Figure 5:
Asymptotic least square fits (solid curves) of the frequency of
the soft mode (data points) for square lattice circular systems with
12, 68, and 492 spins, using the functional form
.
Only the data with 
very close to 
were
used to obtain the fitted curves. This functional form fits well over
the full range of 
only for small systems (
).
For fixed boundary conditions, the soft mode for 
has been
fit to the following functional form, as suggested through the simple Ansatz by
Wysin [23];
where 
and 
are the fitting parameters. Generally,
if all the data where 
are used in the fit, this functional form
produces an accurate fit only for the smaller systems, up to about 24 spins.
More generally, we can use only a limited number of the data points nearest to
the zero frequency point, and apply this form there to estimate 
.
Some typical asymptotic fits are shown in Fig. 5, for systems with 12,
68, and 492 spins. The values of 
determined this way converge to
a limit near 
for the infinite sized system. The
frequency of the soft mode at 
, 
,
gives an indication of the overall frequency scale for this mode, and is shown
in Fig. 6, versus system size (numbers not obtained from any
fitting). The result is compared with an asymptotic fit to the function,
. This is close to a linear dependence on
inverse system length.
Figure 6:
Size dependence of the soft mode for square lattice
circular systems with N sites, using fixed boundary conditions.
The frequency 
of the soft mode at 
is shown on a log-log plot, and
compared with an asymptotic fit to the function, 
. This is close to a linear dependence on inverse
system diameter.
