III Classical and Quantum Solitons - ϕ<sup>4</sup> kinks

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

U(φ)

2π 4π2π4π

Now there are infinitely many distinct vacua. In this case, we find we need to

pick W such that

dφ

= 2 sin

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

= 2 sin

φ.

This can be solved. This involves integrating a

csc

, and ultimately gives us a

solution

φ(x) = 4 tan

−1

x−a

We can check that this solution interpolates between 0 and 2π.

2π

Unlike the

theory, dynamical multi-kink solutions exist here and can be

derived exactly. One of the earlier ways to do so was via B¨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

)

−1/2

For v = 0.01, this looks like

2π

−2π

t = 0

t = ±400

Note that since

(

x, t

) =

(

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

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

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





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

nπ

and

n ±

for some

, 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

. For the

sine-Gordon theory, it is much better to think of this as a field modulo 2

, i.e.

as a function

φ : R → S

In this language, the boundary condition is that

(

) = 0

mod

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

are now the same on

two sides, we can join the ends of the domain

together, and we can think of

as a map

φ : S

→ S

instead. This is a compactification of space.

Topologically, such maps are classified by their winding number, or the degree,

which we denote

. 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

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

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.

2π

Thus, we can think of the Q as the net soliton number.

But this construction we presented is rather specific to maps from

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

2π

µν

∂

φ,

where

µν

is the anti-symmetric tensor in 1 + 1 dimensions, chosen so that

= 1.

In components, this is just

2π

(∂

φ, −∂

φ).

This is conserved because of the symmetry of mixed partial derivatives, so that

∂

2π

µν

∂

φ = 0.

As usual, a current induces a conserved charge

Q =

∞

−∞

dx =

2π

∞

−∞

∂

φ dx =

2π

(φ(∞) −φ(−∞)) = n

− n

−

which is the formula we had earlier.

Note that all these properties do not depend on

satisfying any field equa-

tions! It is completely topological.

Finally, there is also a differential geometry way of defining

. We note that

the target space S

has a normalized volume form ω so that

ω = 1.

For example, we can take

ω =

2π

dφ.

Now, given a mapping

R → S

, we can pull back the volume form to obtain

∗

ω =

2π

dφ

dx.

We can then define the degree of the map to be

Q =

∗

ω =

2π

∞

−∞

dφ

dx.

This is exactly the same as the formula we obtained using the current!

Note that even though the volume form is normalized on

and has integral

1, the integral when pulled back is not 1. We can imagine this as saying if we

wind around the circle

times, then after pulling back, we would have pulled

back

“copies” of the volume form, and so the integral will be

times that of

the integral on S

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.