Solvable Models of Magnetic Skyrmions
539
If these integrals are well-defined, they only depend on global properties of n and on
its boundary behaviour. If the latter is kept fixed, the energy is therefore minimised
when the square in (8) vanishes, i.e. when the Bogomol’nyi equation holds. This is
a gauged version of the Bogomol’nyi equations in the standard Belavin–Polyakov
model [11], but with a definite sign:
n × D 1 n = D 2 n.
(11)
Ths Bogomol’nyi equation implies the variational equation of the energy functional (3), see [10].
In the context of magnetic skyrmions, Eq. (11) first appeared in [7] where it
was noticed that, for a certain family of potentials V , it characterises the energy
minimisers in the Q = −1 sector of the theory with the standard DM term
(n, ∇ × n). The role of this equation in critically coupled magnetic skyrme models
for arbitrary degree Q ≥ −1 was observed and explored in [9]. Its role and
solvability in the more general gauged sigma model and the associated magnetic
skyrmion models are the subject of [10]. Generalised versions of this equation have
been studied in differential geometry as vortex equations for maps from Riemann
surfaces into Kähler manifolds which permit the action of a Lie group, but in that
case the gauge field typically obeys a second, coupled equation [14]. The relation
between these vortex equations and Eq. (11) with a fixed background as proposed
in [10] was clarified in [15].
2.3 Boundary Contributions to the Energy
As far we are aware it is not known for which class of magnetisation fields n the
general energy functional (1) is well-defined and finite. For the standard DM term
(n, ∇ × n) and a certain class of potentials V , this question is answered in [6] and
[7], where it was also pointed that, for analytical reasons, it is preferable to modify
the energy functional by adding the boundary term
E κ,∞ [n] = −κ
R 2
(∂ 1 n 2 − ∂ 2 n 1 )dx 1 dx 2 .
(12)
Adding this term to the energy effectively modifies the DM term: κ(n, ∇ × n) is
replaced by κ((n−e 3 ), ∇ ×n) where e 3 = (0, 0, 1) t , and this is the term considered
in [6, 7].
In the context of gauged sigma models, it was proposed in [10] that one should
more generally add the boundary term
E A,∞ [n] = −4πΩ A [n] =
R 2
(∂ 1 (A 2 , n) − ∂ 2 (A 1 , n))dx 1 dx 2
(13)
539
If these integrals are well-defined, they only depend on global properties of n and on
its boundary behaviour. If the latter is kept fixed, the energy is therefore minimised
when the square in (8) vanishes, i.e. when the Bogomol’nyi equation holds. This is
a gauged version of the Bogomol’nyi equations in the standard Belavin–Polyakov
model [11], but with a definite sign:
n × D 1 n = D 2 n.
(11)
Ths Bogomol’nyi equation implies the variational equation of the energy functional (3), see [10].
In the context of magnetic skyrmions, Eq. (11) first appeared in [7] where it
was noticed that, for a certain family of potentials V , it characterises the energy
minimisers in the Q = −1 sector of the theory with the standard DM term
(n, ∇ × n). The role of this equation in critically coupled magnetic skyrme models
for arbitrary degree Q ≥ −1 was observed and explored in [9]. Its role and
solvability in the more general gauged sigma model and the associated magnetic
skyrmion models are the subject of [10]. Generalised versions of this equation have
been studied in differential geometry as vortex equations for maps from Riemann
surfaces into Kähler manifolds which permit the action of a Lie group, but in that
case the gauge field typically obeys a second, coupled equation [14]. The relation
between these vortex equations and Eq. (11) with a fixed background as proposed
in [10] was clarified in [15].
2.3 Boundary Contributions to the Energy
As far we are aware it is not known for which class of magnetisation fields n the
general energy functional (1) is well-defined and finite. For the standard DM term
(n, ∇ × n) and a certain class of potentials V , this question is answered in [6] and
[7], where it was also pointed that, for analytical reasons, it is preferable to modify
the energy functional by adding the boundary term
E κ,∞ [n] = −κ
R 2
(∂ 1 n 2 − ∂ 2 n 1 )dx 1 dx 2 .
(12)
Adding this term to the energy effectively modifies the DM term: κ(n, ∇ × n) is
replaced by κ((n−e 3 ), ∇ ×n) where e 3 = (0, 0, 1) t , and this is the term considered
in [6, 7].
In the context of gauged sigma models, it was proposed in [10] that one should
more generally add the boundary term
E A,∞ [n] = −4πΩ A [n] =
R 2
(∂ 1 (A 2 , n) − ∂ 2 (A 1 , n))dx 1 dx 2
(13)