Introduction

Magnetic states exhibiting non-zero net magnetization include ferromagnets, ferrimagnets, or weak ferromagnets (including canted antiferromagnets). Due to the non-zero magnetization, those magnetic states exhibit physical phenomena1,2,3,4,5,6,7,8,9,10,11,12,13 such as magnetic attraction, various magneto-optical properties (magneto-optical Kerr (MOKE), Faraday, and magnetic circular dichroism), and anomalous Hall-type effects. The anomalous Hall-type effects manifest anomalous Hall, anomalous Ettingshausen, anomalous Nernst, or anomalous thermal Hall effects7,12,14,15,16,17,18. It turns out that these phenomena that usually occur only in magnetic states with non-zero magnetization can take place in certain antiferromagnets with seemingly zero net magnetization14,17,19,20,21, sometimes in the presence of external perturbations such as stress, electric current, or light ilumination. These cases have been called Trompe L’oeil Ferromagnetism22.

It turns out that broken symmetries can be associated with order parameters and emergent phenomena. Herein, we meticulously outline the precise broken symmetries associated with each of these phenomena: non-zero magnetization, diagonal piezomagnetism2,23, circular dichroism24,25,26, nonreciprocal circular dichroism (such as Faraday effect)27, odd-order (including linear) anomalous Hall effect (AHE), and MOKE13,28. Notably, the constraints imposed by symmetry on equilibrium-property tensors, combined with an in-depth MPGs analysis, have been established. For a more comprehensive understanding, readers are encouraged to the provided references1,2,3,23,29,30,31. In this perspective, instead, we utilize the concept of Symmetry Operational Similarity (SOS)29 for these analyses and classify the corresponding magnetic point groups (MPGs) with those broken symmetries, i.e., each of the above phenomena.

Observable physical phenomena can occur or non-zero measurable can be detected when specimens have SOS with the combination of measurable (or experimental setup for measurable) and specimen environments (such as applied external stress, electric fields, or magnetic fields), or specimens with specimen environments have SOS with measurable. This SOS relationship includes when specimens have more, but not less, broken symmetries than the combination of measurable and specimen environments or specimens with specimen environments have more, but not less, broken symmetries than measurable. In other words, to have a SOS relationship, specimens cannot have higher symmetries than the combination of measurable and specimen environments or specimens with specimen environments cannot have higher symmetries than measurable. Our SOS approach, considering the group symmetry of the whole experimental setup, is in accordance with the well-known Neumann’s Principle1,32, and its power lies in providing simple and physically transparent views of otherwise unintuitive phenomena in complex materials, without considering specific coupling terms or the relevant Hamiltonians. Furthermore, this approach has the capability to connect seemingly unrelated phenomena in systematic ways and can be employed to discover potential materials with potentially desirable properties, as well as uncover exciting phenomena in existing materials.

To find the requirements of broken symmetries for various phenomena, we, first, define the general symmetry operation notations for three orthogonal x, y, and z axes such as Rx = 2-fold rotation around the x-axis, Mx = mirror reflection with mirror perpendicular to the x axis, I = space inversion, T = time reversal, etc. Then, we have these general relationships: Rx⊗Ry = Rz, Mz⊗Ry = Mx, Mx⊗Rz = My, Mx⊗Ry = Mz, My⊗Mz = Rx, Mx⊗Mz = Ry, Mx⊗My = Rz, Mx⊗Rx = My⊗Ry = Mz⊗Rz = I, Mx = Rx⊗I, My = I⊗Ry, and Mz = I⊗Rz. All are commutative operations. Measurables such as magnetization or optical activity setups have translational symmetry, so we can ignore or freely allow any translations, i.e., any translations are considered as a unit operation. Similarly, when we consider one-dimensional (1D) measurables invariant under any rotations along the 1D direction, then we ignore or freely allow any rotations around the axis, i.e., any rotations around the 1D direction are considered as a unit operation. For example, magnetization along z should be invariant under any rotations along z, so we ignore or freely allow any rotations around z33,34.

Symmetry of ferromagnetism

Magnetization (M) along z, which is a 1D object along z, has broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y} and unbroken [1,I,Mz,Rz], and thus has broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y} with free rotation along z. As discussed above, when we consider the symmetry of 1D objects such as M, we always allow any free rotations along the 1D direction. For example, \(\bar{4}\)′ has unbroken C4z⊗I⊗T and broken I⊗T, but when we consider the SOS relationship of \(\bar{4}\)′ with a 1D object along z, then \(\bar{4}\)′ has unbroken I⊗T with free rotation along z, so \(\bar{4}\)′ does not have SOS with Mz. All and also only MPGs, belonging to the ferromagnetic point groups, do have broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y} with free rotation along z or the relevant requirements along x or y. The thirty-one (31) ferromagnetic point groups include 1, \(\bar{1}\), 2, 2′, m, m′, 2/m, 2′/m′, 2′2′2, m′m′m, m′m′2, m′m2′, 4, \(\bar{4}\), 4/m, 42′2′, 4m′m′, \(\bar{4}\)2′m′, 4/mm′m’, 3, \(\bar{3}\), 32′, 3m’, \(\bar{3}\)m′, 6, \(\bar{6}\), 6/m, 62′2′, 6m′m', \(\bar{6}\)m′2′, 6/mm′m′2,31. Note that \(\bar{4}\)′ does not belong to the ferromagnetic point groups.

The presence of non-zero net magnetization in magnetic states in ferromagnetic point groups is sometimes evident, but it is not always. For example, magnetic states depicted in Fig. 1a–d appear to be antiferromagnetic states with 120° spins without any net magnetic moments. However, all belong to ferromagnetic point groups; Fig. 1a; mm′m′, which has been observed in Mn3Sn (Cmc′m′)9, Fig. 1b; m'mm', which has been observed in Mn3(Ge,Ga) (Cm′cm′), Fig. 1c; 2/m; kagome lattice with lattice distortions, shown with solid-line bonds (P21/n)35, and Fig. 1d; \(\bar{3}\)m′, which has been observed in Mn3(Rh,Ir,Pt) (R\(\bar{3}\)m′)7,36. Kagome layer in Fig. 1(d) corresponds to the MPG 6’/m’m’m, which does not belong to the ferromagnetic point group; however, ABC-type stacking of kagome layers results in the MPG \(\bar{3}m^{\prime}\), which now belongs to the ferromagnetic point group. Indeed, small but non-zero magnetization has been observed, at least, in Mn3Sn and Mn3(Ge,Ga)15,28.

Fig. 1: Various specimens and experimental setups for AHE, circular dichroism and MOKE.
figure 1

a–d are for Mn3Sn (Cmc′m′), Mn3(Ge,Ga) (Cm′cm′), kagome lattice with lattice distortions, shown with solid-line bonds (P21/n), and one (6’/m'm'm) of three stacked kagome layers in Mn3(Rh,Ir,Pt) (R\(\bar{3}\)m′), respectively. e–h Four experimental setups to measure AHE, i.e., transverse voltage induced by current. i–k Three experimental setups to measure circular dichroism. l, m Two experimental setups to measure MOKE.

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Symmetry of the odd-order AHE measurements

The Hall effect was discovered by Edwin Hall while he was working on his doctoral degree in 1879, and has been well utilized to measure carrier density as well as detect small magnetic fields37. This so-called ordinary Hall effect contrasts with the anomalous Hall effect (AHE) in ferromagnets, which is sometimes called extraordinary Hall effect or spontaneous Hall effect18. This AHE exists in zero applied magnetic field, and varies linearly with applied electric current, so its sign changes when the current direction is reversed. It was proposed that AHE can exist in truly antiferromagnetic systems such as Mn3(Rh,Ir,Pt) with Kagome lattice7,36, originating from the Berry curvature. In fact, Mn3(Sn,Ge), forming in the same crystallographic structure as that of Mn3(Rh,Ir,Pt), is experimentally reported to exhibit a significant AHE9,14. However, it turns out that Mn3(Sn,Ge) does exhibit a small, but finite net magnetic moment15,28, and the exact experimental situation of Mn3(Rh,Ir,Pt) is presently unclear, partially due to the presence of competing multiple magnetic states in the system, and also the absence of bulk crystal study. Topological Hall effects in skyrmion systems have been reported38,39, and occur typically in the presence of external magnetic fields. Note that AHE of antiferromagnets with zero or small magnetization can be particularly useful for the fast sensing of magnetic fields due to the intrinsic fast dynamics of antiferromagnets40.

Herein, we define AHE as transverse voltage induced by applied current in zero magnetic fields. The sets of (electric current, +/−), (electric current, h/c), (thermal current, +/−), and (thermal current, h/c) correspond to the Hall, Ettingshausen, Nernst, and thermal Hall effects, all of which we call Hall-type effects, respectively (+/- means an induced voltage difference and h/c (hot/cold) means an induced thermal gradient, accumulated on off-diagonal surfaces). In terms of symmetry, there is little difference among these four types of Hall effects, so, for example, the existence of non-zero AHE means the presence of non-zero anomalous Nernst effect. With this multi-faceted nature of Hall-type effect, it is imperative to find the accurate relationship among all different kinds of Hall-type effects, and also the requirements to have non-zero values of various Hall-type effects.

From our SOS analysis, we can tell a certain phenomenon is a zero, odd-order, or even-order effect. Herein, we will discuss the requirement of broken symmetries for odd-order anomalous Hall effects. We can have these transformations for the experimental setup for AHE measurements in Fig. 1e–h: (e) ↔ (e) by Mz, Mx⊗T, and Ry⊗T; (e) ↔ (h) by I, Rz, My⊗T, and Rx⊗T; (e) ↔ (f) by My, Rx, I⊗T, and Rz⊗T; (e) ↔ (g) by T, Mx, Ry, and Mz⊗T. Thus, Odd-order AHEyx means Odd-order AHE with current along x and Hall voltage along y in Fig. 1a, and the experimental setup to measure odd-order AHEyx has unbroken [1,I,Mz,Rz,Mx⊗T,My⊗T,Rx⊗T,Ry⊗T] and broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T}. Now, any specimens having SOS with the above experimental setup will show Odd-order AHEyx when they have broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T}. The rest independent ones can be either broken or unbroken. For example, for broken {I}, Odd-order AHE of the original domain is the same as that of the domain after space inversion. Emphasize that the requirements for anomalous Ettingshausen, anomalous Nernst, and anomalous thermal Hall effects are identical to those for AHE. It turns out that all ferromagnetic point groups23,41 can have non-zero net magnetic moments. These MPGs display a symmetry pattern involving broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y} with free rotation along z when the net magnetic moments align along the z-axis, or broken {I⊗T,T,My,Mz,Ry,Rz,C3y,C3z} with free rotation along x when the net magnetic moments are along x, or broken {I⊗T,T,Mx,Mz,Rx,Rz,C3x,C3z} with free rotation along y when the net magnetic moments are along y – the relevant MPGs are listed in Figs. 2–4.

Fig. 2: Magnetic point groups for various ferromagnetism-like phenomena along z.
figure 2

Odd-order AHEyx is expected to be identical to Odd-order AHExy, except for the possible sign difference. Green: MPGs of diagonal linear magnetoelectric effect; Blue: MPGs of chiral point group; Turquoise: MPGs of linear magnetoelectric effect and chirality.

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Fig. 3: Magnetic point groups for various ferromagnetism-like phenomena along x.
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Only in ferromagnetic point groups, Odd-order AHEzy is expected to be identical to Odd-order AHEyz, except for the possible sign difference. Green: MPGs of diagonal linear magnetoelectric effect; Blue: MPGs of chiral point group; Turquoise: MPGs of both.

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Fig. 4: Magnetic point groups for various ferromagnetism-like phenomena along y.
figure 4

Only in ferromagnetic point groups, Odd-order AHEzx is expected to be identical to Odd-order AHExz, except for the possible sign difference. Green: MPGs of diagonal linear magnetoelectric effect; Blue: MPGs of chiral point group; Turquoise: MPGs of both.

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As discussed earlier, all magnetic states in Fig. 1a–d belong to ferromagnetic point groups, so do exhibit Odd-order AHE. Since all magnetic states in Fig. 1(a–d) do accompany non-zero magnetic moments, the AHEs are, in fact, linear with applied electric current. MPGs for Odd-order AHEyx, requiring broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T}, and those for Odd-order AHEyx,xy, requiring broken {I⊗T,T,Mxy,Myx,Rxy,Ryx,C3xy,Mz⊗T,Rz⊗T}, are listed in Fig. 2. MPGs for Odd-order AHEzy, requiring broken {I⊗T,T,My,Mz,Ry,Rz,C3y,Mx⊗T,Rx⊗T}, and those for Odd-order AHEzx, requiring broken {I⊗T,T,Mx,Mz,Rx,Rz,C3x,My⊗T,Ry⊗T}, are listed in Figs. 3 and 4, respectively. In all ferromagnetic point groups that can have non-zero net moments along z (x or y), Odd-order AHEyx (AHEzy or AHEzx) is the same as Odd-order AHExy (AHEyz or AHExz) except their sign difference (i.e. they are anti-symmetric).

However, there are three types of non-ferromagnetic point groups, which allow Odd-order AHE; (1) MPGs of C3z can have Odd-order AHEzx or AHEzy with broken {I⊗T,T,Mz,My,Rz,Ry,C3y,Mx⊗T,Rx⊗T}, but they have zero Odd-order AHExz or AHEyz due to the unbroken C3z: the examples are 3, \(\bar{3}\), 32, 3m, \(\bar{3}\)m, 6′, \(\bar{6}\)', 6′/m′, 6′22′, 6′mm′, \(\bar{6}\)'m2′, 6'/m'mm' for Odd-order AHEzy, and 3, \(\bar{3}\), 32′, 3m′, \(\bar{3}\)m′, 6′, \(\bar{6}\)', 6′/m′, \(\bar{6}\)'m′2 for Odd-order AHEzx. (2) MPGs of C4z⊗T or C4z⊗I⊗T can have Odd-order AHEyx and Odd-order AHExy with broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y,Mz⊗T,Rz⊗T}: the examples are 4′, \(\bar{4}\)′, 4′/m, 4'2'2, 4′m'm, \(\bar{4}\)′2'm, \(\bar{4}\)′m'2, 4'/mm'm. These MPGs are truly antiferromagnetic without any net magnetic moment but can exhibit odd-order AHE. All of these MPGs do have unbroken 4' or \(\bar{4}\)′, so the odd-order AHEs in these point groups are associated with symmetric tensors, unlike antisymmetric tensors for odd-order AHEs in ferromagnetic space groups. For example, for \(\bar{4}\)′ point group, (+Jx,+Ey) becomes (+Jy,+Ex) under \(\bar{4}\)′, while the point group is invariant, so the relevant conductivity tensor components are symmetric. These Odd-order AHEs in truly antiferromagnetic states occur without non-zero magnetic moment, so must be high-order effects. High odd-order AHEs with symmetric tensors have never been reported and will be an exciting research direction. (3) Cubic MPGs allow Odd-order AHEyx,xy as well as Odd-order AHExy,yx: 23, m\(\bar{3}\), 4'32′, \(\bar{4}\)′3m′, m\(\bar{3}\)m'. Note that those truly antiferromagnetic states mentioned above were omitted20 from the list of MPGs exhibiting a Hall-vector in antiferromagnetic states in Ref. 20. Furthermore, despite our symmetry requirements indicating their prohibition, MPGs mmm, 4/mmm, and 6/mmm were identified42 as possessing 3rd-order AHE responses driven by Berry curvature multipoles in Ref. 42.

It turns out that this Odd-order AHE corresponds to Off-diagonal even-order current-induced magnetization. When we consider the experimental setup for measuring magnetization along z induced in an even-order by current along x, we can readily find out that the relevant requirement for non-zero Off-diagonal even-order current-induced magnetization is broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T}, which is identical with the requirement for Odd-order AHEyx. Thus, we can conclude that Odd-order AHE results from Off-diagonal even-order current-induced magnetization. Zeroth-order current-induced magnetization is considered the cause of Linear AHE in ferromagnetic point groups with non-zero net magnetization in the presence of no current. Linear AHE can exist only in materials whose magnetic point groups belong to the ferromagnetic point group, which is always associated with non-zero net magnetic moments. In other words, there is absolutely no difference between the requirement for non-zero linear AHE and that for non-zero net magnetic moment in terms of symmetry. However, our symmetry argument cannot tell if the linear AHE effect is solely due to the net magnetic moment or not, and recent theories predict that the antiferromagnetic part of those magnetic point groups belonging to the ferromagnetic point group can be responsible for large linear AHE7,14,21. In any case, the 180° flipping of the net magnetic moment in any of those magnets by external magnetic fields is expected to result in the flipping of the entire antiferromagnetic vector and the sign change of linear AHE, even if the net magnetic moment is tiny. Note that k (or J) can represent electric current, light propagation as well as any constant currents such as phonon or magnon motions. Therefore, for example, phonons in any Odd-order AHE systems can accompany induced even-order magnetization that is perpendicular to the phonon k. These phonons can be referred to as 'magnetic phonons'.

We emphasize that our SOS approach can tell if a certain phenomenon is zero, non-zero odd-order, or non-zero even-order effect, and broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T} is, in fact, the requirement for Odd-order AHE. Recently, the concept of altermagnetism was introduced: their ordered spins are truly antiferromagnetic, but can exhibit, for example, non-zero linear AHE due to orbital magnetism through Berry curvature20,43,44. However, it turns out that all those altermagnets, showing linear AHE discussed so far, such as MnTe and RuO2 thin films20,21 belong to ferromagnetic point groups.

Symmetry of piezomagnetism

Piezoelectricity is the phenomenon of induced polarization, i.e., voltage gradient, with external stress. Similarly, piezomagnetism is the phenomenon of induced net magnetic moment with external stress, and there can exist Diagonal or Off-diagonal piezomagnetism. The experimental setup for Diagonal piezomagnetism along z, which is 1D, has broken {I⊗T,T,Mx,My,Rx,Ry} and unbroken [1,I,Mz,Rz], and thus has broken {I⊗T,T,Mx,My,Rx,Ry} with free rotation along z. MPGs for Diagonal piezomagnetism along z, requiring broken {I⊗T,T,Mx,My,Rx,Ry} with free rotation along z, those for Diagonal piezomagnetism along x, requiring broken {I⊗T,T,My,Mz,Ry,Rz} with free rotation along x, and those for Diagonal piezomagnetism along y, requiring broken {I⊗T,T,Mx,Mz,Rx,Rz} with free rotation along y, are listed in Figs. 2, 3 and 4, respectively. For example, ‘broken {I⊗T,T,Mx,My,Rx,Ry} with free rotation along z’ is a subset of ‘broken {I⊗T,T,Mx,My,Rx,Ry,C3x,C3y} with free rotation along z’, which is the requirement for ferromagnetic point groups with magnetization along z.

Therefore, all ferromagnetic point groups do exhibit Diagonal piezomagnetism along the magnetization direction. The only difference between the requirement for, for example, Diagonal piezomagnetism along x and that for ferromagnetic point groups with magnetization along x is broken {C3y,C3z}, and this difference is due to the breaking of both C3y and C3z by external stress along x in the case of Diagonal piezomagnetism along x. C3 symmetry is defined to be along z, all MPGs, showing Diagonal piezomagnetism along z, belong to ferromagnetic point groups, but all MPGs with C3z, showing Diagonal piezomagnetism along x or y, do not belong to ferromagnetic point groups, as shown in Figs. 3 and 4. Breaking C3z by external stress is the essential part of the Diagonal piezomagnetism along x or y in MPGs with C3z, which do not exhibit magnetization along x or y in zero strain.

Off-diagonal piezomagnetismzx with stress along x and induced magnetization along z has broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T} and unbroken [1,I,Mz,Rz,Rx⊗T,Ry⊗T], and thus has broken {I⊗T,T,Mx,My,Rx,Ry,C3x,Mz⊗T,Rz⊗T}, which is, interestingly, identical with the requirement for Odd-order AHEyx. We also remark that in any piezomagnets, electric fields can also induce magnetization, but this electromagnetic effect will be of even order, meaning that the sign of the electric fields does not affect the outcome. In terms of ferromagnetism and piezomagnetism, the magnetic point groups identified via our SOS approach overlap with those obtained from the standard tensorial approach2,23.

Symmetry of Circular Dichroism

An experimental setup to measure Circular Dichroism (CD) is shown in Fig. 1(i–k). Circular Dichroism45 includes the Faraday effect46, natural optical activity, and Circular PhotoGalvanic Effect (CPGE)47,48. Figure 1(i, j) are linked through {I⊗T,Mz⊗T,Mx,My}, but each setup is invariant under [1, I, Mz, Rx⊗T, Ry⊗T] with any spatial rotation around z. Thus, this setup has broken {I⊗T,Mx,My} with any spatial rotation around z is required to have CD along z (CDz). MPGs for CDz, requiring broken {I⊗T,Mx,My}, those for CDx, requiring broken {I⊗T,My,Mz}, and those for CDy, requiring broken {I⊗T,Mx,Mz} are shown in Figs. 2, 3, and 4, respectively. Emphasize that for this symmetry consideration, we allow any spatial rotation around z freely. The broken-symmetry requirement for CD is a subset of those for Odd-order AHE, except that free rotations should be allowed for the symmetry consideration for CD. Thus, most of MPGs for non-zero Odd-order AHE, except 4′/m, 4′m′m, \(\bar{4}\)'2′m, 4′/mmm′, m\(\bar{3}\), m\(\bar{3}\)m′, \(\bar{4}\)', \(\bar{4}\)'2m′,\(\,\bar{4}\)'m2′, \(\bar{4}\)'3m′ for Odd-order AHEyx, allow CDz. In the case of all ferromagnetic point groups, which is a part of MPGs for Odd-order AHE, CD is always along the magnetization direction.

Now, CD can be nonreciprocal if all of {T,Rx,Ry,C3x,C3y} are broken additionally since any of {T,Rx,Ry} can link Fig. 1i and Fig. 1k in the condition of any free rotations along z. Thus, the requirement for Nonreciprocal CD (NCD) along z is broken {I⊗T,T,Mx,My,Rx,Ry} with free rotation along z, which is identical to the requirement for Diagonal piezomagnetism along z. Thus, all MPGs for Diagonal piezomagnetism along one direction do exhibit NCD along the direction. All ferromagnetic point groups do exhibit NCD along the magnetization direction, which is associated with the Faraday effect. Note that breaking C3z by external stress, discussed in the above section, becomes also relevant for CD, since light propagation itself can also break C3z. The following example illustrates how NCD can be linked with Diagonal piezomagnetism in nonferromagnetic point groups. Non-ferromagnetic MPGs, such as 32, 3m, and \(\bar{3}\)m, should exhibit NCDx with zero NCDy or z, and they also exhibit Diagonal piezomagnetism along x (piezoMx). Diagonal piezoMx is the same as Diagonal even-order (e.g. 2nd-order) current (light propagation or electric current)-induced M, and this induced M can result in NCD.

Both space-inversion and time-reversal symmetries are broken in 32 and 3m, and their space-inversion domains do exhibit the identical NCDx, but their time-reversal domains should display the opposite NCDx. We emphasize that light-induced M can result in reciprocal CD as well as NCD. For example, reciprocal CD (i.e., natural optical activity) occurs in chiral point groups, and light propagation in chiral materials does induce M along the light propagation direction in a linear fashion, which, in turn, results in M-induced CD49. Since the induced M flips its sign when the light propagation direction flips by 180° (i.e. M is induced in a linear fashion), this CD effect is reciprocal. In other words, natural optical activity in chiral materials can be understood in terms of the Faraday effect due to light-induced M.

Symmetry of Magneto-Optical Kerr Effect (Moke)

The experimental setup to measure the non-zero Magneto-Optical Kerr Effect (MOKE), i.e. the light-polarization rotation effect of reflected linearly-polarized light, is shown in Fig. 1l, m. First, we note that this experimental setup is also invariant under any spatial rotations around z, so we can ignore or freely allow any rotations around z for symmetry considerations. The MOKE setup has broken {T,Mx,My}, so the requirement for MOKE along z (MOKEz) is broken {T,Mx,My} with freely-allowed rotations around z. MPGs for MOKEz, requiring broken {T,Mx,My} with free rotation around z, those for MOKEx, requiring broken {T,My,Mz} with free rotation around z, and those for MOKEy, requiring broken {T,Mx,Mz} with free rotation around z, are summarized in Figs. 2, 3 and 4, respectively. Note that, for example, in tetragonal point groups, Mx can be different from Mxy, and broken {T,Mx,My} with freely-allowed rotations around z means broken {T,Mx,My,Mxy,Myx} for those tetragonal point groups. Certainly, all ferromagnetic point groups have broken {T,Mx,My}, so they exhibit MOKE.

It turns out that the requirement for diagonal linear magnetoelectric effects along z is broken {T,I,Mx,My,Rx⊗T,Ry⊗T} with free rotation along z, which requires more broken symmetries than MOKEz, so all linear magnetoelectrics along z do exhibit MOKEz. Our symmetry analysis has revealed that MPGs permitting the MOKE effect extend beyond the confines of the 31 ferromagnetic point groups documented in the literature50. Figures 2–4 vividly illustrate non-ferromagnetic MPG examples for the MOKE effect, encompassing those associated with linear magnetoelectric properties (highlighted in green and turquoise), as well as MPGs characterized by diagonal piezomagnetism (central dotted circles) and chiral point groups (highlighted in blue and turquoise). The phenomena of MOKE in all linear magnetoelectrics can be considered as a result of magnetization induced by the presence of a surface in diagonal linear magnetoelectrics, since the presence of a surface is necessary for MOKE and any surface of all diagonal linear magnetoelectrics can have surface magnetization51.

Figures 2–4 provide a concise overview of ferromagnetism-like phenomena in three principal directions: z, x, and y with the required broken symmetries presented in Table 1. The symmetry criteria of Circular Dichroism (CD) and Magneto-Optical Kerr Effect (MOKE) are shown to be interchangeable through the I⊗T ↔ T symmetry transformation. Note that I⊗T ↔ T accompanies, for example, Mx⊗T = Rx⊗I⊗T ↔ Rx⊗T. CDz requires the lack of {I⊗T,Mx,My} symmetries with any spatial rotation around z, which can be transformed into {T,Mx,My} for MOKEz. Figures 2–4 visually represent the interconnection. Figure 2 illustrates the same number, 54, MPGs, enclosed within the large CDz and MOKEz circles, respectively, while also showing the same number, 34, of MPGs at their intersection with odd-order AHEyx. Figures 3 and 4 similarly reflect symmetrical patterns for CDx & MOKEx (Fig. 3) and CDy & MOKEy (Fig. 4). This observation reveals the concept that Circular Dichroism (CD) and Magneto-Optical Kerr Effect (MOKE) can be viewed as conjugate phenomena in the context of underlying symmetry.

Table 1 Required broken symmetries and relevant MPGs for certain measurables or phenomena.
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Candidate materials

Using Figs. 2–4, one can readily identify potential materials suitable for measuring the various phenomena we have discussed. Materials belonging to thirty-one ferromagnetic point groups (1, \(\bar{1}\), 2, 2′, m, m′, 2/m, 2′/m′, 2′2′2, m′m′m, m′m′2, m′m2′, 4, \(\bar{4}\), 4/m, 42′2′, 4m′m′, \(\bar{4}\)2′m′, 4/mm′m′, 3, \(\bar{3}\), 32′, 3m',\(\,\bar{3}\)m′, 6, \(\bar{6}\), 6/m, 62′2′, 6m′m', \(\bar{6}\)m′2′, 6/mm′m′) can exhibit all relevant phenomena. As we have discussed, these materials include seemingly-antiferromagnetic magnets such as Mn3(Sn,Ge,Ga,Rh,Ir,Pt), as shown in Fig. 1a, b and d and the so-called altermagnets such as MnTe and RuO2 with the MPG m′m′m20,21. Another examples are metallic cubic Pd3Mn52 and insulating NaMnFeF653 forming in ferromagnetic 32′ with unbroken [C3z,Rx⊗T]. The 32′ point group, allowing all of these phenomena, e.g. magnetizationz, Odd-order AHEyx, Odd-order AHExy, Odd-order AHEzx, NCDy or z, Diagonal piezomagnetismy or z, and MOKEy or z, so Pd3Mn and NaMnFeF6 can be studied for these phenomena. Note that since magnetization of 32′ is along z, so Odd-order AHEyx, Odd-order AHExy, NCDz, Diagonal piezomagnetismz and MOKEz are entirely expected; however, they can also exhibit Odd-order AHEzx, NCDy, Diagonal piezomagnetismy and MOKEy, even though magnetization along y is zero. These off-magnetization-direction phenomena have never been observed.

Emphasize that the current for AHE can be electric current or other propagating quasiparticles such as thermal current, propagating lights, magnons and phonons. As discussed earlier, Odd-order AHE, associated with symmetric tensors, can occur in true antiferromagnets in the MPGs of 4', \(\bar{4}\)′, 4'/m, 4'2'2 (or 4'22′), 4'm'm (or 4'mm'), \(\bar{4}\)′2'm, \(\bar{4}\)′m'2 (\(\bar{4}\)′2m′), 4'/mmm'. MPG of the magnetic ground state of RuO2 in bulk form is 4'/mmm'19, which does not belong to the ferromagnetic point group. Thus, high-odd-order AHE can be expected in RuO2, which requires future experimental confirmation. The magnetic states of insulating Er2Ge2O7 and Pb(TiO)Cu4(PO4)4 are non-ferromagnetic 4'22′ with unbroken [C4z⊗T,Rz,Rx,Ry,Rxy⊗T,Ryx⊗T], allowing Odd-order AHEyx,xy, CDx, y, xy, yx or z, and MOKEx, y, xy or yx. Thus, it is imperative to measure Odd-order AHEyx,xy, CDx, y, xy, yx or z, and MOKEx, y, xy or yx in Pb(TiO)Cu4(PO4)4 and Er2Ge2O7. Note that this Odd-order AHEyx,xy can be observed as an anomalous thermal Hall effect in those insulating 4'22′ systems without any ferromagnetic moment.

Ferroelectric hexagonal h-HoMnO3 adopts the B1 magnetic state below 70 K, which corresponds to MPG 6'mm'54,55. Fig. 5a depicts the triangular configuration of spins in the ab-plane with MPG 6'mm'. MPG 6'mm' allows Off-diagonal piezomagnetismxy, Odd-order AHEzy, Diagonal piezomagnetismx, NCDx, and MOKEx (Fig. 5b), all of which are the results of our SOS analysis and need to be experimentally verified. The antiferromagnetic state of Fe2Mo3O8 also forms in the same MPG 6'mm'56. Note that when we have Diagonal odd-order (such as linear) piezomagnetismx and Off-diagonal odd-order (such as linear) piezomagnetismxy, applying an electric field or current along x or y will induce magnetization along x, which is an even-order (such as quadratic) with applied electric field/current, so the sign change of applied electric field/current will not change the sign of induced magnetization. Finally, note that the MPG of CsFeCl357 below TN = 4.7 K is \(\bar{6}\)′m2', while the point group of it above TN is centrosymmetric 6/mmm. \(\bar{6}\)′m2' has unbroken [C6z⊗I⊗T,C3z,Mx,Ry⊗T, Mz⊗T], so can exhibit Odd-order AHEzy, NCDx, Diagonal piezomagnetismx, and MOKEx. Thus, insulating CsFeCl3 in a non-ferromagnetic point group with zero magnetization can exhibit NCD such as the Faraday effect, which has been always thought to be confined in ferromagnetic systems.

Fig. 5: Various phenomena in MPG 6ʹmmʹ.
figure 5

a A schematic of antiferromagnetic order in the Mn trimerized triangular layers of polar h-HoMoO3. The B1 phase of the MPG 6ʹmmʹ appears below 70 K. b A schematic of the sample configuration for measurements including high-odd-order AHEzy, MOKEx, NCDx, off-diagonal piezomagnetismxy, and Diagonal piezomagnetismx. Jy the applied electric current and Jz the Hall current. Green double arrows denote stress forces.

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Conclusion

Our SOS concept incorporates the symmetry relationship between the specimen and the experimental setup, encompassing measurable parameters and the sample environment, without considering local couplings or relevant tensorial terms. The SOS approach can tell if a certain measurable relevant to a particular phenomenon is zero, non-zero odd-order, or non-zero even-order of, for example, applied electric current. By employing this SOS approach, we have successfully identified all MPGs relevant for each of ferromagnetism-like phenomena, including magnetic attraction/repulsion, diagonal piezomagnetism, nonreciprocal circular dichroism (such as Faraday effect), odd-order (including linear) anomalous Hall effect, and magneto-optical Kerr effect. The ferromagnetism-like phenomena can manifest only in two ways: first, through non-zero magnetization in ferromagnetic point groups, where symmetry permits non-zero magnetization; and second, through magnetization induced by external perturbations such as electric current flow, light propagation, or stress. Undoubtedly, the categorized MPGs for each ferromagnetism-like phenomenon, along with our SOS approach, will serve as crucial guidance for future advancements in magnetism-related science and technology.