Sigma models on curved space and bubble refraction in doped   antiferromagnets
  The dynamics of bubble solitons in two-dimensional isotropic antiferromagnets, inhomogeneously doped so that the exchange integral J becomes position dependent, is studied. In the usual continuum approximation, the system reduces to a nonlinear sigma model on a spacetime whose geometry depends on J(x). It is shown, both within the geodesic approximation, and by appealing to field theoretic conservation laws, that a bubble incident on a domain-wall inhomogeneity undergoes refraction in accordance with Snell's law, 1/J being identified with the refractive index, and that sufficiently oblique impacts result in total internal reflexion. Possible applications of this phenomenon to the construction of bubble lenses and bubble guides(in analogy with fibre-optic cables) are considered. 
J.M. Speight