But it follows from the picture that there are also "twisted" n-times solutions φ(x) = 2πn x/L which correspond to local minima of energy E_{n} ~ n^{2}. |
Topologically non-equivalent maps S_{x}^{1} → S_{n}^{1} differ by the topological charge Q which determines how many times the circle S_{x}^{1} is "wraped" on the circle S_{n}^{1}. |
To calculate Q one can integrate throughout the chain phase difference
dφ(x) = φ'(x) dx between close vectors s(x) and
s(x+dx)
Q = ^{1}/_{2π} ∫ φ'(x) dx = Δφ/2π . |
VortecesTopoligical "deffects" on a plane appear if one take a contour with Q ≠ 0 . Then there are vorteces within it. In three dimensional space one shall draw a surface on this contour and then vortex threads cross it. |
For the continuum XY-chain with Q ≠ 0 above it is impossible to move upwords smoothly all "blue" arrows rotating spins in the picture plane. But one makes it easy by rotation around the x axis in 3D space (i.e. for the Heisenberg spins). Therefore Heisenberg chains have no topological excitations. It is equivalent to the statement that one can shrink into a point any map of a circle to a sphere (i.e. any contour on a sphere). |