For TIS to be a black-white symmetry requires ${\theta }^2=e$, whereas ${\theta }^2=-1$ for half-integer spin states [11]. The case ${\theta }^2=e$ is relevant for a range of important quantities and phenomena. The present work focuses on macroscopically observable quantities such as electric and magnetic multipoles that must transform according to the integer IRs of the full rotation group $R_{i\times \theta }$ for which $i^2={\theta }^2=e$. Moreover, we are interested in electronic band structures as in Fig. 2. Double-group IRs where ${\theta }^2=-e$ are relevant when analyzing eigenstates like Bloch functions in a spinful theory. However, the operator of spin angular momentum $\mathit{\sigma }$ by itself transforms like a vector ($\ell =1$) [see Eq. (5c)]. Therefore, single-group IRs are sufficient when the relevant effective Hamiltonians for a spinful band theory are derived, e.g., via the theory of invariants [14]. Hence, for the range of quantities that are of interest here, the group theory of the symmetries $i$ and $\theta$ is the same.

Previous work does not offer a universally accepted notation for the four basic types of tensors. In Refs. [59, 60], tensors with signatures $++$, $+-$, $-+$, and $--$ are denoted as i-tensor, s-tensor, t-tensor, and u-tensor, respectively, while Refs. [56, 57, 58] use the symbols $1_e^{+}$, $1_m^{+}$, $1_e^{-}$, and $1_m^{-}$. See also Table 4 in Ref. [62].

If $G$ contains none of the transformations $i$, $\theta$, or $i\theta$ as individual elements, then $\tilde{G}=G$ and $C_{\zeta }\equiv C_1$. If $G$ contains all three of these transformations as elements, then $C_{\zeta }\equiv C_{i\times \theta }$. The only remaining possibility is that $G$ contains exactly one inversion symmetry $\gamma$ as an individual group element, in which case $C_{\zeta }\equiv C_{\gamma }$.

The electrotoroidal tensor $T_1^{++}$ is an interesting exception among all spherical tensors. It cannot be expressed as a compound tensor combining one nontrivial and one or two trivial tensors. This tensor does not arise in Table 5 either. In addition, the tensor component $T_{10}^{++}$ is not an invariant tensor for any continuous dihedral group $G_{d\infty }$; it is only invariant under the continuous cyclic group $G_{c\infty }\times C_{i\times \theta }$, so it cannot be distinguished from even-$\ell$ electric components $T_{g0}^{++}$ (see Appendix pp2-s2). Therefore $T_1^{++}$ is not considered further in this work.

For conceptual clarity, the current analysis is restricted to bands without orbital degeneracies near the $\mathrm{\Gamma }$ point $k=0$ of the Brillouin zone. All arguments can be extended to bands with orbital degeneracies and other expansion points in the Brillouin zone [14].

Different symmetry classifications identifying point and space group symmetries permitting weak ferromagnetism have been developed by Tavger [20] and Turov [110].

See Ref. [46] for a general overview. The relation between modern uses for the term “handedness” and the precise meaning of “chirality” is discussed, e.g., in Ref. [47].

The definition, basic structure, and general properties of magnetic-point-group classes are discussed in Appendix pp1-s3. The application to continuous and discrete point groups is discussed in Appendix pp2.

The chiral charges $T_0^{s{s}^{\prime }}$ are different from the chiral charges encountered in relativistic field theories based on Noether’s theorem, i.e., based on a conservation law reflecting a continuous chiral symmetry [139]. The chiral charges $T_0^{s{s}^{\prime }}$ reflect the discrete symmetries in a crystalline environment.

Systems considered to be chiral according to the definitions of Ref. [45] could be either electrochiral or multichiral. Both the parachiral and magnetochiral categories would conventionally be considered as achiral, and antimagnetochirality is equivalent to false chirality. Magnetic chirality as discussed, e.g., in Ref. [141] constitutes a subset of the electrochiral category.

Recent work has studied chirality-induced orbital-angular-momentum textures [168, 169, 170].

The rows of Tables 11, 13, and 14, listing only one invariant tensor on the right, can be interpreted as sections of the character tables for the $\ell =0$ IRs of the full rotation group $R_{i\times \theta }$ (Table 11) and the $m=0$ IRs of $C_{\infty }\times C_{i\times \theta }$ (Table 13) and the full axial rotation group $D_{\infty }\times C_{i\times \theta }$ (Table 14), respectively. All of these IRs are one dimensional, and “•” (“∘”) indicates that the respective character is $+1$ ($-1$). The invariant tensors on the right then represent basis functions for these IRs.

In Tables 12, 15, and 18, the main block of groups from a given class has the groups of type J, ii-I, i-II, and ii-II from this class at its four corners. See table captions for the discussion of how placement of magnetic groups within a class’s main block illustrates SIS-TIS duality.