Vorteces in the XY model

As we know in the XY model spins rotate in a plane. Interaction energy of nearest neighbours pair is
    Eij = -J (sisj) = -J cos(ji - jj) ,
where i-th spin phase ji is measured e.g. from the horizontal axis in the counter-clockwise direction.
Energy of spin interaction is minimal in ordered state, when all spins are aligned. Therefore on a 3D lattice at low temperatures there is a phase transition in ordered state with non-zero magnetization. However on a 2D lattice alignned spins are unstable with respect to long-wave fluctuations. I.e. small fluctuations are accumulated on infinite lattice and destroy long order at any finite temperature.
XY But in the 2D XY model there are interesting excitations with nontrivial topology - vorteces and anti-vorteces (marked to the left by red and green squares). Under path-tracing around a vortex or anti-vortex spins complete revolution on +-2p . 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.
vertex 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 vorteces and all 3 lower ones are anti-vorteces.
The XY model is dual to the 2D Coulomb model. Vorteces and anti-vorteces are dual to electrical charges with different signs. Therefore vortex and anti-vortex attract each other and annihilate. For periodic boundary conditions the phase difference under tracing along the border is zero, therefore the number of vorteces is equal to the number of anti-vorteces (the total charge is 0).

Birth and annihilation of the vortex - antivortex pair

vertex2 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 > Tc = 0.893 dissociation of bounded pais takes place - it is the Kosterlitz - Thouless phase transiton.
3D 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 vorteces in the local copy of of the Rongfeng Sun's applet (I only emphasized verteces).
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updated 20 Apr 2004