540
B. Schroers
to obtain a well-defined variational problem. Clearly, (13) reduces to (12) for the
simple gauge field (7).
Adding the term (13) to the energy (3) has a number of advantages, at least from
an analytical point of view. It does not change the Euler–Lagrange equation one
obtains for variations which vanish rapidly at infinity, but its inclusion means that
one can allow for variation with a slower fall-off. We refer the reader to [10] for
details. Furthermore, the study of solutions of arbitrary degree in [9] shows that the
modified energy is well-defined for some solutions for which the unmodified energy
integral (3) is not.
Geometrically, the unmodified energy (3) has a natural interpretation when
evaluated on a solution of the Bogomol’nyi equation as the equivariant degree of
that solution [15]. By contrast, the modified energy evaluated on a solution n of the
Bogomol’nyi equation is equal to the integral expression for the degree:
E A [n] + E A,∞ [n] = 4πQ[n] if n × D 1 n = D 2 n.
(14)
3 Exact Magnetic Skyrmions
3.1 The General Solution in Four Easy Steps
Since the magnetic skyrmion energy functional (1) with the potential (6) is a
particular example of the energy for a gauged sigma model of the form (3), we can
obtain an infinite family of solutions of the variational equations by solving (11).
Here we focus on the formula needed for magnetic skyrmions, so for constant gauge
fields. In that case, the solution of (11) can be obtained via the following recipe. For
details we again refer to [10].
(I) Complex coordinate for the magnetisation: In order to write down the
solution, one needs to work in terms of a complex stereographic coordinate
for the magnetisation field n. It is given by stereographic projection from the
south pole, or algebraically by
w =
n 1 + in 2
1 + n 3
.
(15)
(II) Complexified gauge field: Next, one needs to write the gauge field explicitly
as an su(2) matrix-valued gauge field on R 2 according to
A i =
3
a=1
A
a
i t a , i = 1, 2,
(16)
B. Schroers
to obtain a well-defined variational problem. Clearly, (13) reduces to (12) for the
simple gauge field (7).
Adding the term (13) to the energy (3) has a number of advantages, at least from
an analytical point of view. It does not change the Euler–Lagrange equation one
obtains for variations which vanish rapidly at infinity, but its inclusion means that
one can allow for variation with a slower fall-off. We refer the reader to [10] for
details. Furthermore, the study of solutions of arbitrary degree in [9] shows that the
modified energy is well-defined for some solutions for which the unmodified energy
integral (3) is not.
Geometrically, the unmodified energy (3) has a natural interpretation when
evaluated on a solution of the Bogomol’nyi equation as the equivariant degree of
that solution [15]. By contrast, the modified energy evaluated on a solution n of the
Bogomol’nyi equation is equal to the integral expression for the degree:
E A [n] + E A,∞ [n] = 4πQ[n] if n × D 1 n = D 2 n.
(14)
3 Exact Magnetic Skyrmions
3.1 The General Solution in Four Easy Steps
Since the magnetic skyrmion energy functional (1) with the potential (6) is a
particular example of the energy for a gauged sigma model of the form (3), we can
obtain an infinite family of solutions of the variational equations by solving (11).
Here we focus on the formula needed for magnetic skyrmions, so for constant gauge
fields. In that case, the solution of (11) can be obtained via the following recipe. For
details we again refer to [10].
(I) Complex coordinate for the magnetisation: In order to write down the
solution, one needs to work in terms of a complex stereographic coordinate
for the magnetisation field n. It is given by stereographic projection from the
south pole, or algebraically by
w =
n 1 + in 2
1 + n 3
.
(15)
(II) Complexified gauge field: Next, one needs to write the gauge field explicitly
as an su(2) matrix-valued gauge field on R 2 according to
A i =
3
a=1
A
a
i t a , i = 1, 2,
(16)