Relations between defects in the bulk and on the surface of an ordered medium: a topological investigation

Abstract

Starting from a proposal of Volovik (1978) we investigate surface point, line, and wall defects with the aid of relative homotopy groups. The exact sequence of homotopy groups is used in interpreting the relations between surface and bulk singularities. In particular, we consider the possibilities for surface singularities to move into the bulk, for bulk singularities to leave the medium through the surface, and for singular loops to be broken apart by the surface. The theory is extended to the case where the surface induces a thermodynamic phase differing from that in the bulk. An example is given of a biaxial nematic surface with the Klein bottle as the order-parameter space. Due to the fundamental theorem for homotopy groups of fibre bundles, the classification scheme of surface defects is identical for all pairs of bulk and surface order-parameter spaces which are related by an inverse-bundle projection.