Toroidal moment

A toroidal moment is an independent term in the multipole expansion of electromagnetic fields besides magnetic and electric multipoles. While electric dipoles can be understood as separated charges and magnetic dipoles as circular currents, axial (or electric) toroidal dipoles describes toroidal charge arrangements whereas polar (or magnetic) toroidic dipole (also called anapole) correspond to the field of a solenoid bent into a torus.

Symmetry properties of dipole moments

All dipoles moments are vectors with different symmetries concerning spatial inversion (I: r -> -r) and time reversal (T: t -> -t). Either the dipole moment stays invariant under the symmetry transformation ("1") or it changes its direction ("-1"):

Dipole moment	I	T
axial toroidal dipole moment	1	1
electric dipole moment	-1	1
magnetic dipole moment	1	-1
polar toroidal dipole moment	-1	-1

History

The existence of a magnetic toroidal dipole, or so-called anapole moment, in the multipole expansion of electromagnetic fields was discovered by Zel’dovich in 1957. The nuclear toroidal moment of Cesium had been measured in 1997 by Wood et al.

File:Anapole.tif

Solenoid currents j (blue) inducing a toroidal magnetic moment (red).

Magnetic toroidal moments in condensed matter physics

In condensed matter magnetic toroidal order can be induced by different mechanisms:

• Order of localized spins breaking spatial inversion and time reversal. The resulting toroidal moment is described by a sum of cross products of the spins S_i of the magnetic ions and their positions r_i within the magnetic unit cell: T = Σ_i r_i x S_i
• Formation of vortices by delocalized magnetic moments.
• On-site orbital currents (as found in multiferroic CuO).

Magnetic toroidal moment and its relation to the magnetoelectric effect

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 P_i = ε_ijkT_jH_k (with ε being the Levi-Civita symbol). The resulting magnetoelectric tensor describing the cross-correlated response is thus antisymmetric.

Ferrotoroidicity in condensed matter physics

A phase transition to spontaneous long-range order of microscopic magnetic toroidal moments has been termed "ferrotoroidicity". It is expected to fill the symmetry schemes of primary ferroics (phase transitions with spontaneous point symmetry breaking) with a space-odd, time-odd macroscopic order parameter. A ferrotoroidic material would exhibit domains which could be switched by an appropriate field, e.g. a magnetic field curl.

The existence of ferrotoroidicity is still under debate and clear-cut evidence has not been presented yet—mostly due to the difficulty to distinguish ferrotoroidicity from antiferromagnetic order, as both have no net magnetization and the order parameter symmetry is the same.