# Octupolar Tensors for Liquid Crystals

## TL;DR

This paper provides an algebraic characterization of octupolar tensors in liquid crystals, revealing possible intra-octupolar transitions and offering a quantitative framework for understanding molecular orientations.

## Contribution

It introduces a closed-form algebraic expression for the dome-shaped and separatrix surfaces of octupolar tensors, advancing the understanding of phase transitions in liquid crystals.

## Key findings

- Derived algebraic expressions for key tensor surfaces.
- Identified potential intra-octupolar transition points.
- Enhanced the theoretical framework for liquid crystal phases.

## Abstract

A third-order three-dimensional symmetric traceless tensor, called the \emph{octupolar} tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with \emph{three} maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a \emph{separatrix} surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible \emph{intra-octupolar} transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.