The XY model is the simplest O(N) model, having **N=2**, the O(N) model
being a set of rotors (N-component continuous valued spins) on an
N-sphere. For , this model is asymptotically free
[Polyakov:75a], and for **N=3**, there exist so called instanton
solutions. Some of these properties are analogous to those of gauge
theories in four dimensions; hence, these models are interesting. In
particular, the O(3) model in two dimensions should
shed some light on the asymptotic freedom of QCD (SU(3)) in four
dimensions. The predictions of the renormalization group for the
susceptibility and inverse correlation length (i.e., mass gap)
**m** in the O(3) model are [Brezin:76a]

and

respectively. If **m** and vary according to these equations,
without the correction of order , they are said to follow
asymptotic scaling. Previous work was able
to confirm that this picture is qualitatively correct, but was not able
to probe deep enough in the area of large correlation lengths to obtain
good agreement.

The combination of the over-relaxed algorithm and the computational power of the FPS T-Series allowed us to simulate lattices of sizes up to . We were thus able to simulate at coupling constants that correspond to correlation lengths up to 300, on lattices where finite-size effects are negligible. We were also able to gather large statistics and thus obtain small statistical errors. Our simulation is in good agreement with similar cluster calculations [Wolff:89b;90a]. Thus, we have validated and extended these results in a regime where our algorithm is the only known alternative to clustering.

**Table 4.8:** Coupling Constant, Lattice Size, Autocorrelation Time, Number of
Overrelaxed Sweeps, Susceptibility, and Correlation Length for the O(3)
Model

We have made extensive runs at 10 values of the coupling constant. At the
lowest , several hundred thousand sweeps were collected, while for the
largest values of , between **50,000** and **100,000** sweeps were made.
Each sweep consists of between 10 iterations through the lattice at the
former end and 150 iterations at the latter. The statistics we have gathered
are equivalent to about 200 days, use of the full 128-node FPS machine.

Our results for the correlation length and susceptibility for each coupling and lattice size are shown in Table 4.8. The autocorrelation times are also shown. The quantities measured on different-sized lattices at the same agree, showing that the infinite volume limit has been reached.

To compare the behavior of the correlation length and susceptibility with the asymptotic scaling predictions, we use the ``correlation length defect'' and ``susceptibility defect'' , which are defined as follows: , , so that asymptotic scaling is seen if , go to constants as . These defects are shown in Figures 4.22 and 4.23, respectively. It is clear that asymptotic scaling does not set in for , but it is not possible to draw a clear conclusion for -though the trends of the last two or three points may be toward constant behavior.

**Figure 4.22:** Correlation Length Defect Versus the Coupling Constant for the O(3)
Model

**Figure 4.23:** Susceptibility Defect Versus the Coupling Constant for the O(3) Model

**Figure 4.24:** Decorrelation Time Versus Number of Over-relations Sweeps
for Different Values of

We gauged the speed of the algorithm in producing statistically independent
configurations by measuring the autocorrelation time . We used this to
estimate the dynamical critical exponent **z**, which is defined by . For constant , our fits give . However,
we discovered that by increasing in rough proportion to , we
can improve the performance of the algorithm significantly. To compare the
speed of decorrelation between runs with different , we define a new
quantity, which we call ``effort,'' . This measures
the computational effort expended to obtain a decorrelated configuration. We
define a new exponent from , where
is chosen to keep constant. We also found that the behavior of
the decorrelation time can be approximated over a good range by

A fit to the set of points (, , ) gives
, . Thus, is
significantly lower than **z**. Figure 4.24 shows versus
, with the fits shown as solid lines.

Wed Mar 1 10:19:35 EST 1995