Biaxial escape in nematics at low temperature

TL;DR

This paper proves that at low temperatures, minimizers of the Landau-de Gennes free energy in nematic liquid crystals do not vanish and must become strongly biaxial, revealing a new topological behavior.

Contribution

It demonstrates that low-temperature minimizers are non-vanishing and must exhibit biaxial escape, contrasting with simplified models and advancing understanding of nematic topologies.

Findings

Minimizers do not vanish at low temperature.

Minimizers must become strongly biaxial.

They lie in a different homotopy class from uniaxial configurations.

Abstract

In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain $\Omega\subset \mathbb{R}^3$. We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and H\'elein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in $\Omega$. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class.