# "Hedgehog" zoo

# Instantons and anti-instantons

You see below two instantons with different phases (you will see others
when you change the phase value).

*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).
Anti-instanton differs from instanton by conjugation *z* = x - iy*.
Therefore you can get it from corresponding instanton
by exchanging of two vectors *(x, y)* and
*(x, -y)* symmetric with respect to the
*x* axis. Anti-instanton's pole is rotated as
*z*_{o} = e^{-iα}/2 (i.e. in reverse order with
respect to the instanton's one).

You see that spins (arrows) in the horizontal cross-section
*y = 0* (right above the text) can be aligned by rotation
around the *x* axis. It is easy to align field along the
*x = 0.5* cross-section. But it is impossible to align
all spins simultaneously, therefore instantons are not topologically
equivalent to the trivial map **n**(x,y) = **n**_{o}
with *Q = 0* .

# Two-instantons solitons

It is not very difficult to imagine two remote instantons.
It is easy too to calculate fields in the limit case when two identical
instantons are in the same point. You see below such anti-instantons for
*n = 2* .

Contents
Topological solitons in the Heisenberg
ferromagnetic
Geometric phase transition

*updated* 10 June 2004