Microscopic models have been used to reveal the existence of an order parameter that is associated with many complex dipolar structures in magnets and ferroelectrics. This order parameter involves a double cross product of the local dipoles with their positions. It provides a measure of subtle microscopic features, such as the helicity of the two domains inherent to onion states, curvature of the dipolar pattern in flower states, or characteristics of sets of vortices with opposite chirality (e.g., distance between the vortex centers and/or the magnitude of their local dipoles).

• Received 13 November 2007

DOI:https://doi.org/10.1103/PhysRevB.77.060101