where

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.

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.
| |

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. |

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 dissociation of
bounded pais takes place - it is the Kosterlitz - Thouless phase transiton.
_{c} = 0.893 |

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. |

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