As we know that materials having more than one order parameters are called multiferroics. For ferromagnetic materials it is "spontaneous magnetization". Like this what is the order parameter for ferrotoroidic materials.
Ferrotoroidic is also an another order parameter co-exist independently along with ferromagnetic/anti-ferro domains and can be used to define multiferroic materials as well after a long debatable controversies. The magnetic vortices arrangement (ferrotoroid) are included as independent parameter co-exist along with other parameters like spontaneous magnetization/polarization/elasicity/etc. This can be spatially distinguished by an optical SHG.
The loss of spatial inversion symmetry results in ferroelectricity whereas the loss of time reversal symmetry results in ferromagnetism. Ferroelasticity is a result of broken rotational symmetry although it remains invarient under both spatial inversion and time reversal symmetries. There is a fourth possibility which corresponds to when both spatial inversion and time reversal symmetries are simultaneously broken. This is the case for ferrotoroidic materials where the long rage order is related to to an ordering of magnetic vortex like structures characterized by a toroidal dipolar moment which can be identified as toroidal order parameter.
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Dear Tapan Sir
It is very nice description. In what type of material this order has been seen predominately. Material with centro-symmetry or non-centro-symmetry. I recently read one article that it is observed in manganese oxide, which is highly centro-symmetry. But theoretically, when symmetries are broken, then it would be a non-centro symmetry. Is I am correct?
Ferrotoroidic domain walls have been observed in LiCo(PO4)3 using second harmonic generation technique [8] and a symmetry classification of such domains has been given recently [9]. Spontaneous toroidal moments have been attributed to exist in the multiferroic phase of Ba2CoGe2O7 (BCG) [10] and may be related to theobserved unusual magnetoelectric effects [11, 12]. Single crystalline thin films of MnTiO3 with an ilmenite struc- ture also exhibit a ferrotoroidic structure [13]. Neutron polarimetry indicates that the magnetoelectric MnPS3 is a viable candidate for ferrotoroidicity [14]. The magnetic phase transition in BiFeO3 implies the appearance of a toroidal moment [15] Ab initio calculations suggest that the olivine material Li4MnFeCoNiP4O16 is possibly fer- rotoroidic [16].
A sequence of possibly two ferrotoroidal phase transi- tions has been considered phenomenologically for Ni-Br boracites [17]. Another physical realization of toroidal or- der has been considered as an interacting system of discs with a triangle of spins on each disc [18]. Both charge and spin currents can lead to a toroidal state [19]. Recent observation of orbital currents in CuO through resonant x-ray diffraction provides a direct evidence of antiferro- toroidic ordering [20] Ferrotoroidic materials exhibit lin- ear magnetoelectric effect; in fact, the toroidic moment is related to an antisymmetric component of the mag- netoelectric tensor (αij ≠ αji). Moreover, the toroidic moment can be viewed as a quantum geometric phase [21]. Beyond the magnetoelectric applications toroidic materials can also act as novel metamaterials [22].
Note that the references are listed in my answer before.
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