Abstract
The characteristics of ferro(ferri)magnetism with nonzero magnetization include magnetic attraction, magnetic circular dichroism, and magnetooptical Kerr (MOKE), Faraday, and various anomalous Halltype (Hall, Ettingshausen, Nernst, and thermal Hall) effects. Nonmagnetic or antiferromagnetic materials in external electric fields or other environments (called specimen constituents) can share symmetry operational similarity (SOS) with magnetization (\(\boldsymbol{\mathcal{M}}\)) in relation to broken symmetries. These specimen constituents can be associated with nonzero magnetization and/or show ferromagnetismlike behaviors, so we say that they exhibit Trompe L’oeil Ferromagnetism. Examples include linear magnetoelectric materials such as Cr_{2}O_{3} under electric fields, Faraday effect in chiral materials such as tellurium with current flow, magnetic field induced by the motion of Neel or Blochtype ferroelectric walls, and magnetooptical Kerr (MOKE), Faraday effect, and/or anomalous Halltype effects in certain antiferromagnets such as Cr_{2}O_{3}, MnPSe_{3}, Mn_{4}(Nb,Ta)_{2}O_{9}, and Mn_{3}(Sn,Ge,Ga). A large number of new specimen constitutes having SOS with \(\boldsymbol{\mathcal{M}}\) are proposed, and require future experimental verification of their ferromagnetismlike behaviors, and also theoretical understanding of possible microscopic mechanisms.
Introduction
Ferro(ferri)magnets with nonzero magnetic moments has been of great fascination and practical value ever since a piece of lodestone (magnetite, Fe_{3}O_{4}) was used as a compass more than 2000 years ago^{1,2,3}. In fact, the invention and further development of storing information with ferro(ferri)magnets has been a key component of the microelectronics revolution for the last century^{3}. In addition to their magnetic attraction, ferro(ferri)magnets can show unique physical phenomena such as various magnetooptical properties (magnetooptical Kerr (MOKE), Faraday, and magnetic circular dichroism) and anomalous Halltype effects^{4,5,6,7,8,9,10,11,12,13,14,15,16}. The anomalous Halltype effects include anomalous Hall, anomalous Ettingshausen, anomalous Nernst, or anomalous thermal Hall effects^{11,12,13,14,15,16}. It turns out that these phenomena that traditionally thought to occur only in ferro(ferri)magnets with nonzero magnetic moments can take place in certain specimens with zero magnetic moment, sometimes in the presence of external electric/strain fields or with time evolution. We call these cases as Trompe L’oeil Ferromagnetism.
It turns out that observable physical phenomena can occur when specimen constituents (i.e., lattice distortions or spin arrangements in external fields or other environments, etc., and also their time evolution) and measuring probes/quantities (i.e., propagating light, electrons or other particles in various polarization states, including light or electrons with spin or orbital angular momentum, bulk polarization or magnetization, etc., and also experimental setups to measure, e.g., Halltype effects) share symmetry operational similarity (SOS) in relation to broken symmetries^{17}. This SOS relationship includes when specimen constituents have more, but not less, broken symmetries than measuring probes/quantities do. In other words, in order to have a SOS relationship, specimen constituents “cannot have higher symmetries” than measuring probes/quantities do. The power of the SOS approach lies in providing simple and physically transparent views of otherwise unintuitive phenomena in complex materials. Furthermore, this approach can be leveraged to identify new materials that exhibit potentially desired properties as well as new phenomena in known materials.
In this paper, we discuss that certain specimen constituents with nonmagnetic or antiferromagnetic materials can exhibit SOS with magnetization (\(\boldsymbol{\mathcal{M}}\)), and what kinds of ferromagnetismlike behaviors these specimen constituents can exhibit. Here, we define R = π (i.e. twofold) rotation operation with the rotation axis perpendicular to the \(\boldsymbol{\mathcal{M}}\) direction, R = π (i.e. twofold) rotation operation with the rotation axis along the \(\boldsymbol{\mathcal{M}}\) direction, I = space inversion, M = mirror operation with the mirror perpendicular to the \(\boldsymbol{\mathcal{M}}\) direction, M = mirror operation with the mirror plane containing the \(\boldsymbol{\mathcal{M}}\) direction, T = time reversal operation. (In the standard crystallographic notations, “R, R, I, M, M, and T” are “C_{2⊥}, C_{2ll}, \(\bar 1\), m_{⊥}, m_{ll}, 1′, respectively. Our notations are intuitive to consider onedimensional (1D) objects such as \(\boldsymbol{\mathcal{M}}\).) Evidently, +\(\boldsymbol{\mathcal{M}}\) becomes \(\boldsymbol{\mathcal{M}}\) by R symmetry operation, i.e. \(\boldsymbol{\mathcal{M}}\) has broken R symmetry. Similarly, +\(\boldsymbol{\mathcal{M}}\) becomes \(\boldsymbol{\mathcal{M}}\) by M or T symmetry operation. Note that + \(\boldsymbol{\mathcal{M}}\) remains to be +\(\boldsymbol{\mathcal{M}}\) under I symmetry operation. In fact, {R,M,T} is the set of all “independent” broken symmetries of \(\boldsymbol{\mathcal{M}}\). Emphasize that since we consider the 1D nature of \(\boldsymbol{\mathcal{M}}\), translational symmetry is ignored. The complete sets of broken symmetries for four vectors discussed in this paper are listed in Fig. 1. Electric polarization (P) has broken {R,I,M}, and velocity vector (k, linear momentum or wave vector) has broken {R,I,M,T}. A uniform strain gradient (SG) along the strain gradient direction behaves like P in terms of symmetry operations; in other words, SG has broken {R,I,M}. The act of applying an external electric field (E_{ext}), inducing electric current (induced J), for various transport measurements where quasiequilibrium processes are involved has broken {R,I,M,T}. Note that +E_{ext} (induced +J) becomes −E_{ext} (induced −J) under any of {R,I,M}, but under T symmetry operation, “induced +J” may not become “induced −J” while +E_{ext} becomes −E_{ext}. This is because T operation on “the act of applying an external electric field +E_{ext}” results in “the act of applying an external electric field −E_{ext}”, which does not necessarily accompany “induced −J”, and the magnitude of induced J may change when the direction of E_{ext} is switched. Therefore, T operation on +E_{ext} (induced +J) results in −E_{ext} (induced −J′), in general^{18,19}. Also note that periodic crystallographic or magnetic lattices can have two, three, four, and sixfold rotational symmetries (C_{2}, C_{3}, C_{4}, and C_{6} symmetries in the standard crystallographic notations, respectively), but 1D measuring probes/quantities such as \(\boldsymbol{\mathcal{M}}\), P, and k do not have C_{3}, C_{4}, and C_{6} symmetries around the rotation axis perpendicular to \(\boldsymbol{\mathcal{M}}\), P, or k. Thus, specimen constituents having SOS with \(\boldsymbol{\mathcal{M}}\), P, and k should not have any of C_{3}, C_{4}, and C_{6} symmetries around the rotation axis perpendicular to \(\boldsymbol{\mathcal{M}}\), P, or k. This consideration becomes important when we discuss antiferromagnetic states having SOS with \(\boldsymbol{\mathcal{M}}\).
Results and discussion
Linear magnetoelectricity
Linear magnetoelectrics are antiferromagnetic materials that do not exhibit SOS with P with broken {R,I,M} in zero applied magnetic field (H), but do show SOS with P in nonzero H. These linear magnetoelectrics exhibit reciprocal magnetoelectric effects, i.e., they do not exhibit SOS with \(\boldsymbol{\mathcal{M}}\) with broken {R,M,T} in zero applied electric field (E), but do show SOS with \(\boldsymbol{\mathcal{M}}\) in nonzero E, and indeed exhibit nonzero \(\boldsymbol{\mathcal{M}}\) in nonzero E. The lefthandside specimen constituents in Fig. 2a–c in zero E show only a part of broken {R,M,T}, but they in nonzero E have now broken all of {R,M,T}, so show SOS with \(\boldsymbol{\mathcal{M}}\), which is consistent with, for example, the linear magnetoelectric effects in Cr_{2}O_{3}; diagonal linear magnetoelectric effect, corresponding to Fig. 2a, and offdiagonal linear magnetoelectric effect, corresponding to Fig. 2b, before and after spin flop transition, respectively^{20}. The specimen constituents in Fig. 2c are for magnetic monopoles (1st and 2nd), magnetic toroidal moment (3rd), and magnetic quadrupole (4th and 5th). None of the cases in Fig. 2c in zero E has SOS with \(\boldsymbol{\mathcal{M}}\), but all of them do have SOS with \(\boldsymbol{\mathcal{M}}\) in nonzero E; thus, all can exhibit linear magnetoelectricity. Note that the 1st and 2nd cases in Fig. 2c correspond to Ntype magnetic monopole, and Stype magnetic monopole in the presence of the same E will result in − \(\boldsymbol{\mathcal{M}}\). Similarly, the 3rd case in Fig. 2c corresponds to counterclockwise magnetic toroidal moment, and clockwise magnetic toroidal moment in the presence of the same E will result in − \(\boldsymbol{\mathcal{M}}\). We can also consider various spin configurations on (buckled) honeycomb lattice as shown in Fig. 3a–c. None of these spin configurations in zero E has SOS with \(\boldsymbol{\mathcal{M}}\), but all of them in nonzero E do have SOS with \(\boldsymbol{\mathcal{M}}\), which indicates that all can be linear magnetoelectrics. Most of these cases have not been experimentally observed in real compounds, and it will be highly demanding to verify these symmetrydriven predictions in real materials^{21}. These antiferromagnetic states in Fig. 3 have been reported in various compounds with (buckled) honeycomb lattice such as (Mn,Fe,Co,Ni)P(S,Se)_{3}, BaNi_{2}V_{2}O_{8}, (Ca,Sr)Mn_{2}Sb_{2}, Na_{2}Ni_{2}TeO_{6}, and (Mn,Co)_{4}(Nb,Ta)_{2}O_{9}, but their linear magnetoelectricity has mostly not been reported yet^{21,22,23,24,25,26,27,28}. For example, the 2nd case in Fig. 3b corresponds to the Ising antiferromagnetic state in, e.g., Mn_{4}(Nb,Ta)_{2}O_{9} with buckled honeycomb lattice, and the 1st case in Fig. 3c represents the inplane antiferromagnetic order in buckled honeycomb lattice with anions in, e.g., MnPSe_{3}.
One exemplary system relevant to Fig. 2c is hexagonal (h) R(Mn,Fe)O_{3} (R = rare earths), which is an improper ferroelectric with the simultaneous presence of Mn/Fe trimerization in the ab plane and ferroelectric polarization along the c axis. A number of different types of magnetic order with inplane Mn spins have been identified in hR(Mn,Fe)O_{3}. The socalled A1type magnetic order in hR(Mn,Fe)O_{3} combined with Mn trimerization can induce a net toroidal moment, and the socalled A2type magnetic order in hR(Mn,Fe)O_{3} combined with Mn trimerization can accompany a net magnetic monopole^{29,30}. The linear magnetoelectric effect associated with these magnetic monopole and toroidal moment can be a topic for the future investigation. Note that reversing the E direction in Figs. 2a–c and 3a–c (for example, by R operation) should accompany the reversal of \(\boldsymbol{\mathcal{M}}\), which is consistent with the nature of linear magnetoelectricity. E like P has broken {R,I,M}, and similarly a strain gradient vector (SG, a uniform strain gradient along a particular direction) has broken {R,I,M}. Thus, when E is replaced by SG, the lefthandside specimen constituents in Figs. 2a–c and 3a–c have broken {R,M,T}, so do have SOS with \(\boldsymbol{\mathcal{M}}\), which means that all linear magnetoelectrics can exhibit flexomagnetism, i.e. the induction of a net magnetic moment by a strain gradient (the converse may not be true.). We emphasize that there exists a sign ambiguity in all linear magnetoelectrics discussed above as well as all cases having SOS with 1D objects in this paper. In other words, the symmetry arguments cannot determine the absolute sign of \(\boldsymbol{\mathcal{M}}\), e.g., in Figs. 2 and 3. However, all antiferromagnetic spins in, e.g., the speciment consitituent of Fig. 1a flip by 180° through, e.g., T, then the sign of \(\boldsymbol{\mathcal{M}}\) also changes. Note that while symmetry does not fix the absolute sign, but microscopc mechanism determines it. For example, the sign of spinorbital coupling can fix the absolute sign of induced \(\boldsymbol{\mathcal{M}}\).
Dynamic or quasiequilibrium mechanisms for Trompe L’oeil Ferromagnetism
It is also interesting to consider how to induce magnetization in nonmagnetic materials in nontrivial manners. Note that since any static configuration of P or structural distortions cannot break T, a time component has to be incorporated into P configurations or structural distortions to induce a SOS relationship with \(\boldsymbol{\mathcal{M}}\). Two structural examples having SOS with \(\boldsymbol{\mathcal{M}}\) are shown in Fig. 4a, b. When electric current is applied to a tellurium crystal with a screwtype (so called monoaxial) chiral lattice, corresponding to the lefthandside cartoon of Fig. 4a, \(\boldsymbol{\mathcal{M}}\) can be induced in a linear fashion^{31}. Consistently, Faraday rotation of linearlypolarized THz light propagating along the chiral axis of a tellurium crystal is also observed in the presence of quasiequilibrium electric current flow along the chiral axis, and this induced Faraday rotation effect is linearly proportional to the electric current^{32}. This effect is supposed to occur even when light propagation and electric current are parallel, but perpendicular to the chiral axis, as displayed in the righthandside cartoon of Fig. 4a, which needs to be verified experimentally. Interestingly, the transfer of photocurrent in chiral DNA turns out to be highly spin polarized^{33}. All of these highly nontrivial effects correspond to the SOS relationship in Fig. 4a. Figure 4b represent rotating P having SOS with \(\boldsymbol{\mathcal{M}}\), and can be realized when a Neel or Blocktype ferroelectric wall moves in the direction perpendicular to the ferroelectric wall^{34,35}, even though any motion of Isingtype ferroelectric walls will not induce \(\boldsymbol{\mathcal{M}}\). These effects can be utilized to flip the magnetization of a ferro(ferri)magnetic island sitting on the top of a ferroelectric with Neel or Blocktype walls, but have not been experimentally realized yet.
Anomalous Halltype effects, involving quasiequilibrium processes, can be also understood in terms of SOS with \(\boldsymbol{\mathcal{M}}\). The sets of (E_{ext}, +, −), (E_{ext}, h, c), (ΔT, +, −), and (ΔT, h, c) of Fig. 4c correspond to the Hall, Ettingshausen, Nernst, and thermal Hall effects, respectively. In the case of, e.g., Hall effect, switching both E_{ext} and (+,) simultaneously can be achieved by any of {R,I} (here, symmetry operations are defined with respect to \(\boldsymbol{\mathcal{M}}\) in the following), so when a specimen with broken {R,I} can exhibit a Hall effect that is nonlinear with E_{ext}^{17}. Consider, for example, a Hall effect experiment on “a specimen with notbroken R or I” in external H or with internal \(\boldsymbol{\mathcal{M}}\),. Then, the notbroken R or I operation on the entire experimental setup including the specimen results in flipping the sign of E_{ext} and (+, −) Hall voltage without changing H/\(\boldsymbol{\mathcal{M}}\) direction, meaning that the flipping the sign of E_{ext} leads to flipping the Hall voltage sign without changing the magnitude, so the leading term of Hall voltage proportional to E_{ext} is allowed, but a (E_{ext})^{2} term is not permitted in terms of symmetry. By the way, this works only for R or I operation. Thus, if both R or I are broken, then no symmetry limits the Hall effect linear in E_{ext}, which opens up the possibility of a nonlinear effect such as Hall voltage varying like (E_{ext})^{2} in terms of symmetry. We here consider only specimens with notbroken {R,I}, i.e. the Haltype effects linear with E_{ext}. Then, one can show readily that the entire experimental setup with a specimen with no broken symmetry in the lefthandside cartoon in Fig. 4c have broken {R,M,T}, so do have SOS with \(\boldsymbol{\mathcal{M}}\). This leads to a highly nontrivial theorem that any specimen with broken {R,M,T}, but notbroken {R,I} (i.e., having SOS with \(\boldsymbol{\mathcal{M}}\)) can show all of anomalous Hall, anomalous Ettingshausen, anomalous Nernst, and anomalous thermal Hall effects, which are linear with E_{ext}. This important theorem is closely relevant to the presence of anomalous Halltype effects in antiferromagnets without any significant net moments, but with SOS with \(\boldsymbol{\mathcal{M}}\), which we will discuss later.
MOKE and Faradaytype effects
When an object with angular momentum transmits through a specimen constituent, its behavior can depend on the sign of the angular momentum. This is, in general, called circular dichroism, and the angular momentum can be spin angular momentum (in, e.g., circularlypolarized light) or orbital angular momentum (in, e.g., vortex beams)^{17,36,37,38}. This angular momentumsigndependent (circular in the case of circularlypolarized light) dichroism can be linear (frequency conserving) or nonlinear (frequency changing)^{4,5,6,7,8,9,10}. Two particular examples of linear circular dichroism are the rotation of light polarization in the transmission of linearly polarized light through a chiral material (called natural optical activity), and the rotation of light polarization in the transmission of linearlypolarized light through a ferro(ferri)magnet (called Faraday effect), and these effects result from the different refractive indices or speeds of light of leftcircularlypolarized and rightcircularlypolarized lights. The nonlinear circular dichroism is associated quasiequilibrium states with inelastic (dissipative) processes, and the circular dichroism in ferro(ferri)magnetic materials or nonmagnetic (or antiferromagnetic) materials in magnetic fields is called magnetic circular dichroism. Xray magnetic circular dichroism (XMCD) is a nonlinear effect, is associated with electronic transitions among atomic orbital states, and has been heavily studied to explore elementspecific magnetism of ferro(ferri)magnets. A giant optical rotation at a THzfrequencyrange antiferromagnetic resonance can occur in the collinear antiferromagnetic state of chiral Ni_{3}TeO_{6}, which exhibits a significant natural optical activity in visible optical range at temperatures even far above an antiferromagnetic transition temperature^{39,40}. It turns out that the symmetry requirement for all of the above linear or nonlinear effects in transmission or absorptiontype experiments in chiral materials, ferro(ferri)magnets or even certain antiferromagnets is broken {M,M⊗R,I⊗T}^{17}. Therefore, in this paper, all of the above effects, including, e.g., natural optical activity in chiral materials, and Faraday effect and XMCD in magnetic materials, will be called “Faradaytype effects” or just “Faraday effects” for the sake of simplicity.
On the other hand, Magnetooptical Kerr effect (MOKE) refers the lightpolarization rotation of linearly polarized light when it is “reflected” on, e.g., ferro(ferri)magnetic surfaces^{17}. The symmetry requirement for MOKE is broken {M,M⊗R,T}^{17}. Certainly, ferro(ferri)magnets with \(\boldsymbol{\mathcal{M}}\) does have broken{M,M⊗R,T}, so do exhibit MOKE for light propagation along the \(\boldsymbol{\mathcal{M}}\) direction. Chiral material without magnetism can exhibit Faraday effect (in fact, natural optical activity), but does not exhibit MOKE since T is not broken in chiral materials (see Table 1). Interestingly, antiferromagnetic Cr_{2}O_{3} in Fig. 2a in zero H has broken {M,M⊗R,T}, so does show MOKE. However, both T and I are broken in Cr_{2}O_{3}, but I⊗T is not broken, so Cr_{2}O_{3} does not exhibit Faraday effect (see Table 1)^{41}.
Now, we can prove a second theorem that breaking all of {M,M⊗R,I⊗T} and {M,M⊗R,T} in a specimen constituent, where T and I⊗T are broken in the same way, is identical with breaking {R,M,T}, so the specimen constituent shows SOS with \(\boldsymbol{\mathcal{M}}\). We consider the specimen constituents where T and I⊗T are broken in the same way, so I = I⊗T⊗T = T⊗T, so I is not broken. When all {M,M⊗R,T,I⊗T} are broken, but I is not broken, R = I⊗M is broken. Furthermore, M⊗R = I⊗R⊗R = R^{1} = R. Thus, {R,M,T} is the set of all independent broken symmetries. Therefore, we can conclude this nontrivial theorem: all materials having SOS with \(\boldsymbol{\mathcal{M}}\) exhibit both Faraday effect and MOKE, and any inversionsymmetric materials exhibiting both Faraday and MOKE do show SOS with \(\boldsymbol{\mathcal{M}}\). The combination of two theorems in this paper leads to the conclusion that any specimens having SOS with \(\boldsymbol{\mathcal{M}}\) can exhibit all of Faraday effect, MOKE, and anomalous Halltype effects.
A number of antiferromagnetic states without any net (significant) magnetic moments showing MOKE and/or Faraday effect are illustrated in Fig. 5. The antiferromagnetic states in Fig. 5a–c have broken {M,M⊗R,T}, but notbroken {M,M⊗R,I⊗T} (here, symmetry operations are defined along the leftright direction in Fig. 5a and the outofpageplane direction in Fig. 5b–d), so are supposed to exhibit MOKE, but no Faraday effect. Figure 5a, identical with Fig. 2a when E is zero, corresponds to the antiferromagnetic state of Cr_{2}O_{3}. The spin configurations in Fig. 5c, d are reported to be realized in Mn_{4}(Nb,Ta)_{2}O_{9} with buckled honeycomb lattice and MnPSe_{3} with buckled honeycomb lattice, respectively^{22,23,28}. Figure 5b represents Ising Atype antiferromagnetic order in a honeycomblattice bilayer, which can be realized in a bilayer of CrI_{3} or CrBr_{3}. MOKE in Cr_{2}O_{3} has been observed, but none of other effects have been reported^{41}. The antiferromagnetic states in Fig. 5e–g have broken {M,M⊗R,T}, and also broken {M,M⊗R,I⊗T} (here, symmetry operations are defined along the leftright direction in Fig. 5e and the outofpageplane direction in Fig. 5f, g), so should exhibit both MOKE and Faraday effect. Figure 5e corresponds to an inversionsymmetric antiferromagnetic state in kagome lattice, which is realized in, for example, Mn_{3}Sn, Fig. 5f can be Ising Atype antiferromagnetic order in a heterogenous honeycomblattice bilayer with, e.g., one CrI_{3} layer and the other CrBr_{3} layer, and Fig. 5g represents Ising Atype antiferromagnetic order in a heterogenous honeycomblattice bilayer. MOKE in Mn_{3}Sn has been observed, but any other effects have not been reported yet^{42,43,44,45}. Note that according to the first theorem discussed earlier, all cases in Fig. 5e–g have broken {M,M⊗R,T} and M,M⊗R,I⊗T}, so do have SOS with \(\boldsymbol{\mathcal{M}}\) (along the leftright direction in Fig. 5e and the outofpageplane direction in Fig. 5f, g)), and all can also exhibit anomalous Halltype effects.
Faradaytype effect with xray illumination can be observed in xray absorption nearedge spectra (XANES) with circularly polarized xray. By recording the XANES spectra with leftcircularlypolarized σ^{+} and rightcircularlypolarized σ^{−} xray with a magnetic field applied either parallel (H^{+}) or antiparallel (H^{−}) to the xray wavevector, we could obtain the three relevant types of dichroism:
where µ (σ,H) stands for the absorption measured for the indicated polarization and sign of the magnetic field. XM(N)CD indicate xray magnetic (natural) circular dichroism signals and XMχD is xray magnetochiral dichroism. The XNCD signal is independent of the applied magnetic field, whereas XMCD changes its sign when the direction of the applied magnetic field is reversed. XMχD does not require polarized light, and manifests as only changes in absorption for two directions of magnetic field. These three dichroism effects are pictorially depicted in Fig. 6. Here, \(\boldsymbol{\mathcal{M}}\) is magnetization induced by applied H, and golden springs represent left or righttype structural chirality of a specimen. Note that XMCD becomes zero when \(\boldsymbol{\mathcal{M}}\) (or an object having SOS with \(\boldsymbol{\mathcal{M}}\)) vanishes. If Left=Right (i.e. no chirality), then XNCD vanishes. XMχD is nonzero only if both Left≠Right (i.e., having chirality) and also \(\boldsymbol{\mathcal{M}}\) ≠0. It is anticipated that new types of XMCD, XNCD, and XMχD experiments can be performed on some of specimen constituents discussed in this paper. For example, XMCD is expected in antiferromagnetic specimens having SOS with \(\boldsymbol{\mathcal{M}}\).: examples include Mn_{3}(Sn,Ge,Ga) (see below for Mn_{3}(Ge,Ga)) and the specimens in Fig. 5f, g. XMχD is anticipated in conical spin states or helical spin states in H.
Antiferromagnetic states with Trompe L’oeil Ferromagnetism
Herein, we discuss further antiferromagnetic states having SOS with \(\boldsymbol{\mathcal{M}}\). The 1st spin configuration in kagome lattice in Fig. 7a, identical with that in Fig. 5e, has broken {R,M,T}, so has SOS with \(\boldsymbol{\mathcal{M}}\). In fact, both cases in Fig. 7a do have broken {R,M,T}, so has SOS with \(\boldsymbol{\mathcal{M}}\) and can exhibit MOKE, Faraday effect, and also anomalous Halltype effects. Indeed, MOKE, anomalous Hall effect (AHE), anomalous Nernst effect, and anomalous thermal Hall effect have been observed in Mn_{3}Sn where the 1st antiferromagnetic state in Fig. 7a occurs^{44,45,46,47,48,49,50,51,52,53}. The microscopic mechanism for these effects has been proposed to be Berry curvature, providing a fictitious magnetic field, in the unique antiferromagnetic state. Note that anomalous Ettingshausen effect can occur in Mn_{3}Sn, but has not been reported yet as far as the author is aware of. The antiferromagnetic state on kagome lattice in Fig. 7b has inversion symmetry, but also has threefold rotation symmetry around the outofpageplane axis, so does not have SOS with \(\boldsymbol{\mathcal{M}}\). However, in the presence of uniform strain (green double arrow), which breaks the threefold rotational symmetry, the Fig. 7b state now shows SOS with \(\boldsymbol{\mathcal{M}}\). Similarly, the allinallout magnetic states in kagome lattice and pyrochlore lattice in Fig. 7c, d, respectively, in the presence of uniaxial uniform strain (green double arrow) exhibit SOS with \(\boldsymbol{\mathcal{M}}\). In the absence of green arrows, all of Fig. 7b–d have broken {R,M,T}, but Fig. 7b, c have threefold rotational symmetry around the axis perpendicular to the page plane, and Fig. 7d has threefold rotational symmetry around the axis perpendicular to one of the triangular planes of tetrahedra, so they do not have SOS with \(\boldsymbol{\mathcal{M}}\). However, these threefold rotational symmetries are broken in the presence of the greendoublearrow strain, so now all of them have SOS with \(\boldsymbol{\mathcal{M}}\). These straininduced SOS relationships with \(\boldsymbol{\mathcal{M}}\) indicate that they are a kind of piezomagnets. Note that the twoinoneout (or oneintwoout) spin arrangement in a triangle, which can be a part of antiferromagnetic kagome lattice, and the threeinoneout (or oneinthreeout) spin arrangement in a tetrahedron, which can be a part of antiferromagnetic pyrochlore lattice, in Fig. 7e do have SOS with P with broken {R,I,M}^{54,55}. In fact, this SOS relationship with P holds on a triangular lattice as long as two bonds have 60° spin order and one bond with 120° spin order, as shown in Fig. 7e. Also note that when antiferromagnetic states with no external perturbations having SOS with \(\boldsymbol{\mathcal{M}}\) tend to, in fact, show tiny net magnetic moments (on the order of 0.001 µB/magnetic ion), possibly due to the contribution of orbital magnetic moment, originating from the SOS relationship with \(\boldsymbol{\mathcal{M}}\)^{45}. Figure 7f, g depict particular types of planar antiferromagnetic order in ABstacked triangular lattices, having SOS with \(\boldsymbol{\mathcal{M}}\)^{56,57}. This magnetic lattice is common in hcp systems; however, these types of planar antiferromagnetic order have never been observed experimentally or discussed theoretically, as far as the author is aware of. Anyhow, if these inversionsymmetric antiferromagnetic states are realized, then they will exhibit MOKE, Faraday effect and also anomalous Halltype effects.
We would like to clarify some of confusing and conflicting issues in literatures. Numerous different kinds of noncolinear and coplanar antiferromagnetic order have been realized in kagome lattice, and many of them have notbroken R (or C_{3}) symmetry, so cannot have SOS with \(\boldsymbol{\mathcal{M}}\). In addition, some of them (e.g., the Fig. 7b case without green double arrow (i.e. strain) and the 3rd case without green double arrow in Fig. 7c) do have broken R symmetry, but also maintain threefold rotational symmetry around the outofpageplane axis, so cannot have SOS with \(\boldsymbol{\mathcal{M}}\). The original prediction of the presence of anomalous Hall effect and MOKE in noncolinear antiferromagnets through a Berry curvature mechanism was done on Mn_{3}Ir and Mn_{3}(Rh,Ir,Pt), respectively, with the magnetic state of the 3rd case of Fig. 7c without green double arrow^{44,46,58,59}. However, this magnetic state has threefold rotational symmetry, so cannot have SOS with \(\boldsymbol{\mathcal{M}}\). Indeed, the experimental observation of anomalous Hall effect and MOKE (and anomalous Nernst and thermal Hall effects) in noncollinear antiferromagnets was made on Mn_{3}Sn, Mn_{3}Ge, and Mn_{3}Ga^{45,47,48,49,50,51,60,61,62}. The magnetic states of Mn_{3}Sn and Mn_{3}Ge(Ga) turn out to be the 1st and 2nd cases of Fig. 7a, respectively. Both cases of Fig. 7a do have SOS with \(\boldsymbol{\mathcal{M}}\), but the directions of the relevant \(\boldsymbol{\mathcal{M}}\)’s are along different directions with respect to the underlying kagome lattice.
Conclusion
In general, the SOS approach can be applied to numerous physical phenomena in solids such as nonreciprocity, magnetisminduced ferroelectricity, linear magnetoelectricity, optical activities (including MOKE and Faraday effect), photogalvanic effects, second harmonic generation, and anomalous Halltype effects, and can be leveraged to identify new materials that potentially exhibit the desired physical phenomena. We have proven two relevant and important theorems: (1) Any materials with broken {R,M,T}, but notbroken {R,I} (i.e., having SOS with \(\boldsymbol{\mathcal{M}}\)) can show all of anomalous Hall, anomalous Ettingshausen, anomalous Nernst, and anomalous thermal Hall effects that are linear with E_{ext}, and (2) all materials having SOS with \(\boldsymbol{\mathcal{M}}\) exhibit both Faraday effect and MOKE, and any inversionsymmetric materials exhibiting both Faraday and MOKE do show SOS with \(\boldsymbol{\mathcal{M}}\). In this paper, we have deliberated a large number of specimen constituents having SOS with \(\boldsymbol{\mathcal{M}}\). Some of these do exhibit a significant measurable magnetization, and others exhibit ferromagnetlike behaviors without any significant magnetization. The former includes linear magnetoelectrics in the presence of E, electric current flow in chiral materials, and moving Neel or Blochtype ferroelectric walls. The latter includes particular types of antiferromagnetic states exhibiting various ferromagnetlike behaviors such as MOKE, Faraday effect, and anomalous Halltype effects. Some other types of antiferromagnetic states with zero net magnetic moment in, e.g., Cr_{2}O_{3} can show MOKE without any Faraday effect nor anomalous Halltype effects. We have presented numerous new cases for these three categories within the concept of SOS – new theoretically and experimentally. Certainly, these propositions need to be experimentally confirmed, and also the potential microscopic mechanisms for these new phenomena ought to be theoretically explored.
Methods
The relationships between specimen constituents (i.e., lattice distortions or spin arrangements in external fields or other environments, etc., and also their time evolution) and measuring probes/quantities (i.e., propagating light, electrons, or other particles in various polarization states, including light or electrons with spin or orbital angular momentum, bulk polarization or magnetization, etc., and also experimental setups to measure, e.g., Halltype effects) are analyzed in terms of the characteristics under various symmetry operations of rotation, space inversion, mirror reflection, and time reversal. When specimen constituents and measuring probes/quantities share the same broken symmetries, except translation symmetry, they are said to exhibit symmetry operation similarities (SOS), and the corresponding phenomena can occur. In terms of SOS, we have considered the specimen constituents for linear magnetoelectric effects, production of effective magnetic fields with moving ferroelectric walls, and also production of net magnetization from antiferromagnetic states. In addition, we also discuss the requirements for the observation of MOKE, Faradaytype effects, and/or anomalous Halltype effects in terms of broken symmetries.
Data availability
The datasets generated during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4413 to the Rutgers Center for Emergent Materials. We have greatly benefited from discussions with Valery Kiryukhin, FeiTing Huang, Christian Batista, and Alessandro Stroppa.
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Cheong, SW. Trompe L’oeil Ferromagnetism. npj Quantum Mater. 5, 37 (2020). https://doi.org/10.1038/s4153502002353
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Permutable SOS (symmetry operational similarity)
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