Magnetic toroidicity

Directional non-reciprocity refers to the phenomenon where the motion in one direction differs from the motion in the opposite direction. This behavior is observed across various systems, such as one-way traffic and materials displaying electronic/optical directional dichroism, characterized by the symmetry of velocity vectors. Magnetic toroidal moments (MTMs), which typically arise from rotational spin arrangements, also possess the symmetry of velocity vectors, making them inherently directionally non-reciprocal. In this paper, we examine magnetic point groups (MPGs) that exhibit MTMs, subsequently leading to off-diagonal linear magnetoelectricity. Our focus is on the induction of MTMs through electric fields, magnetic fields, or shear stress, while enumerating the relevant MPGs. The findings of our study will serve as valuable guidance for future investigations on directional non-reciprocity, MTMs, and off-diagonal linear magnetoelectric effects.


Introduction
In the classification of breaking spatial inversion and time reversal symmetries or not, there exists two types of magnetic ferroic orderings.A ferromagnetic ordering breaks time reversal and does not break spatial inversion, while a magnetic toroidic ordering breaks both [1].The magnetic toroidal moment refers to a property exhibited by certain physical systems, where there is a unique circulation of magnetic field lines within the system, forming a torus-like structure.This moment arises when there are closed loops of magnetic field within the system, distinct from the more common magnetic dipole moment associated with a pair of magnetic poles.The concept of the magnetic toroidal moment is important in various areas of physics, such as in the study of exotic materials and topological insulators, where it plays a key role in understanding novel electronic states and their potential applications [2][3][4].
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The toroidal moment may even arise from non-magnetic systems.For example, considering a current flow, the multipole expansion reveals a toroidal dipole term in (1), which suggests a fundamental similarity between a motion and a toroidal moment.To unveil the intrinsic behaviors of a toroidal moment under transformation, a comprehensive symmetry analysis is desired.Here are the symbols of symmetry operations we used in the present discussion: T-time reversal, I-spatial inversion, R x,y,z -two-fold rotation along x, y, or z, M x,y,z -mirror reflection about the plane perpendicular to x, y, or z, C nx,ny,nz -n-fold rotation along x, y, or z, and ⊗-combination of two operations.Note that, x, y, and z are defined to be orthogonal to each other in this manuscript.The symmetry requirement for a toroidal moment along z is to have broken {T, I, R x , R y , M x ⊗ T, M y ⊗ T, C 3x , C 3y } with free rotation along z (FR z ), which is equivalent to a constant velocity vector k along z.FR z means that any rotation along z is freely allowed.For example, with FR z , I is considered to be unbroken in a system with broken I but unbroken C 4z ⊗ I. Constant velocity corresponds to constant current of any kind of quasi-particles such as electrons, holes, phonons, photons and magnons.Therefore, this symmetry similarity brings up an intrinsic non-reciprocal behavior, i.e. a non-reciprocal directional dichroism (NDD) effect, of a toroidal moment.For instance, an optical diode effect has been observed in polar materials such as FeZnMo 3 O 8 [5] and BiFeO 3 [6] with applied magnetic field perpendicular to the lattice polarization, which creates a toroidal vector P × M broken.Note that P x × M y has broken {T, I, R x , R y , M x ⊗ T, M y ⊗ T, C 3x , C 3y } with FR z , and NDD with P x × M y is along z.Directional dichroism of magnetic excitation has been reported in Ba 2 CoGe 2 O 7 , which hosts an intrinsic magnetic toroidicity in its spin texture [7].All those discoveries shed light on magnetically controllable directional optics and electronics devices.
A straightforward physical picture of a toroidal moment is a magnetic vortex with a head-to-tail spin arrangement, but in real crystalline materials with diverse symmetries, the types of a toroidal moment can vary.There are several literatures that discussed the definition of toroidal moment in bulk crystalline materials [8,9].Nevertheless, some important questions, such as what symmetry criteria of a spin texture with a net toroidal moment, how to create a toroidal moment through external perturbations such as magnetic/electric fields or various strains, and what observable physical phenomena are related with a magnetic toroidal moment, are still not systematically examined yet.Herein, we summarize real examples of toroidal moment in condensed matter physics and revisit the magnetic point groups (MPGs) that allow a toroidal moment.Based on the dot product relationship of 1D objects [10], we propose three routes to create toroidal moment by the combination of objects in different symmetry categories.

Examples of magnetic toroidal moment
Figure 1 lists a few examples of toroidal moments.Figure 1(a) shows that a magnetic vortex has toroidal moment perpendicular to the spin plane, and figure 1(b) depicts that the cross product of polarization and magnetization P × M is a toroidal moment.Figure 1(c) displays that a dimerized 1D chain of antiferromagnetic (AFM) spins has a toroidal moment perpendicular to the spin plane, and figure 1(d) shows that a Neeltype AFM ordering of spins parallel to the x axis in a honeycomb lattice has a toroidal moment along the z axis.Figure 1(e) depicts that a Neel-type AFM ordering of spin parallel to the z axis in a honeycomb lattice dimerized along the y axis has a toroidal moment along the x axis.The dimerization distortion is necessary for toroidal moment in figure 1(e) because it breaks the rotation symmetry along the z axis, which does not allow a toroidal moment perpendicular to it.According to figure 1(b), in any materials, applying magnetic field and electric field perpendicular to each other could induce a toroidal moment and an NDD effect.All of these toroidal objects mentioned above have MPG mmm ′ (or m ′ mm), and the toroidal moment direction is perpendicular to the broken mirror plane, i.e. the mirror for m ′ .Some symmetries may also spontaneously exhibit toroidal moments along multiple directions.For example, the lowsymmetry 1 ′ MPG allows toroidal moments along any of x, y, and z.Real material candidates for this case include the magnetic states of NaCrSi 2 O 6 [11], BaNi 2 (PO 4 ) 2 [12], CaMnGe 2 O 6 [13], MnPSe 3 [14], CaMn 2 Sb 2 [15], and YbMn 2 Sb 2 [16].It turns out that the requirement for offdiagonal linear magnetoelectricity along x and y is broken FR z in this case).All these 1 ′ magnetic states exhibit all non-zero elements in their off-diagonal linear magnetoelectric tensors.Note that these 1 ′ magnetic states happen to have also non-zero diagonal linear magnetoelectric tensors.

MPGs allowing toroidal moments
As shown in figure 2, a trinity diagram of MPGs has been established in terms of their transformation properties under various symmetry operations [17].There are 31 MPGs which are compatible with ferromagnetism (M MPGs), there are This figure is regenerated from [17].Reproduced from [17].© IOP Publishing Ltd.All rights reserved.31 MPGs compatible with ferroelectricity (P MPGs), and there are also 31 MPGs compatible with ferro-toroidicity (k MPGs).The symmetry operational similarity (SOS) means that an object has equal or lower, but not higher symmetries, compared with a measurable.We argue that a P MPG has SOS with an electric field along z (E z ), if it has broken {I ⊗ T, I, R x , R y , R x ⊗ T, R y ⊗ T, C 3x , C 3y } with FR z , an M MPG has SOS with a magnetization along z (M z ), if it has broken {I ⊗ T, T, M x , M y , R x , R y , C 3x , C 3y } with FR z , and an k MPG has SOS with a toroidal moment (k z ), if it has broken {T, I, R x , R y , M x ⊗ T, M y ⊗ T, C 3x , C 3y } with FR z .We emphasize that all the 'k' point groups transform like a velocity vector, thus are allowed to host toroidal moments.Consequentially, all the 'k' point groups show NDD effects and off-diagonal magnetoelectricity [18,19].Note that some magnetic structures having off-diagonal magnetoelectricity (such as a magnetic quadrupole in Pb(TiO)Cu 4 (PO 4 ) 4 , which corresponds to the MPG of 4 ′ 22 ′ [20]) do not belong to 'k' point groups, and do not show regular NDD effects.Toroidal moments may also co-exist with magnetization or polarization or both.An interesting example is multiferroic BaCoSiO 4 [21,22], which spontaneously has a toroidal moment, a ferromagnetic moment, and an electric polarization, and its magnetic point group is 6, which belongs to the 'MPk' groups in the diagram.Note that a toroidal moment could possibly exist even in lattices with a high crystallographic symmetry, such as a cubic lattice.For example, Cu 3 TeO 6 has a cubic Ia3 space group, but the magnetic ordering below 61 K has 3 ′ magnetic point group, which allows a toroidal moment along the three-fold axis, i.e. the body diagonals of the cubic matrix.Thus, a multiple-k toroidal domain configuration is expected if there is no (spontaneous) poling.Note that an experimental observation of off-diagonal magnetoelectricity has been reported in Cu 3 TeO 6 recently [23].In addition, in magnets belonging to 'k' point groups, phonons propagating along directions parallel or antiparallel to the toroidal moment will exhibit discernible differences, which can be readily detected, for example, in Raman scattering experiments.

Electric-and magnetic-field tunable toroidal moments
In the beginning of this section, we would like to recapture the methodology of the definition, classification, and operation of 1D objects.A system belongs to a 1D object when it possesses at least a C 2 rotational symmetry around the z axis but does not have any rotational symmetry other than possibly C 2 around the x or y axis.Additionally, any symmetry breaking that occurs within the system should be unidirectional, ensuring that the combination of two broken symmetry operations always leads to an unbroken state.Based on broken or unbroken {R, I, T}, there exist eight types of 1D objects (four, A, P, A ′ , P ′ ; vector-like, the other four, D, C, D ′ , C ′ ; director-like).A dot product (X • Y = Z) between two 1D objects (X and Y) along a spatial direction couples them to behave as one 1D object (Z) along the same direction.The dot product relationships of 1D objects established in [10] form a Z 2 × Z 2 × Z 2 group with Abelian additive operation.In terms of symmetry, a toroidal moment belongs to a P ′ object, and there are three pathways to induce a toroidal moment through a combination of other 1D objects based on the dot product relationships: The first relationship A ′ • C = P ′ means that a magnetization on a chiral object can induce a toroidal moment and NDD effects (figure 3(e)).This phenomenon is known as the magneto-chiral effect, and it has been experimentally demonstrated in chiral magnets such as Ni 3 TeO 6 [24,25].Note that in general, a chiral object C does not have a well-defined direction, in other words, a chiral object is chiral in any direction, so the applied magnetic field and the induced NDD direction can be along any axis, but always parallel to each other.Therefore, A ′ • C = P ′ represents a diagonal H-induced toroidal moment.In principle, this mechanism works for all chiral (C) materials, which belong to the 'H-induced k' point groups listed in figure 6.The requirement for 'diagonal H-induced k' along the z axis is broken {I ⊗ T, I, M x , M y , M x ⊗ T, M y ⊗ T} with FR z , and all C point groups have broken {I ⊗ T, I} with any FR [26].
The second relationship C ′ • A = P ′ can be illustrated as a ferro-rotation structural distortion on a C ′ object could induce a toroidal moment.A C ′ object should have broken {T,I} with any FR [10], and it should show non-zero diagonal magnetoelectricity along all three directions, xx, yy, and zz.The magnetic monopole is an example of a C ′ object.Then, we use the example of ilmenite MnTiO 3 and corundum Cr 2 O 3 to illustrate C ′ • A = P ′ .Figures 3(a) and (b) show schematics of the crystal and magnetic structure of Cr 2 O 3 [27], which has been proven to be a C ′ object [10].The counterclockwise and clockwise ferro-rotations cancel out in each unit cell.Figure 3(c  displays an equivalent magnetic structure of MnTiO 3 but without the ferro-rotation, and it falls into the symmetry classification of a C ′ object as well, i.e. broken {T, I} with any FR [10].The real structure of MnTiO 3 has a space group R3 allowing a ferro-rotation distortion.The magnetic structure has Ising spins parallel/antiparallel to the three-fold axis (z), and a magnetic point group 3 ′ .Therefore, as shown in figure 3(d), the magnetic structure of MnTiO 3 can be regarded as a combination of a ferro-rotation A on top of a C ′ (figure 3(c)).Consistent with the argument of C ′ • A = P ′ , MnTiO 3 hosts a toroidal moment [28][29][30], and the toroidal domains have been visualized experimentally [31].Also, the linear magnetoelectric tensor of MnTiO 3 (3 ′ ) shown in (2) allows a non-zero off-diagonal element σ 12 arising from the toroidal moment along z, whereas the C ′ object Cr 2 O 3 (3 ′ m ′ ) only allows diagonal linear magnetoelectric responses (shown in (3)).Overall, the comparison between MnTiO 3 and Cr 2 O 3 gives a precise example of C ′ • A = P ′ .Note that applying an electric field parallel to the axial vector of a ferro-rotational object gives rise to a chirality, i.e.P ′ • A = C ′ .Then, applying a magnetic field to the induced chirality can produce a toroidal moment through A ′ • C = P ′ .Thus, a combination of electric field and magnetic field on top of a ferro-rotational object can produce a toroidal moment as well as an NDD effect [32]  ( Ferro-rotation A can also be illustrated as external shear stress.Figure 4 depicts two types of in-plane shearing forces applied to an z plane with the z-axis pointing out of the plane.Note that the shear stress must come in pairs, otherwise the object will rotate.A shear stress in the sense indicated in figure 4(a) produces a rotational arrangement with two-fold symmetry about z axis and all broken mirrors parallel to z axis, i.e. ferro-rotation along z.Changing the sign of shear stress will change the rotational sense, giving the situation as figure 4(b).The shearing force on the z-plane behaviors as ferro-rotation A along z.In the relationship C ′ • A = P ′ , shear stress acts as ferro-rotation A on a C ′ object, which results in a toroidal moment and NDD effect.For example, the absence NDD effect along x or y of C ′ object Cr 2 O 3 (3 ′ m ′ ) described above can be activated in the presence of shear stress along x/y directions.Moreover, LiMnPO 4 is reported as a diagonal linear magnetoelectric along x, y, and z with MPG of m ′ m ′ m ′ below the antiferromagnetic transition (T N = 33.7 K) [33].Collinear antiferromagnetic spinel MnAl 2 O 4 (T N = 42 K) reveals a diagonal linear magnetoelectric effect along x and y, which is associated with the MPG 4 ′ /m ′ m ′ m [34].Both MPGs m ′ m ′ m ′ and 4 ′ /m ′ m ′ m do not belong to any groups listed in figure 6, i.e. no E/Minduced NDD effect.However, we expect an NDD effect when one applies corresponding shear stress to those linear magnetoelectrics.Specifically, shear stress along x (y or z) on LiMnPO 4 below T N will induce NDD along x (y or z) and shear stress along x (y) on MnAl 2 O 4 below T N will induce NDD along x(y).
The third relationship D ′ • P = P ′ refers to that applying electric field on a D ′ object can induce a toroidal moment.The symmetry requirement for a D ′ object is to have broken {I ⊗ T, T} with any FR [10].Based on that, we have carried out the completed magnetic point group set for D ′ object: 1, 1, 2, m, 2/m, mmm, 222, mm2, 4, 4, 4/m, 422, 4mm, 42m, 4/mmm, 3, 3, 32, 3m, 3m, 6, 6, 6/m, 622, 6mm, 6m2, 6/mmm, 23, m3, 432, 43m, and m3m.Consistently, all of those D ′ groups show all three non-zero diagonal elements in their electro-toroidic (electric-field-induced toroidal moment) tensors.Note that we have considered symmetries that are along only those basis vectors of the conventional crystallographic coordinate systems.For example, we discuss the symmetries along x, y, z, xy, and yx directions of the tetragonal and cubic MPGs while only symmetries along x, y, z directions in orthorhombic MPGs are considered in our analysis.Figure 5 demonstrates the physics picture of that applying electric field on a D ′ object induces a toroidal moment.Using a spin texture with magnetic point group mmm as an example, it has broken {I ⊗ T, T} with FR and belongs to D ′ .The magnetic ions are located at corner-sharing oxygen rectangular coordinates.As shown in figures 5(a) and (b), the ground state of mmm has zero toroidal moment, because the toroidal moments of two sublattices are equal in magnitude and opposite in direction, so they cancel out and there is no net toroidal moment.Applying an out-of-plane electric field will cause an out-ofplane displacement of oxygen atoms.This oxygen displacement brings Dzyaloshinskii-Moriya interactions and additional canting to the spins (cyan arrows in figure 5(c)), and the toroidal moments of those two sublattices in the final spin structure have different magnitude and do not cancel out anymore (|T 1 | > |T 2 | due to lattice a > b).Therefore, applying electric field on mmm induces a net tunable toroidal moment (and an NDD effect) parallel to the electric field.
When dealing with field-induced effects, it is important to consider that the applied electric or magnetic fields bring additional symmetry breaking.For example, the requirement to Then, summarizing magnetic point group classifications in terms of hosting toroidal moments, diagonal E-induced toroidal moment, and diagonal H-induced toroidal moments, a trinity diagram (figure 6) is carried out.Note that, in the overlapping regimes, some objects having toroidal moments are also allowed to have E-induced and H-induced toroidal moment in addition to the original toroidal moment, either along the same direction or a different direction.It is similar to the case that a piezoelectric material with an original polarization can have an additional polarization under an electric field.However, since a polarization and an electric field have the same symmetry, applying an electric field on any materials can induce additional polarization.Differently, an electric field or a magnetic field have distinct symmetry to a toroidal moment, therefore, they can induce additional toroidal moments only if the material symmetries fall into the requirements discussed above.Some real material candidates for diagonal E-induced toroidal moments and directions are summarized in table 1.Again, most of them are D ′ but some are not, e.g.Pb 2 MnO 4 (4 ′ 2m ′ ), because E brings additional symmetry breaking.Also, the directions working for diagonal E-induced toroidal moment are more than the D ′ direction of those MPGs.The total MPG number for "diagonal E-induced k" and that for "diagonal H-induced k" are identical.This is due to a conjugate relationship between those two phenomena.The requirement for "diagonal E-induced k" is broken {I ⊗ T, T, R x ⊗ T, R y ⊗ T, M x ⊗ T, M y ⊗ T} with FRz, and that for "diagonal H-induced k" along the z axis is broken {I ⊗ T, I, M x , M y , M x ⊗ T, M y ⊗ T} with FRz.These two requirements are interchangeable with the exchange of I ↔ T [35,36].Note that, for example, with I ↔ T, M x = R x ⊗ I becomes R x ⊗ T, and M x ⊗ T = R x ⊗ I ⊗ T remains to be M x ⊗ T.

Conclusion
In summary, the presence of a magnetic toroidal moment along z requires the breaking of {T, I, R x , R y , M x ⊗ T, M y ⊗ T, C 3x , C 3y } with FR z , and this symmetry condition is equivalent to a constant velocity vector 'k' along z.Consequently, any system exhibiting a toroidal moment also displays an NDD effect.All MPGs in the 'k' point group can host toroidal moments, NDD effects, and off-diagonal magnetoelectricity.We propose three approaches to induce toroidal moments: diagonal H-induced toroidal moments (most C objects), ferro-rotation distortions on C ′ objects, and diagonal E-induced toroidal moments (most D ′ objects).We discuss completed sets of MPGs and a number of real material candidates for induced toroidal moments.However, many of these phenomena have yet to be experimentally demonstrated, offering significant opportunities for future research.Furthermore, current research on the NDD effects primarily focuses on the non-reciprocal electric transport, optics, and magnons.Nevertheless, it is important to note that the 'k' vector could represent any types of velocities, including phonons and heat flow.Therefore, exploring new types of 'k' in the context of the NDD effects through experimental investigation could be a promising avenue for future studies.

Figure 1 .
Figure 1.(a) A spin vortex has toroidal moment out-of-plane.(b) A spin plus a polarization has toroidal moment perpendicular to them.(c) A dimerized chain has toroidal moment perpendicular to the spins.(d) A honeycomb lattice with in-plane antiferromagnetic (e) is dimerized honeycomb lattice.Electric polarization, spin, and toroidal moment are displayed by red, blue, and black arrows, respectively.

Figure 2 .
Figure 2. Classification of magnetic point groups in terms of compatible ferroic orders.P, M, and k allow spontaneous polarization, magnetization, and magnetic toroidicity, respectively.This figure is regenerated from[17].Reproduced from[17].© IOP Publishing Ltd.All rights reserved. )

Figure 3 .
Figure 3. (a) A structural diagram of Cr 2 O 3 .(b) Cr 2 O 3 is C ′ .C and CC represent clockwise and counter-clockwise ferro-rotations, respectively.(c) The object consisting of anti-parallel spins and polarizations is C ′ .(d) Combination of a ferro-rotation and the C ′ object in (c) produce a P ′ toroidal moment, e.g.MnTiO 3 .(e) Applying magnetic field on a chiral object C produces a P ′ toroidal moment.(f) Applying electric field on a D ′ object produces a P ′ toroidal moment.

Figure 5 .
Figure 5. (a) A magnetic state in a rectangular lattice with mmm magnetic point group.(b) Two sublattices with toroidal moment cancel out each other.(c) Applying out-of-plane electric field induces oxygen distortions that result in DM interactions, and the additional spin components due to the DM interactions are displayed by cyan arrows.(d) Toroidal moments in two sublattices do not cancel each other out anymore in the presence of an out-of-plane electric field.

Figure 6 .
Figure 6.Classification of magnetic point groups (MPGs) in terms of showing toroidal moments (P ′ ), electric-field-induced toroidal moments, and magnetic-field-induced toroidal moments, respectively.Blue: non-D ′ ; red: non-C; purple: non-D ′ nor C. The external electric-field or magnetic-field induced mirror (Mxy) breaking in those colored MPGs can give rise to diagonal E-induced toroidal moments, and diagonal H-induced toroidal moments, respectively.4 ′ m ′ m shows the NDD effects along xy/yx.

Table 1 .
Examplary material candidates for diagonal E-induced toroidal moment effect.