# The invariant theory of chiral order parameters

## TL;DR

This paper explores how chiral order parameters influence phase transitions using invariant theory to define chiral structural features and their energy contributions.

## Contribution

The paper introduces a framework for analyzing chiral order parameters through invariant polynomials in phase transitions.

## Key findings

- Invariant polynomials can describe chiral structural features in distorted crystal structures.
- Chiral order parameters contribute uniquely to the free energy in phase transitions.
- The method clarifies the role of chirality in low-symmetry phases.

## Abstract

In the Landau theory of phase transitions, the free energy of a system is expanded as a sum over the ring of invariant polynomials in the components of order parameters (e.g. atomic displacements, magnetic moments, lattice strains, polyhedral rotations, order-disorder parameters, etc.) belonging to irreducible representation of the parent symmetry group. The physical characteristics of the phase transition can then be understood in terms of the non-zero coefficients of the expansion. Features of Landau theory can also be applied to understand the properties of the low-symmetry phase when the parent phase employed is not physically accessible. In the present work, we explore the invariant polynomials that arise in phase transitions and otherwise-distorted crystal structures involving chiral order parameters, which can then be exploited to clearly define chiral structural features and their contributions to the free energy.

---
Source: https://tomesphere.com/paper/PMC12585436