Vortices 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(φi - φj ) ,
where i-th spin phase φi 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 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.
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 vortices and all 3 lower ones are anti-vortices.
The XY model is dual to the 2D Coulomb model. Vortices and anti-vortices 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 vortices is equal to the number of anti-vortices (the total charge is 0).
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.
You can watch random spin distributions cooling and vortices formation at Vortices in the XY model (WebCL and GPU based MC simulations).

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.

Vortices in 2D antiferromagnets

You see below vortex and antivortex in antiferromagnet model (J < 0 and T = 0). Pictures from the paper Vortex dynamics in two-dimensional antiferromagnets S.Komineas and N.Papanicolaou, Nonlinearity 11 (1998) 265-290 arxiv.org/abs/cond-mat/9612043.
 
Further you see vortex-vortex, vortex-antivortex pairs and a vortex with Δφ = 4π in ferromagnet.
 
 

You can play with 2D vortices in the local copy of of the Rongfeng Sun's applet (I only emphasized verteces).


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updated 31 Oct 2010