Trompe L'oeil Ferromagnetism: magnetic point group analysis

Ferromagnetism can be characterized by various unique phenomena such as non-zero magnetization (inducing magnetic attraction/repulsion), diagonal piezomagnetism, nonreciprocal circular dichroism (such as Faraday effect), odd-order (including linear) anomalous Hall effect, and magneto-optical Kerr effect. We identify all broken symmetries requiring each of the above phenomena, and also the relevant magnetic point groups (MPGs) with those broken symmetries. All of ferromagnetic point groups, relevant for ferromagnets, ferri-magnets and weak ferromagnets, can certainly exhibit all of these phenomena, including non-zero magnetization. Some of true antiferromagnets, which are defined as magnets with MPGs that do not belong to ferromagnetic point groups, can display these phenomena through magnetization induced by external perturbations such as applied current, electric fields, light illumination, and strain. Such MPGs are identified for each external perturbation. Since high-density and ultrafast spintronic technologies can be enabled by antiferromagnets, our findings will be an essential guidance for the future magnetism-related science as well as technology.

with non-zero magnetization can take place in certain antiferromagnets with seemingly zero net magnetization, sometime in the presence of external perturbations such as stress or electric fields or with time evolution.These cases have been called Trompe L'oeil Ferromagnetism 18 .
It turns out that broken symmetries can be associated with new order parameters and emergent phenomena.Herein, we identify the exact broken symmetries associated with each of these phenomena: non-zero magnetization, diagonal piezomagnetism 19 , circular dichroism 2,6,20,21 , nonreciprocal circular dichroism (such as Faraday effect), odd-order (including linear) anomalous Hall effect (AHE), and MOKE 22 .For these analyses, we utilize the concept of Symmetry Operational Similarity (SOS), and also 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, in order 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.The power of the SOS approach 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 can be leveraged to identify new materials that exhibit potentially desired properties as well as new phenomena in known 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,
All of our 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 z. 23,24 SYMMETRY OF FERROMAGNETISM 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. 1(a)-(d) appear to be antiferromagnetic states with 120 spins without any net magnetic moments.

SYMMETRY OF THE ODD-ORDER AHE MEASUREMENTS
The Hall effect, a hallmark of how Maxell's equations work in materials, 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 fields 27 .This so-called ordinary Hall effect contrasts with the anomalous Hall effect (AHE) in "ferromagnets", which is sometime called extraordinary Hall effect or spontaneous Hall effect 17 .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 lattice 7,26 , originating from the Berry curvature.In fact, Mn3(Sn,Ge), forming in the same crystallographic structure with that of Mn3(Rh,Ir,Pt), is experimentally reported to exhibit a significant AHE 9,13 .However, it turns out that Mn3(Sn,Ge) does exhibit a small, but finite net magnetic moment 14,22 , 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 reported 28,29 , 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 antiferromagnets 30 .
Herein, we define AHE as "Transverse voltage induced by applied current in zero magnetic field".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 these multi-faceted nature of Hall-type effects, 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 evenorder 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 The rest independent ones can be either broken or unbroken.For example, for broken {I}, basically, Odd-order AHE of the original domain is the same with 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 groups 31,32 can have non-zero net magnetic moments, and do have broken {IT,T,Mx,My,Rx,Ry,C3x,MzT,RzT} with free rotation along z when the net magnetic moments are along z, broken {IT,T,My,Mz,Ry,Rz,C3z,MxT,RxT} with free rotation along x when the net magnetic moments are along x, and broken {IT,T,Mx,Mz,Rx,Rz,C3x,MyT,RyT} with free rotation along y when the net magnetic moments are along ythe relevant MPGs are listed in Figs.2-4.As discussed earlier, all magnetic states in Fig. 1(a)-(d) belong to ferromagnetic point groups, so do exhibit Odd-order AHE.Since all magnetic states in Fig. 1 It turns out that this Odd-order AHE corresponds to Off-diagonal even-order currentinduced 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 currentinduced 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.
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 curvature [33][34][35] .However, it turns out that all those altermagnets, showing linear AHE discussed so far, such as MnTe and RuO2 thin films 35,36 belong to ferromagnetic point groups.

SYMMETRY OF DIAGONAL PIEZOMAGNETISM
Piezoelectricity is the phenomenon of induced polarization, i.

SYMMETRY OF MAGNETO-OPTICAL KERR EFFECT (MOKE)
The experimental setup to measure non-zero Magneto-Optical Kerr Effect (MOKE), i.e.
the light-polarization rotation effect of reflected linearly-polarized light, is shown in Fig. 1(l)&(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.but not belonging to ferromagnetic point groups, are diagonal linear magnetoelectrics.This new 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 magnetization 41 .

CANDIDATE MATERIALS
Using Figures 2-4 phenomena.As we have discussed, these materials include seemingly-antiferromagnetic magnets such as Mn3(Sn,Ge,Ga,Rh,Ir,Pt), as shown in Fig. 1(a), (b) and (d) and the so-called altermagnets such as MnTe and RuO2 with the MPG m′m′m. 35,36 nother examples are metallic cubic Pd3Mn 42 and insulating NaMnFeF6 43 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  45 which can exhibit Off-diagonal piezomagnetismxy, Odd-order AHEzy, Diagonal piezomagnetismx, NCDx, and MOKEx (see Fig. 2), which need to be experimentally verified.Note that when we have Diagonal odd-order (such as linear) piezomagnetismx and Off-diagonal odd-order (such as linear) piezomagnetismxy, applying 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 CsFeCl3 below TN=4.7 K is 6 ̅ 'm2', while the point group of it above TN is centrosymmetric 6/mmm.
6 ̅ 'm2' has unbroken {C6zIT,Mx,Mxy,My,MzT,C3z}, 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.

CONCLUSION
Our SOS concept incorporates the symmetry relationship between specimen and experimental setup, encompassing measurables and sample environment, without considering local coupling 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.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, electric fields, light propagation, or strain.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.AHEyx is expected to be identical to Odd-order AHExy, except for the possible sign difference.
Green: MPGs of diagonal linear magnetoelectric; Blue: MPGs of chiral point group; Turquoise: MPGs of linear magnetoelectric and chirality.
PhotoGalvanic Effect (CPGE)39,40 .Fig.1(i) and (j) are linked through {Mx,My}, but each setup is invariant under {1,TRx,TRy} and any spatial rotation around z. Thus, this setup has unbroken {TRx,TRy} and broken {Mx,My}, so "broken {Mx,My} + unbroken {TRx,TRy}broken {Mx,My}" = broken {IT,Mx,My} 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 ′m m , 4′ ̅ 2 ′m m , 4′ / mmm′ , m3 ̅ , m3 ̅ m m ′ , 4′ ̅ , 4′ ̅ 2 m′ , 4′ ̅ 3 m′ 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.
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.In fact, it turns out that all MPGs showing MOKE,

Fig. 2
Fig. 2 Magnetic point groups for various ferromagnetism-like phenomena along z.Odd-order

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

Fig. 4 Table 1 ,
Fig. 4 Magnetic point groups for various ferromagnetism-like phenomena along y.Only in ferromagnetic point groups, Odd-order AHEzx is expected to be identical to Odd-order AHExz, except the possible sign difference.Green: MPGs of diagonal linear magnetoelectric; Blue: MPGs of chiral point group; Turquoise: MPGs of both.
groups are associated with symmetric tensors, unlike antisymmetric tensors for oddorder AHEs in ferromagnetic space groups.For example, for 4 ̅ ' point group, (+Jx,+Ey) becomes (+Jy,+Ex) under 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.Odd-order AHEs with symmetric tensors have never been reported and will be an exciting new research direction.(3) Cubic MPGs allow Odd-order AHEyx,xy as well as Odd-order AHExy,yx: 23, m3 ̅ , 4'32', 4 ̅ '3m', m3 ̅ m'.