On topological spin excitations on a rigid torus

TL;DR

This paper investigates topological spin excitations on a toroidal surface, analyzing how their configurations and energies change with the torus's geometry, including limits to cylinders and spheres, revealing diverse vortex and soliton behaviors.

Contribution

It introduces a comprehensive study of classical spin configurations on a torus, connecting geometric limits to different topological excitations and their energetic properties.

Findings

Fractional soliton solutions on a torus.

Vortex configurations vary with torus geometry, including singularities.

Limits to cylinder and sphere geometries recover known topological states.

Abstract

We study Heisenberg model of classical spins lying on the toroidal support, whose internal and external radii are $r$ and $R$, respectively. The isotropic regime is characterized by a fractional soliton solution. Whenever the torus size is very large, $R\to\infty$, its charge equals unity and the soliton effectively lies on an infinite cylinder. However, for R=0 the spherical geometry is recovered and we obtain that configuration and energy of a soliton lying on a sphere. Vortex-like configurations are also supported: in a ring torus ($R>r$) such excitations present no core where energy could blow up. At the limit $R\to\infty$ we are effectively describing it on an infinite cylinder, where the spins appear to be practically parallel to each other, yielding no net energy. On the other hand, in a horn torus ($R=r$) a singular core takes place, while for $R<r$ (spindle torus) two such singularities appear. If $R$ is further diminished until vanish we recover vortex configuration on a sphere.