Toroidal moment

A complex expression allows the current density J to be written as a sum of electric, magnetic, and toroidal moments using Cartesian[1] or spherical[2] differential operators.

Its magnitude along direction i is given by Since this term arises only in an expansion of the current density to second order, it generally vanishes in a long-wavelength approximation.

where d, μ, and a are the electric, magnetic, and anapole moments, respectively, and σ is the vector of Pauli matrices.

Either the dipole moment stays invariant under the symmetry transformation ("+1") or it changes its direction ("−1"): In condensed matter magnetic toroidal order can be induced by different mechanisms:[7] The presence of a magnetic toroidic dipole moment T in condensed matter is due to the presence of a magnetoelectric effect: Application of a magnetic field H in the plane of a toroidal solenoid leads via the Lorentz force to an accumulation of current loops and thus to an electric polarization perpendicular to both T and H. The resulting polarization has the form Pi = εijkTjHk (with ε being the Levi-Civita symbol).

A phase transition to spontaneous long-range order of microscopic magnetic toroidal moments has been termed ferrotoroidicity.

For this reason, heavy Majorana fermions have been suggested as plausible candidates for cold dark matter.