Where are the Hedgehogs in Nematics?

TL;DR

This paper explains why point defects, or 'hedgehogs', are rarely observed in nematic liquid crystals after rapid phase transitions, due to topological constraints reducing their expected density.

Contribution

It provides a topological analysis showing the extremely low expected density of hedgehog defects in nematics, clarifying experimental observations.

Findings

Point defect density is approximately 10^{-8} per initial domain.

Topological constraints explain the scarcity of hedgehogs.

No point defects are observed until the defect network coarsens significantly.

Abstract

In experiments which take a liquid crystal rapidly from the isotropic to the nematic phase, a dense tangle of defects is formed. In nematics, there are in principle both line and point defects (``hedgehogs''), but no point defects are observed until the defect network has coarsened appreciably. In this letter the expected density of point defects is shown to be extremely low, approximately $10^{-8}$ per initially correlated domain, as result of the topology (specifically, the homology) of the order parameter space.