Topological solitons in the Heisenberg
ferromagnetic
The nfield theory
In the Heisenberg model spin is a 3D vector n = (n_{1},
n_{2}, n_{3}) with unit length
n^{2} = 1 . We consider n(x,y) field
on 2D plane. In continuum limit the system energy
E = 1/2 ∫ dx dy
(n_{x}· n_{x} +
n_{y}· n_{y} ) ,
n_{x}= ∂n/∂x .
It is necessary that n(∞) → n_{o}
in order to the total energy remains finite. The trivial field
n(x,y) = n_{o} corresponds to the global
energy minimum E = 0 .

By means of the stereogrphic projection one can map a point on the sphere
n = (sin θ cos φ, sin θ sin φ, cos θ)
into a point on the plane w = (u, v)
w = u + i v =
(n_{1} + i n_{2})/(1  n_{3}) =
ctg(θ/2) e^{iφ}.
Note that all infinite points are mapped into one point  the "north
pole" of the sphere. By means of substitution

n_{1} = 2u/(1 + u^{2} + v^{2}),
n_{2} = 2v/(1 + u^{2} + v^{2}),
n_{3} = 1  2/(1 + u^{2} + v^{2}),
we get (see also continuum limit)
E = 2 ∫ dx dy
(u_{x}^{2} + v_{x}^{2} +
u_{y}^{2} + v_{y}^{2}) /
(1 + u^{ 2} + v^{ 2})^{2} =
2 ∫ dx dy
[(u_{x}  v_{y} )^{2} +
(u_{y} + v_{x} )^{2} +
2(u_{x} v_{y}  u_{y} v_{x} )] /
(1 + u^{ 2} + v^{ 2})^{2}. (*)
In the spherical coordinate system
u = ctg θ/2 cos φ , v = ctg θ/2 sin φ
the third term in the bracket is reduced to the topological charge
∫ sin θ
(φ_{x}θ_{y} 
φ_{y}θ_{x}) dx dy =
∫ sin θ dφ dθ =
∫ dΩ = 4π Q .
Thus from (*) it follows that
E ≥ 4π Q
and minimum is reached under the KoshiRiemannian condishions
u_{x}  v_{y} = 0,
u_{y} + v_{x} = 0 ,
i.e. when w(z) = u(x,y) + iv(x,y) is an analitic function of
z = x + iy.
Instantons and antiinstantons
If w(∞) → 1 , then ninstantons and nantiinstantons
solutions are
w = ∏_{i=1,n} (z  a_{i} )/(z 
b_{i} )
è w = ∏_{i=1,n} (z*  a_{i}
)/(z*  b_{i} ) ,
where a_{i} and α_{i} = Arg(a_{i} )
are radius and phase of ith instanton, and complex number
b_{i} determines its position.
Oneinstanton solution is
w = u + i v =
1  e^{iα} / z = ctg(θ/2) e^{iφ} ,
As since E is inariant under scaling transphormations x' = ax,
then instanton energy do not depend on its radius and a = 1 ,
b = (0 + 0i) are used. Antiinstanton differs by conjugation z*.
3D VRML field models (see
Why VRML?):
instanton with α=0, antiinstantons with
α=0, α=1.5,
"chupachups". 3D are usefull for introduction but
2D Java visualization seems me more informaive.
Applet below makes view from above on the vector of antiinstanton with
the phase α = 0 . At the right and bottom borders you see
crosssections along the vertical and horizontal grey lines. Red arrows are
directed to observer and blue ones  backwards.
w(∞) → 1 corresponds to the direction towards observer.
The backwards direction corresponds to w(z_{o}) = 1 .
Therefore the "pole" (the blue point) is placed at
z_{o} = e^{iα}/2 . At last for vectors placed
in the picture plane Re(w) = 0 , it corresponds to the circle centered
in the pole with radius 1/2 and passing through the coordinate
origine. Inside the circle (blue) arrows are directed towards observer
and outside the circle (red) arrows are directed backwards.
Controls:
Click mouse with Alt (Ctrl) to zoom in (out) the picture two times.
Draw the gray crossed lines to move them. You can see coordinates of the
crossing in the Status bar (below).
Press Enter to set new phase (a field).
Below different HUE colors are used for different spin phases φ.
"Top" spins (θ = 0) are white and "bottom" ones
(θ = π) are black.
Instantons (topological solitons) are looked as color round defects
in these pictures.
The sphere to sphere maps
By means of the stereographic projection one can "wrap" the plane
(x_{1}, x_{2}) into the sphere S_{x}^{2}
(all infinite spins are parallel as since all infinite points of the plane
are mapped into the "north pole" of the sphere).
Thus nfield makes a sphere to sphere map
S_{x}^{2} → S_{n}^{2} .
As like circle to circle maps, nonequivalent maps differ by topological
charge, i.e. how much times the sphere
S_{x}^{2} is wraped on the sphere
S_{n}^{2}.

Three vectors n(x),
n(x+dx_{1}) and
n(x+dx_{2}) "occupy" on the sphere
S_{n}^{2} an area (or space angle)
dΩ. For small dx_{1}, dx_{2}
the space angle is proportional to the volume between these vectors
(n ·[∂_{1}n ,
∂_{2}n]) dx_{1} dx_{2} .

Therefore the total topological charge is
Q =
^{1}/_{4π} ∫ dΩ(x) =
^{1}/_{4π} ∫ sin θ
dθ(x)dφ(x) = ^{1}/_{8π}
∫ d^{2}x ε_{μν}(n
[∂_{μ}n, ∂_{μ}n]) .
I'm grateful to D.E.Burlankov for discussions.
Contents
Excitations in 1D spin chain
Hedgehog zoo
updated 12 June 2004