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

TL;DR

This paper introduces a tensor density measure for topological charge in three-dimensional nematic phases, enabling the identification and characterization of line and point defects through a path-independent integral and a disclination density tensor.

Contribution

It presents a novel tensor density measure and a disclination density tensor for topological defects in nematic phases, linking defect charge to tensor properties and behavior.

Findings

The measure yields topological charge via path-independent integration.

The disclination density tensor locates and characterizes defects.

The tensor behaves consistently in defect-free and defect-rich configurations.

Abstract

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}$.