Umbilic Lines in Orientational Order

TL;DR

This paper identifies umbilic lines as geometric singularities in three-dimensional chiral order fields, providing a new way to detect and analyze Skyrmion structures and their topological properties.

Contribution

It introduces a geometric framework for characterizing umbilic lines in chiral materials, linking them to topological invariants like the Hopf invariant.

Findings

Umbilic lines are natural geometric singularities in orientational order.

These lines can be used to localize and identify Skyrmion distortions.

The linking of umbilic lines relates to the Hopf invariant of the texture.

Abstract

Three-dimensional orientational order in systems whose ground states possess non-zero, chiral gradients typically exhibits line-like structures or defects: $\lambda$ lines in cholesterics or Skyrmion tubes in ferromagnets for example. Here we show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalisation of the umbilic points of a surface. We characterise these lines in terms of the natural vector bundles that the order defines and show that they give a way to localise and identify Skyrmion distortions in chiral materials -- in particular that they supply a natural representative of the Poincar\'{e} dual of the cocycle describing the topology. Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.