Computational Physics

502 Computational methods for lattice field theories

free energy which is needed to change the orientation of the spins on the right hand
column. This excess free energy scales as

F∝δ^2 (15.96)

for small anglesδ. At the KT temperature the force needed to pull the handle
drops to zero, as the vortex system has melted, which is noticeable through the
proportionality constantdropping to zero.
The quantityis called thespin-wave stiffness[32]orhelicity modulus. It can
be calculated in a system with periodic boundary conditions using the following
formula[32]:

=

J

2 L^2







〈

∑

〈ij〉

cos(θi−θj)

〉

−

J

kBT

〈[

∑

i

sin(θi−θi+ˆex)

] 2 〉

−

J

kBT

〈[

∑

i

sin(θi−θi+ˆey)

] 2 〉



. (15.97)

From the Kosterlitz-Thouless theory [29, 33, 34] it follows that the helicity mod-
ulus has a universal value= 2 kBTKT/πat the KT transition. The drop to zero is
smooth for finite lattices but it becomes steeper and steeper with increasing lattice
size.Figure 15.4shows/Jas a function ofkBT/J. The line/J= 2 (kBT/J)/π
is also shown and it is seen that the intersection of the helicity modulus curve with
this line gives the value from which the helicity modulus drops to zero. You can
check your program by reproducing this graph.
Edwards and Sokal have found that for theXY model the dynamic critical
exponent in the low-temperature phase is zero or almost zero [28].

15.5.3 Geometric cluster algorithms

The cluster algorithms described so far flips or rotates spins on a lattice. In fact,
Wolff’s version of the algorithm for theXYmodel boils down to flipping an Ising
spin. Cluster algorithms strongly rely on a reflection symmetry of the Hamiltonian:
flipping all spins does not affect the Hamiltonian. This is the reason that a simple
distinction can be made for pairs which may be frozen and those which certainly
will not. The same holds for theq-state Potts model: there we have a symmetry
under permutation of all spin values. Another way of looking at this is that flipping
a large cluster in an Ising model with a magnetic field, yields an energy loss or gain
proportional to the volume (surface in two dimensions), which leads to very low
acceptance rates in phases with a majority spin. Hence cluster algorithms will be
less efficient for such systems.