3Skyrmions
III Classical and Quantum Solitons
3.4 Asymptotic field and forces for B = 1 hedgehogs
We now consider what happens when we put different
B
= 1 hedgehogs next
to each other. To understand this, we look at the profile function
f
, for
m
= 0.
For large r, this has the asymptotic form
f(r) ∼
C
r
2
.
To obtain this, we linearize the differential equation for
f
and see how it behaves
as
r → ∞
and
f →
0. The linearized equation doesn’t determine the coefficient
C
, but the full equation and boundary condition at
r
= 0 does. This has to be
worked out numerically, and we find that C ≈ 2.16.
Thus, as
σ ∼
1, we find
π ∼ C
x
r
3
. So the
B
= 1 hedgehog asymptotically
looks like three pion dipoles. Each pion field itself has an axis, but because we
have three of them, the whole solution is spherically symmetric.
We can roughly sketch the Skyrmion as
− +
−
+
−
+
Note that unlike in electromagnetism, scalar dipoles attract if oppositely oriented.
This is because the fields have low gradient. So the lowest energy arrangement
of two B = 1 Skyrmions while they are separated is
− +
−
+
−
+
+ −
−
+
+
−
The right-hand Skyrmion is rotated by 180
◦
about a line perpendicular to the
line separating the Skyrmions.
These two Skyrmions attract! So two Skyrmions in this “attractive channel”
can merge to form the B = 2 torus, which is the true minimal energy solution.
+
−
+
+
−−
−
− +
+
The blue and the red fields have no net dipole, even before they merge. There
is only a quadrupole. However, the field has a strong net green dipole. The whole
field has toroidal symmetry, and these symmetries are important if we want to
think about quantum states and the possible spins these kinds of Skyrmions
could have.
For B = 4 fields, we can begin with the arrangement
− +
−
+
−
+
− +
+
−
+
−
+ −
+
−
−
+
+ −
−
+
+
−
To obtain the orientations, we begin with the bottom-left, and then obtain the
others by rotating along the axis perpendicular to the faces of the cube shown
here.
This configuration only has a tetrahedral structure. However, the Skyrmions
can merge to form a cubic B = 4 Skyrmion.