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.
Highlights
Three-dimensional systems described by orientational order, and more generally vector fields, are often seen to contain linelike geometrical features
In nematic liquid crystals as escaped disclinations and thick lines [1,2], cholesteric liquid crystals as λ lines or dislocations [3,4,5,6,7], and as double twist cylinders in blue phases [8,9], in superfluids they arise as the cores of vortices [10,11,12,13], and they can be identified in the Skyrmion [14] textures of ferromagnets [15,16,17,18,19,20,21,22], Bose-Einstein condensates [23,24,25,26,27,28], and liquid crystals [3,29,30,31,32,33]
In this paper we show that a large number of these linelike structures can be identified with a set of natural geometric singularities in a vector field, which we call umbilic lines
Summary
Three-dimensional systems described by orientational order, and more generally vector fields, are often seen to contain linelike geometrical features. In this paper we show that a large number of these linelike structures can be identified with a set of natural geometric singularities in a vector field, which we call umbilic lines These lines, which correspond to local minima of several free energies and Hamiltonians, attain topological significance through their correspondence to Skyrmions, giving a natural relationship between the geometric and topological features of orientational order. For arbitrary three-dimensional textures there is no general theoretical formalism for identifying localized linelike structures in liquid crystals, magnetic materials, or any other system with orientational order This may be contrasted with an analogous situation in optics where linelike degeneracies in the electromagnetic field known as C lines (points where the polarization is circular rather than elliptical) can be both observed experimentally and identified, in full generality, from theoretical expressions given in terms of the electric and magnetic fields [38,39,40,41,42,43,44]. The final section gives the global extension to the case of umbilic loops, which we see as having the greatest potential for future developments, both theoretical and experimental
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