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      Umbilic Lines in Orientational Order

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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: \(\lambda\) 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.

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          Author and article information

          Journal
          2015-07-31
          Article
          10.1103/PhysRevX.6.011033
          1507.08875
          00e40451-7bbd-47b6-bc3f-dc13a73d4962

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          Phys. Rev. X 6, 011033 (2016)
          14 pages, 9 figures
          cond-mat.soft

          Condensed matter
          Condensed matter

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