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A '''toroidal moment''' is an independent term in the [[multipole expansion]] of [[electromagnetic field]]s besides magnetic and electric [[multipole]]s.  In the electrostatic multipole expansion, all charge and current distributions can be expanded into a complete set of electric and magnetic multipole coefficients.  However, additional terms arise in an electrodynamic multipole expansion.  The coefficients of these terms are given by the toroidal multipole moments as well as time derivatives of the electric and magnetic multipole moments.  While [[electric dipole]]s can be understood as separated charges and [[magnetic dipole]]s as circular currents, axial (or electric) toroidal dipoles describes toroidal charge arrangements whereas polar (or magnetic) toroidal dipole (also called '''anapole''') correspond to the field of a [[solenoid]] bent into a [[torus]].

== Classical toroidal dipole moment ==

A complicated expression allows the [[current density]] '''J''' to be written as a sum of electric, magnetic, and toroidal moments using Cartesian<ref>{{Citation
 | last = Radescu
 | first = E., Jr.
 | last2 = Vaman
 | first2 = G.
 | title = Cartesian multipole expansions and tensorial identities
 | journal = Progress In Electromagnetics Research B
 | volume = 36
 | pages = 89-111
 | year = 2012
 | url = http://www.jpier.org/pierb/pier.php?paper=11090702
 | doi = 10.2528/PIERB11090702 }}</ref> or spherical<ref>{{Citation
 | last = Dubovik
 | first = V. M.
 | last2 = Tugushev
 | first2 = V. V.
 | title = Toroid moments in electrodynamics and solid-state physics
 | journal = Physics Reports
 | volume = 187
 | issue = 4
 | pages = 145–202
 |date=March 1990
 | doi = 10.1016/0370-1573(90)90042-Z }}</ref> differential operators.  The lowest order toroidal term is the toroidal dipole.  Its magnitude along direction ''i'' is given by

:<math>T_i=\frac{1}{10c} \int [r_i(\mathbf{r}\cdot\mathbf{J})-2r^2J_i] \mathrm{d}^3x.</math>

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

== Quantum toroidal dipole moment ==
In 1957, [[Yakov Zel'dovich]] found that because the [[weak interaction]] [[weak interaction#Violation of symmetry|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.<ref>Ya. B. Zel'dovich, Zh. Eksp. Teor. Fiz. 33, 1531 (1957)[JETP 6, 1184 (1957)].</ref>  The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is

:<math>H \propto -d(\mathbf{\sigma}\cdot\mathbf{E}) -\mu(\mathbf{\sigma}\cdot\mathbf{B}) -a(\mathbf{\sigma}\cdot \nabla\times\mathbf{B}),</math>

where ''d'', ''μ'', and ''a'' are the electric, magnetic, and anapole moments, respectively, and '''σ''' is the vector of [[Pauli matrices]].<ref>{{Citation
 | last = Dubovik
 | first = V. M.
 | last2 = Kuznetsov
 | first2 = V. E.
 | title = The Toroid Moment of Majorana Neutrino
 | journal = Int. J. Mod. Phys.
 | volume = A13
 | pages = 5257-5278
 | year = 1998
 | arxiv = hep-ph/9606258
 | doi = 10.1142/S0217751X98002419 }}</ref>

The nuclear toroidal moment of [[cesium]] had been measured in 1997 by Wood ''et al''.<ref>C.S. Wood, S.C. Bennett, D. Cho, B.P. Masterson, J.L. Roberts, C.E. Tanner and C.E. Wieman; Measurement of Parity Nonconservation and an Anapole Moment in Cesium; Science '''275''', 1759–1763 (1997).</ref>
[[File:Solenoid currents inducing a toroidal magnetic moment.tif|thumb| Solenoid currents '''j''' (blue) inducing a toroidal magnetic moment (red).]]

== Symmetry properties of dipole moments ==

All dipole moments are vectors which can be distinguished by their differing symmetries under [[Parity (physics)|spatial inversion]] (P: '''r''' ↦ −'''r''') and [[T-symmetry|time reversal]] (T: ''t'' ↦ −''t''). Either the dipole moment stays invariant under the symmetry transformation ("+1") or it changes its direction ("−1"):

{| class="wikitable"
|-
! Dipole moment 
! style="font-weight:normal" |P
! style="font-weight:normal" |T
|-
| axial toroidal dipole moment || +1 || +1
|-
| [[electric dipole moment]] || −1 || +1
|-
| [[magnetic dipole moment]] || +1 || −1
|-
| polar toroidal dipole moment || −1 || −1
|}

== Magnetic toroidal moments in condensed matter physics ==
In [[condensed matter]] magnetic toroidal order can be induced by different mechanisms:<ref>N.A. Spaldin, M. Fiebig and M. Mostovoy;
The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect; J. Phys. Cond. Mat. '''20''', 434203 (2008).</ref>

* Order of localized [[Spin (physics)|spins]] breaking spatial inversion and time reversal. The resulting toroidal moment is described by a sum of cross products of the spins '''S'''<sub>''i''</sub> of the magnetic ions and their positions '''r'''<sub>''i''</sub> within the magnetic unit cell:<ref>C. Ederer and N.A. Spaldin; Towards a microscopic theory of toroidal moments in bull periodic crystals; Phys. Rev. B '''76''', 214404 (2007).</ref> '''T''' = ∑<sub>''i''</sub>&nbsp;'''r'''<sub>''i''</sub>&#8239;×&#8239;'''S'''<sub>''i''</sub>
* Formation of vortices by delocalized magnetic moments.
* On-site [[Atomic orbital|orbital]] currents (as found in [[multiferroic]] [[CuO]]).<ref>V. Scagnoli, U. Staub, Y. Bodenthin, R. de Souza, M. Garcia-Fernandez, M. Garganourakis, A. Boothroyd, D. Prabhakaran, D. and S. Lovesey;
Observation of Orbitals Currents in CuO; Science '''332''', 696 (2011)</ref>
* Orbital loop currents have been proposed in copper oxides superconductors  <ref> C.M. Varma; Theory of the pseudogap state of the cuprates; Phys. Rev. B 73, 155113 (2006) </ref> 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. <ref> B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud and P. Bourges; Magnetic Order in the Pseudogap Phase of High-TC Superconductors; Phys. Rev. Lett. 96, 197001 (2006)</ref>

== 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''<sub>''i''</sub> = ε<sub>''ijk''</sub>''T''<sub>''j''</sub>''H''<sub>''k''</sub> (with ε being the [[Levi-Civita symbol]]). The resulting magnetoelectric [[tensor]] describing the cross-correlated response is thus [[antisymmetric]].

== Ferrotoroidicity in condensed matter physics ==
{{or section|date=June 2013}}
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.

== Anapole dark matter ==
All [[CPT symmetry|CPT self-conjugate]] particles, in particular the [[Majorana fermion]], are forbidden from having any multipole moments other than toroidal moments.<ref>{{Citation
 | last = Boudjema
 | first = F.
 | last2 = Hamzaoui
 | first2 = C.
 | last3 = Rahal
 | first3 = V.
 | last4 = Ren
 | first4 = H. C.
 | title = Electromagnetic Properties of Generalized Majorana Particles
 | journal = Phys. Rev. Lett.
 | volume = 62
 | issue = 8
 | pages = 852–854
 | year = 1989
 | doi = 10.1103/PhysRevLett.62.852 }}</ref>
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]].<ref>{{Citation
 | last = Ho
 | first = C. M.
 | last2 = Scherrer
 | first2 = R. J.
 | title = Anapole Dark Matter
 | journal = Phys. Lett. B
 | volume = 722
 | issue = 8
 | pages = 341-346
 | year = 2013
 | arxiv = 1211.0503
 | doi = 10.1103/PhysRevLett.62.852 }}</ref><ref>[http://news.vanderbilt.edu/2013/06/dark-matter/ "New, simple theory may explain mysterious dark matter"]</ref>

==References==
{{reflist}}

[[Category:Electromagnetism]]