next up previous
Next: Final Comments Up: Discussion and Conclusions Previous: The Broken Symmetry

Finite Size and Boundary Effects

There are some finite size and boundary effects in these results that cannot be avoided, but that do not invalidate the results. For example, the choice of fixed boundary conditions eliminates the Goldstone mode related to global rotation of the spins in the XY plane that is present for free boundary conditions. However, the fixed boundary condition has the advantage that it reduces the spin fluctuations at the boundary, whereas the free boundary condition artificially enhances those boundary fluctuations. These are minor differences. The effect of the finite sized system, for the most part, can be understood to produce a finite frequency spacing between the modes, that becomes smaller as the reciprocal system length. This causes the frequency scale of the soft mode (a) at , in Fig. 6, to go to zero for the infinite sized system, which is partly an artifact of the calculation, because this is the lowest mode for fixed boundary conditions. For the free boundary conditions, this soft mode lies higher up in the spectrum. In a real system of physical interest at some temperature above the Kosterlitz-Thouless temperature, we could not consider an isolated vortex and its normal modes, because entropic effects would always produce a length scale (i.e., correlation length) at which the nearest neighboring vortex would be found. Thus, it may not be necessary to consider the infinite sized system limit, because the neighboring vortices will produce an effective finite length scale over which we might think that the vortex is restricted.

Gary M Wysin
Mon Sep 11 12:02:10 CDT 1995