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# NUMERICAL APPLICATION

First, for a given value of , the static vortex structure was found using the relaxation procedure described in Section ii. This static structure acquires nonzero out-of-plane spin components for . Then, the eigenvalue problem [Eq. (2.14)] was solved numerically for the eigenvectors and corresponding eigenvalues, for systems with hundreds of sites. Because there are two variables per site, the size of the matrix to diagonalize is , where N is the number of sites in the lattice. Calculations were performed for approximately circular shaped systems, with the vortex centered in the system. For these finite systems, there is a choice of either a free boundary condition or a fixed boundary condition. For the free boundary condition, the lattice is cut off along a circular boundary, and then the sites on the edge of the system simply have a lower coordination number than those in the interior, and have a lower effective stiffness as a result. For the fixed boundary condition, the set of spins on the boundary of the system is coupled to an extra set of spins that are outside the system, still on the same lattice, but held fixed in the directions that the static in-plane vortex would give them. In this way, even the spins at the edge of the system have the same coordination number as those in the interior, however, they are coupled to spins that do not move that are outside the system. Therefore, this fixed boundary condition, which is stiffer than the free boundary condition, tends to give higher eigenfrequencies. Additionally, the fixed boundary condition results in values for that converge faster to a limit with increasing system size, and for this reason, most of the results reported here were produced with fixed boundary conditions.

For a given system, the eigenspectrum was determined for a sequence of closely spaced values of between 0 and 1. An eigenvector for one value of was projected onto the eigenvectors for the previous value of , and then identified with the one with which the overlap was the greatest. This allowed the eigenfrequencies for the different modes to be tracked as a function of . For a system with N sites, N modes of positive frequency, corresponding to the operators, resulted, along with an equivalent set of N modes of negative frequency, corresponding to the conjugate operators. Double precision was used so that degenerate pairs of eigenmodes could be unambiguously identified, a necessity for performing the eigenvector overlaps.

Next: RESULTS: SQUARE LATTICE Up: NORMAL MODES OF VORTICES Previous: NORMALIZATIONSPIN EXPECTATIONS

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