The invariant theory of chiral order parameters

When studying phase transitions with Landau theory, the free energy of a system is expanded as a sum over the ring of invariant polynomials in the components of order parameters. Order parameters include atomic displacements, magnetic moments, lattice strains, polyhedral rotations, order-disorder parameters, etc., and order parameters belong to irreducible representations of the parent symmetry group. The physical characteristics belonging to the phase transition can be understood in terms of the non-zero coefficients of the expansion. Low-symmetry phase properties can be understood using features of Landau theory even when the parent phase is not accessible. In the present work, we explore invariant polynomials that arise in phase transitions and otherwise distorted crystal structures involving chiral order parameters.