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.

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.