3Skyrmions
III Classical and Quantum Solitons
3.3 Other Skyrmion structures
There are other ways of trying to get Skyrmion solutions.
Product Ansatz
Suppose
U
1
(
x
) and
U
2
(
x
) are Skyrmions of baryon numbers
B
1
and
B
2
. Since
the target space is a group SU(2), we can take the product
U(x) = U
1
(x)U
2
(x).
Then the baryon number is
B
=
B
1
+
B
2
. To see this, we can consider the
product when
U
1
and
U
2
are well-separated, i.e. consider
U
(
x − a
)
U
2
(
x
) with
|a|
large. Then we can see the baryon number easily because the baryons are
well-separated. We can then vary
a
continuously to 0, and
B
doesn’t change
as we make this continuous deformation. So we are done. Alternatively, this
follows from an Eckmann–Hilton argument.
This can help us find Skyrmions with baryon number
B
starting with
B
well-separated
B
= 1 hedgehogs. Of course, this will not be energy-minimizing,
but we can numerically improve the field by letting the separation vary.
It turns out this is not a good way to find Skyrmions. In general, it doesn’t
give good approximations to the Skyrmion solutions. They tend to lack the
desired symmetry, and this boils down to the problem that the product is
not commutative, i.e.
U
1
U
2
6
=
U
2
U
1
. Thus, we cannot expect to be able to
approximate symmetric things with a product ansatz.
The product ansatz can also be used for several
B
= 4 subunits to construct
configurations with baryon number 4
n
for
n ∈ Z
. For example, the following is
a B = 31 Skyrmion:
B = 31 Skyrmion by P. H. C. Lau and N. S. Manton
This is obtained by putting eight
B
= 4 Skyrmions side by side, and then cutting
off a corner.
This strategy tends to work quite well. With this idea, we can in fact find
Skyrmion solutions with baryon number infinity! We can form an infinite cubic
crystal out of
B
= 4 subunits. For
m
= 0, the energy per baryon is
≈
1
.
038
×
12
π
2
.
This is a very close to the lower bound!
We can also do other interesting things. In the picture below, on the left, we
have a usual
B
= 7 Skyrmion. On the right, we have deformed the Skyrmion
into what looks like a
B
= 4 Skyrmion and a
B
= 3 Skyrmion. This is a cluster
structure, and it turns out this deformation doesn’t cost a lot of energy. This
two-cluster system can be used as a model of the lithium-7 nucleus.
B = 7 Skyrmions by C. J. Halcrow