The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect

TLDR

The toroidal moment is closely linked to the antisymmetric magnetoelectric effect, yet ferrotoroidic systems exhibit intrinsic limits on magnetoelectric response compared to other ferroic materials. This review surveys toroidal moments and ferrotoroidic ordering, identifies open questions, and highlights prospective applications of ferrotoroidic materials. The authors define the local toroidal moment, discuss its symmetry properties and periodic extension, and outline experimental probes such as magnetoelectric susceptibility, X‑ray diffraction, optical techniques, and bulk toroidization measurements. Ferrotoroidicity is shown to arise as a primary ferroic order that can be described both microscopically through multipole expansion and macroscopically via symmetry‑based free‑energy expansion.

Abstract

The concept of toroidal moments in condensed-matter physics and their long-range ordering in a so-called ferrotoroidic state is reviewed. We show that ferrotoroidicity as a form of primary ferroic order can be understood both from microscopic (multipole expansion) and macroscopic (symmetry-based expansion of the free energy) points of view. The definition of the local toroidal moment and its transformation properties under the space-inversion and time reversal operations are highlighted and the extension to periodic bulk systems is discussed. Particular attention is paid to the relationship between the toroidal moment and the antisymmetric magnetoelectric effect and to limitations of the magnetoelectric response in ferrotoroidic systems and ferroic materials in general. Experimental access to the ferrotoroidic state by magnetoelectric susceptibility measurements, x-ray diffraction and optical techniques or direct measurement of the bulk toroidization is discussed. We outline the pertinent questions that should be clarified for continued advancement of the field and mention some potential applications of ferrotoroidic materials.