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].
