1205.0287 (Vagson L. Carvalho-Santos et al.)

Geometry dependent energy of topological vortices on surfaces with
negative and variable curvature    [PDF]

Vagson L. Carvalho-Santos, Felipe A. Apolonio

We have applied the Anisotropic Heisenberg Model on the surfaces of the catenoid and hyperboloid, which present negative and variable curvature. Two kinds of topological excitations were considered. The first one is given when we take $\lambda=0$ (isotropic model), which yields to the sine-Gordon equation and we have obtained a $\pi$-soliton like solution, so corresponding to the first homotopy class of the second homotopy group of the mapping of the spin sphere. The second one is given by the $\lambda=-1$ case, that consists in the XY model and a vortex turning around the surface appears. The results show that the vortex energy depends of the length scale of the underlying geometry and, for small central radius (CR), the hyperboloid presents lower vortex energy than that on a catenoid, which, as well as the cylinder case, has its vortex energy varying with the characteristic length $1/\rho$. We also have shown that for any CR value, the lowest value to the vortex energy occurs on the polar hyperboloid surface.

View original: http://arxiv.org/abs/1205.0287