Φ
1
dΦ
2
) =
1
2
d(Φ
1
dΦ
2
− Φ
2
dΦ
1
).
By the divergence theorem, we find that
Q =
1
2π
I
∂D
Φ
1
dΦ
2
− Φ
2
dΦ
1
.
We then use that on the boundary,
Φ
1
= cos χ, Φ
2
= sin χ,
so
Q =
1
2π
I
∂D
(cos
2
χ + sin
2
χ) dχ =
1
2π
I
∂D
dχ = N.
Thus, the charge is just the winding number of g!
Now notice that our derivation didn’t really depend on our domain being
D
. It could have been any region bounded by a simple closed curve in
R
2
. In
particular, we can take it to be a disk D
R
of arbitrary radius R.
What we are actually interested in is a field
Φ : R
2
→ D.
We then impose asymptotic boundary conditions
Φ ∼ g = e
iχ
as |x| → ∞. We can still define the charge or degree by
Q =
1
π
Z
R
2
j
0
dx
1
dx
2
=
1
π
lim
R→∞
Z
D
R
j
0
dx
1
dx
2
This is then again the winding number of g.
It is convenient to rewrite this in terms of an inner product.
R
2
itself has an
inner product, and under the identification
R
2
∼
=
C
, the inner product can be
written as
(a, b) =
¯ab + a
¯
b
2
.
Use of this expression allows calculations to be done efficiently if one makes use
of the fact that for real numbers a and b, we have
(a, b) = (ai, bi) = ab, (ai, b) = (a, bi) = 0.
In particular, we can evaluate
(iΦ, dΦ) = (iΦ
1
− Φ
2
, dΦ
1
+ idΦ
2
) = Φ
1
dΦ
2
− Φ
2
dΦ
1
.
This is just (twice) the current V we found earlier! So we can write our charge
as
Q =
1
2π
lim
R→∞
I
|x|=R
(iΦ, dΦ).
We will refer to (
i
Φ
,
dΦ) as the current, and the corresponding charge density is
j
0
=
1
2
d(iΦ, dΦ).
This current is actually a familiar object from quantum mechanics: recall
that for the Schr¨odinger’s equation
i
∂Φ
∂t
= −
1
2m
∆Φ + V (x)Φ, ∆ = ∇
2
.
the probability
R
|
Φ
|
s
d
x
is conserved. The differential form of the probability
conservation law is
1
2
∂
t
(Φ, Φ) +
1
2m
∇ · (iΦ, ∇Φ) = 0.
What appears in the flux term here is just the topological current!