Computational Physics

15.5 Reducing critical slowing down 501

Figure 15.3. A pair of vortices, one with positive and one with negative vorticity.

Figure 15.3. The vortices can be assigned avorticitywhich is roughly the ‘winding’
number of the spins along a closed path around the vortex – the vorticity assumes
values 2π,− 2 π(for higher temperatures 4πetc. can also occur). An isolated vortex
requires an amount of energy which scales logarithmically with the lattice size and
is thus impossible in a large lattice at finite temperature. However, vortex pairs of
opposite vorticity are possible; their energy depends on the distance between the
vortices and is equal to the Coulomb interaction (which is proportional to lnRfor
two dimensions) for separationsRlarger than the lattice constant. The system can
only contain equal numbers of positive and negative vortices. At low temperatures
the vortices occur in bound pairs of opposite vorticity (to be compared with electrical
dipoles), and the spin-waves dominate the behaviour of the model in this phase. It
turns out that the correlations are long-ranged:

〈θi−θj〉∝

1

|ri−rj|xT

, (15.95)

for large separation|ri−rj|, with a critical exponentxTwhich varies with tem-
perature. At the KT transition, the dipole pairs unbind and beyond the transition
temperatureTKTwe have a fluid of freely moving vortices (to be compared with a
plasma).
Imagine you have anXYlattice with fixed boundary conditions: the spins have
orientationθ=0 on the left hand side of the lattice and you have a handle which
enables you to set the fixed valueδof the spin orientation of the rightmost column
ofXY-spins. Turning the handle fromδ=0 at low temperatures, you will feel a
resistance as if it is attached to a spring. This is due to a nonvanishing amount of