# Umbilic Lines in Orientational Order

Thomas Machon, Gareth P. Alexander

Research output: Contribution to journalArticle (Academic Journal)peer-review

25 Citations (Scopus)

## 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.
Original language Undefined/Unknown Physical Review X https://doi.org/10.1103/PhysRevX.6.011033 Published - 28 Mar 2016

### Bibliographical note

14 pages, 9 figures

## Keywords

• cond-mat.soft