Permutable SOS (symmetry operational similarity)

Based on symmetry consideration, quasi-one-dimensional (1D) objects, relevant to numerous observables or phenomena, can be classified into eight different types. We provide various examples of each 1D type and discuss their symmetry operational similarity (SOS) relationships, which are often permutable. A number of recent experimental observations, including current-induced magnetization in polar or chiral conductors, non-linear Hall effect in polar conductors, spin-polarization of tunneling current to chiral conductors, and ferro-rotational domain imaging with linear gyration are discussed in terms of (permutable) SOS. In addition, based on (permutable) SOS, we predict a large number of new phenomena in low symmetry materials that can be experimentally verified in the future.


INTRODUCTION
The concept of SOS (symmetry operational similarity) 1,2 between specimen constituents (i.e., periodic crystallographic or magnetic lattices in external fields or other environments, etc., and also their time evolution) and measuring probes/quantities (observables such as 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., Hall-type effects) in relation to broken symmetries is powerful to understand and predict observable physical phenomena in low symmetry materials. 3 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 complicated and unintuitive phenomena in complex materials.
In conjunction with the earlier publications, 1-3 we here discuss (quasi-)one-dimensional objects (simply referred as 1D objects): objects can be specimen constituents (periodic lattices in the presence of external perturbations) or measuring probes/quantities (observables), and (quasi-)1D refers objects that has a certain rotation symmetry around the 1D direction, and all rotational symmetry, possibly except C2 symmetry, is broken around any directions perpendicular to the 1D direction. Note that periodic crystallographic or magnetic lattices can have 2-, 3-, 4-, or 6fold rotational symmetries (C2, C3, C4, or C6 symmetries in the standard crystallographic notations, respectively), 1D specimen constituents must have at least, C2 or C3 symmetry around the 1D direction, and broken all of C3, C4, and C6 symmetries around any directions perpendicular to the 1D direction. One example of 1D objects is cycloidal spins having SOS with polarization (P), and cycloidal spins do have C2 symmetry around P, but no rotation symmetry, including C2, around any direction perpendicular to the P direction. 1 Here, we define R= (i.e. 2-fold) rotation operation with the rotation axis perpendicular to the 1D object direction, R= (i.e. 2-fold) rotation operation with the rotation axis along the 1D object direction, I=space inversion, M=mirror operation with the mirror perpendicular to the 1D object direction, M= mirror operation with the mirror plane containing the 1D object direction, T=time reversal operation. (In the standard crystallographic notations, "R, R, I, M, M, and T" are "C2⊥, C2ll,1, m⊥, mll, 1, respectively. 1 Our notations are intuitive to consider 1D objects. In some cases, we specify 2-fold rotation and mirror operations further using the notations of R•, R_, R, M, M, and M_ with the relevant axes/planes, defined with respect to the page plane, in the subscriptions. Note that since we consider the 1D objects, translational symmetry is ignored. For example, T symmetry in a simple 1D antiferromagnet is not broken since we ignore translation. Hlinka has proposed that there exist eight kinds of vectorlike physical quantities in term of their invariance under spatiotemporal symmetry operations. 4 In fact, our 1D objects are exactly like the vectorlike physical quantities in terms of symmetry, so there are eight kinds of 1D objects. We will call them as "D, C, P, A, D, C, P, A". D, C, P, and A refer to Director, Chirality, Polarization, and rotational Axial (electric toroidal) vector, respectively, and  refers to the time reversal broken version. All left-hand-side 1D objects (D, C, D, C) in Fig. 1 have notbroken R (i.e. they are director-like) and all right-hand-side 1D objects (P, R, P, R) have broken R (i.e. they are vector-like).

RESULTS AND DISCUSION
We have attempted to identify various exemplary 1D objects that can be classified into each of the eight Hlinka's classifications 4 , which are listed in Fig. 1

Permutable SOS of P x P  A
The usefulness of the SOS approach is that it can be commonly applied to various physical quantities that share the same set of broken symmetries. P can be polarization (P) or an electric field (E), but it can be also a temperature gradient, strain gradient, or surface effective electric field, P can be velocity/wave vector (k) of electric current, spin wave or thermal current, or toroidal moment, and A is magnetization (M) or magnetic field (H). Moreover, the SOS relationships established among them can sometimes be permutable. Fig. 3(a), (b) and (c) shows pictorially the permutable SOS relationships of P x A  P, A x P  P, P x P  A, respectively. Note that all of P, P, and A are vector-like.
It turns out that the motion of quasi-particles such as electrons, spin waves, phonons, and photons in a specimen or the motion of the specimen itself in specimen constituents can be non-reciprocal if the specimen constituents have SOS with k. When +k becomes −k under a symmetry operation, while a specimen constituent where quasiparticles are moving with ±k is invariant under the symmetry operation, then the experimental situation becomes reciprocal. On the other hand, when a specimen constituent has SOS with k, then there is no symmetry operation that can connect these two experimental situations: one specimen constituent with +k and the identical specimen constituent with −k. Thus, the experimental situation can become non-reciprocal, even though the magnitude of the non-reciprocal effects cannot be predicted.
Thus, Fig. 3 Fig. 3(a). We anticipate that applying in-plane H to polar Fe(Mn,Ni,Zn)2Mo3O8 or polar (and chiral) Ni3TeO6 9 will induce non-reciprocal spin wave along the in-plane direction perpendicular to H. The monolayer of 2H-MoS2, shown in Fig. 4(a), is non-centrosymmetric, but non-polar due to the presence of C3 symmetry. However, uniaxial uniform strain along a particular in-plane direction breaks the C3 symmetry and induces P in the direction perpendicular to the strain.
When electric current is applied to an in-plane direction perpendicular to the strain-induced P, then M develops along the out-of-plane direction, which is depicted in Fig. 4  We also note that the so-called circular dichroism Hall effect can be understood on a similar footing. 16 Illumination of circularly-polarized light on a chiral material can induce electric current, which is called a circular photogalvanic effect (CPGE). [17][18][19] In replaced by H and P, respectively, it corresponds to [5]. This demonstrates a permutable SOS relationship as well as the permutable (and duality) nature of linear magnetoelectric effects.
Permutable SOS relationships of [2], [5], and [7] can be also well exemplified in hexagonal rare-earth ferrites 20 . The A2 phase of the system carries P (P) along the c axis and inplane magnetic monopole (C), and a net magnetic moment M (A) along the c axis is, indeed, observed experimentally, which is neatly consistent with [2]. Through a spin reorientation transition, the magnetic state transforms into the A1 phase with magnetic toroidal moments (P), where the net magnetic moment M (A) is no longer allowed according to P • P D. This is another premium example of how these permutable SOS relationships can be used to explain and predict physical properties in real materials. Figure 5(b) corresponds to [6], which explains longitudinal and also transverse magnetochiral effects, i.e non-reciprocal electronic transport or optical effects in chiral materials in the presence of external magnetic fields. 21 Note that a twisted graphene bilayer is chiral, so exhibits a natural optical activity, 22 and should exhibit non-reciprocal optical and also transport effects in the presence of magnetic fields, which need to be experimentally verified. Their inverse effects in Fig. 5(c), corresponding to [1], include magnetization induced by electric current in chiral materials and spin-polarized tunneling in chiral materials. The magnetization induced by electric current in chiral Te single crystals has been reported in a Faraday-type optical experiment as well as a direct magnetization pulse measurement. [23][24] The chiralitydependent spin-polarization of tunneling current was recently observed in chiral Co1/3NbS2 25 Fig. 5(c), flowing electric current across the chiral structure induces a magnetization along the current direction, which should work even for local probes such as scanning tunneling microscopy (STM). Fig. 6(b) shows the spin-polarized STM (SP-STM) measurement around a topological vortex that clearly reveals the alternating domain contrast locked to the chirality of each domain. 25 It is noteworthy that the induced domain contrast does not require long range ordered magnetic moments, in contrast to the conventional SP-STM.
Instead, the specimen constituent comprised of the chiral structure and the tunneling current gives the same set of broken symmetries as magnetization, as shown in Fig. 5(c).
We would like to note finally that ferro-rotation is rather common in numerous materials, 27-31 but does not break inversion symmetry, so difficult to be coupled with measuring probes, and thus has been studied only in a limited degree. The traditional way to study ferrorotational domains is atomic-resolved transmission electron microscopy (TEM) such as highresolution TEM or high-angle annular dark-field (HAADF)-STEM to observe the ferro-rotational distortions or ferro-rotational domains 26 . Fig. 6(e) depicts a pair of ferro-rotational domains, corresponding to a clockwise and counterclockwise rotation of the FeO6 octahedra in RbFe(MoO4)2. Utilizing cryogenic dark-field TEM technique (Fig. 6f) In terms of SOS, we have considered the specimen constituents for light-or current-induced magnetization, NLHE, spin polarization of electric current in non-magnetic or paramagnetic states, linear magneto-electric effects, and natural optical activity or non-reciprocal effects in the presence of an electric field. In addition, we also discuss the requirements for the observation of MOKE, Faraday-type effects, and/or anomalous Hall-type effects in terms of broken symmetries.       Cheong et al. Fig. 6