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 ;
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.