542
B. Schroers
V A (n) =
κ 2
2
1 − (n, ˆ
A)
2 =
κ 2
8
(n − ˆ
A)
4 .
(21)
The DM term (5) and the integrand of the boundary term (13) combine neatly into
3
a=1
2
i=1 D ai (∂ i n × (n − ˆ
A)) a in this case. For the simple case (7) with DM term
κ(n, ∇ × n) and potential V (n) =
κ 2
2 (1 − n 3 ) 2 , one checks that the matrix (18) is
g =
1 −
i
2 κ ¯
z
0
1
.
(22)
The solution (19) is best written in terms of the inverse coordinate v = 1/w as
v = −
i
2
κ ¯
z + h,
(23)
where h = 1/f is, like f , an arbitrary holomorphic map C → CP
1 . The simplest
choice h = 0 leads to the Bloch hedgehog skyrmion with Q = −1 and the
Belavin–Polyakov profile function θ(r) = 2 arctan
r
2κ
, as already noticed in [7].
Many other solutions are discussed in [9], and our Fig. 1 shows the solutions one
obtains for different choices of rational functions h. It follows from the calculations
in [9] that the degree of a skyrmion configuration depends on the parameter
L = lim |z|→∞ |(2h)/(κ ¯
z)|. For configurations determined by (23) with rational
h(z) = p(z)/q(z), where p and q are polynomials of degree M and N , it is
Q[n] =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
M
if L > 1
N
if L = 1
N − 1 if L < 1.
(24)
This shows in particular that in this model there are infinitely many solutions of
the Bogomol’nyi equation (11) for each integer degree Q ≥ −1. The modified
energy (14) takes the values 4πQ on these solutions.
Rank One DM Interactions The spiralization tensor has rank one when A 1 and A 2
are linearly dependent, so A 1 × A 2 = 0. In this case, the curvature F 12 vanishes
and the gauge field (and therefore the DM interaction) can be removed by an SU (2)
gauge transformation. The solvable model with the potential (6) can therefore be
mapped into the standard Belavin–Polyakov O(3) sigma model [11]. It follows
immediately that solutions of (11) exist for any integer degree Q in these models,
and that their energy (14) is 4π |Q|. In particular it follows that skyrmions and
anti-skyrmions of equal and opposite degree have the same energy. This result was
derived in [8] for |Q| = 1 in more general rank one models. Example solutions of
solvable rank one models and their properties are discussed in [10] and also [16]. We
note that a similar reformulation in terms of a flat gauge field was recently applied
to a rather different ferromagnetic model in [17].
B. Schroers
V A (n) =
κ 2
2
1 − (n, ˆ
A)
2 =
κ 2
8
(n − ˆ
A)
4 .
(21)
The DM term (5) and the integrand of the boundary term (13) combine neatly into
3
a=1
2
i=1 D ai (∂ i n × (n − ˆ
A)) a in this case. For the simple case (7) with DM term
κ(n, ∇ × n) and potential V (n) =
κ 2
2 (1 − n 3 ) 2 , one checks that the matrix (18) is
g =
1 −
i
2 κ ¯
z
0
1
.
(22)
The solution (19) is best written in terms of the inverse coordinate v = 1/w as
v = −
i
2
κ ¯
z + h,
(23)
where h = 1/f is, like f , an arbitrary holomorphic map C → CP
1 . The simplest
choice h = 0 leads to the Bloch hedgehog skyrmion with Q = −1 and the
Belavin–Polyakov profile function θ(r) = 2 arctan
r
2κ
, as already noticed in [7].
Many other solutions are discussed in [9], and our Fig. 1 shows the solutions one
obtains for different choices of rational functions h. It follows from the calculations
in [9] that the degree of a skyrmion configuration depends on the parameter
L = lim |z|→∞ |(2h)/(κ ¯
z)|. For configurations determined by (23) with rational
h(z) = p(z)/q(z), where p and q are polynomials of degree M and N , it is
Q[n] =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
M
if L > 1
N
if L = 1
N − 1 if L < 1.
(24)
This shows in particular that in this model there are infinitely many solutions of
the Bogomol’nyi equation (11) for each integer degree Q ≥ −1. The modified
energy (14) takes the values 4πQ on these solutions.
Rank One DM Interactions The spiralization tensor has rank one when A 1 and A 2
are linearly dependent, so A 1 × A 2 = 0. In this case, the curvature F 12 vanishes
and the gauge field (and therefore the DM interaction) can be removed by an SU (2)
gauge transformation. The solvable model with the potential (6) can therefore be
mapped into the standard Belavin–Polyakov O(3) sigma model [11]. It follows
immediately that solutions of (11) exist for any integer degree Q in these models,
and that their energy (14) is 4π |Q|. In particular it follows that skyrmions and
anti-skyrmions of equal and opposite degree have the same energy. This result was
derived in [8] for |Q| = 1 in more general rank one models. Example solutions of
solvable rank one models and their properties are discussed in [10] and also [16]. We
note that a similar reformulation in terms of a flat gauge field was recently applied
to a rather different ferromagnetic model in [17].