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
= -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.
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).
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
You can watch random spin distributions cooling and vortices formation at
Vortices in the XY model (WebCL and GPU
based MC simulations).
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 > Tc = 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
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
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).
Next: Excitations in 1D spin chain
updated 31 Oct 2010