Umbilic Lines in Orientational Order

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 Poincaré 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.

14 pages, 9 figures

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• Moving frames and compatibility conditions for three-dimensional director fields
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• Screw Symmetry, Chiral Hydrodynamics and Odd Instability in Active Cholesterics
• Holonomy-based Diagnostic of Strain Compatibility in Birefringence Imaging of Stress-induced Ferroelectric SrTiO${}_3$