The key to pinning down the existence and type of transition is a study of correlation length and in-plane susceptibility, because their divergences constitute the most direct evidence of a phase transition. These quantities are much more difficult to measure, and large lattices are required in order to avoid finite size effects. These key points are lacking in the previous works, and are the focus of our study. By extensive use of the Mark IIIfp Hypercube, we are able to measure spin correlations and thermodynamic quantities accurately on very large lattices (). Our work [Ding:90h;92a] provides convincing evidence that a phase transition does occur at a finite temperature in the extreme quantum case, spin-. At transition point, , the correlation length and susceptibility diverge exactly according to the form of Kosterlitz-Thouless (Equation 6.18).
We plot the correlation length, , and the susceptibility, , in Figures 6.18 and 6.19. They show a tendency of divergence at some finite . Indeed, we fit them to the form predicted by Kosterlitz and Thouless for the classical model
The fit is indeed very good ( per degree of freedom is 0.81), as shown in Figure 6.18. The fit for correlation length gives
A similar fit for susceptibility, is also very good ():
as shown in Figure 6.19. The good quality of both fits and the closeness of 's obtained are the main results of this work. The fact that these fits also reproduce the expected scaling behavior with
is a further consistency check. These results strongly indicate that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at . We note that this is consistent with the trend of the ``twist energy'' [Loh:85a] and that the rapid increase of vortex density near is due to the unbinding of vortex pairs. Figures 6.18 and 6.19 also indicate that the critical region is quite wide (), which is very similar to the spin-1/2 Heisenberg model, where the behavior holds up to . These two-dimensional phenomena are in sharp contrast to the usual second-order transitions in three dimensions.
Figure 6.18: Correlation Length and Fit. (a) versus T. The vertical line indicates diverges at ; (b) versus . The straight line indicates .
Figure: (a) This figure repeats the plot of Figure 6.18(a) showing on a coarser scale both the high temperature expansion (HTE) and the Kosterlitz-Thouless fit (KT). (b) Susceptibility and Fit
The algebraic exponent is consistent with the Ornstein-Zernike exponent at higher T. As , shifts down slightly and shows signs of approaching 1/4, the value at for the classical model. This is consistent with Equation 6.21.
Figure 6.20: Specific Heat . For , the lattice size is .
We measured energy and specific heat, (for we used a lattice). The specific heat is shown in Figure 6.20. We found that has a peak above , at around . The peak clearly shifts away from on the much smaller lattice. DeRaedt, et al. [DeRaedt:84a] suggested a logarithmic divergent in their simulation, which is likely an artifact of their small m values. One striking feature in Figure 6.20 is a very steep increase of at . The shape of the curve is asymmetric near the peak. These features of the curve differ from that in the classical XY model [Gupta:88a].