Incommensurate toroidal phase transitions in magnetic classes

Abstract

Three-dimensional and two-dimensional representations of magnetic crystal point groups, according to which a toroidal moment transforms and which admit a Lifshitz invariant, are considered. Independent invariants necessary for a description of a sequence of initial-incommensurate-commensurate phase transitions are presented. Symmetry groups of commensurate phases, having a spontaneous toroidal moment and hence a magnetoelectric effect, are listed.

Key Words:

• toroidal moment
• Lifshitz invariant
• magnetoelectric tensor