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. The lowest order toroidal term is the toroidal dipole. Its magnitude along direction i is given by

${\displaystyle T_{i}={\frac {1}{10c}}\int [r_{i}(\mathbf {r} \cdot \mathbf {J} )-2r^{2}J_{i}]\mathrm {d} ^{3}x.}$

Since this term arises only in an expansion of the current density to second order, it generally vanishes in a long-wavelength approximation.

However, a recent study comes to the result that the toroidal multipole moments are not a separate multipole family, but rather higher order terms of the electric multipole moments.[3]

In 1957, Yakov Zel'dovich found that because the weak interaction violates parity symmetry, a spin ​¹⁄₂ Dirac particle must have a toroidal dipole moment, also known as an anapole moment, in addition to the usual electric and magnetic dipoles.[4] The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is

${\displaystyle H\propto -d(\mathbf {\sigma } \cdot \mathbf {E} )-\mu (\mathbf {\sigma } \cdot \mathbf {B} )-a(\mathbf {\sigma } \cdot \nabla \times \mathbf {B} ),}$

where d, μ, and a are the electric, magnetic, and anapole moments, respectively, and σ is the vector of Pauli matrices.[5]

The nuclear toroidal moment of cesium was measured in 1997 by Wood et al..[6]

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

All dipole moments are vectors which can be distinguished by their differing symmetries under spatial inversion (P: 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"):

In condensed matter magnetic toroidal order can be induced by different mechanisms:[7]

• 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:[8] T = ∑_i r_i × S_i
• Formation of vortices by delocalized magnetic moments.
• On-site orbital currents (as found in multiferroic CuO).[9]
• Orbital loop currents have been proposed in copper oxides superconductors[10] that might be important to understand high-temperature superconductivity. Experimental verification of symmetry-breaking by such orbital currents has been claimed in cuprates through polarized neutron-scattering.[11]

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.

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. Both of these hallmark properties of a ferroic state have been demonstrated in an artificial ferrotoroidic model system based on a nanomagnetic array[12]

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.

All CPT self-conjugate particles, in particular the Majorana fermion, are forbidden from having any multipole moments other than toroidal moments.[13] At tree level, an anapole-only particle interacts only with external currents, not with free-space electromagnetic fields, and the interaction cross-section diminishes as the particle velocity slows. For this reason, heavy Majorana fermions have been suggested as plausible candidates for cold dark matter.[14][15]

• Stefan Nanz: Toroidal Multipole Moments in Classical Electrodynamics. Springer 2016. ISBN 978-3-658-12548-6