1ϕ4 kinks

III Classical and Quantum Solitons



1.5 Sine-Gordon kinks
We end the section by briefly talking about kinks in a different theory, namely
the sine-Gordon theory. In this theory, kinks are often known as solitons instead.
The sine-Gordon theory is given by the potential
U(φ) = 1 cos φ.
Again, we suppress coupling constants, but it is possible to add them back.
The potential looks like
φ
U(φ)
2π 4π2π4π
Now there are infinitely many distinct vacua. In this case, we find we need to
pick W such that
dW
dφ
= 2 sin
1
2
φ.
Static sine-Gordon kinks
To find the static kinks in the sine-Gordon theory, we again look at the Bogomolny
equation. We have to solve
dφ
dx
= 2 sin
1
2
φ.
This can be solved. This involves integrating a
csc
, and ultimately gives us a
solution
φ(x) = 4 tan
1
e
xa
.
We can check that this solution interpolates between 0 and 2π.
x
φ
0
2π
a
Unlike the
φ
4
theory, dynamical multi-kink solutions exist here and can be
derived exactly. One of the earlier ways to do so was via acklund transforms,
but that was very complicated. People later invented better methods, but they
are still not very straightforward. Nevertheless, it can be done. Ultimately, this
is due to the sine-Gordon equation being integrable. For more details, refer to
the IID Integrable Systems course.
Example. There is a two-kink solution
φ(x, t) = 4 tan
1
v sinh γx
cosh γvt
,
where, as usual, we have
γ = (1 v
2
)
1/2
.
For v = 0.01, this looks like
x
φ
2π
2π
t = 0
t = ±400
Note that since
φ
(
x, t
) =
φ
(
x, t
), we see that this solution involves two
solitons at first approaching each other, and then later bouncing off. Thus, the
two kinks repel each other. When we did kinks in
φ
4
theory, we saw that a kink
and an anti-kink attracted, but here there are two kinks, which is qualitatively
different.
We can again compute the force just like the
φ
4
theory, but alternatively,
since we have a full, exact solution, we can work it out directly from the solution!
The answers, fortunately, agree. If we do the computations, we find that the
point of closest approach is 2 log
2
v
if v is small.
There are some important comments to make. In the sine-Gordon theory, we
can have very complicated interactions between kinks and anti-kinks, and these
can connect vastly different vacua. However, static solutions must join 2
and
2(
n ±
1)
π
for some
n
, because if we want to join vacua further apart, we will
have more than one kink, and they necessarily interact.
If we have multiple kinks and anti-kinks, then each of these things can have
their own velocity, and we might expect some very complicated interaction
between them, such as annihilation and pair production. But remarkably, the
interactions are not complicated. If we try to do numerical simulations, or use
the exact solutions, we see that we do not have energy loss due to “radiation”.
Instead, the solitons remain very well-structured and retain their identities. This,
again, is due to the theory being integrable.
Topology of the sine-Gordon equation
There are also a lot of interesting things we can talk about without going into
details about what the solutions look like.
The important realization is that our potential is periodic in
φ
. For the
sine-Gordon theory, it is much better to think of this as a field modulo 2
π
, i.e.
as a function
φ : R S
1
.
In this language, the boundary condition is that
φ
(
x
) = 0
mod
2
π
as
x ±∞
.
Thus, instead of thinking of the kink as joining two vacua, we can think of it as
“winding around the circle” instead.
We can go further. Since the boundary conditions of
φ
are now the same on
two sides, we can join the ends of the domain
R
together, and we can think of
φ
as a map
φ : S
1
S
1
instead. This is a compactification of space.
Topologically, such maps are classified by their winding number, or the degree,
which we denote
Q
. This is a topological (homotopy) invariant of a map, and
is preserved under continuous deformations of the field. Thus, it is preserved
under time evolution of the field.
Intuitively, the winding number is just how many times we go around the
circle. There are multiple (equivalent) ways of making this precise.
The first way, which is the naive way, is purely topological. We simply have
to go back to the first picture, where we regard
φ
as a real value. Suppose the
boundary values are
φ(−∞) = 2n
π, φ() = 2n
+
π.
Then we set the winding number to be Q = n
+
n
.
Topologically, we are using the fact that
R
is the universal covering space of
the circle, and thus we are really looking at the induced map on the fundamental
group of the circle.
Example. As we saw, a single kink has Q = 1.
x
φ
0
2π
Thus, we can think of the Q as the net soliton number.
But this construction we presented is rather specific to maps from
S
1
to
S
1
.
We want something more general that can be used for more complicated systems.
We can do this in a more “physics” way. We note that there is a topological
current
j
µ
=
1
2π
ε
µν
ν
φ,
where
ε
µν
is the anti-symmetric tensor in 1 + 1 dimensions, chosen so that
ε
01
= 1.
In components, this is just
j
µ
=
1
2π
(
x
φ,
t
φ).
This is conserved because of the symmetry of mixed partial derivatives, so that
µ
j
µ
=
1
2π
ε
µν
µ
ν
φ = 0.
As usual, a current induces a conserved charge
Q =
Z
−∞
j
0
dx =
1
2π
Z
−∞
x
φ dx =
1
2π
(φ() φ(−∞)) = n
+
n
,
which is the formula we had earlier.
Note that all these properties do not depend on
φ
satisfying any field equa-
tions! It is completely topological.
Finally, there is also a differential geometry way of defining
Q
. We note that
the target space S
1
has a normalized volume form ω so that
Z
S
1
ω = 1.
For example, we can take
ω =
1
2π
dφ.
Now, given a mapping
φ
:
R S
1
, we can pull back the volume form to obtain
φ
ω =
1
2π
dφ
dx
dx.
We can then define the degree of the map to be
Q =
Z
φ
ω =
1
2π
Z
−∞
dφ
dx
dx.
This is exactly the same as the formula we obtained using the current!
Note that even though the volume form is normalized on
S
1
and has integral
1, the integral when pulled back is not 1. We can imagine this as saying if we
wind around the circle
n
times, then after pulling back, we would have pulled
back
n
“copies” of the volume form, and so the integral will be
n
times that of
the integral on S
1
.
We saw that these three definitions gave the same result, and different
definitions have different benefits. For example, in the last two formulations, it
is not a priori clear that the winding number has to be an integer, while this is
clear in the first formulation.