Solvable Models of Magnetic Skyrmions
537
2 Magnetic Skyrmions and Gauged Sigma Models
2.1 Formulating Magnetic Skyrme Models as Gauged Sigma
Models
The most general energy functional for the magnetisation field n : R 2 → S 2 which
we consider has the form
E[n] =
R 2
1
2
(∇n)
2
+
3
a=1
2
i=1
D ai (∂ i n × n) a + V (n)
dx 1 dx 2 ,
(1)
where V is a potential which may include a Zeeman term and anisotropy terms, and
D is the spiralization tensor parametrising the DM interaction.
In the following we will use the complex coordinate z = x 1 + ix 2 in the plane,
and define associated derivatives in the standard way, so ∂ z =
1
2 (∂ 1 − i∂ 2 ) and
∂ ¯
z =
1
2 (∂ 1 + i∂ 2 ). As shown in [10], one can view the expression (1) as the energy
functional of a gauged sigma model with a fixed SU (2) background gauge field. To
see this, it is convenient to think of an SU (2) gauge field in the plane simply as a
pair of vectors A 1 and A 2 in R 3 , one for each Cartesian direction in the plane, which
act on the magnetisation field n ∈ R 3 via the vector product (the commutator of the
su(2) Lie algebra) so that the covariant derivative and curvature are
D i n = ∂ i n + A i × n, i = 1, 2, F 12 = ∂ 1 A 2 − ∂ 2 A 1 +
1
2
A 1 × A 2 .
(2)
The energy functional of the gauged non-linear sigma models studied in [10] is
E A [n] =
R 2
1
2
|D 1 n|
2
+
1
2
|D 2 n|
2
− (F 12 , n)
dx 1 dx 2 .
(3)
In the application to magnetic skyrmions, the gauge field is fully determined
by the spiralization tensor: the Cartesian components A 1 and A 2 are simply the
negatives of the column vectors which make up the 3 × 2 spiralization matrix D. In
symbols
(A i ) a = −D ai ,
(4)
so that the DM term can be written as
3
a=1
2
i=1
D ai (∂ i n × n) a =
2
i=1
(A i × n, ∂ i n).
(5)
537
2 Magnetic Skyrmions and Gauged Sigma Models
2.1 Formulating Magnetic Skyrme Models as Gauged Sigma
Models
The most general energy functional for the magnetisation field n : R 2 → S 2 which
we consider has the form
E[n] =
R 2
1
2
(∇n)
2
+
3
a=1
2
i=1
D ai (∂ i n × n) a + V (n)
dx 1 dx 2 ,
(1)
where V is a potential which may include a Zeeman term and anisotropy terms, and
D is the spiralization tensor parametrising the DM interaction.
In the following we will use the complex coordinate z = x 1 + ix 2 in the plane,
and define associated derivatives in the standard way, so ∂ z =
1
2 (∂ 1 − i∂ 2 ) and
∂ ¯
z =
1
2 (∂ 1 + i∂ 2 ). As shown in [10], one can view the expression (1) as the energy
functional of a gauged sigma model with a fixed SU (2) background gauge field. To
see this, it is convenient to think of an SU (2) gauge field in the plane simply as a
pair of vectors A 1 and A 2 in R 3 , one for each Cartesian direction in the plane, which
act on the magnetisation field n ∈ R 3 via the vector product (the commutator of the
su(2) Lie algebra) so that the covariant derivative and curvature are
D i n = ∂ i n + A i × n, i = 1, 2, F 12 = ∂ 1 A 2 − ∂ 2 A 1 +
1
2
A 1 × A 2 .
(2)
The energy functional of the gauged non-linear sigma models studied in [10] is
E A [n] =
R 2
1
2
|D 1 n|
2
+
1
2
|D 2 n|
2
− (F 12 , n)
dx 1 dx 2 .
(3)
In the application to magnetic skyrmions, the gauge field is fully determined
by the spiralization tensor: the Cartesian components A 1 and A 2 are simply the
negatives of the column vectors which make up the 3 × 2 spiralization matrix D. In
symbols
(A i ) a = −D ai ,
(4)
so that the DM term can be written as
3
a=1
2
i=1
D ai (∂ i n × n) a =
2
i=1
(A i × n, ∂ i n).
(5)