Solvable Models of Magnetic Skyrmions
541
where t a = −
i
2 τ a , and τ a are the Pauli matrices. In fact we require the complex
linear combination
A ¯
z =
1
2
(A 1 + iA 2 ).
(17)
In the case at hand, this is a constant, complex and traceless 2 × 2 matrix, so
generically an element of sl(2, C).
(III) Solution in complex coordinates: The solution of the Bogomol’nyi equation
is given in terms of the exponential
g(¯ z) = exp
−
1
2
(A 1 + iA 2 )¯ z
=
a(¯ z) b(¯ z)
c(¯ z) d(¯ z)
,
(18)
which is a 2×2 matrix function of ¯
z with determinant one. The general solution
of (11) in stereographic coordinates is
w(z, ¯
z) =
c(¯ z) + d(¯ z)f (z)
a(¯ z) + b(¯ z)f (z)
,
(19)
where f is an arbitrary holomorphic function from C into CP
1
C ∪ {∞}
(in particular it is allowed to take the value ∞).
(IV) Translating back into Cartesian coordinates: Substitution of the general
solution (19) into the inverse of (15)
n 1 + in 2 =
2w
1 + |w| 2 , n 3 =
1 − |w| 2
1 + |w| 2 ,
(20)
yields an explicit (but possibly complicated) formula for the magnetisation
field.
The energy density of Bogomol’nyi solutions is either the degree density or the
sum of the degree density and the vorticity, depending on the choice of energy
functional, see our discussion in Sect. 2.3. Expressions for both directly in terms
the stereographic coordinates are given in [9, 10].
3.2 Examples
Axisymmetric DM Terms As discussed in [10], the DM term is invariant under
rotations in the plane and simultaneous rotations of the magnetisation field about
a suitable axis if and only if A 1 and A 2 are orthogonal and have the same length.
In that case (A 1 , A 2 , A 1 × A 2 ) is an oriented and (up to scaling) orthonormal basis
of R 3 . With |A 1 | = |A 2 | = κ, the potential for the solvable model is conveniently
expressed in terms of ˆ
A := A 1 × A 2 /κ 2 as
541
where t a = −
i
2 τ a , and τ a are the Pauli matrices. In fact we require the complex
linear combination
A ¯
z =
1
2
(A 1 + iA 2 ).
(17)
In the case at hand, this is a constant, complex and traceless 2 × 2 matrix, so
generically an element of sl(2, C).
(III) Solution in complex coordinates: The solution of the Bogomol’nyi equation
is given in terms of the exponential
g(¯ z) = exp
−
1
2
(A 1 + iA 2 )¯ z
=
a(¯ z) b(¯ z)
c(¯ z) d(¯ z)
,
(18)
which is a 2×2 matrix function of ¯
z with determinant one. The general solution
of (11) in stereographic coordinates is
w(z, ¯
z) =
c(¯ z) + d(¯ z)f (z)
a(¯ z) + b(¯ z)f (z)
,
(19)
where f is an arbitrary holomorphic function from C into CP
1
C ∪ {∞}
(in particular it is allowed to take the value ∞).
(IV) Translating back into Cartesian coordinates: Substitution of the general
solution (19) into the inverse of (15)
n 1 + in 2 =
2w
1 + |w| 2 , n 3 =
1 − |w| 2
1 + |w| 2 ,
(20)
yields an explicit (but possibly complicated) formula for the magnetisation
field.
The energy density of Bogomol’nyi solutions is either the degree density or the
sum of the degree density and the vorticity, depending on the choice of energy
functional, see our discussion in Sect. 2.3. Expressions for both directly in terms
the stereographic coordinates are given in [9, 10].
3.2 Examples
Axisymmetric DM Terms As discussed in [10], the DM term is invariant under
rotations in the plane and simultaneous rotations of the magnetisation field about
a suitable axis if and only if A 1 and A 2 are orthogonal and have the same length.
In that case (A 1 , A 2 , A 1 × A 2 ) is an oriented and (up to scaling) orthonormal basis
of R 3 . With |A 1 | = |A 2 | = κ, the potential for the solvable model is conveniently
expressed in terms of ˆ
A := A 1 × A 2 /κ 2 as