Due to E(φ) periodicity in the XY model there are interesting excitations with nontrivial topology - vortices and anti-vortices (marked to the left by red and green squares). Under path-tracing around a vortex or anti-vortex spins complete revolution on ±2π . Under movement in the direction of phase growth (i.e. if spins rotate in the counter-clockwise direction) vertex is traced in the counter-clockwise direction and anti-vortex is traced in the clockwise direction. | |
As since spin rotation depends on the difference of spin directions along a path, therefore it is not changed if we turn all spins together on the same angle. In Fig.2. all 3 uper pictures are vortices and all 3 lower ones are anti-vortices. |
Similar to the Ising model spins whit different φ can be painted in different colors Hue(φ/2π) (see the picture to the left). It is evident that vortices and anti-vortices correspond to the singular points (defects) where all colors meet together. |
At low temperature all spins are aligned locally in the same direction. Inversion of a spin by thermal fluctuations generates a vortex - anti-vortex pair (see Fig.3). Reverse fluctuation results in anihilation of this pair. Due to vortex - anti-vortex attraction at low temperature they make bounded pairs. At T > T_{c} = 0.893 dissociation of bounded pais takes place - it is the Kosterlitz - Thouless phase transiton. |
On a 3D lattice vortex (antivortex) is represented as an arrow. Its direction is determined by the right-hand screw rule when an elementary placket is traced (moving in the direction of phase growth). These arrows make vortex threads and rings. You see that the fluctuation of spin inversion generates a vortex ring. As since opposite sides of a ring are attracted (similar to a vortex - anti-vortex pair), therefore it tries to collapse and annihilate. |
You can play with 2D vortices in the local copy of of the Rongfeng Sun's applet (I only emphasized verteces).