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]
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.