Hypertoroidal moment in complex dipolar structures

The very recent use of atomistic simulations to investigate low-dimensional ferroelectrics and ferromagnets has led to the discovery of a new order parameter that is associated with the formation and evolution of many complex dipolar structures (such as onion and flower states or double vortices). Such new order parameter has been named as the hypertoidal moment, involves a double cross product of the local dipoles with the vectors locating their positions, and provides a measure of subtle microscopic features. Here, the recent studies devoted to the discovery of such order parameter and how to control it in zero-dimensional systems are reviewed. We also give additional information, such as the symmetry, conjugate field and associated susceptibility of the electric and magnetic hypertoidal moments. A discussion about the existence of the hypertoidal moment and its evolution as a function of temperature and applied field, as well as its possible multi-values, is also provided for complex states (such as nanostripes and nanobubbles) in periodic dipolar systems.