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arXiv · 1507.08875

Umbilic Lines in Orientational Order

Thomas Machon·Gareth P. Alexander
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Abstract

Three-dimensional orientational order in systems whose ground states possess non-zero, chiral gradients typically exhibits line-like structures or defects: $λ$ 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.

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Thomas Machon, Gareth P. Alexander. 2015-07-31. Umbilic Lines in Orientational Order. https://doi.org/10.1103/physrevx.6.011033

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