"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 zo = 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) = no 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