The geodesic rule for higher codimensional global defects

TL;DR

This paper extends the geodesic rule to higher codimensional global defects, proposing geometric and stochastic approaches to predict defect formation, with applications to systems like nematic liquid crystals.

Contribution

It introduces a generalized geodesic rule for higher codimensional defects using geometric and diffusion methods, linking convex hulls to defect formation.

Findings

Geometric structures are totally geodesic submanifolds.

Diffusion equations lead to the same defect density predictions.

Probability calculation for vortex formation in nematic liquid crystals.

Abstract

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On the other hand, stochastic arguments lead to a diffusion equation approach, from which the geodesic rule is deduced. It turns out that the most appropriate geometric structure that one should consider is the convex hull of the values of the order parameter on the causal volumes whose collision gives rise to the defect. We explain why these two approaches lead to similar results when calculating the density of global defects by using a theorem of Cheeger and Gromoll. We present a computation of the probability of formation of strings/vortices in the case of a system, such as nematic liquid crystals, whose vacuum is $\mathbb{R}P^2$.