Linking emergent phenomena and broken symmetries through one-dimensional objects and their dot/cross products

Abstract

The symmetry of the whole experimental setups, including specific sample environments and measurables, can be compared with that of specimens for observable physical phenomena. We, first, focus on one-dimensional (1D) experimental setups, independent from any spatial rotation around one direction, and show that eight kinds of 1D objects (four; vector-like, the other four; director-like), defined in terms of symmetry, and their dot and cross products are an effective way for the symmetry consideration. The dot products form a Z2xZ2xZ2 group with Abe-lian additive operation, and the cross products form a Z2xZ2 group with Abelian additive operation or Q8, a non-abelian group of order eight, depending on their signs. Those 1D objects are associated with characteristic physical phenomena. When a 3D specimen has Symmetry Operational Similarity (SOS) with (identical or lower, but not higher, symmetries than) an 1D object with a particular phenomenon, the 3D specimen can exhibit the phenome-non. This SOS approach can be a transformative and unconventional avenue for symmetry-guided materials designs and discoveries.

Ferrorotational Selectivity in Ilmenites

“…These objects lack reflective symmetry elements, such as mirror planes and inversion points. As a well-comprehended concept integral to modern chemistry and crystallography, chirality could offer us insight into the less explored yet emerging field of ferrorotational (FR) order (known as ferro-axial order), a recently defined ferroic order. − …”

“…For detailed symmetry consideration, ferrorotational order along z, FR z with A z , is defined as "broken twofold rotations around x (C 2x ) and y (C 2y ) axes, and unbroken space-inversion (I) and time-reversal symmetries (T)" when any rotations around z are allowed freely. 3 Ferrorotational orders are common, yet unlike other ferroic orders described by vector parameters such as ferroelectricity and ferromagnetism, they have received limited attention. This is partially because both space-inversion and time-reversal symmetries are preserved in FR materials.…”

“…Ferrorotation can be interpreted as a two-dimensional form of “chirality,” with the inversion symmetry preserved. For detailed symmetry consideration, ferrorotational order along z , FR z with A z , is defined as “broken twofold rotations around x (C 2 x ) and y (C 2 y ) axes, and unbroken space-inversion ( I ) and time-reversal symmetries ( T )” when any rotations around z are allowed freely …”

“…Ferrorotational orders are common, yet unlike other ferroic orders described by vector parameters such as ferroelectricity and ferromagnetism, they have received limited attention. This is partially because both space-inversion and time-reversal symmetries are preserved in FR materials. ,, At first glance, FR orders might appear featureless due to their lack of strong conjugate vector fields with unbroken { I , T }, resulting in no apparent electromagnetic activities. However, as proposed by Cheong et al, when an electric field is applied simultaneously along the FR axis, FR orders exhibit behavior similar to chirality.…”

Electric Ferro-Axial Moment as Nanometric Rotator and Source of Longitudinal Spin Current

“…Recently, an electric axiality, which has the opposite time-reversal parity to the magnetization, has attracted growing interest [46,47], since the direct observation of its electronic ordering termed as ferro-axial (or ferro-rotational) ordering in RbFe(MoO 4 ) 2 [48,49] and NiTiO 3 [49][50][51]. Owing to the different time-reversal parity, a ferro-axial ordered state exhibits qualitatively different physical phenomena from the conventional ferromagnetic ordering [52][53][54][55], such as antisymmetric thermopolarization [56], longitudinal spin current generation [57,58], and nonlinear transverse magnetization [59]. However, materials to be identified as ferro-axial ordering in experiments are much smaller than those as ferromagnetic ordering: Co 3 Nb 2 O 8 [60], CaMn 7 O 12 [61], Ca 5 Ir 3 O 12 [62][63][64][65], BaCoSiO 4 [66], K 2 Zr(PO 4 ) 2 [67], Na 2 Hf(BO 3 ) 2 [68], and Na-superionic conductors [69].…”