The XY (or O(2)) model consists of a set of continuous valued spins regularly arranged on a two-dimensional square lattice. Fifteen years ago, Kosterlitz and Thouless (KT) predicted that this system would undergo a phase transition as one changed from a low-temperature spin wave phase to a high-temperature phase with unbound vortices. KT predicted an approximate transition temperature, , and the following unusual exponential singularity in the correlation length and magnetic susceptibility:
where and the correlation function exponent is defined by the relation .
Our simulation [Gupta:88a] was done on the 128-node FPS (Floating Point Systems) T-Series hypercube at Los Alamos. FPS software allowed the use of C with a software model similar (communication implemented by subroutine call) to that used on the hypercubes at Caltech. Each FPS node is built around Weitek floating-point units, and we achieved per node in this application. The total machine ran at , or at about twice the performance of one processor of a CRAY X-MP for this application. We use a 1-D torus topology for communications, with each node processing a fraction of the rows. Each row is divided into red/black alternating sites of spins and the vector loop is over a given color. This gives a natural data structure of () words for lattices of size . The internode communications, in both lattice update and measurement of observables, can be done asynchronously and are a negligible overhead.
Figure 4.19: Autocorrelation Times for the XY Model
Previous numerical work was unable to confirm the KT theory, due to limited statistics and small lattices. Our high-statistics simulations are done on , , , and lattices using a combination of over-relaxed and Metropolis algorithms which decorrelates as . (For comparison, a Metropolis algorithm decorrelates as .) Each configuration represents over-relaxed sweeps through the lattice followed by Metropolis sweeps. Measurement of observables is made on every configuration. The over-relaxed algorithm consists of reflecting the spin at a given site about , where is the sum of the nearest-neighbor spins, that is,
This implementation [Creutz:87a], [Brown:87a] of the over-relaxed algorithm is microcanonical, and it reduces critical slowing down even though it is a local algorithm. The ``hit'' elements for the Metropolis algorithm are generated as , where is a uniform random number in the interval , and is adjusted to give an acceptance rate of 50 to 60 percent. The Metropolis hits make the algorithm ergodic, but their effectiveness is limited to local changes in the energy. In Figure 4.19, we show the autocorrelation time vs. the correlation length ; for , we extract , and for , we get .
Table 4.7: Results of the XY Model Fits: (a) in T, and (b) in T Assuming the KT Form. The fits KT1-3 are pseudominima while KT4 is the true minimum. All data points are included in the fits and we give the for each fit and an estimate of the exponent .
We ran at 14 temperatures near the phase transition and made unconstrained fits to all 14 data points (four parameter fits according to Equation 4.24), for both the correlation length (Figure 4.20) and susceptibility (Figure 4.21). The key to the interpretation of the data is the fits. We find that fitting programs (e.g., MINUIT, SLAC) move incredibly slowly towards the true minimum from certain points (which we label spurious minima), which, unfortunately, are the attractors for most starting points. We found three such spurious minima (KT1-3) and the true minimum KT4, as listed in Table 4.7.
Figure 4.20: Correlation Length for the XY Model
Figure 4.21: Susceptibility for the XY Model
Thus, our data was found to be in excellent agreement with the KT theory and, in fact, this study provides the first direct measurement of from both and data that is consistent with the KT predictions.