The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect

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Multiferroic materials and magnetoelectric physics: symmetry, entanglement, excitation, and topology (2015, background): “Besides the ferroelectric order and ferromagnetic order, there exist two more primary ferroic orders [71]: ferroelastic order which breaks neither the time-reversal symmetry nor the space-inversion symmetry [69]; and ferrotoridal order T ∼M × r which breaks both the time-reversal symmetry and the sp”
Magnetoelectric resonance with electromagnons in a perovskite helimagnet (2011, background): “On the other hand, the magnetoelectric resonance observed at present may be viewed as the dynamical resonance of toroidal momen”
The evolution of multiferroics (2016, background): “Ferroelectricity originates from two Bi 3+ electrons that shift away from the Bi ion and towards the FeO6 octahedra, giving rise to a spontaneous polarization P along the [111] direction.”
Ginzburg-Landau theory for skyrmions in inversion-symmetric magnets with competing interactions (2015, unknown): “A measurable quantity associated with the helicity φ0 is the toroidal moment T (φ0) ≡ ∫ dr3r × S ∝ sin φ0 [54].”