A tensor density measure of topological charge in three dimensional nematic phases

A path independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor $\mathbf{D}$, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor $\mathbf{D}$ may be nonzero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behavior there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from $\mathbf{D}$.