III Classical and Quantum Solitons

2Vortices

2.5 Physics of vortices

Recall we began with the ungauged Ginzburg–Landau theory, and realized the

solitons didn’t have finite energy. We then added a gauge field, and the problem

went away — we argued the coupling to the gauge field “gave mass” to the trans-

verse component, thus allowing the existence of finite energy soliton solutions. In

the book of Jaffe and Taubes there are results on the exponential decay of gauge

invariant combinations of the fields which are another expression of this effect

— the Higgs mechanism. However, there is a useful and complementary way of

understanding how gauge fields assist in stabilizing finite energy configurations

against collapse, and this doesn’t require any detailed information about the

theory at all — only scaling. We now consider this technique, which is known

either as the Derrick or the Pohozaev argument.

Suppose we work in

space dimensions. Then a general scalar field Φ :

→

has energy functional given by



|∇Φ|

+ U(Φ)



for some

. In the following we consider smooth finite energy configurations

for which the energy is stationary. To be precise, we require that the energy is

stationary with respect to variations induced by rescaling of space (as is made

explicit in the proof); we just refer to these configurations as stationary points.

Theorem

(Derrick’s scaling argument)

Consider a field theory in

-dimensions

with energy functional

E[Φ] =



|∇Φ|

+ U(Φ)



x = T + W,

with

the integral of the gradient term and

the integral of the term involving

– If d = 1, then any stationary point must satisfy

T = W.

– If d = 2, then all stationary points satisfy W = 0.

– If d ≥ 3, then all stationary points have T = W = 0, i.e. Φ is constant.

This forbids the existence of solitons in high dimensions for this type of

energy functional.

Proof.

Suppose Φ were such a stationary point. Then for any variation Φ

of Φ

such that Φ = Φ

, we have

dλ



λ=1

E[Φ

] = 0.

Consider the particular variation given by

(x) = Φ(λx).

Then we have

W [Φ

] =

U(Φ

(x)) d

x = λ

−d

U(Φ(λx)) d

(λx) = λ

−d

W [Φ].

On the other hand, since

contains two derivatives, scaling the derivatives as

well gives us

T [Φ

] = λ

2−d

T [Φ].

Thus, we find

E[Φ

] = λ

2−d

T [Φ] + λ

−d

W [Φ].

Differentiating and setting λ = 1, we see that we need

(2 − d)T [Φ] − dW [Φ] = 0.

Then the results in different dimensions follow.

The

= 2 case is rather interesting. We can still get interesting soliton

theories if we have sufficiently large space of classical vacua {Φ : W (Φ) = 0}.

Example. In d = 2, we can take ` = 3 and

W (φ) = (1 − |φ|

)

Then the set

= 0 is given by the unit sphere

⊆ R

. With

constrained

to this 2-sphere, this is a

-model, and there is a large class of such maps

(

) which minimize the energy (for a fixed topology) — in fact they are just

rational functions when stereographic projection is used to introduce complex

coordinates.

Derrick’s scaling argument is not only a mathematical trick. We can also

interpret it physically. Increasing

corresponds to “collapsing” down the field.

Then we see that in

d ≥

3, both the gradient and potential terms favour collapsing

of the field. However, in

= 1, the gradient term wants the field to expand, and

the potential term wants the field to collapse. If these two energies agree, then

these forces perfectly balance, and one can hope that stationary solitons exist.

We will eventually want to work with theories in higher dimensions, and

Derrick’s scaling argument shows that for scalar theories with energy functionals

as above this isn’t going to be successful for three or more dimensions, and

places strong restrictions in two dimensions. There are different ways to get

around Derrick’s argument. In the Skyrme model, which we are going to study

in the next chapter, there are no gauge fields, but instead we will have some

higher powers of derivative terms. In particular, by introducing fourth powers of

derivatives in the energy density, we will have a term that scales as

4−d

, and

this acts to stabilize scalar field solitons in three dimensions.

Now let’s see how gauge theory provides a way around Derrick’s argu-

ment without having to depart from employing only energy densities which

are quadratic in the derivatives (as is highly desirable for quantization). To

understand this, we need to know how gauge fields transform under spatial

rescaling.

One way to figure this out is to insist that the covariant derivative D

must scale as a whole in the same way that ordinary derivatives scale in scalar

field theory. Since

∂

= λ(∂

Φ)(λx),

we would want A

to scale as

)

(x) = λA

(λx).

We can also see this more geometrically. Consider the function

: R

→ R

x 7→ λx.

Then our previous transformations were just given by pulling back along

Since A is a 1-form, it pulls back as

∗

) = λA

(λx) dx

Then since B = dA, the curvature must scale as

(x) = λ

B(λx).

Thus, we can obtain a gauged version of Derrick’s scaling argument.

Since we don’t want to develop gauge theory in higher dimensions, we will

restrict our attention to the Ginzburg–Landau model. Since we already used the

letter λ, we will denote the scaling parameter by µ. We have

, Φ

) =



B(µx)

+ µ

|DΦ(µx)|

(1 − |Φ(µx)|

)



(µx)



(y) + |DΦ(y)|

4µ

(1 − |Φ(y)|

)



Again, the gradient term is scale invariant, but the magnetic field term counter-

acts the potential term. We can find the derivative to be

dµ



µ=1

, Φ

) =



−

(1 − |Φ|

)



Thus, for a soliton, we must have

x =

(1 − |Φ|

)

Such solutions exist, and we see that this is because they are stabilized by the

magnetic field energy in the sense that a collapse of the configuration induced

by rescaling would be resisted by the increase of magnetic energy which such a

collapse would produce. Note that in the case

= 1, this equation is just the

integral form of one of the Bogomolny equations!

Scattering of vortices

Derrick’s scaling argument suggests that vortices can exist if

λ >

0. However, as

we previously discussed, for

λ 6

= 1, there are forces between vortices in general,

and we don’t get static, separated vortices. By doing numerical simulations,

we find that when

λ <

1, the vortices attract. When

λ >

1, the vortices repel.

Thus, when

λ >

1, the symmetric vortices with

N >

1 are unstable, as they

want to break up into multiple single vortices.

We can talk a bit more about the

= 1 case, where we have static multi-

vortices. For example, for

= 2, the solutions are parametrized by pairs of

points in C, up to equivalence

, Z

) ∼ (Z

, Z

We said the moduli space is

, and this is indeed true. However,

and

are

not good coordinates for this space. Instead, good coordinates on the moduli

space

are given by some functions symmetric in

and

. One valid

choice is

Q = Z

+ Z

, P = Z

In general, for vortex number

, we should use the elementary symmetric

polynomials in Z

, ··· , Z

as our coordinates.

Now suppose we set our vortices in motion. For convenience, we fix the

center of mass so that Q(t) = 0. We can then write P as

P = −Z

If we do some simulations, we find that in a head-on collision, after they collide,

the vortices scatter off at 90

◦

. This is the 90

◦

scattering phenomenon, and holds

for other λ as well.

In terms of our coordinates,

is smoothly evolving from a negative to a

positive value, going through 0. This corresponds to

7→ ±iZ

−Z

Note that at the point when they collide, we lose track of which vortex is which.

Similar to the

kinks, we can obtain effective Lagrangians for the dynamics

of these vortices. However, this is much more complicated.