Polarization |  | $e$ | $i$ | $\theta$ | $i\theta$ | $C_{\zeta}$ |  even $++$ |  odd $-+$ |  even $--$ |  odd $+-$ |  even $-+$ |  odd $++$ |  even $+-$ |  odd $--$ | point-group types
parapolar | (PP) | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $C_{i\times\theta}$ | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\bullet$ | $\circ$ | $\circ$ | LABEL:gcat:ii-II
electropolar | (EP) | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $C_{\theta}$ | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ |  LABEL:gcat:i-II, LABEL:gcat:iii-II
magnetopolar | (MP) | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ | $C_{i}$ | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $\circ$ | $\bullet$ | $\bullet$ | $\circ$ |  LABEL:gcat:ii-I, LABEL:gcat:ii-III
antimagnetopolar | (AMP) | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $C_{i\theta}$ | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $\circ$ | $\bullet$ |  LABEL:gcat:JJ, LABEL:gcat:JJp
multipolar | (MuP) | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $C_{1}$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ |  LABEL:gcat:J, LABEL:gcat:JJJpp, LABEL:gcat:JJJ, LABEL:gcat:i-III, LABEL:gcat:iii-I, LABEL:gcat:JJJp, LABEL:gcat:iiip-III, LABEL:gcat:iii-IIIp
Chirality |  | $C$ | $iC$ | $\theta C$ | $i\theta C$ | $n$ | $T_{0}^{++}$ |  | $T_{0}^{--}$ |  | $T_{0}^{-+}$ |  | $T_{0}^{+-}$ |  | point-group types
parachiral | (PC) | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $1$ | $\bullet$ |  | $\circ$ |  | $\circ$ |  | $\circ$ |  |  LABEL:gcat:ii-II, LABEL:gcat:JJJpp, LABEL:gcat:iii-II, LABEL:gcat:ii-III, LABEL:gcat:JJp, LABEL:gcat:JJJp, LABEL:gcat:iiip-III, LABEL:gcat:iii-IIIp
electrochiral | (EC) | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $2$ | $\bullet$ |  | $\circ$ |  | $\bullet$ |  | $\circ$ |  |  LABEL:gcat:i-II, LABEL:gcat:i-III
magnetochiral | (MC) | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ | $2$ | $\bullet$ |  | $\circ$ |  | $\circ$ |  | $\bullet$ |  |  LABEL:gcat:ii-I, LABEL:gcat:iii-I
antimagnetochiral | (AMC) | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $2$ | $\bullet$ |  | $\bullet$ |  | $\circ$ |  | $\circ$ |  |  LABEL:gcat:JJ, LABEL:gcat:JJJ
multichiral | (MuC) | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $4$ | $\bullet$ |  | $\bullet$ |  | $\bullet$ |  | $\bullet$ |  | LABEL:gcat:J

The columns of Table 1 define the electric, magnetic, and toroidal IRs
$D_{\ell}^{ss^{\prime}}$ of the full rotation group $R_{i\times\theta}$. The
IRs $D_{\ell}^{ss^{\prime}}$ and associated irreducible spherical tensors
$T_{\ell}^{ss^{\prime}}$ fall into eight fundamentally distinct _families_
that are defined via three parities associated with $s$, $s^{\prime}$, and
$\ell$ [63].

As discussed above, a physical quantity transforming according to an IR
$D_{\ell}^{ss^{\prime}}$ of $R_{i\times\theta}$ becomes permitted by symmetry
in a system with symmetry group $G\subset R_{i\times\theta}$ if mapping
$D_{\ell}^{ss^{\prime}}$ onto the IRs of $G$ includes the identity
representation $\Gamma_{1}$ of $G$. Following Refs. [4, 5, 22] and exploiting
the general correspondence discussed above between irreducible spherical
tensors $T_{\ell}^{ss^{\prime}}$ under $R_{i\times\theta}$ and
crystallographic tensors $T_{\Gamma}$ under crystallographic point groups $G$,
in the present work we _define_ electric, magnetic, and toroidal order via
Table 1, i.e., the presence of multipole order is characterized via the
condition that mapping $D_{\ell}^{ss^{\prime}}$ onto the IRs of $G$ includes
the identity representation $\Gamma_{1}$ of $G$. This general criterion is
independent of a microscopic model for multipole order. It applies to finite
and infinite systems. Also, it applies to insulators and metals.

In crystalline solids, multipole order is represented by macroscopic multipole
_densities_. Examples include the electric dipolarization and the
magnetization. Analogous to the multipoles in finite systems, the
corresponding multipole densities transform as spherical tensors
$T_{\ell}^{ss^{\prime}}$ introduced in the preceding Sec. IV.1. According to
Neumann’s principle, the relevant symmetry group governing the appearance of
these multipole densities is the crystallographic point group associated with
the space group of a crystal structure [4, 5].

The basic classification of electromagnetic multipole densities permitted by
magnetic point groups derives from the possibility to decompose each group $G$
as

$G=\tilde{G}\times C_{\zeta}\quad,$ (17)

where $C_{\zeta}$ is the inversion group that can be formed from inversion
symmetries $\gamma$ that are contained as group elements in $G$, and
$\tilde{G}$ is the subgroup of $G$ that contains none of the inversion
symmetries as an individual group element [70]. The five different inversion
groups $C_{\zeta}$ that may appear in Eq. (17) define five _categories_ of
multipole order [22], see the upper part of Table 11 for the association
between categories and their corresponding inversion groups. Groups $G$ whose
decompositions (17) contain the same inversion group $C_{\zeta}$ belong to the
same category of multipole order. As shown in Table 11, the five categories
are each characterized by a certain pattern of allowed and forbidden electric,
magnetic, electrotoroidal and magnetotoroidal multipole densities. More
specifically, Tables 18 and 19 in Appendix D list the lowest order of
multipole densities permitted by each crystallographic point group $G$.

The _parapolar_ category subsumes crystal structures that have both SIS and
TIS, which therefore permit only even/̄$\ell$ electric multipole densities [7]
(also called _parapolarizations_ [22]). Even/̄$\ell$ electric multipole
densities are also allowed in all other categories discussed below.
_Electropolar_ crystals have only TIS and, thus, odd/̄$\ell$ electric
multipole densities (_electropolarizations_ [22]) are allowed in addition.
Pyroelectric and ferroelectric media with their bulk electric dipolarization
($\ell=1$) [13] are familiar examples from the electropolar category, as are
the zincblende materials with their bulk electric octupolarization ($\ell=3$)
[22]. Having only SIS, the _magnetopolar_ category permits odd/̄$\ell$
magnetic multipole densities (_magnetopolarizations_ [22]) and therefore
includes both ferromagnets ($\ell=1$) [14, 15, 16] and altermagnets ($\ell=3$)
[38, 39, 71, 22]. The _antimagnetopolar_ category is characterized by SIS and
TIS both being broken but CIS still being a good symmetry, which allows
even/̄$\ell$ magnetic multipole densities (_antimagnetopolarizations_ [22]) to
exist. The $i\theta$/̄symmetric antiferromagnets [72, 73, 74, 75] belong to
the antimagnetopolar category. Crystal structures without SIS, TIS and CIS are
in the _multipolar_ category. The low symmetry of these systems allows
electropolarizations, magnetopolarizations, and antimagnetopolarizations to
coexist.

While the parapolar category is synonymous with type LABEL:gcat:ii-II, all
other categories of multipole order subsume more than one of the magnetic-
point-group types introduced in Sec. II.3. See the upper part of Table 11 for
a listing of types belonging to each category. Within the layout of Tables 5,
8 and 9, groups associated with the different categories can be found as per
the scheme given in the center part of Table 3.

###  IV.3 Irreducible tensors representing multipole order

In position space, order/̄$\ell$ multipoles are associated with the $\ell$th
power of Cartesian components of position, leading to alternating even and odd
transformation behavior under SIS as $\ell$ is increased; see Table 1. In
contrast, the behavior under TIS is the same for all multipoles of a given
kind (electric, magnetic, electrotoroidal or magnetotoroidal) and coincides
with the transformation property of the corresponding charge. Electric
multipoles in finite systems thus can be written as a power expansion in
Cartesian components of position (in the following collectively denoted $r$)
[56, 66], and magnetic multipoles can be expressed using polynomials in $r$
and components of spin (collectively denoted $\sigma$) representing magnetic
dipole moments. Similarly, multipole densities in infinite, periodic systems
can be written as polynomials in components of the wave vector (collectively
denoted $k$) and $\sigma$, see Table 12. The spin-dependent band structure
$E_{n\sigma}({\bm{\mathrm{k}}})$ thus directly reflects the presence of
multipolar order in solids, as previously discussed in detail in Ref. [22] and
briefly reviewed below.

For conceptual clarity, in this work the analysis of electronic bands is
restricted to bands without orbital degeneracies near the $\Gamma$ 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 [8]. In this
work, $\sigma$ is a generic place holder for a magnetic dipole moment. It can,
but need not be realized as a spin magnetic moment. It may also represent an
orbital magnetic moment.

Table 12:  Powers of Cartesian components of position (collectively denoted $r$), components of the wave vector (collectively denoted $k$) and components of spin (collectively denoted $\sigma$) required for a polynomial representation of irreducible spherical tensors $T_{\ell}^{ss^{\prime}}$ transforming according to the IRs $D_{\ell}^{ss^{\prime}}$ of $R_{i\times\theta}$. Polynomials involving $r$ (upper part) [$k$ (lower part)] are generally suited for finite (infinite periodic) systems. The symbol $n$ denotes a non-negative integer. For the even/̄$\ell$ magnetotoroidal IRs $D_{\ell}^{+-}$, polynomials in $r,\sigma$ or $k,\sigma$ can only be formed for $\ell>0$, but no such polynomials exist that transform according to $D_{0}^{+-}$. A dash indicates that no such polynomials can be constructed for any $\ell$. Terms $2n$ in the exponents reflect the fact that we can always multiply a tensor $T_{\ell}^{ss^{\prime}}$ with a scalar $r^{2n}$ or $k^{2n}$ without changing its transformational properties. | electric | magnetic | electrotoroidal | magnetotoroidal
---|---|---|---|---
$\ell$ even |  $++$ : | $r^{\ell+2n}$ |  $--$ : | $r^{\ell\pm 1+2n}\sigma$ |  $-+$ : | $\hskip 8.00003pt-$ |  $+-$ : | $r^{\ell+2n}\sigma\quad\mbox{(only $\ell>0$)}$
$\ell$ odd |  $-+$ : | $r^{\ell+2n}$ |  $+-$ : | $r^{\ell\pm 1+2n}\sigma$ |  $++$ : | $\hskip 8.00003pt-$ |  $--$ : | $r^{\ell+2n}\sigma$
$\ell$ even |  $++$ : | $k^{\ell+2n}$ |  $--$ : | $\hskip 8.00003pt-$ |  $-+$ : | $k^{\ell\pm 1+2n}\sigma$ |  $+-$ : | $k^{\ell+2n}\sigma\quad\mbox{(only $\ell>0$)}$
$\ell$ odd |  $-+$ : | $k^{\ell+2n}\sigma$ |  $+-$ : | $k^{\ell\pm 1+2n}\sigma$ |  $++$ : | $\hskip 8.00003pt-$ |  $--$ : | $k^{\ell+2n}$

Explicit expressions for the irreducible spherical tensors in Table 12 can be
derived from the rank/̄1 tensors

$\displaystyle T_{1}^{-+}$
$\displaystyle=\bigl{[}-{\textstyle\frac{1}{\sqrt{2}}}(x+iy),z,{\textstyle\frac{1}{\sqrt{2}}}(x-iy)\bigr{]}$
(18a) $\displaystyle T_{1}^{--}$
$\displaystyle=\bigl{[}-{\textstyle\frac{1}{\sqrt{2}}}(k_{x}+ik_{y}),k_{z},{\textstyle\frac{1}{\sqrt{2}}}(k_{x}-ik_{y})\bigr{]}$
(18b) $\displaystyle T_{1}^{+-}$
$\displaystyle=\bigl{[}-{\textstyle\frac{1}{\sqrt{2}}}(\sigma_{x}+i\sigma_{y}),\sigma_{z},{\textstyle\frac{1}{\sqrt{2}}}(\sigma_{x}-i\sigma_{y})\bigr{]}$
(18c)

using Eq. (16). The resulting tensors $T_{\ell}^{ss^{\prime}}$ can also be
interpreted as basis functions for the IRs $D_{\ell}^{ss^{\prime}}$ of
$R_{i\times\theta}$. However, it turns out that using the rank/̄1 tensors
(18), for both finite systems (upper part of Table 12) and infinite periodic
systems (lower part of Table 12), such tensors can be realized for only six
out of the eight multipole families. The missing two families reflect the fact
that the sets of building blocks ($r,\sigma$ or $k,\sigma$) are incomplete. To
fill these gaps in Table 12, we would need, e.g., (fictitious or engineered)
electric dipole moments (signature $-+$) that complement the magnetic dipole
moments $\sigma$ (signature $+-$) as building blocks [23].

Table 13:  Powers of Cartesian components of position (collectively denoted $r$), components of the wave vector (collectively denoted $k$) and components of spin (collectively denoted $\sigma$) required for a polynomial representation of irreducible crystalline tensors with signature $ss^{\prime}$. Polynomials involving $r$ (upper part) [$k$ (lower part)] are generally suited for finite (infinite periodic) systems. The symbol $n$ denotes a non-negative integer. | electric | magnetic
---|---|---
[$\ell$ even] |  $++$ : | $r^{2n+2}$ |  $--$ : | $r^{2n+1}\sigma$
[$\ell$ odd] |  $-+$ : | $r^{2n+1}$ |  $+-$ : | $r^{2n}\sigma$
[$\ell$ even] |  $++$ : | $k^{2n+2}$ |  $--$ : | $k^{2n+1}$
[$\ell$ odd] |  $-+$ : | $k^{2n+1}\sigma$ |  $+-$ : | $k^{2n}\sigma$

Similar to the construction of irreducible spherical tensors using Eqs. (16)
and (18), one can construct irreducible crystallographic tensors in $k,\sigma$
using the Clebsch-Gordan coefficients for the crystallographic point groups
tabulated in Ref. [7]. This is known as the theory of invariants [8]. In the
present context, this was discussed in more detail in Ref. [22]. The essential
difference between irreducible spherical tensors and irreducible
crystallographic tensors lies in the fact that the index $\ell$ represents a
good quantum number for the full rotation group $R_{i\times\theta}$, but not
for the crystallographic point groups $G$, i.e., under point groups $G$ the
number of parities characterizing the multipole densities is reduced from
three to two. Therefore, in crystalline systems, the four toroidal families
each must be merged with the respective electromagnetic family that has the
same signature $ss^{\prime}$. In crystalline systems, we thus have only four
distinct families of tensors, see Table 13. This result reflects the fact that
the electromagnetic and toroidal tensors are fundamentally distinct only under
the full rotation group $R_{i\times\theta}$. In a crystalline environment
characterized by a finite crystallographic point group $G$, electromagnetic
and toroidal tensors cannot be distinguished anymore, as only the signature
$ss^{\prime}$ remains a distinguishing feature of tensors [22]. This is
related to the fact discussed above that any linear combination of tensors
transforming according to the same IR $\Gamma$ of $G$ yield a tensor that
likewise transforms according to the IR $\Gamma$ of $G$. The generic exponents
listed in Table 13 for the crystallographic tensors summarize the different
exponents of the corresponding electromagnetic and toroidal spherical tensors
in Table 12 with the same signature. The first column in Table 13 indicates in
square brackets the parity of the index $\ell$ of the spherical electric and
magnetic tensors (as defined in Table 1) that the respective crystallographic
tensors correspond to.

###  IV.4 Multipole densities and band structure

According to the theory of invariants [8], the irreducible crystallographic
tensor operators listed in Table 13 arise in a Taylor expansion of the spin-
dependent electronic band structure $E_{n\sigma}({\bm{\mathrm{k}}})$. As
discussed in more detail in Ref. [22], the spin-dependent band structure
$E_{n\sigma}({\bm{\mathrm{k}}})$ thus directly reflects the presence of
multipole order in solids. The essence of this analysis is given by Fig. 1 in
Ref. [22], showing the unique patterns of band dispersions
$E_{\sigma}({\bm{\mathrm{k}}})$ associated with each category.

For example, the familiar Rashba term reflects the electric dipolarization
($\ell=1$) in the electropolar hexagonal wurtzite structure [76]. Similarly,
the Dresselhaus term in the likewise electropolar cubic zincblende structure
represents an electric octupolarization ($\ell=3$) [77].

Generally, the electronic band structure yields measurable _indicators_ for
multipole order in a crystal [22].

###  IV.5 Compound multipoles

The tensors $T_{\ell}^{ss^{\prime}}$ listed in Table 12 are the observable
manifestations of electric, magnetic, and toroidal order, as discussed in Sec.
IV.3. It may appear counterintuitive that the absence of polynomials in
$k,\sigma$ transforming like even/̄$\ell$ magnetic multipoles indicates that
these multipole densities, which represent antimagnetopolar order, do not have
_direct_ observable manifestations in the electronic band structure associated
with them. However, Eq. (16) implies that electric and magnetic multipoles can
be combined to form _compound multipoles_ that are likewise associated with
tensors in Table 12. We will find below that these compound tensors can act as
mediators for the observability of electric and magnetic multipoles.

When constructing compound multipoles from electric and magnetic multipoles,
we can view even/̄$\ell$ electric multipoles as _trivial_ building blocks
because such multipoles exist in any crystal structure (see Sec. IV.2). On the
other hand, odd/̄$\ell$ electric and all magnetic multipoles are _nontrivial_
as they arise only for certain categories of polarized matter. We will show
now that all toroidal tensors (except $T_{1}^{++}$ [78]) can be obtained as
products (16) combining one nontrivial and one or two trivial electromagnetic
tensors such that the compound tensor inherits the signature of the nontrivial
tensor. Of course, similar to electric and magnetic multipole densities,
whether or not specific compound multipole densities are realized in a
specific crystal structure depends also on the point group $G$ of that
structure. See Tables 18 and 19 in Appendix D. Also, compound multipoles can
manifest themselves in the electronic band structure if the respective tensors
can be formed as polynomials in $k,\sigma$ consistent with Table 12 and the
specific point group $G$.

Table 14:  Multiplication table for electric and magnetic IRs $D_{\ell}^{ss^{\prime}}$ of the full rotation group $R_{i\times\theta}$ with $\ell\leq 3$. We have $D_{1,2,3}^{ss^{\prime}}\equiv D_{1}^{ss^{\prime}}+D_{2}^{ss^{\prime}}+D_{3}^{ss^{\prime}}$ etc. Bold subscripts $\ell$ indicate that an IR contained in the product representation represents a toroidal multipole. $\rule{0.0pt}{12.05553pt}D_{0}^{++}$ | $D_{0}^{--}$ |  $D_{1}^{-+}$ | $D_{1}^{+-}$ |  $D_{2}^{++}$ | $D_{2}^{--}$ |  $D_{3}^{-+}$ |  $D_{3}^{+-}$ | $D_{\ell}^{ss^{\prime}}$
---|---|---|---|---|---|---|---|---
$\rule{0.0pt}{11.19443pt}D_{0}^{++}$ | $D_{0}^{--}$ |  $D_{1}^{-+}$ | $D_{1}^{+-}$ |  $D_{2}^{++}$ | $D_{2}^{--}$ |  $D_{3}^{-+}$ |  $D_{3}^{+-}$ | $D_{0}^{++}$
| $D_{0}^{++}$ |  $D_{1}^{+-}$ | $D_{1}^{-+}$ |  $D_{2}^{--}$ | $D_{2}^{++}$ |  $D_{3}^{+-}$ |  $D_{3}^{-+}$ | $D_{0}^{--}$
|  |  $D_{0,1,2}^{++}$ | $D_{0,\bm{1},2}^{--}$ |  $D_{1,\bm{2},3}^{-+}$ | $D_{1,\bm{2},3}^{+-}$ |  $D_{2,\bm{3},4}^{++}$ |  $D_{2,\bm{3,}4}^{--}$ | $D_{1}^{-+}$
|  |  | $D_{0,1,2}^{++}$ |  $D_{1,\bm{2},3}^{+-}$ | $D_{1,\bm{2},3}^{-+}$ |  $D_{2,\bm{3},4}^{--}$ |  $D_{2,\bm{3},4}^{++}$ | $D_{1}^{+-}$
|  |  |  |  $D_{0,1,2,3,4}^{++}$ | $D_{0,\bm{1},2,\bm{3},4}^{--}$ |  $D_{1,\bm{2},3,\bm{4},5}^{-+}$ |  $D_{1,\bm{2},3,\bm{4},5}^{+-}$ | $D_{2}^{++}$
|  |  |  |  | $D_{0,1,2,3,4}^{++}$ |  $D_{1,\bm{2},3,\bm{4},5}^{+-}$ |  $D_{1,\bm{2},3,\bm{4},5}^{-+}$ | $D_{2}^{--}$
|  |  |  |  |  |  $D_{0,1,2,3,4,5,6}^{++}$ |  $D_{0,\bm{1},2,\bm{3},4,\bm{5},6}^{--}$ | $D_{3}^{-+}$
|  |  |  |  |  |  |  $D_{0,1,2,3,4,5,6}^{++}$ | $D_{3}^{+-}$

To illustrate how compound tensors can be formed from electric and magnetic
tensors, Table 14 gives the multiplication table for electric and magnetic IRs
$D_{\ell}^{ss^{\prime}}$ of the full rotation group $R_{i\times\theta}$ with
$\ell\leq 3$. The range of ranks $\ell$ contained in a product representation
is given by Eq. (15).

For example, Table 14 shows that a magnetic monopole $T_{0}^{--}$ can arise as
a product of an electric and a magnetic multipole with the same $\ell>0$.
These multipoles with $\ell>0$ break spherical symmetry so that the symmetry
group of a system realizing such a synthetic magnetic monopole cannot be the
spherical magnetic group $R_{i\theta}$ (that would be the symmetry group of an
elementary magnetic monopole if it existed). But a product of electric and
magnetic multipoles with large $\ell$ can be arbitrarily close to spherical
symmetry, which indicates an avenue for realizing “realistic” synthetic
magnetic monopoles [79, 80].

Products (16) of electric and magnetic tensors yield not only electric and
magnetic compound tensors, but they also yield electrotoroidal and
magnetotoroidal compound tensors (marked with bold subscripts $\ell$ in Table
14). For example, an electric quadrupole $T_{2}^{++}$ and an electric dipole
$T_{1}^{-+}$ jointly give rise to an electrotoroidal quadrupole $T_{2}^{-+}$.

Note that an electrotoroidal compound tensor $T_{1}^{++}$ does not appear in
Table 14. Electrotoroidal compound tensors $T_{\ell}^{++}$ with odd $\ell$ are
realized by antisymmetric (noncommuting) products
$(T_{\ell_{1}}^{ss^{\prime}}\times T_{\ell_{1}}^{ss^{\prime}})_{\mathrm{a}}$
of the same tensor $T_{\ell_{1}}^{ss^{\prime}}$ or by products
$T_{\ell_{1}}^{ss^{\prime}}\times\tilde{T}_{\ell_{2}}^{ss^{\prime}}$ of two
distinct tensors $T_{\ell_{1}}^{ss^{\prime}}$ and
$\tilde{T}_{\ell_{2}}^{ss^{\prime}}$ with the same signature $ss^{\prime}$
[56]. However, odd/̄$\ell$ IRs $D_{\ell}^{++}$ appearing off-diagonally in the
multiplication table have $\ell\geq 3$. The IR $D_{1}^{++}$ arises only on the
diagonal. Assuming that for each $\ell$ a system has only one electric and one
magnetic multipole $T_{\ell}^{ss^{\prime}}$, second-order products of these
multipoles thus cannot realize an electrotoroidal compound tensor
$T_{1}^{++}$. Similarly, toroidal scalars $T_{0}^{\mp\pm}$ do not arise as
second-order products of electric and magnetic tensors, see Table 14.

Toroidal compound tensors $T_{0}^{\mp\pm}$ and $T_{1}^{++}$ arise in third-
order products of electric and magnetic tensors, for example:

$\rule{0.0pt}{30.1388pt}\left.\mbox{\begin{tabular}[]{nLn}\raisebox{0.0pt}[0.0pt]{$\left.\mbox{\begin{tabular}[]{nLn}T_{2}^{++}\\\\[4.30554pt]
T_{4}^{++}\end{tabular}}\right\\}$}\;T_{3}^{++}\\\\[17.22217pt]
T_{3}^{\mp\pm}\end{tabular}}\right\\}\;T_{0}^{\mp\pm},\hskip
20.00003pt\left.\mbox{\begin{tabular}[]{nLn}\raisebox{0.0pt}[0.0pt]{$\left.\mbox{\begin{tabular}[]{nLn}T_{2}^{++}\\\\[4.30554pt]
T_{4}^{++}\end{tabular}}\right\\}$}\;T_{6}^{++}\\\\[17.22217pt]
T_{6}^{++}\end{tabular}}\right\\}\;T_{1}^{++},\hskip
20.00003pt\left.\mbox{\begin{tabular}[]{nLn}\raisebox{0.0pt}[0.0pt]{$\left.\mbox{\begin{tabular}[]{nLn}T_{2}^{--}\\\\[4.30554pt]
T_{4}^{--}\end{tabular}}\right\\}$}\;T_{3}^{++}\\\\[17.22217pt]
T_{2}^{++}\end{tabular}}\right\\}\;T_{1}^{++}.$ (19)

Similar to magnetic scalars $T_{0}^{--}$, the toroidal scalars $T_{0}^{-+}$
and $T_{0}^{+-}$ can only arise in electromagnetic media that break spherical
symmetry.

Inspection of Table 14 and Eq. (19) shows that all nontrivial compound tensors
(except $T_{1}^{++}$ [78]) can be obtained as products combining one
nontrivial and one or two trivial tensors. By definition, trivial tensors have
the signature $++$. Therefore, toroidal compound tensors inherit the signature
of the nontrivial electromagnetic tensor they are composed of, while the index
$\ell$ of toroidal compound tensors has the opposite parity as the index
$\ell$ of the nontrivial tensor they are composed of. Consistent with Tables
12 and 13, we thus interpret toroidal compound tensors as manifestations of
the nontrivial electromagnetic tensor they are composed of.

In physical terms, compound tensors generally have no immediate microscopic
source associated with them. Compound tensors representing products of
electric and magnetic tensors describe the combined effect of electric and
magnetic multipoles as commonly studied in quantum-mechanical perturbation
theory. Interestingly, this contrasts with the equations of classical
electromagnetism that are linear [66] so that, within the framework of
classical electromagnetism, a superposition of electric and magnetic
multipoles cannot give rise to such qualitatively new terms.

The above group-theoretical analysis of compound tensors is independent of
specific realizations of the tensors $T_{\ell}^{ss^{\prime}}$ as discussed in
Table 12. It is equally applicable to finite and infinite periodic systems.

Consistent with our definition of categories of multipole order based on the
signature $ss^{\prime}$, the terms electropolarization (signature $-+$),
magnetopolarization ($+-$), and antimagnetopolarization ($--$) refer jointly
to the respective odd/̄$\ell$ electric, odd/̄$\ell$ magnetic and even/̄$\ell$
magnetic multipoles as well as the corresponding toroidal compound moments
with the same signature, unless we want to emphasize the composition of
toroidal compound tensors.

###  IV.6 Examples of compound moments

Reference [22] contains a detailed study of how electric and magnetic
multipoles manifest themselves via characteristic terms in the electronic band
structure of a crystal, using variations of lonsdaleite and diamond as
examples. A careful inspection shows that a number of examples discussed in
that work rely on compound moments.

For instance, we can have three qualitatively distinct homogenous polynomials
in $k,\sigma$ of degree 2 in a crystal, two of which are compound moments. (i)
Crystal structures with point groups $T$ and $O$ permit the crystalline tensor
${\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{k}}}$. This term transforms like an
electrotoroidal compound scalar $T_{0}^{-+}$ that arises as a product (19) of
a nontrivial odd/̄$\ell$ electric multipole and two trivial even/̄$\ell$
electric multipoles. See also Sec. V. (ii) Electropolar crystal structures
such as wurtzite (point group $C_{6v}$, ignoring TIS) permit an electric
dipole density that manifests itself via the Rashba term
$k_{x}\sigma_{y}-k_{y}\sigma_{x}$ [76]. Under $R_{i\times\theta}$, this term
transforms like the component $T_{10}^{-+}$ of an electric dipole density.
(iii) The cubic zincblende structure (point group $T_{d}$) permits an electric
octupole density ($\ell=3$) that manifests itself via the Dresselhaus term
$\sigma_{x}k_{x}(k_{y}^{2}-k_{z}^{2})+\mbox{cp}$, where cp denotes cyclic
perutation [77]. Under $R_{i\times\theta}$, this term transforms like the
component $T_{3,-2}^{-+}+T_{3,2}^{-+}$ of an electric octupole density
($\ell=3$). When uniaxial strain is applied to a zincblende structure (in
[001] direction, thus reducing the crystal symmetry from $T_{d}$ to $D_{2d}$),
we get the crystalline tensor $k_{x}\sigma_{x}-k_{y}\sigma_{y}$. Under
$R_{i\times\theta}$, this term transforms like the component
$T_{2,-2}^{-+}+T_{2,2}^{-+}$ of an electrotoroidal quadrupole density that
arises as a product (16) of the nontrivial electric octupole density
($\ell=3$) that exists already in unstrained zincblende and a trivial strain-
induced electric quadrupole density ($\ell=2$). The electrotoroidal quadrupole
density represents the lowest even-rank electrotoroidal moment density
permitted in a structure with point group $D_{2d}$, see Table 18.

Pristine lonsdaleite has the crystallographic point group $D_{6h}=D_{6}\times
C_{i}$ (ignoring TIS). An electric octupolarization ($\ell=3$) reduces the
point-group symmetry to $D_{3h}=D_{6}[D_{3}]$, and it gives rise to two new
invariants (to lowest order in ${\bm{\mathrm{k}}}$) [22]

$\displaystyle H_{1}$
$\displaystyle=c_{1}\,\sigma_{z}\,k_{y}\left(3k_{x}^{2}-k_{y}^{2}\right)$
(20a) $\displaystyle H_{2}$
$\displaystyle=c_{2}\,k_{z}\left[\sigma_{x}k_{x}k_{y}+{\textstyle\frac{1}{2}}\sigma_{y}(k_{x}^{2}-k_{y}^{2})\right]\,,$
(20b)

where $c_{1}$ and $c_{2}$ are material-specific prefactors. These terms can be
rearranged as

$\displaystyle H^{(3)}$ $\displaystyle={\textstyle\frac{5}{8}}(H_{1}-2H_{2})$
(21a) $\displaystyle H^{(4)}$
$\displaystyle={\textstyle\frac{3}{8}}(H_{1}+6H_{2})\,.$ (21b)

Here, the first term transforms like the component
$T_{3,-3}^{-+}+T_{3,3}^{-+}$ of an electric octupole density ($\ell=3$), while
the second term transforms like the component $T_{4,-3}^{-+}+T_{4,3}^{-+}$ of
an electrotoroidal hexadecapole density ($\ell=4$). The latter represents the
lowest even-rank electrotoroidal moment density permitted in a structure with
point group $D_{3h}$, see Table 18. It is realized as a compound moment
density that combines the nontrivial electric octupole density ($\ell=3$) with
the trivial electric quadrupole density ($\ell=2$) present already in pristine
lonsdaleite.

An important example of compound moments arises in the context of
antimagnetopolarizations. Very generally, unlike electropolarizations and
magnetopolarizations, even/̄$\ell$ antimagnetopolarizations $T_{\ell}^{--}$
have no indicators in the band structure associated with them, see Table 12.
However, antimagnetopolarizations combined with trivial even/̄$\ell$
parapolarizations give rise to odd/̄$\ell$ magnetotoroidal densities
$T_{\ell}^{--}$ that manifest themselves in the band structure via terms
proportional to odd powers in $k$ (Table 12). In the following, we thus
associate antimagnetopolarizations with such terms $k^{2n+1}$. While
odd/̄$\ell$ electropolarizations and odd/̄$\ell$ magnetopolarizations permit a
simple correspondence between the rank $\ell$ and a minimal-degree polynomial
representation of the associated tensor operators (Table 12), no such
correspondence exists for even/̄$\ell$ antimagnetopolarizations.

For example, in a diamond crystal structure, a magnetic quadrupole density
($\ell=2$) gives rise to a term $k_{z}(k_{x}^{2}-k_{y}^{2})$ [81, 22]. Under
$R_{i\times\theta}$, this term transforms like the component
$T_{3,-2}^{--}+T_{3,2}^{--}$ of a magnetotoroidal octupole density ($\ell=3$).
This term arises as a product (16) of a nontrivial magnetic quadrupole density
($\ell=2$) and a trivial electric hexadecapole density ($\ell=4$) present
already in pristine diamond.

A complete review of the examples in Ref. [22] is beyond the scope of the
present work. Tables 18 and 19 in Appendix D give the lowest ranks of
electromagnetic multipoles and toroidal compound multipoles in crystal
structures with different crystallographic point groups $G$.

##  V Chirality in solids

Chirality literally means “handedness”, referring to one of the most
fundamental and ubiquitous types of asymmetry in nature [82]. On a
conceptional level, chirality is associated with the existence of, at least,
two inequivalent versions of a physical entity—called _enantiomorphs_ —that
cannot be superposed by proper rotations and/or translations [83, 27, 84, 85,
86, 87, 88]. Instead, interconversion of enantiomorphs requires $i$/̄improper
rotations $iC$, i.e., transformations that involve space inversion $i$ such as
a mirror reflection (which is a $\pi$ rotation about an axis perpendicular to
the reflection plane followed by space inversion $i$). Equivalently, an object
has been called _chiral_ if its symmetry group does not contain any
$i$/̄improper rotation. An _achiral_ object can be mapped onto itself by an
$i$/̄improper rotation, i.e., it shows no enantiomorphism [27].

The importance of TIS for discussing physical implications of chirality has
been noted early on [83, 84]. In particular, a distinction was made between
two different types of chirality: so-called _true chirality_ requires that
only $i$/̄improper rotations but not $\theta$/̄improper rotations can
interconvert between enantiomorphs, whereas systems for which both
$i$/̄improper and $\theta$/̄improper rotations but not $i\theta$/̄improper
rotations interconvert the enantiomorphs have been called _false chiral_ [27,
83, 84].

The space groups of chiral crystal structures allow two distinct cases. Either
an enantiomorphic pair of crystal structures transforms according to an
enantiomorphic pair of distinct space groups, or the pair of crystal
structures transforms according to the same space group. In total, 65
nonmagnetic space groups describe chiral structures. These are the 65 Sohncke
groups that were identified early on in crystallography and that represent a
subset of the 230 nonmagnetic space groups [51]. However, in agreement with
Neumann’s principle, the macroscopically observable features of chirality only
depend on the crystallographic point group $G$ of the crystal class of a
crystal structure [5]. The 65 Sohncke groups are exactly the space groups for
which the respective point groups $G$ are one of the 11 (nonmagnetic) chiral
point groups, i.e., the distinction between the two cases described above is
not relevant at the level of Neumann’s principle. The Sohncke groups include
symmorphic and nonsymmorphic space groups. Ignoring TIS, the 11 chiral point
groups $G$ are precisely the proper point groups $G_{\mathrm{p}}$ that form
the roots for the 11 Laue classes $L(G_{\mathrm{p}})$ of crystallographic
point groups identified in Sec. III.2.

Optical activity is often viewed as a unique hallmark of chirality. While this
is correct for molecules in solution, crystal structures can be optically
active though they are not chiral (point groups $C_{s}$, $C_{2v}$, $S_{4}$ and
$D_{2d}$) [5, 27].

###  V.1 Categories of chirality

In the present work, we treat SIS, TIS, and CIS on an equal footing to obtain
a more systematic understanding of chirality and to acquire a unified
description of associated physical phenomena. In the lower part of Table 11,
we identify five categories of chirality that are distinguished by which
$\gamma$/̄improper rotations are present in the point group of a system. The
_parachiral_ category refers to systems that have all three kinds of
$\gamma$/̄improper rotations among their symmetries. These systems that have
no enantiomorphism. The _electrochiral_ , _magnetochiral_ and
_antimagnetochiral_ systems have $\theta$/̄improper, $i$/̄improper, and
$i\theta$/̄improper rotations, respectively, among their symmetries so that
each of these categories have two distinct enantiomorphs. Finally,
_multichiral_ systems have only proper rotations among their symmetries so
that these systems have four distinct enantiomorphs.

Within the symmetry classification of magnetic point groups presented in Sec.
II.3 and summarized in Table 2, each group type is associated with one
category of multipole order and one category of chirality. The five categories
of multipole order (polarizations) and the five categories of chirality
(introduced in the upper and lower parts of Table 11, respectively) represent
two complementary classifications based on the presence or absence of the
inversion symmetries $i$, $\theta$, and $i\theta$. Polarizations reflect the
presence or absence of $i$, $\theta$, and $i\theta$ as _independent_ elements
in the symmetry group $G$ of a physical system. Chiralities, on the other
hand, reflect the presence or absence in $G$ of any _composite_ symmetry
elements $iC$, $\theta C$, and $i\theta C$ (i.e., presence or absence of
$\gamma$/̄improper rotations $\gamma C$). See Table 11. This implies that we
have an intricate connection between multipole order and chirality that may
exist in a system. For example, very generally $\Lambda$/̄polarity is a
necessary, though not sufficient, requirement for $\Lambda$/̄chirality, where
$\Lambda$ may stand for electro, magneto, antimagneto, and multi, see Table 2.

Specifically, the absence of $\gamma$/̄improper rotations in chiral systems
implies that the respective monopoles $T_{0}^{ss^{\prime}}\neq T_{0}^{++}$
become symmetry-allowed, see Table 11. These monopoles thus represent _chiral
charges_. They are signatures of chiral order similar to how electric and
magnetic multipoles are signatures of polar order, though we will find below
that these monopole densities are a sufficient, but not a necessary criterion
for chirality. Our naming of the five categories of chirality reflects the
nature of the allowed scalars other than the universally allowed $T_{0}^{++}$
(see Table 11). Compared with the existing nomenclature [27] comprising (true)
chirality, false chirality and achirality, the five categories constitute a
more comprehensive and systematic classification of chirality in solids.
Specifically, electrochiral and multichiral systems are chiral as per the
definition in Ref. [27], parachiral and magnetochiral systems are both
achiral, and antimagnetochirality is equivalent to false chirality.

Figure 2:  Categories of chirality illustrated for the unipolar-unichiral and
multipolar-multichiral groups in the class $L(C_{\infty})$. As a visual aid,
different enantiomorphs are distinguished by color. Straight (curved) colored
arrows indicate translational (rotational) motion of an object. Dark (light)
gray double-arrows connect two objects that transform into each other under
the specified symmetry operation, which is (is not) a good symmetry of the
object. $C_{2}$ represents a $\pi$ rotation about any axis perpendicular to
the high-symmetry $z$ axis. (b) In electrochiral systems, space inversion $i$
interconverts the enantiomorphs, but time inversion $\theta$ does not. (c) The
situation is reversed for magnetochiral systems. (d) Both $i$ and $\theta$
separately interconvert enantiomorphs in antimagnetochiral systems, but their
combination $i\theta$ leaves them invariant. In contrast, $i\theta$
interconverts the two enantiomorphs present in electro- and magnetochiral
systems. (e) All three inversions $i$, $\theta$ and $i\theta$ interconvert the
four enantiomorphs of a multichiral system. The chiral objects in panels (b)
to (d) transform like tensor components $T_{\ell 0}^{ss^{\prime}}$ as
indicated in panel (a). These tensor components $T_{\ell 0}^{ss^{\prime}}$ can
be derived either by reducing the symmetry of tensor components
$T_{u0}^{ss^{\prime}}$ or by reducing the symmetry of tensor components
$T_{\ell 0}^{++}$. Figure 3:  Categories of chirality illustrated for the
unipolar-unichiral and multipolar-multichiral groups in the class
$L(D_{\infty})$. The schematic representation of polarizations
$T_{g0}^{ss^{\prime}}$ is based on the multiplication table for irreducible
tensors of the full axial rotation group $D_{\infty}\times C_{i\times\theta}$
(Table 16). Other conventions are the same as in Fig. 2. (a) The chiral
objects in panels (b) to (d) transform like tensor components
$T_{g0}^{ss^{\prime}}$ that can be derived by reducing the symmetry of tensor
components $T_{g0}^{++}$. Figure 4:  Categories of chirality illustrated for
the multipolar-unichiral groups in the class $L(D_{\infty})$. The multipolar-
multichiral group $C_{\infty}$ illustrated in panel (e) belongs to
$L(C_{\infty})$. Same conventions as in Fig. 2. The chiral objects in panels
(b) to (e) can be viewed as superpositions of the parachiral objects in panel
(a).

We now provide a brief overview of each category. Figures 2, 3 and 4 provide
illustrative examples for these chiralities based on the continuous axial
classes $L(C_{\infty})$ and $L(D_{\infty})$. These figures consider the
symmetry transformations in the full axial rotation group $D_{\infty}\times
C_{i\times\theta}$ to explore how different axially symmetric objects are
either invariant (i.e., mapped onto themselves), mapped onto a rotated copy of
themselves, or mapped onto distinct enantiomorphs. Materials candidates for
the different categories of chirality are listed in Appendix F.

####  V.1.1 Parachirality

Parachiral point groups contain $\gamma$/̄improper rotations $\gamma C$ for
all three inversions $\gamma=i$, $\theta$ and $i\theta$. Therefore, such
systems have just one enantiomorph, i.e., the parachiral groups do not allow
any kind of enantiomorphism. No scalars of $R_{i\times\theta}$ other than
$T_{0}^{++}$ are permitted under parachiral groups. In particular, the scalar
$T_{0}^{-+}$ is forbidden, and parachiral systems are therefore achiral as per
the definition in Ref. [27]. Eight among the 15 group types defined in Table 2
are parachiral.

Examples of parachiral groups include the full axial rotation groups
$C_{\infty}\times C_{i\times\theta}$ and $D_{\infty}\times C_{i\times\theta}$
permitting scalars $T_{\ell 0}^{++}$ and $T_{g0}^{++}$, respectively, as
illustrated in Figs. 2(a), 3(a) and 4(a).

####  V.1.2 Electrochirality

Electrochiral groups contain $\theta$/̄improper rotations $\theta C$, but no
$i$/̄improper or $i\theta$/̄improper rotations, i.e., an inequivalent
enantiomorph is generated when a transformation $iC$ or $i\theta C$ is applied
to the crystal. Electrochiral systems permit the pseudoscalar $T_{0}^{-+}$.
See Figs. 2(b), 3(b), and 4(b) for illustrative examples.

Electrochiral groups are either electropolar (type LABEL:gcat:i-II) or
multipolar (type LABEL:gcat:i-III). Electrochiral magnets belong to the latter
type. Examples of nonmagnetic electrochiral systems are shown in Figs. 2(b)
and 3(b). In these, chirality arises from the interplay of a parapolarization
with an electropolarization, i.e., between even-$\ell$ and odd-$\ell$
electric-multipole densities. The scenario depicted in Fig. 4(b) is equivalent
to an electrochiral ferromagnet, where electrochirality emerges from the
presence of collinear magnetic and magnetotoroidal dipolarizations, which is
the crystal analog of electrochirality arising when a magnetic field is
applied parallel to a beam of light [83].

Electrochiral systems are optically active and they show a linear current-
induced magnetization [89, 90, 91], though electrochirality is sufficient but
not necessary for these effects [5, 27, 92].

####  V.1.3 Magnetochirality

Magnetochiral groups contain $i$/̄improper rotations $iC$, but
$\theta$/̄improper and $i\theta$/̄improper rotations are absent. Therefore,
any $\theta$/̄improper or $i\theta$/̄improper rotation generates an
inequivalent enantiomorph of a system described by such a group. See Figs.
2(c), 3(c), and 4(c) for illustrative examples.

The nontrivial scalar allowed in magnetochiral groups is $T_{0}^{+-}$, which
has no indicator associated with it in the electronic band structure, see
Table 12 and Sec. V.2. Some physical consequences arising from the presence of
$T_{0}^{+-}$ have recently been conjectured in Ref. [93]. As the scalar
$T_{0}^{-+}$ is forbidden under magnetochiral groups, magnetochiral systems
would be considered achiral according to Ref. [27]. However, the
enantiomorphism arising under $\theta$/̄improper and $i\theta$/̄improper
rotations makes magnetochiral systems fundamentally different from parachiral
ones, as enantiomorphism is completely absent in the latter.

Magnetochiral groups are either magnetopolar (type LABEL:gcat:ii-I) or
multipolar (type LABEL:gcat:iii-I). Examples of magnetopolar systems are shown
in Figs. 2(c) and 3(c). Here enantiomorphism arises from the interplay of a
parapolarization with a magnetopolarization, i.e., even/̄$\ell$ electric and
odd/̄$\ell$ magnetic multipole densities. The scenario in Fig. 4(c) is an
example of the multipolar type, which equivalently represents magnetochirality
arising from collinear electric and magnetotoroidal dipolarizations. The same
magnetochirality characterizes the situation where an electric field is
applied parallel to a light beam.

####  V.1.4 Antimagnetochirality

Antimagnetochiral groups contain $i\theta$/̄improper rotations, but
$i$/̄improper and $\theta$/̄improper rotations are absent. Thus,
$\gamma$/̄improper rotations $\gamma C$, where $\gamma=i$ or $\theta$,
generate an inequivalent enantiomorph of an antimagnetochiral structure. See
the examples depicted in Figs. 2(d), 3(d), and 4(d).

The antimagnetochiral category subsumes what has previously been referred to
as false chirality [83, 84]: although such a system exhibits enantiomorphism,
it does not permit a scalar $T_{0}^{-+}$ and is therefore not considered to be
truly chiral. Instead, a scalar $T_{0}^{--}$ is allowed in antimagnetochiral
systems, though this scalar has no indicator associated with it in the
electronic band structure, see Table 12 and Sec. V.2.

Antimagnetochiral groups are either antimagnetopolar (type LABEL:gcat:JJ) or
multipolar (type LABEL:gcat:JJJ). Figures 2(d) and 3(d) show examples of
antimagnetopolar antimagnetochiral systems, where antimagnetochirality arises
from the combination of a parapolarization with an antimagnetopolarization.
Figure 4(d) shows a multipolar system, representing also the situation where
antimagnetochirality arises from the presence of collinear electric
polarization and magnetization, or, equivalently, from parallel electric and
magnetic fields [83].

####  V.1.5 Multichirality

Multichiral groups do not contain $\gamma$/̄improper rotations $\gamma C$ for
any of the inversions $\gamma=i$, $\theta$ or $i\theta$. Hence, these groups
coincide with the proper (type-LABEL:gcat:J) point groups. See Figs. 2(e),
3(e), and 4(e) for illustrative examples.

By the usual definition based on whether a scalar $T_{0}^{-+}$ is allowed
[27], groups in the multichiral category have been classified as showing true
chirality. However, they could equally be considered to exhibit false
chirality [94]. Our more systematic analysis elucidates how the multichiral
category differs fundamentally from both of these previously discussed
chiralities. Firstly, this category has four distinct enantiomorphs, as all
three improper rotations $iC$, $\theta C$ and $i\theta C$ yield distinct
versions of the given object. See Figs. 2(e), 3(e), and 4(e). Secondly, all
four scalars $T_{0}^{ss^{\prime}}$ are permitted under the multichiral groups,
which thus combine the characteristics of electrochiral, magnetochiral and
antimagnetochiral systems discussed above.

Multichiral systems are necessarily multipolar because coexisting chiralities
require the interplay of a parapolarization with two different polarizations.

Table 15:  Lowest-degree invariant tensors realizing chiralities in the continuous classes $L(R)$ and $L(D_{\infty})$, the cubic classes $L(O)$ and $L(T)$, and the axial classes $L(D_{n})$ and $L(C_{n})$ with $n=2,3,4,6$. We have $R\supset O\supset T$ and $D_{\infty}\supset D_{n}\supset C_{n}$, so that terms allowed under the respective supergroup are also allowed under the respective subgroup. cp denotes cyclic permutation of the preceding term and $(x\leftrightarrow y)$ denotes the preceding term with $x$ and $y$ interchanged. For the crystallographic classes, we denote in square brackets the degree $\ell$ of multipolar order represented by the respective terms as well as the maximal group that leaves this term invariant. For electrochirality and magnetochirality, we only consider electric and magnetic tensors with odd $\ell$; we do not consider toroidal compound tensors with even $\ell$. For antimagnetochirality, we consider the lowest-degree odd/̄$\ell$ magnetotoroidal compound tensors representing an antimagnetopolarization. class $L(G_{\mathrm{p}})$ | $L(R)$ | $L(O)$ | $L(T)$
---|---|---|---
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{i\times\theta}$ | PC |  | $k^{2}$ | $[\ell=4,O\times C_{i\times\theta}]$ | $k_{y}^{2}k_{z}^{2}+\mbox{cp}$ |  |
|  |  |  | $[\ell=6,O\times C_{i\times\theta}]$ | $\left[k_{x}^{4}(k_{y}^{2}+k_{z}^{2})+\mbox{cp}\right]-8k_{x}^{2}k_{y}^{2}k_{z}^{2}$ | $[\ell=6,T\times C_{i\times\theta}]$ | $(k_{y}^{2}-k_{z}^{2})(k_{z}^{2}-k_{x}^{2})(k_{x}^{2}-k_{y}^{2})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{\theta}$ | EC |  | ${\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{k}}}$ | $[\ell=9,O\times C_{\theta}]$ |  | $\sigma_{x}k_{x}[k_{y}^{2}k_{z}^{2}(3k_{y}^{4}-10k_{y}^{2}k_{z}^{2}+3k_{z}^{4})$
---
$+k_{x}^{2}(k_{x}^{2}-k_{y}^{2}-k_{z}^{2})(k_{y}^{4}-6k_{y}^{2}k_{z}^{2}+k_{z}^{4})]+\mbox{cp}$
$[\ell=3,O[T]\times C_{\theta}]$ | $\sigma_{x}k_{x}(k_{y}^{2}-k_{z}^{2})+\mbox{cp}$
$G_{\mathrm{p}}\times C_{i}$ | MC |  |  | $[\ell=9,O\times C_{i}]$ | $\sigma_{x}k_{y}k_{z}(k_{y}^{2}-k_{z}^{2})[5k_{x}^{4}-3k_{x}^{2}(k_{y}^{2}+k_{z}^{2})+k_{y}^{2}k_{z}^{2}]+\mbox{cp}$ | $[\ell=3,O(T)\times C_{i}]$ | $\sigma_{x}k_{y}k_{z}+\mbox{cp}$
$G_{\mathrm{p}}\times C_{i\theta}$ | AMC |  |  | $[\ell=9,O\times C_{i\theta}]$ | $k_{x}k_{y}k_{z}(k_{y}^{2}-k_{z}^{2})(k_{z}^{2}-k_{x}^{2})(k_{x}^{2}-k_{y}^{2})$ | $[\ell=3,O(T)\times C_{i\theta}]$ | $k_{x}k_{y}k_{z}$
class $L(G_{\mathrm{p}})$ | $L(D_{\infty})$ | $L(D_{6})$ | $L(C_{6})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{i\times\theta}$ | PC |  | $k_{x}^{2}+k_{y}^{2};k_{z}^{2}$ | $[\ell=6,D_{6}\times C_{i\times\theta}]$ | $k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})^{2};\;k_{y}^{2}(k_{y}^{2}-3k_{x}^{2})^{2}$ | $[\ell=6,C_{6}\times C_{i\times\theta}]$ | $k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{\theta}$ | EC |  |  | $\sigma_{x}k_{x}+\sigma_{y}k_{y};$
---
$\sigma_{z}k_{z}$
$[\ell=7,D_{12}[D_{6}]\times C_{\theta}]$ |  | $\sigma_{x}k_{x}[k_{y}^{2}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})+3k_{z}^{2}(k_{x}^{4}-10k_{x}^{2}k_{y}^{2}+5k_{y}^{4})]$
---
$-(x\leftrightarrow
y)-3\sigma_{z}k_{z}(k_{x}^{2}-k_{y}^{2})(k_{x}^{4}-14k_{x}^{2}k_{y}^{2}+k_{y}^{4})$
$[\ell=1,D_{\infty}[C_{\infty}]\times C_{\theta}]$ | $\sigma_{x}k_{y}-\sigma_{y}k_{x}$
$G_{\mathrm{p}}\times C_{i}$ | MC |  |  | $[\ell=7,D_{12}(D_{6})\times C_{i}]$ |  | $3\sigma_{x}k_{y}k_{z}(5k_{x}^{4}-10k_{x}^{2}k_{y}^{2}+k_{y}^{4})+(x\leftrightarrow y)$
---
$-\sigma_{z}k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})$
$[\ell=1,D_{\infty}(C_{\infty})\times C_{i}]$ | $\sigma_{z}$
$G_{\mathrm{p}}\times C_{i\theta}$ | AMC |  |  | $[\ell=7,D_{12}(D_{6})\times C_{i\theta}]$ | $k_{x}k_{y}k_{z}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i\theta}]$ | $k_{z}$
class $L(G_{\mathrm{p}})$ |  | $L(D_{3})$ | $L(C_{3})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{i\times\theta}$ | PC |  |  | $[\ell=4,D_{3}\times C_{i\times\theta}]$ | $k_{y}k_{z}(k_{y}^{2}-3k_{x}^{2})$ | $[\ell=4,C_{3}\times C_{i\times\theta}]$ | $k_{x}k_{z}(k_{x}^{2}-3k_{y}^{2})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{\theta}$ | EC |  |  | $[\ell=3,D_{6}[D_{3}]\times C_{\theta}]$ | $2\sigma_{x}k_{z}k_{x}k_{y}+\sigma_{y}k_{z}(k_{x}^{2}-k_{y}^{2})+\sigma_{z}k_{y}(k_{y}^{2}-3k_{x}^{2})$ | $[\ell=1,D_{\infty}[C_{\infty}]\times C_{\theta}]$ | $\sigma_{x}k_{y}-\sigma_{y}k_{x}$
$G_{\mathrm{p}}\times C_{i}$ | MC |  |  | $[\ell=3,D_{6}(D_{3})\times C_{i}]$ | $\sigma_{x}(k_{x}^{2}-k_{y}^{2})-2\sigma_{y}k_{x}k_{y}$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i}]$ | $\sigma_{z}$
$G_{\mathrm{p}}\times C_{i\theta}$ | AMC |  |  | $[\ell=3,D_{6}(D_{3})\times C_{i\theta}]$ | $k_{x}(k_{x}^{2}-3k_{y}^{2})$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i\theta}]$ | $k_{z}$
class $L(G_{\mathrm{p}})$ |  | $L(D_{4})$ | $L(C_{4})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{i\times\theta}$ | PC |  |  | $[\ell=4,D_{4}\times C_{i\times\theta}]$ | $k_{x}^{2}k_{y}^{2}$ | $[\ell=4,C_{4}\times C_{i\times\theta}]$ | $k_{x}k_{y}(k_{x}^{2}-k_{y}^{2})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{\theta}$ | EC |  |  | $[\ell=5,D_{8}[D_{4}]\times C_{\theta}]$ |  | $\sigma_{x}k_{x}[k_{x}^{2}(k_{y}^{2}-k_{z}^{2})-k_{y}^{2}(k_{y}^{2}-3k_{z}^{2})]+(x\leftrightarrow y)$
---
$+\sigma_{z}k_{z}(k_{x}^{4}-6k_{x}^{2}k_{y}^{2}+k_{y}^{4})$
$[\ell=1,D_{\infty}[C_{\infty}]\times C_{\theta}]$ | $\sigma_{x}k_{y}-\sigma_{y}k_{x}$
$G_{\mathrm{p}}\times C_{i}$ | MC |  |  | $[\ell=5,D_{8}(D_{4})\times C_{i}]$ | $\sigma_{x}k_{y}k_{z}(k_{y}^{2}-3k_{x}^{2})-(x\leftrightarrow y)-\sigma_{z}k_{x}k_{y}(k_{x}^{2}-k_{y}^{2})$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i}]$ | $\sigma_{z}$
$G_{\mathrm{p}}\times C_{i\theta}$ | AMC |  |  | $[\ell=5,D_{8}(D_{4})\times C_{i\theta}]$ | $k_{x}k_{y}k_{z}(k_{x}^{2}-k_{y}^{2})$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i\theta}]$ | $k_{z}$
class $L(G_{\mathrm{p}})$ |  | $L(D_{2})$ | $L(C_{2})$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{i\times\theta}$ | PC |  |  | $[\ell=2,D_{2}\times C_{i\times\theta}]$ | $k_{x}^{2}-k_{y}^{2};\;k_{y}^{2}-k_{z}^{2};\;k_{z}^{2}-k_{x}^{2}$ | $[\ell=2,C_{2}\times C_{i\times\theta}]$ | $k_{x}k_{y}$
$\rule{0.0pt}{11.62494pt}G_{\mathrm{p}}\times C_{\theta}$ | EC |  |  | $[\ell=3,O[T]\times C_{\theta}]$ | $\sigma_{x}k_{x}(k_{y}^{2}-k_{z}^{2})+\mbox{cp}$ | $[\ell=1,D_{\infty}[C_{\infty}]\times C_{\theta}]$ | $\sigma_{x}k_{y}-\sigma_{y}k_{x}$
$G_{\mathrm{p}}\times C_{i}$ | MC |  |  | $[\ell=3,O(T)\times C_{i}]$ | $\sigma_{x}k_{y}k_{z}+\mbox{cp}$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i}]$ | $\sigma_{z}$
$G_{\mathrm{p}}\times C_{i\theta}$ | AMC |  |  | $[\ell=3,O(T)\times C_{i\theta}]$ | $k_{x}k_{y}k_{z}$ | $[\ell=1,D_{\infty}(C_{\infty})\times C_{i\theta}]$ | $k_{z}$

###  V.2 Chirality and band structure

As discussed in Sec. IV and Ref. [22], the spin-dependent electronic band
structure $E_{n\sigma}({\bm{\mathrm{k}}})$ directly reflects the categories of
polarizations in solids. These categories manifest themselves via
characteristic terms in the band structure that transform like irreducible
tensors $T_{\ell}^{ss^{\prime}}$. In the following, we argue that the band
structure $E_{n\sigma}({\bm{\mathrm{k}}})$ likewise reflects the categories of
chirality in solids via an interplay of parapolar and unipolar (i.e.,
electropolar, magnetopolar, or antimagnetopolar) terms.

The categories of chirality can be defined via the criterion which monopoles
$T_{0}^{ss^{\prime}}$ are permitted in a system, see Table 11. For example, in
electrochiral crystal structures, the electrotoroidal compound scalar
$T_{0}^{-+}$ becomes symmetry-allowed. In a band structure, such a scalar is
realized by a term ${\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{k}}}$. The
presence of this scalar in a band structure has thus been identified as a key
feature of electrochiral systems [88]. It can be viewed as a minimal model of
an effective low-energy Hamiltonian for electrochirality in solids.

In contrast, using the same building blocks $k,\sigma$, the charges
$T_{0}^{+-}$ and $T_{0}^{--}$ have no band-structure indicators associated
with them (see Table 12). We can envision in magnetochiral systems (fictitious
or engineered) compound scalars
$T_{0}^{+-}\propto{\bm{\mathrm{d}}}\cdot{\bm{\mathrm{k}}}$ and in
antimagnetochiral systems compound scalars
$T_{0}^{--}\propto{\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{d}}}$, where
${\bm{\mathrm{d}}}$ is an electric dipole density (signature $-+$) similar to
the magnetic dipole density ${\bm{\mathrm{\sigma}}}$ (signature $+-$) [23].

However, for a crystal structure to be chiral, it is not decisive whether and
how chiral charges $T_{0}^{ss^{\prime}}$ are actually realized. A monopole
$T_{0}^{ss^{\prime}}$ is a sufficient, but not a necessary criterion for
chirality in crystals. Electropolarizations, magnetopoloarizations, and
antimagnetopolarizations $T_{\ell}^{ss^{\prime}}$ with $\ell>0$ combined with
parapolarizations $T_{\ell}^{++}$ with $\ell>0$ can likewise make a system
chiral. We show in the following how tensors $T_{\ell}^{ss^{\prime}}$ with
$\ell>0$ can be used to define minimal models of effective low-energy
Hamiltonians for electrochirality, magnetochirality and antimagnetochirality
for all Laue classes $L(G_{\mathrm{p}})$.

Our theory is illustrated in Table 15 listing the lowest-degree invariant
tensors realizing chiralities in the continuous classes $L(R)$ and
$L(D_{\infty})$, the cubic classes $L(O)$ and $L(T)$, and the axial classes
$L(D_{n})$ and $L(C_{n})$ with $n=2,3,4,6$.

Figure 5:  Chirality exhibited in the band structure for unipolar-unichiral
groups in the classes $L(R)$ (left columns), $L(O)$ (center columns), and
$L(T)$ (right columns). The first row shows energy surfaces of a parachiral
system, whereas the second, third and fourth row show energy surfaces for the
two enantiomorphs allowed under the indicated electrochiral, magnetochiral and
antimagnetochiral groups. Figure 6:  Unipolarizations giving rise to
unichirality in the classes $L(T)$ (Fig. 5) and $L(C_{4})$ (Fig. 7) may also
exist in systems with point groups in the classes $L(O)$ and $L(D_{4})$,
respectively, where these terms, as shown here, _do not_ give rise to
unichirality. Figure 7:  Chirality exhibited in the band structure for
unipolar-unichiral groups in the classes $L(D_{\infty})$ (left columns),
$L(D_{4})$ (center columns), and $L(C_{4})$ (right columns). The first row
shows energy surfaces of a parachiral system, whereas the second, third and
fourth row show energy surfaces for the two enantiomorphs allowed under the
indicated electrochiral, magnetochiral and antimagnetochiral groups.

####  V.2.1 Spherical class $L(R)$

The full rotation group $R_{i\times\theta}$ is parachiral. The only invariant
tensor permitted by this group is $T_{0}^{++}$ that is realized by the scalar
$k^{2}$ and functions thereof [Fig 5(a)]. The electrochiral spherical group
$R_{\theta}$ also permits a tensor
$T_{0}^{-+}\propto{\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{k}}}$ (and functions
thereof). The latter term gives rise to two distinct enantiomorphs [95, 22,
96], see Fig 5(d). As discussed before, the magnetochiral group $R_{i}$ and
the antimagnetochiral group $R_{i\theta}$ have no band structure indicators
associated with them.

####  V.2.2 Cubic crystal system [classes $L(O)$ and $L(T)$]

Beyond the isotropic terms already permitted by the groups in the spherical
class $L(R)$, all groups in the cubic class $L(O)$ also permit an electric
hexadecapolarization ($\ell=4$) that manifests itself in lowest order via a
term [22]

$H^{(\mathrm{e},4)}\propto
k_{x}^{2}k_{y}^{2}+k_{y}^{2}k_{z}^{2}+k_{z}^{2}k_{x}^{2}\,.$ (22)

This term represents a warping of the energy dispersion
$E_{\sigma}({\bm{\mathrm{k}}})$ of band electrons in cubic crystal structures
[97, 98], see Fig 5(b).

The group $O\times C_{\theta}$ is electrochiral. In lowest order $\ell$, it
permits an electropolarization with $\ell=9$ that gives rise to a term

$\displaystyle H^{(\mathrm{e},9)}\propto{}$
$\displaystyle\sigma_{x}k_{x}[k_{y}^{2}k_{z}^{2}(3k_{y}^{4}-10k_{y}^{2}k_{z}^{2}+3k_{z}^{4})$
$\displaystyle+k_{x}^{2}(k_{x}^{2}-k_{y}^{2}-k_{z}^{2})(k_{y}^{4}-6k_{y}^{2}k_{z}^{2}+k_{z}^{4})]+\mbox{cp}\,,$
(23)

see Fig. 5(e). The electrotoroidal scalar
${\bm{\mathrm{\sigma}}}\cdot{\bm{\mathrm{k}}}$ that is generally likewise
present in such systems can be interpreted as a compound moment density (16)
due to the combined effect of the electropolarization (23) and
parapolarizations including the hexadecapolarization (22).

By SIS-TIS duality, the group $O\times C_{i}$ is magnetochiral. In lowest
order $\ell$, it permits a magnetopolarization with $\ell=9$ that gives rise
to a term

$\displaystyle H^{(\mathrm{m},9)}\propto{}$
$\displaystyle\sigma_{x}k_{y}k_{z}(k_{y}^{2}-k_{z}^{2})[5k_{x}^{4}-3k_{x}^{2}(k_{y}^{2}+k_{z}^{2})+k_{y}^{2}k_{z}^{2}]$
$\displaystyle\hskip 10.00002pt+\mbox{cp}\,,$ (24)

see Fig. 5(g). Finally, the group $O\times C_{i\theta}$ is antimagnetochiral.
It permits an antimagnetopolarization ($\ell=0$) that manifests itself, in
lowest order $\ell$, via a magnetotoroidal density with $\ell=9$ that gives
rise to a term

$\displaystyle H^{(\mathrm{am},9)}\propto{}$ $\displaystyle
k_{x}k_{y}k_{z}(k_{y}^{2}-k_{z}^{2})(k_{z}^{2}-k_{x}^{2})(k_{x}^{2}-k_{y}^{2})\,,$
(25)

see Fig. 5(i).

Besides the class $L(O)$, the cubic crystal system also includes the class
$L(T)$. As $T\subset O$, the polarizations permitted by the point groups
$G_{O}$ in $L(O)$ are also permitted by the respective groups $G_{T}\subset
G_{O}$ in $L(T)$. Beyond that, the groups in $L(T)$ permit polarizations not
present in $L(O)$, as listed in Table 15.

Specifically, the electrochiral group $T\times C_{\theta}$ permits an electric
octupolarization $\ell=3$ that manifests itself via a Dresselhaus term

$H^{(\mathrm{e},3)}\propto\sigma_{x}k_{x}(k_{y}^{2}-k_{z}^{2})+\mbox{cp}\,.$
(26)

As is well known [77], this term is already allowed for systems with the
parachiral point group $T_{d}\times C_{\theta}=O[T]\times C_{\theta}$ [Fig.
6(a)] realized, e.g., by the zincblende structure, i.e., by itself, this term
is not sufficient as an indicator for electrochirality in the class $L(T)$.
The Dresselhaus term (26) becomes an indicator for electrochirality [Fig.
5(f)] when combined with the parapolarization [$\ell=6$, Fig. 5(c)]

$H^{(\mathrm{e},6)}\propto(k_{y}^{2}-k_{z}^{2})(k_{z}^{2}-k_{x}^{2})(k_{x}^{2}-k_{y}^{2})\,.$
(27)

The parapolarization (27) is forbidden by the groups in the class $L(O)$, but
it is allowed in the class $L(T)$. A parapolarization with $\ell=6$ that is
allowed already in the class $L(O)$ is listed in Table 15.

Similarly, a magnetopolarization ($\ell=3$)

$H^{(\mathrm{m},3)}\propto\sigma_{x}k_{y}k_{z}+\mbox{cp}$ (28)

may exist in systems with magnetochiral group $T\times C_{i}$, but also in
parachiral systems with point group $O(T)\times C_{i}$. The
magnetopolarization (28) becomes an indicator for magnetochirality when
combined with the parapolarization (27), see Fig. 5(h). Likewise, an
antimagnetopolarization yields a magnetotoroidal density with $\ell=3$

$H^{(\mathrm{am},3)}\propto k_{x}k_{y}k_{z}\,.$ (29)

This density may exist in systems with antimagnetochiral group $T\times
C_{i\theta}$, but also in parachiral systems with point group $O(T)\times
C_{i\theta}$. The density (29) becomes an indicator for antimagnetochirality
when combined with the parapolarization (27), see Fig. 5(j).

To summarize, in the class $L(T)$ the unipolar terms (26), (28), and (29)
become indicators of electrochirality, magnetochirality, and
antimagnetochirality, respectively, only when each of these terms is combined
with the parapolarization (27), i.e., jointly the terms define minimal models
for effective low-energy Hamiltonians realizing electrochirality (point group
$T\times C_{\theta}$), magnetochirality ($T\times C_{i}$), and
antimagnetochirality ($T\times C_{i\theta}$), respectively. This is discussed
further in Sec. V.2.4.

####  V.2.3 Axial systems [classes $L(D_{n})$ and $L(C_{n})$]

Similar to the preceding discussion of the cubic classes $L(O)$ and $L(T)$,
Table 15 also lists the lowest-order parapolar and unipolar tensors permitted
by the class $L(D_{\infty})$, as well as the classes $L(D_{n})$ and $L(C_{n})$
with $n=2,3,4,6$. Figure 7 shows the chiral band structure for
$L(D_{\infty})$, $L(D_{4})$, and $L(C_{4})$. For the dihedral classes
$L(D_{6})$, $L(D_{3})$, $L(D_{4})$, and $L(D_{2})$ unichirality is realized by
combining suitable parapolar terms with unipolar terms with $\ell=7$, 3, 5,
and 3, respectively. Note that $D_{2}\subset T$ so that the class $L(D_{2})$
largely reproduces the features of the class $L(T)$.

The electropolar cyclic groups $C_{n}\times C_{\theta}$ permit a spontaneous
electric dipolarization ($\ell=1$) that manifests itself via a Rashba term

$H^{(\mathrm{e},1)}\propto\sigma_{x}k_{y}-\sigma_{y}k_{x}\,.$ (30)

As is well known, this term is already allowed for systems with parachiral
point groups $C_{nv}\times C_{\theta}=D_{n}[C_{n}]\times C_{\theta}$ realized,
e.g., by the wurtzite structure ($C_{6v}\times C_{\theta}$) [76]. See also
Fig. 6(b). When the Rashba term is combined with a suitable parapolar term
forbidden by the class $L(D_{n})$, but allowed by the class $L(C_{n})$ (see
Table 15), jointly these terms describe an electrochiral system [Fig. 7(f)].

Similarly, the cyclic groups $C_{n}\times C_{i}$ are magnetopolar, i.e., they
permit a spontaneous magnetization ($\ell=1$) that manifests itself via an
exchange (Zeeman) term

$H^{(\mathrm{m},1)}\propto\sigma_{z}\,.$ (31)

Such a term is already allowed for systems with the parachiral point groups
$D_{n}(C_{n})\times C_{i}$, see Fig. 6(d). This term becomes an indicator of
magnetochirality when combined with the respective parapolar terms listed for
the classes $L(C_{n})$ in Table 15 [Fig. 7(h)].

Finally, the cyclic groups $C_{n}\times C_{i\theta}$ are antimagnetopolar,
i.e., they permit a spontaneous antimagnetopolarization that manifests itself
via a magnetotoroidal term ($\ell=1$)

$H^{(\mathrm{am},1)}\propto k_{z}\,.$ (32)

Again, such a term is already allowed for systems with the parachiral point
groups $D_{n}(C_{n})\times C_{i\theta}$, see Fig. 6(f). This term becomes an
indicator of antimagnetochirality when combined with the respective parapolar
terms listed for the classes $L(C_{n})$ in Table 15 [Fig. 7(j)].

From a symmetry point of view, an electropolar system with spontaneous
electric dipolarization (30) is equivalent to a parapolar system placed into
an external electric field. In parapolar systems with crystallographic point
group $C_{n}\times C_{i\times\theta}$, electrochirality can therefore be
induced by placing the system into an external electric field [24]. Similarly,
magnetochirality can be induced by placing such a system into an external
magnetic field. Finally, parallel electric and magnetic fields make such
systems multichiral, i.e., not only induce the fields a Rashba term (30) and a
Zeeman term (31), but jointly the fields also induce a magnetotoroidal term
(32).

####  V.2.4 Minimal effective models for chirality

Table 15 lists for each tabulated tensor its maximal group, i.e., the largest
group that leaves the tensor invariant. For the odd/̄$\ell$ electropolar,
magnetopolar and antimagnetopolar tensors, the respective maximal group is
always an improper group beyond the class $L(G_{\mathrm{p}})$, so that these
tensors, by themselves, cannot serve as minimal effective models for chirality
in the class $L(G_{\mathrm{p}})$, as illustrated in Fig. 6.

Quite generally, a proper point group $G_{\mathrm{p}}$ is an invariant
subgroup of $i$/̄minor groups $\tilde{G}_{\mathrm{p}}[G_{\mathrm{p}}]$

$G_{\mathrm{p}}\subset\tilde{G}_{\mathrm{p}}[G_{\mathrm{p}}]\;,$ (33a) see
Sec. II.2. Specifically, we have ($n=1,2,3,\ldots$) $\displaystyle T$
$\displaystyle\subset T_{d}=O[T]$ (33b) $\displaystyle D_{2n}$
$\displaystyle\subset D_{(2n)d}=D_{4n}[D_{2n}]$ (33c) $\displaystyle D_{2n+1}$
$\displaystyle\subset D_{(2n+1)h}=D_{4n+2}[D_{2n+1}]$ (33d) $\displaystyle
C_{2n}$
$\displaystyle\subset\left\\{\begin{array}[]{rl}S_{4n}&=C_{4n}[C_{2n}]\\\
C_{(2n)v}&=D_{2n}[C_{2n}]\end{array}\right.$ (33g) $\displaystyle C_{2n+1}$
$\displaystyle\subset\left\\{\begin{array}[]{rl}C_{(2n+1)h}&=C_{4n+2}[C_{2n+1}]\\\
C_{(2n+1)v}&=D_{2n+1}[C_{2n+1}]\end{array}\right.$ (33j)

and similar for the supergroups $\tilde{G}_{\mathrm{p}}(G_{\mathrm{p}})$ and
$\tilde{G}_{\mathrm{p}}\\{G_{\mathrm{p}}\\}$ of $G_{\mathrm{p}}$. Of course,
while a group like $D_{4}$ is a crystallographic point group, the supergroup
$D_{4d}=D_{8}[D_{4}]\supset D_{4}$ is not a crystallographic point group.
Nonetheless, the improper point group $D_{4d}$ is, ignoring TIS, the maximal
group for the lowest-degree electropolar term permitted by the
crystallographic point group $D_{4}$, see Table 15, so that this electropolar
term, by itself, cannot serve as a minimal effective model for
electrochirality in the class $L(D_{4})$.

Among the finite point groups, the octahedral group $O$ is special as it does
not appear as a subgroup of a finite group outside the class $L(O)$.
Therefore, the unipolar tensors with $\ell=9$ listed in Table 15 for the class
$L(O)$ represent, by themselves, minimal effective models for unichirality in
the class $L(O)$.

While the maximal groups for the unipolar tensors listed in Table 15 generally
go beyond the respective class $L(G_{\mathrm{p}})$, for each of the parapolar
tensors in Table 15 the maximal group is the full group $G_{\mathrm{p}}\times
C_{i\times\theta}$ of the respective class. Therefore, _jointly_ the unipolar
tensors combined with the respective parapolar tensors represent minimal,
effective models for unichirality in the class $L(G_{\mathrm{p}})$.
Multichirality is realized by combining, at least, two different unipolar
tensors in a class $L(G_{\mathrm{p}})$. Table 15 thus provides a complete
theory for minimal effective models describing unipolar-unichiral and
multipolar-multichiral systems [99].

The cooperation of a parapolarization and a unipolarization to realize
unichirality can also be recognized in Fig. 2. In Fig. 2(a) we can envision
that polar properties have been added to the parapolar nonchiral object
$T_{\ell 0}^{++}$. Alternatively, we could imagine that the unipolar nonchiral
objects $T_{u0}^{-+}$, $T_{u0}^{+-}$, and $T_{u0}^{--}$ have acquired a second
nonpolar property so that jointly these properties realize chirality. These
examples illustrate that we can generally have multiple pathways to realize
chirality.

The multipolar-unichiral case in Fig. 4 is different from the unipolar-
unichiral case in Figs. 2 and 3. The objects shown in Fig. 4(a) visualize the
irreducible tensors $T_{g0}^{++}$ and $T_{u0}^{-+}$, $T_{u0}^{+-}$, and
$T_{u0}^{--}$ as indicated. While individual objects $T_{u0}^{ss^{\prime}}$ in
Fig. 4(a) are unipolar but not chiral, it is the superposition of their
polarizations that yields unichirality [Figs. 4(b) to (d)] and multichirality
[Fig. 4(e)].

This rule applies also to minimal effective models for chirality in
multipolar-unichiral systems. For example, the multipolar-electrochiral group
$D_{4}(C_{4})$ permits the electropolar term

$\displaystyle H^{(\mathrm{e},5)}\propto{}$
$\displaystyle\sigma_{x}k_{x}[k_{x}^{2}(k_{y}^{2}-k_{z}^{2})-k_{y}^{2}(k_{y}^{2}-3k_{z}^{2})]+(x\leftrightarrow
y)$
$\displaystyle+\sigma_{z}k_{z}(k_{x}^{4}-6k_{x}^{2}k_{y}^{2}+k_{y}^{4})\,,$
(34)

[because $D_{4}(C_{4})\subset D_{4}\times C_{\theta}\subset D_{8}[D_{4}]\times
C_{\theta}$, see Table 15 and Fig. 1], the magnetopolar term (31) [because
$D_{4}(C_{4})\subset D_{4}(C_{4})\times C_{i}$, see Table 15], and the
antimagnetopolar term (32) [because $D_{4}(C_{4})\subset D_{4}(C_{4})\times
C_{i\theta}$]. Any two of these terms are sufficient as a minimal model for
electrochirality under $D_{4}(C_{4})$ because the largest subgroup common to
any two of the three groups $D_{4}\times C_{\theta}$, $D_{4}(C_{4})\times
C_{i}$, and $D_{4}(C_{4})\times C_{i\theta}$ is $D_{4}(C_{4})$. Of course,
$D_{4}(C_{4})$ permits not only the electropolar term (34), but also the
parapolar term

$H^{(\mathrm{e},4)}\propto k_{x}^{2}k_{y}^{2}$ (35)

because $D_{4}(C_{4})\subset D_{4}\times C_{i\times\theta}$, see Table 15.
Jointly, the terms (34) and (35) can serve not only as a minimal model for
electrochirality under $D_{4}\times C_{\theta}$, but also under
$D_{4}(C_{4})\subset D_{4}\times C_{\theta}$.

Using Table 15, minimal models for other multipolar-unichiral systems can be
constructed in a similar way. We thus have a complete theory for minimal
effective models describing chirality in crystals [99]. Our analysis shows
that chirality in solid-state systems is qualitatively different from
chirality in molecules and elementary-particle physics.

##  VI Conclusions and outlook

Using symmetry as a fundamental guiding principle, we have developed a
comprehensive theory of electromagnetism and chirality in crystalline solids.
We have identified five categories of electric and magnetic multipole order
and five categories of chirality that provide a complementary, comprehensive
classification of crystal structures and their physical properties. The
categories yield 15 types of crystal structures with distinctive electric and
magnetic properties. Multiple versions of chirality that were previously
overlooked, including multichirality with four distinct enantiomorphs are a
novel criterion underlying our comprehensive classification. Our work shows
explicitly how chirality is linked to electromagnetic order and band
structure.

Clearly, the concept of SIS-TIS duality, Eq. (7), can be extended from
abstract group theory to a duality of physical quantities and effects
characterizing crystal structures. For example, an electric field
${\bm{\mathcal{E}}}$ and a magnetic field ${\bm{\mathcal{B}}}$ can be
considered duals of each other in the sense that the groups according to which
fields ${\bm{\mathcal{E}}}$ and ${\bm{\mathcal{B}}}$ transform are related by
$i\leftrightarrow\theta$ duality; see Table 10. This implies, in turn, that
physical effects associated with fields ${\bm{\mathcal{E}}}$ and
${\bm{\mathcal{B}}}$ can likewise be related by $i\leftrightarrow\theta$
duality. A simple example is the electric response of a medium to an electric
field. The dual effect is the magnetic response of a medium to a magnetic
field. The magnetoelectric effect can be considered a self-dual effect. From a
different perspective, pairs of material tensors can be related by
$i\leftrightarrow\theta$ duality if they are permitted under dual pairs of
magnetic groups, see Tables 2, 3, 5, 8 and 9 and Appendix E. Examples of pairs
of dual tensors are given by the rank/̄1 pyroelectric and pyromagnetic tensors
and the rank/̄3 piezoelectric and piezomagnetic tensors, while the rank/̄2
magnetoelectric tensor is self-dual [100].

A more thorough analysis of duality-related physical effects also needs to
consider Onsager’s reciprocal relations accounting for the symmetry of
fluctuations under time inversion $\theta$ when studying irreversible
processes [101]. In the context of magnetic systems breaking TIS such an
analysis is not trivial. For example, following early work [102], the symmetry
of the conductivity tensor describing the anomalous Hall effect in
magnetically ordered systems was discussed intensely till a consensus was
reached. See Ref. [103]. A proper discussion of this important point is beyond
the scope of the present work. It will be the subject of a future publication.

###### Acknowledgements.

RW and UZ acknowledge stimulating discussions with N. Spaldin. RW also
benefited from discussions with R. Cavanaugh, A. Hoffmann, L. Lurio, and M.
Norman. UZ thanks S. Marsland for useful comments on an early version of the
manuscript. Work at Argonne was supported by DOE BES under Contract No. DE-
AC02-06CH11357.

##  Appendix A Refined symmetry classifications

This Appendix contains a general discussion of the group-theoretical framework
underlying the refined symmetry classifications used in this work. Appendix
A.2 comments briefly on spin groups.

###  A.1 General formalism

Starting from the 11 crystallographic point groups $G_{\mathrm{p}}$ of proper
rotations, symmetry classifications of crystal structures can be
systematically refined by permitting a wider range of symmetry elements in the
symmetry groups for these systems. Formally, the construction of nonmagnetic
point groups, magnetic point groups, and even spin point groups by combining
groups $G_{\mathrm{p}}$ of proper rotations with space inversion $i$ [6], time
inversion $\theta$ [14] and spin rotations [104] follows a similar scheme,
where each set of additional symmetries permits a more fine-grained
classification of the symmetry groups $H$ that can be realized by physical
systems compared with the symmetry groups $G$ that arise in a coarse-grained
symmetry analysis [105]. The additional symmetries are generally characterized
by a symmetry group $\bar{G}$, and we want to explore the possibilities how
the symmetries in $\bar{G}$ can be combined with the symmetries in $G$ to form
new symmetry groups $H$.

Two limiting cases are trivial. First, the symmetry group remains $G$ if all
symmetries in $\bar{G}$ are broken. Second, the symmetry group becomes the
product group $\bar{G}\times G$ if all symmetries in $\bar{G}$ are by
themselves good symmetries of the system. Nontrivial scenarios arise if
symmetries represented by elements in $\bar{G}$ are by themselves broken, but
these elements must be combined with symmetry elements in $G$ to obtain new
good symmetries of the system. To identify and describe these scenarios in a
systematic way [6, 14, 104], we consider all subgroups $\bar{U}_{1}$ and
$\bar{U}_{2}$ of $\bar{G}$ with $\bar{U}_{1}\cap\bar{U}_{2}=\\{e\\}$. We
assume that the group $\bar{U}_{1}$ contains the symmetry elements in
$\bar{G}$ that are good symmetries of a system independent of the symmetries
in $G$, whereas $\bar{U}_{2}$ contains the symmetry elements in $\bar{G}$ that
must be combined with elements in $G$ to form a group
$\bar{H}(\bar{U}_{2},G)$. The symmetry groups $H$ then become

$H=\bar{U}_{1}\times\bar{H}(\bar{U}_{2},G)\,.$ (36)

We obtain the admissible groups $\bar{H}(\bar{U}_{2},G)$ via the homomorphism
theorem for groups [106]. Given $G$ and $\bar{U}_{2}$, we identify all
invariant subgroups $U$ of $G$ such that $\bar{U}_{2}$ is isomorphic to the
factor group $G/U$. Then we obtain $\bar{H}(\bar{U}_{2},G)$ by combining the
cosets for the invariant subgroup $U$ of $G$ with the elements in
$\bar{U}_{2}$, consistent with the isomorphism relating $\bar{U}_{2}$ and
$G/U$. By definition, $\bar{H}$ is isomorphic to $G$. The order of the group
$H$ equals the order of $G$ times the order of $\bar{U}_{1}$.

If we combine the 11 groups $G=G_{\mathrm{p}}$ of proper rotations with the
space inversion group $\bar{G}=C_{i}$, the symmetry groups $H$ are the 32
nonmagnetic crystallographic point groups. More specifically, the two-element
group $\bar{G}=C_{i}$ permits three cases. (i) When
$\bar{U}_{1}=\bar{U}_{2}=\\{e\\}$, Eq. (36) yields the type/̄i groups
$G_{\mathrm{p}}$. (ii) When $\bar{U}_{1}=C_{i}$ and $\bar{U}_{2}=\\{e\\}$, we
get the type/̄ii groups $C_{i}\times G_{\mathrm{p}}$. (iii) Finally,
$\bar{U}_{1}=\\{e\\}$ and $\bar{U}_{2}=C_{i}$ yields the type/̄iii groups,
consistent with the discussion in Sec. II.2 and Eq. (5b).

In the same way, combining the 32 nonmagnetic crystallographic point groups
with the time inversion group $\bar{G}=C_{\theta}$ yields the 122 magnetic
point groups (types I, II, and III) that are further classified in Tables 2
and 9.

###  A.2 Spin groups

Equation (36) can also be used to derive spin groups [104, 107, 108] that
recently have received renewed interest [38, 39]. While the following
discussion can readily be generalized to space groups, for clarity we focus on
point groups only. Generally, the symmetry elements $g$ in nonmagnetic
crystallographic point groups can be associated with operators $\hat{g}$ that
represent transformations $C_{g}$ of the position vector ${\bm{\mathrm{r}}}$

${\bm{\mathrm{r}}}^{\prime}=\hat{g}\,{\bm{\mathrm{r}}}=C_{g}\,{\bm{\mathrm{r}}}\,.$
(37)

These operators $\hat{g}$ also describe the transformation of scalar property
functions of a crystal structure such as the charge density
$\rho({\bm{\mathrm{r}}})$

$\hat{g}\,\rho({\bm{\mathrm{r}}}^{\prime})=\rho({\bm{\mathrm{r}}})\qquad\mathrm{or}\qquad\hat{g}\,\rho({\bm{\mathrm{r}}})=\rho(C_{g}^{-1}{\bm{\mathrm{r}}})\,.$
(38)

Then we associate a nonmagnetic symmetry group $G$ with a crystallographic
structure if for all $g\in G$ the operators $\hat{g}$ leave the scalar
function $\rho({\bm{\mathrm{r}}})$ _invariant_

$\hat{g}\,\rho({\bm{\mathrm{r}}})=\rho(C_{g}^{-1}{\bm{\mathrm{r}}})=\rho({\bm{\mathrm{r}}})\,.$
(39)

Magnetic crystal structures have a magnetization density
${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$ that is characterized not only by a
nontrivial behavior under the black-white symmetry time inversion $\theta$.
But similar to Eq. (38) we have

$\hat{h}\,{\bm{\mathcal{m}}}({\bm{\mathrm{r}}})=\bar{C}_{h}\,{\bm{\mathcal{m}}}(C_{h}^{-1}{\bm{\mathrm{r}}})\,,$
(40)

that is, the symmetry group becomes an extended group $H$ as in Eq. (36),
where we associate with each $h\in H$ a transformation $C_{h}$ of the position
vector ${\bm{\mathrm{r}}}$ and a transformation $\bar{C}_{h}$ of the vector
${\bm{\mathcal{m}}}$. We have $C_{h}=\mathcal{C}\mathcal{I}$ where
$\mathcal{C}\in R$ is a proper rotation and $\mathcal{I}\in C_{i}$, whereas
$\bar{C}_{h}=\bar{\mathcal{C}}\mathcal{T}$ with $\bar{\mathcal{C}}\in R$ and
$\mathcal{T}\in C_{\theta}$. This is consistent with the signature $-+$ ($+-$)
of ${\bm{\mathrm{r}}}$ (${\bm{\mathcal{m}}}$). The invariance condition for
${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$ then becomes

$\hat{h}\,{\bm{\mathcal{m}}}({\bm{\mathrm{r}}})=\bar{C}_{h}\,{\bm{\mathcal{m}}}(C_{h}^{-1}{\bm{\mathrm{r}}})={\bm{\mathcal{m}}}({\bm{\mathrm{r}}})\,.$
(41)

Magnetic groups combine nonmagnetic groups $G$ with $\bar{G}=C_{\theta}$ so
that the transformations $\bar{C}_{h}$ and $C_{h}$ in Eq. (41) have the same
proper rotational parts, $\bar{\mathcal{C}}=\mathcal{C}$. Spin groups, on the
other hand, combine nonmagnetic groups $G$ (that are subgroups of $R_{i}\equiv
R\times C_{i}$) with groups
$\bar{U}_{1},\bar{U}_{2}\subset\bar{G}=R_{\theta}\equiv R\times C_{\theta}$ to
obtain a more fine-grained classification of the symmetries of a magnetization
density ${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$. Here, the decomposition
$\bar{U}_{1}\times\bar{U}_{2}\subset R_{\theta}$ with
$\bar{U}_{1}\cap\bar{U}_{2}=\\{e\\}$ yields five cases [104]. (i) The trivial
case $\bar{U}_{1}=\bar{U}_{2}=\\{e\\}$ yields spin groups that are equivalent
to the magnetic type/̄I groups. (ii) Similarly, $\bar{U}_{1}=R_{\theta}$ and
$\bar{U}_{2}=\\{e\\}$ is equivalent to the magnetic type II. (iii)
$\bar{U}_{1}=D_{\infty}(C_{\infty})$ (which is the dual of
$D_{\infty}[C_{\infty}]=C_{\infty v}$) implies $\bar{U}_{2}=C_{\theta}$. Here
the groups $\bar{H}(C_{\theta},G)$ are the type/̄III magnetic groups. This
case represents collinear magnetization densities
${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$. (iv) $\bar{U}_{1}=C_{2}(C_{1})$
(which is the dual of $C_{2}[C_{1}]=C_{s}$) implies that $\bar{U}_{2}$ is a
subgroup of $D_{\infty}(C_{\infty})$, i.e., $\bar{U}_{2}$ is the dual of one
of the nonmagnetic crystallographic groups $G\subset
D_{\infty}[C_{\infty}]=C_{\infty v}$. This case represents planar
magnetization densities ${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$. (v) Finally,
we may have $\bar{U}_{1}=\\{e\\}$ and $\bar{U}_{2}\neq\\{e\\}$, i.e.,
$\bar{U}_{2}$ is the dual of one of the 32 nonmagnetic crystallographic
groups. This case represents spatial magnetization densities
${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$. For the group $G=O_{h}$, the spin
point groups are tabulated in Ref. [104]. Reference [108] tabulates all spin
point groups for cases (iii), (iv) and (v), though without referencing the
cases they apply to. More recently, spin groups have also been tabulated in
Ref. [109].

Clearly, spin groups can be further generalized using
$\bar{G}=R_{i\times\theta}$ [110] which is appropriate for magnetic structures
with not only a magnetization density ${\bm{\mathcal{m}}}({\bm{\mathrm{r}}})$
($\ell=1$ and signature $+-$) but also, e.g., a magnetic quadrupole density
($\ell=2$ and signature $--$) [17, 22]. The general case
$\bar{G}=R_{i\times\theta}$ also represents electropolar and multipolar
structures with electric multipole densities.

##  Appendix B Form of subclasses $\boldsymbol{L(G_{0}^{i\theta},i\theta)}$

The concept of subclasses of magnetic point groups is introduced in Sec. II.5.
While the form of subclasses $L(G_{0}^{i},i)$ and $L(G_{0}^{\theta},\theta)$
is straightforward, the form of subclasses $L(G_{0}^{i\theta},i\theta)$
discussed in the following is more subtle.

Every magnetic point group $G\in L(G_{\mathrm{p}})$ has the general form [31]

$G=\tilde{G}_{\mathrm{p}}+i\\{g_{\nu}^{(i)}\\}+\theta\\{g_{\nu}^{(\theta)}\\}+i\theta\\{g_{\nu}^{(i\theta)}\\}\,\,.$
(42)

Here $\tilde{G}_{\mathrm{p}}\subseteq G_{\mathrm{p}}$ is a proper point group,
and $\\{g_{\nu}^{(\gamma)}\\}$ denotes the set of operations in the class root
$G_{\mathrm{p}}$ that appears in $G$ in combination with the black-white
symmetry element $\gamma\in\\{i,\theta,i\theta\\}$. In the following, we
assume $G\supsetneq G_{\mathrm{p}}$, i.e., at least one set
$\\{g_{\nu}^{(\gamma)}\\}$ is nonempty. The relation $\gamma^{2}=e$ and
general group properties of $G$ imply the relations

$\displaystyle g_{\nu}^{(\gamma)}g_{\mu}^{(\gamma)}$
$\displaystyle\in\tilde{G}_{\mathrm{p}}$ $\displaystyle\mbox{for any fixed
$\gamma$}\,\,,$ (43a) $\displaystyle
g_{\nu}^{(\gamma)}g_{\mu}^{(\gamma^{\prime})}$
$\displaystyle\in\\{g_{\nu^{\prime}}^{(\gamma\gamma^{\prime})}\\}$
$\displaystyle\mbox{for any pair $\gamma\neq\gamma^{\prime}$}\,\,.$ (43b)

It follows from Eq. (43b) that either only one set $\\{g_{\nu}^{(\gamma)}\\}$
is nonempty, or all three sets are nonempty.

To understand how the mapping $i\theta\mapsto e$ transforms the group $G$
given in Eq. (42), we first note

$\theta=i^{2}\,\theta=i\,i\theta\;\stackrel{{\scriptstyle i\theta\mapsto
e}}{{\longmapsto}}\;i$ (44a) to find $G\;\stackrel{{\scriptstyle
i\theta\mapsto
e}}{{\longmapsto}}\;G^{i\theta}_{0}\equiv\tilde{G}_{\mathrm{p}}+\\{g_{\nu}^{(i\theta)}\\}+i\big{(}\\{g_{\nu}^{(i)}\\}+\\{g_{\nu}^{(\theta)}\\}\big{)}\,,$
(44b)

where clearly $G^{i\theta}_{0}$ is again a group. Alternatively, using

$i=i\,\theta^{2}=i\theta\,\theta\;\stackrel{{\scriptstyle i\theta\mapsto
e}}{{\longmapsto}}\;\theta$ (45a) yields $G\;\stackrel{{\scriptstyle
i\theta\mapsto
e}}{{\longmapsto}}\;G_{0\mathrm{d}}^{i\theta}\equiv\tilde{G}_{\mathrm{p}}+\\{g_{\nu}^{(i\theta)}\\}+\theta\big{(}\\{g_{\nu}^{(i)}\\}+\\{g_{\nu}^{(\theta)}\\}\big{)}\,\,.$
(45b)

Thus we can identify two equivalent subclass roots
$G_{0}^{i\theta}\leftrightarrow G_{0\mathrm{d}}^{i\theta}$ that are related
via the duality transformation $i\leftrightarrow\theta$, and

$L(G_{0}^{i\theta},i\theta)\equiv L(G_{0\mathrm{d}}^{i\theta},i\theta)\;.$
(46)

Applying the duality transformation $i\leftrightarrow\theta$ to $G$ in Eq.
(42) yields the dual partner

$G\;\leftrightarrow\;G_{\mathrm{d}}=\tilde{G}_{\mathrm{p}}+\theta\\{g_{\nu}^{(i)}\\}+i\\{g_{\nu}^{(\theta)}\\}+i\theta\\{g_{\nu}^{(i\theta)}\\}\,\,.$
(47)

Equations (44a) and (45a) show that $G_{\mathrm{d}}$ maps to the same subclass
roots $G^{i\theta}_{0}\leftrightarrow G_{0\mathrm{d}}^{i\theta}$ as $G$, i.e.,
dual partners $G$ and $G_{\mathrm{d}}$ belong to the same subclass
$L(G_{0}^{i\theta},i\theta)\equiv L(G_{0\mathrm{d}}^{i\theta},i\theta)$.

Specializing the composition of a subclass $L(G_{0}^{\gamma},\gamma)$
discussed generally in Sec. II.5 to the case $\gamma=i\theta$, we find that
$L(G_{\mathrm{p}},i\theta)$ contains besides $G_{\mathrm{p}}$ also
$G_{\mathrm{\ref{gcat:JJ}}}=G_{\mathrm{p}}\times C_{i\theta}$ and all the
type/̄LABEL:gcat:JJJ groups $G_{\mathrm{p}}\\{\tilde{G}_{\mathrm{p}}\\}\in
L(G_{\mathrm{p}})$. It follows from Eqs. (44a) and (45a) that the subclass
$L(G_{\mathrm{p}}\times C_{i},i\theta)\equiv L(G_{\mathrm{p}}\times
C_{\theta},i\theta)$ contains the groups of type LABEL:gcat:ii-II,
LABEL:gcat:ii-I and LABEL:gcat:i-II as well as all type-LABEL:gcat:ii-III and
type-LABEL:gcat:iii-II groups in $L(G_{\mathrm{p}})$. Groups in a subclass
$L(G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}],i\theta)\equiv
L(G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}}),i\theta)$ contain, besides
$G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}]$ and
$G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}})$, also the type/̄LABEL:gcat:JJp group
$G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}})\times C_{i\theta}$. Furthermore,
$G_{\mathrm{\ref{gcat:JJJp}}}=G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}^{(1)}]\,(\tilde{G}_{\mathrm{p}}^{(1^{\prime})}[\accentset{\approx}{G}_{\mathrm{p}}])$
is in $L(G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}^{(2)}],i\theta)\equiv
L(G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}}^{(2)}),i\theta)$, while
$G_{\mathrm{\ref{gcat:iii-
IIIp}}}=G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}^{(1)}]\,(\tilde{G}_{\mathrm{p}}^{(2)}[\accentset{\approx}{G}_{\mathrm{p}}])$
and $G_{\mathrm{\ref{gcat:iiip-
III}}}=G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}}^{(1)})\,[\tilde{G}_{\mathrm{p}}^{(2)}(\accentset{\approx}{G}_{\mathrm{p}})]$
both belong to $L(G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}^{(1)}],i\theta)\equiv
L(G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}}^{(1)}),i\theta)$.

The combination of $L(G_{\mathrm{p}},i\theta)$ with $L(G_{\mathrm{p}}\times
C_{i},i\theta)\equiv L(G_{\mathrm{p}}\times C_{\theta},i\theta)$ and all
subclasses $L(G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}],i\theta)\equiv
L(G_{\mathrm{p}}(\tilde{G}_{\mathrm{p}}),i\theta)$ yields the class
$L(G_{\mathrm{p}})$ [Eq. (12)].

##  Appendix C IRs of the continuous axial groups

This Appendix reviews the IRs of the continuous axial groups that we refer to
repeatedly in this work. The cyclic groups $G_{c\infty}$ are Abelian so that
all their IRs are one-dimensional. Under these groups, the transformation
behavior of tensor components $T_{\ell m}^{ss^{\prime}}$ of
$R_{i\times\theta}$ depends only on $m$ but not on $\ell$; for fixed $m$ but
different values of $\ell$ (with $|m|\leq\ell$), the components $T_{\ell
m}^{ss^{\prime}}$ transform the same way under groups $G_{c\infty}$. The
invariant tensors of groups $G_{c\infty}$ are then the tensor components
$T_{\ell 0}^{ss^{\prime}}$ (Table 6). Similar to $T_{00}^{++}$ under the
spherical groups, tensor components $T_{\ell 0}^{++}$ are invariant under all
cyclic groups $G_{c\infty}$.

The dihedral groups $G_{d\infty}$ are nonabelian. Under groups $G_{d\infty}$,
the tensor components $T_{\ell 0}^{ss^{\prime}}$ of $R_{i\times\theta}$
transform according to one-dimensional IRs, whereas tensor components
$T_{\ell,\pm m}^{ss^{\prime}}$ with $m\neq 0$ pairwise transform according to
two-dimensional IRs. More specifically, the transformation properties depend
on the parity of $\ell$. For even $\ell$ (“$g$”), tensor components
$T_{g0}^{ss^{\prime}}$ are even under $\pi$ rotations $C_{2}$ about an axis
perpendicular to the main axis of the dihedral groups (chosen as the $z$ axis
in Table 7), whereas for odd $\ell$ (“$u$”), tensor components
$T_{u0}^{ss^{\prime}}$ are odd under rotations $C_{2}$. The invariant tensor
components $T_{\ell 0}^{ss^{\prime}}$ under the dihedral groups $G_{d\infty}$
thus distinguish between $\ell$ even or odd, see Table 7. But similar to
tensor components $T_{\ell 0}^{ss^{\prime}}$ under the cyclic groups, tensor
components with different even or different odd values of $\ell$ behave the
same.

According to the general definition of group types given in Table 2, the
continuous dihedral groups have the group types listed in the first column of
Table 7. A comparison of Tables 7 and 11 shows that the range of multipole
densities $T_{\ell}^{ss^{\prime}}$ permitted by these continuous dihedral
groups falls short of the range of densities generally permitted by the
categories of polarization associated with these group types. For example, the
group $D_{\infty}\times C_{\theta}$ has the type LABEL:gcat:i-II that belongs
to the electropolar category. But the group $D_{\infty}\times C_{\theta}$ does
not allow odd/̄$\ell$ electric multipoles ($-+$). This result can be traced
back to the fact that the groups in the class $L(D_{\infty})$ can also be
viewed as limiting groups $n\rightarrow\infty$ for the groups in the class
$L(D_{n})$. The lowest degree $\ell_{\min}^{(e,u)}$ of odd/̄$\ell$ electric
multipoles permitted by groups $D_{n}\times C_{\theta}$ grows monotonically
with increasing $n$, with $\ell_{\min}^{(e,u)}\rightarrow\infty$ for
$n\rightarrow\infty$ (see Table 18). This does not affect our classification
of categories of polarized matter as crystals are characterized by finite
crystallographic point groups.

Table 16 gives the multiplication table for the eight one-dimensional IRs of
$D_{\infty}\times C_{i\times\theta}$ with $m=0$. Our notation for these IRs
used in Table 16 indicates that we may use as $1\times 1$ representation
matrices the $m=0$ diagonal elements
$d_{\ell}^{ss^{\prime}}\equiv(D_{\ell}^{ss^{\prime}})_{00}$ of the
representation matrices $D_{\ell}^{ss^{\prime}}$ of $R_{i\times\theta}$ with
$\ell$ even (“g”) and $\ell$ odd (“u”). More specifically, Table 16 reflects
the fact that the Clebsch-Gordan coefficients
$(\ell_{1}0,\ell_{2}0|\ell_{1}\ell_{2},\ell 0)$ in Eq. (16) are nonzero only
if $\ell_{1}+\ell_{2}+\ell$ is even [56]. Note that $d_{g}^{++}\equiv\openone$
becomes the identity representation of $D_{\infty}\times C_{i\times\theta}$.

The dihedral groups $D_{\infty}\langle C_{\infty}\rangle_{\gamma}$ leave
tensor components $T_{\ell 0}^{ss^{\prime}}$ with even and odd $\ell$
invariant. Note that it follows from the multiplication table 16 that, in
these groups, the chiral charges $T_{g0}^{ss^{\prime}}$ arise as compound
tensors that are products of the respective invariant tensor components
$T_{u0}^{ss^{\prime}}$.

Table 16:  Multiplication table for the IRs with $m=0$ of the full axial rotation group $D_{\infty}\times C_{i\times\theta}$. Our notation for these IRs indicates that we may use as $1\times 1$ representation matrices the $m=0$ diagonal elements $d_{\ell}^{ss^{\prime}}\equiv(D_{\ell}^{ss^{\prime}})_{00}$ of the representation matrices $D_{\ell}^{ss^{\prime}}$ of $R_{i\times\theta}$ with $\ell$ even (“$g$”) and $\ell$ odd (“$u$”). Note that $d_{g}^{++}\equiv\openone$ becomes the identity representation of $D_{\infty}\times C_{i\times\theta}$. $\rule{0.0pt}{11.62494pt}d_{g}^{++}$ | $d_{g}^{-+}$ | $d_{g}^{+-}$ | $d_{g}^{--}$ | $d_{u}^{++}$ | $d_{u}^{-+}$ | $d_{u}^{+-}$ | $d_{u}^{--}$ |
---|---|---|---|---|---|---|---|---
$\rule{0.0pt}{11.62494pt}d_{g}^{++}$ | $d_{g}^{-+}$ | $d_{g}^{+-}$ | $d_{g}^{--}$ | $d_{u}^{++}$ | $d_{u}^{-+}$ | $d_{u}^{+-}$ | $d_{u}^{--}$ | $d_{g}^{++}$
| $d_{g}^{++}$ | $d_{g}^{--}$ | $d_{g}^{+-}$ | $d_{u}^{-+}$ | $d_{u}^{++}$ | $d_{u}^{--}$ | $d_{u}^{+-}$ | $d_{g}^{-+}$
|  | $d_{g}^{++}$ | $d_{g}^{-+}$ | $d_{u}^{+-}$ | $d_{u}^{--}$ | $d_{u}^{++}$ | $d_{u}^{-+}$ | $d_{g}^{+-}$
|  |  | $d_{g}^{++}$ | $d_{u}^{--}$ | $d_{u}^{+-}$ | $d_{u}^{-+}$ | $d_{u}^{++}$ | $d_{g}^{--}$
|  |  |  | $d_{g}^{++}$ | $d_{g}^{-+}$ | $d_{g}^{+-}$ | $d_{g}^{--}$ | $d_{u}^{++}$
|  |  |  |  | $d_{g}^{++}$ | $d_{g}^{--}$ | $d_{g}^{+-}$ | $d_{u}^{-+}$
|  |  |  |  |  | $d_{g}^{++}$ | $d_{g}^{-+}$ | $d_{u}^{+-}$
|  |  |  |  |  |  | $d_{g}^{++}$ | $d_{u}^{--}$

The cyclic groups in Table 6 are subgroups of the dihedral groups in Table 7.
The maximal groups of tensor components $T_{\ell 0}^{ss^{\prime}}$ are thus
generally the dihedral groups that provide a more fine-grained classification
of the symmetries of $T_{\ell 0}^{ss^{\prime}}$ based on the parity of $\ell$,
see Table 7.

Interestingly, inspection of Table 7 shows that a tensor component
$T_{u0}^{++}$ is not invariant under any of the continuous dihedral groups. It
is only invariant under continuous cyclic groups $G_{c\infty}$, including
$C_{\infty}\times C_{i\times\theta}$, that do not distinguish between tensor
components $T_{u0}^{++}$ and $T_{g0}^{++}$. We note that all tensor components
$T_{\ell m}^{ss^{\prime}}$ are unambiguously defined via the IRs of the
spherical group $R_{i\times\theta}$, even though discussing these tensor
components in conjunction with smaller groups like $C_{i\times\theta}$ [31,
68] and cyclic or dihedral groups [47] can be instructive.

Axial quantities were previously discussed, e.g., in Refs. [47] and [50]. To
ease cross-referencing between these works and ours, we provide Table 17
relating respective notations for axial quantities. The multiplication table
16 is consistent with Table 1 of Ref. [50], but it is at variance with Table V
of Ref. [47].

Table 17:  Notation for axial quantities in the literature. Ref. [47] uses the symbols from Ref. [68] for tensor components $T_{u0}^{ss^{\prime}}$. This work | $T_{g0}^{++}$ | $T_{g0}^{-+}$ | $T_{g0}^{+-}$ | $T_{g0}^{--}$ | $T_{u0}^{++}$ | $T_{u0}^{-+}$ | $T_{u0}^{+-}$ | $T_{u0}^{--}$
---|---|---|---|---|---|---|---|---
Ref. [47] | ${\bm{\mathrm{N}}}$ | ${\bm{\mathrm{C}}}$ | ${\bm{\mathrm{L}}}$ | ${\bm{\mathrm{F}}}$ | ${\bm{\mathrm{G}}}$ | ${\bm{\mathrm{P}}}$ | ${\bm{\mathrm{M}}}$ | ${\bm{\mathrm{T}}}$
Ref. [50] | $\mathcal{D}$ | $\mathcal{C}$ | $\mathcal{D}^{\prime}$ | $\mathcal{C}^{\prime}$ |  $\displaystyle\mathcal{a}$ | $\mathcal{P}$ | $\scalebox{1.2}{\mbox{$\displaystyle\mathcal{a}$}}^{\prime}$ | $\mathcal{P}^{\prime}$

##  Appendix D Spherical tensors and multipole order permitted by the 122
crystallographic magnetic point groups

This Appendix provides explicit details about which spherical tensors
$T_{\ell}^{ss^{\prime}}$ are permitted by the 122 crystallographic magnetic
point groups. The information given here is directly relevant for discussing
electric, magnetic and toroidal multipole order in crystals, see Sec. IV.2. As
discussed below, the information is furthermore relevant for materials and
property tensors that are commonly represented by Cartesian rank/̄$N$ tensors.

Table 18:  Number of independent components of $\theta$/̄even spherical tensors $T_{\ell}^{s+}$ permitted by type/̄II magnetic point groups (considering ranks $\ell\leq 3$). Tensors $T_{\ell}^{++}$ represent parapolarizations and $T_{\ell}^{-+}$ are electropolarizations. A “0” indicates that the tensor is forbidden by symmetry. The electric monopolarization $T_{0}^{++}$ has been omitted, as it is trivially always allowed. The quantities $\ell_{\mathrm{min}}^{(\mathrm{e/et,g/u})}$ indicate, respectively, for each group, the lowest symmetry-allowed even (“$g$”) and odd (“$u$”) orders $\ell$ of electric (“e”) and electrotoroidal (“et”) multipole densities, again ignoring the electric monopolarization, except for the spherical groups for which it is the only allowed parapolarization. Both the Schönflies and Hermann-Mauguin notations [111] are given for each group, as well as the group’s type according to the classification in Table 2. |  |  |  |  | parapolar | electropolar
---|---|---|---|---|---|---
Crystal system | Schönflies |  Hermann- Mauguin | type | $\ell_{\min}^{(e,g)}$ | $\ell_{\min}^{(et,u)}$ | $T_{1}^{++}$ | $T_{2}^{++}$ |  $T_{3}^{++}$ | $\ell_{\min}^{(et,g)}$ | $\ell_{\min}^{(e,u)}$ | $T_{0}^{-+}$ | $T_{1}^{-+}$ | $T_{2}^{-+}$ | $T_{3}^{-+}$
Triclinic | $C_{1}\times C_{\theta}$ | $=C_{1}\times C_{\theta}$ | $1^{\prime}$ | i-II | $2$ | $1$ | $3$ | $5$ |  $7$ | $0$ | $1$ | $1$ | $3$ | $5$ | $7$
| $C_{1}\times C_{i\times\theta}$ | $=C_{i}\times C_{\theta}$ | $\bar{1}1^{\prime}$ | ii-II | $2$ | $1$ | $3$ | $5$ |  $7$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Monoclinic | $C_{2}\times C_{\theta}$ | $=C_{2}\times C_{\theta}$ | $21^{\prime}$ | i-II | $2$ | $1$ | $1$ | $3$ |  $3$ | $0$ | $1$ | $1$ | $1$ | $3$ | $3$
| $C_{2}[C_{1}]\times C_{\theta}$ | $=C_{s}\times C_{\theta}$ | $m1^{\prime}$ | iii-II | $2$ | $1$ | $1$ | $3$ |  $3$ | $2$ | $1$ | $0$ | $2$ | $2$ | $4$
| $C_{2}\times C_{i\times\theta}$ | $=C_{2h}\times C_{\theta}$ | $2/m1^{\prime}$ | ii-II | $2$ | $1$ | $1$ | $3$ |  $3$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Orthorhombic | $D_{2}\times C_{\theta}$ | $=D_{2}\times C_{\theta}$ | $2221^{\prime}$ | i-II | $2$ | $3$ | $0$ | $2$ |  $1$ | $0$ | $3$ | $1$ | $0$ | $2$ | $1$
| $D_{2}[C_{2}]\times C_{\theta}$ | $=C_{2v}\times C_{\theta}$ | $mm21^{\prime}$ | iii-II | $2$ | $3$ | $0$ | $2$ |  $1$ | $2$ | $1$ | $0$ | $1$ | $1$ | $2$
| $D_{2}\times C_{i\times\theta}$ | $=D_{2h}\times C_{\theta}$ | $mmm1^{\prime}$ | ii-II | $2$ | $3$ | $0$ | $2$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Tetragonal | $C_{4}\times C_{\theta}$ | $=C_{4}\times C_{\theta}$ | $41^{\prime}$ | i-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{4}[C_{2}]\times C_{\theta}$ | $=S_{4}\times C_{\theta}$ | $\bar{4}1^{\prime}$ | iii-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $2$ | $3$ | $0$ | $0$ | $2$ | $2$
| $C_{4}\times C_{i\times\theta}$ | $=C_{4h}\times C_{\theta}$ | $4/m1^{\prime}$ | ii-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{4}\times C_{\theta}$ | $=D_{4}\times C_{\theta}$ | $4221^{\prime}$ | i-II | $2$ | $5$ | $0$ | $1$ |  $0$ | $0$ | $5$ | $1$ | $0$ | $1$ | $0$
| $D_{4}[C_{4}]\times C_{\theta}$ | $=C_{4v}\times C_{\theta}$ | $4mm1^{\prime}$ | iii-II | $2$ | $5$ | $0$ | $1$ |  $0$ | $4$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{4}[D_{2}]\times C_{\theta}$ | $=D_{2d}\times C_{\theta}$ | $\bar{4}2m1^{\prime}$ | iii-II | $2$ | $5$ | $0$ | $1$ |  $0$ | $2$ | $3$ | $0$ | $0$ | $1$ | $1$
| $D_{4}\times C_{i\times\theta}$ | $=D_{4h}\times C_{\theta}$ | $4/mmm1^{\prime}$ | ii-II | $2$ | $5$ | $0$ | $1$ |  $0$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Trigonal | $C_{3}\times C_{\theta}$ | $=C_{3}\times C_{\theta}$ | $31^{\prime}$ | i-II | $2$ | $1$ | $1$ | $1$ |  $3$ | $0$ | $1$ | $1$ | $1$ | $1$ | $3$
| $C_{3}\times C_{i\times\theta}$ | $=C_{3i}\times C_{\theta}$ | $\bar{3}1^{\prime}$ | ii-II | $2$ | $1$ | $1$ | $1$ |  $3$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{3}\times C_{\theta}$ | $=D_{3}\times C_{\theta}$ | $321^{\prime}$ | i-II | $2$ | $3$ | $0$ | $1$ |  $1$ | $0$ | $3$ | $1$ | $0$ | $1$ | $1$
| $D_{3}[C_{3}]\times C_{\theta}$ | $=C_{3v}\times C_{\theta}$ | $3m1^{\prime}$ | iii-II | $2$ | $3$ | $0$ | $1$ |  $1$ | $4$ | $1$ | $0$ | $1$ | $0$ | $2$
| $D_{3}\times C_{i\times\theta}$ | $=D_{3d}\times C_{\theta}$ | $\bar{3}m1^{\prime}$ | ii-II | $2$ | $3$ | $0$ | $1$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Hexagonal | $C_{6}\times C_{\theta}$ | $=C_{6}\times C_{\theta}$ | $61^{\prime}$ | i-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{6}[C_{3}]\times C_{\theta}$ | $=C_{3h}\times C_{\theta}$ | $\bar{6}1^{\prime}$ | iii-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $4$ | $3$ | $0$ | $0$ | $0$ | $2$
| $C_{6}\times C_{i\times\theta}$ | $=C_{6h}\times C_{\theta}$ | $6/m1^{\prime}$ | ii-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{6}\times C_{\theta}$ | $=D_{6}\times C_{\theta}$ | $6221^{\prime}$ | i-II | $2$ | $7$ | $0$ | $1$ |  $0$ | $0$ | $7$ | $1$ | $0$ | $1$ | $0$
| $D_{6}[C_{6}]\times C_{\theta}$ | $=C_{6v}\times C_{\theta}$ | $6mm1^{\prime}$ | iii-II | $2$ | $7$ | $0$ | $1$ |  $0$ | $6$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{6}[D_{3}]\times C_{\theta}$ | $=D_{3h}\times C_{\theta}$ | $\bar{6}m21^{\prime}$ | iii-II | $2$ | $7$ | $0$ | $1$ |  $0$ | $4$ | $3$ | $0$ | $0$ | $0$ | $1$
| $D_{6}\times C_{i\times\theta}$ | $=D_{6h}\times C_{\theta}$ | $6/mmm1^{\prime}$ | ii-II | $2$ | $7$ | $0$ | $1$ |  $0$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Cubic | $T\times C_{\theta}$ | $=T\times C_{\theta}$ | $231^{\prime}$ | i-II | $4$ | $3$ | $0$ | $0$ |  $1$ | $0$ | $3$ | $1$ | $0$ | $0$ | $1$
| $T\times C_{i\times\theta}$ | $=T_{h}\times C_{\theta}$ | $m31^{\prime}$ | ii-II | $4$ | $3$ | $0$ | $0$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $O\times C_{\theta}$ | $=O\times C_{\theta}$ | $4321^{\prime}$ | i-II | $4$ | $9$ | $0$ | $0$ |  $0$ | $0$ | $9$ | $1$ | $0$ | $0$ | $0$
| $O[T]\times C_{\theta}$ | $=T_{d}\times C_{\theta}$ | $\bar{4}3m1^{\prime}$ | iii-II | $4$ | $9$ | $0$ | $0$ |  $0$ | $6$ | $3$ | $0$ | $0$ | $0$ | $1$
| $O\times C_{i\times\theta}$ | $=O_{h}\times C_{\theta}$ | $m3m1^{\prime}$ | ii-II | $4$ | $9$ | $0$ | $0$ |  $0$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Number of crystallographic groups allowing tensor  | $13$ | $27$ |  $21$ |  |  | $11$ | $10$ | $13$ | $18$
Axial | $C_{\infty}\times C_{\theta}$ | $=C_{\infty}\times C_{\theta}$ | $\infty 1^{\prime}$ | i-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{\infty}\times C_{i\times\theta}$ | $=C_{\infty h}\times C_{\theta}$ | $\infty/m1^{\prime}$ | ii-II | $2$ | $1$ | $1$ | $1$ |  $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{\infty}\times C_{\theta}$ | $=D_{\infty}\times C_{\theta}$ | $\infty 21^{\prime}$ | i-II | $2$ | $\infty$ | $0$ | $1$ |  $0$ | $0$ | $\infty$ | $1$ | $0$ | $1$ | $0$
| $D_{\infty}[C_{\infty}]\times C_{\theta}$ | $=C_{\infty v}\times C_{\theta}$ | $\infty m1^{\prime}$ | iii-II | $2$ | $\infty$ | $0$ | $1$ |  $0$ | $\infty$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{\infty}\times C_{i\times\theta}$ | $=D_{\infty h}\times C_{\theta}$ | $\infty/mm1^{\prime}$ | ii-II | $2$ | $\infty$ | $0$ | $1$ |  $0$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
Spherical | $R\times C_{\theta}$ | $=R_{\theta}$ | $\infty\infty 1^{\prime}$ | i-II | $0$ | $\infty$ | $0$ | $0$ |  $0$ | $0$ | $\infty$ | $1$ | $0$ | $0$ | $0$
| $R\times C_{i\times\theta}$ | $=R_{i\times\theta}$ | $\infty\infty m1^{\prime}$ | ii-II | $0$ | $\infty$ | $0$ | $0$ |  $0$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$

Table 18 is devoted to $\theta$/̄even tensors $T_{\ell}^{s+}$. It lists the
number of independent components such tensors with rank $\ell\leq 3$ have
under the nonmagnetic (type/̄II) groups belonging to the 11 Laue classes
$L(G_{\mathrm{p}})$. For comparison, the same information is also provided for
type/̄II groups in the three classes of continuous cyclic, dihedral and
spherical point groups. To streamline the presentation, the universally
allowed electric monopolarization $T_{0}^{++}$ has been omitted. Basic
physical properties can be read off from this table. For example, groups
allowing $T_{0}^{-+}$ represent electrochiral systems, see Sec. V.1.2, and
groups allowing $T_{1}^{-+}$ represent pyroelectric and ferroelectric systems
[5].

Table 18 together with Table 9 can be used to infer the properties of tensors
$T_{\ell}^{s+}$ in all 122 magnetic point groups. Specifically, Table 18 gives
the properties of tensors for the crystallographic point groups
$G_{0}^{\theta}\times C_{\theta}$ that appear in Table 9 in the last row of
the main block [52] for a class $L(G_{\mathrm{p}})$. The structure of tensors
$T_{\ell}^{s+}$ allowed under $G_{0}^{\theta}\times C_{\theta}$ is repeated
for all members of the subclass $L(G_{0}^{\theta},\theta)\subset
L(G_{\mathrm{p}})$, which are the groups positioned in the column headed by
$G_{0}^{\theta}$ within the same class’s main block. The formal basis for
these congruences is elaborated in detail in Appendix E.

Table 18 also tabulates the lowest even (“$g$”) and odd (“$u$”) orders $\ell$
of electric (“e”) and electrotoroidal (“et”) multipole densities permitted in
each crystal class. The electric monopolarization $T_{0}^{++}$ is trivially
allowed under all point groups and, therefore, is not considered for
determining $\ell_{\mathrm{min}}^{(\mathrm{e,g})}$ (except for spherical
groups where it is the only allowed parapolarization). Instead, to convey
information that is specific to each group, we tabulate the lowest _nonzero_
even order of parapolarizations allowed under each group.

Table 19:  Number of independent components of $\theta$/̄odd spherical tensors $T_{\ell}^{s-}$ permitted by type/̄III magnetic point groups (considering ranks $\ell\leq 3$). Tensors $T_{\ell}^{+-}$ represent magnetopolarizations and $T_{\ell}^{--}$ are antimagnetopolarizations. A “0” indicates that the corresponding tensor is forbidden by symmetry. The magnetic monopolarization $T_{0}^{+-}$ has been omitted, as it is universally forbidden under type/̄III groups. The quantities $\ell_{\mathrm{min}}^{(\mathrm{m/mt,g/u})}$ indicate, respectively, for each group, the lowest symmetry-allowed even (“$g$”) and odd (“$u$”) orders $\ell$ of magnetic (“m”) and magnetotoroidal (“mt”) multipole densities. Both the Schönflies and Hermann-Mauguin notations [111] are given for each group, as well as the group’s type according to the classification in Table 2. |  |  |  |  | magnetopolar | antimagnetopolar
---|---|---|---|---|---|---
Crystal system | Schönflies |  Hermann- Mauguin | type | $\ell_{\min}^{(mt,g)}$ | $\ell_{\min}^{(m,u)}$ | $T_{1}^{+-}$ | $T_{2}^{+-}$ | $T_{3}^{+-}$ | $\ell_{\min}^{(m,g)}$ | $\ell_{\min}^{(mt,u)}$ | $T_{0}^{--}$ | $T_{1}^{--}$ | $T_{2}^{--}$ | $T_{3}^{--}$
Triclinic | $C_{1}\times C_{i\theta}$ | $=C_{i}(C_{1})$ | $\bar{1}^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $3$ | $5$ | $7$
Monoclinic | $C_{2}(C_{1})$ | $=C_{2}(C_{1})$ | $2^{\prime}$ | i-III | $2$ | $1$ | $2$ | $2$ | $4$ | $2$ | $1$ | $0$ | $2$ | $2$ | $4$
| $C_{2}\\{C_{1}\\}$ | $=C_{s}(C_{1})$ | $m^{\prime}$ | JJJ | $2$ | $1$ | $2$ | $2$ | $4$ | $0$ | $1$ | $1$ | $1$ | $3$ | $3$
| $C_{2}(C_{1})\times C_{i}$ | $=C_{2h}(C_{i})$ | $2^{\prime}/m^{\prime}$ | ii-III | $2$ | $1$ | $2$ | $2$ | $4$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $C_{2}\times C_{i\theta}$ | $=C_{2h}(C_{2})$ | $2/m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $3$ | $3$
| $C_{2}(C_{1})\times C_{i\theta}$ | $=C_{2h}(C_{s})$ | $2^{\prime}/m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $2$ | $1$ | $0$ | $2$ | $2$ | $4$
Orthorhombic | $D_{2}(C_{2})$ | $=D_{2}(C_{2})$ | $2^{\prime}2^{\prime}2$ | i-III | $2$ | $1$ | $1$ | $1$ | $2$ | $2$ | $1$ | $0$ | $1$ | $1$ | $2$
| $D_{2}\\{C_{2}\\}$ | $=C_{2v}(C_{2})$ | $m^{\prime}m^{\prime}2$ | JJJ | $2$ | $1$ | $1$ | $1$ | $2$ | $0$ | $3$ | $1$ | $0$ | $2$ | $1$
| $D_{2}[C_{2}](C_{2}[C_{1}])$ | $=C_{2v}(C_{s})$ | $2^{\prime}m^{\prime}m$ | JJJ” | $2$ | $1$ | $1$ | $1$ | $2$ | $2$ | $1$ | $0$ | $1$ | $1$ | $2$
| $D_{2}(C_{2})\times C_{i}$ | $=D_{2h}(C_{2h})$ | $mm^{\prime}m^{\prime}$ | ii-III | $2$ | $1$ | $1$ | $1$ | $2$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{2}\times C_{i\theta}$ | $=D_{2h}(D_{2})$ | $m^{\prime}m^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $3$ | $1$ | $0$ | $2$ | $1$
| $D_{2}(C_{2})\times C_{i\theta}$ | $=D_{2h}(C_{2v})$ | $mmm^{\prime}$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $2$ | $1$ | $0$ | $1$ | $1$ | $2$
Tetragonal | $C_{4}(C_{2})$ | $=C_{4}(C_{2})$ | $4^{\prime}$ | i-III | $2$ | $3$ | $0$ | $2$ | $2$ | $2$ | $3$ | $0$ | $0$ | $2$ | $2$
| $C_{4}\\{C_{2}\\}$ | $=S_{4}(C_{2})$ | $\bar{4}^{\prime}$ | JJJ | $2$ | $3$ | $0$ | $2$ | $2$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{4}(C_{2})\times C_{i}$ | $=C_{4h}(C_{2h})$ | $4^{\prime}/m$ | ii-III | $2$ | $3$ | $0$ | $2$ | $2$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $C_{4}\times C_{i\theta}$ | $=C_{4h}(C_{4})$ | $4/m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{4}(C_{2})\times C_{i\theta}$ | $=C_{4h}(S_{4})$ | $4^{\prime}/m^{\prime}$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $2$ | $3$ | $0$ | $0$ | $2$ | $2$
| $D_{4}(D_{2})$ | $=D_{4}(D_{2})$ | $4^{\prime}22^{\prime}$ | i-III | $2$ | $3$ | $0$ | $1$ | $1$ | $2$ | $3$ | $0$ | $0$ | $1$ | $1$
| $D_{4}(C_{4})$ | $=D_{4}(C_{4})$ | $42^{\prime}2^{\prime}$ | i-III | $4$ | $1$ | $1$ | $0$ | $1$ | $4$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{4}[C_{4}](D_{2}[C_{2}])$ | $=C_{4v}(C_{2v})$ | $4^{\prime}mm^{\prime}$ | iii’-III | $2$ | $3$ | $0$ | $1$ | $1$ | $2$ | $3$ | $0$ | $0$ | $1$ | $1$
| $D_{4}\\{C_{4}\\}$ | $=C_{4v}(C_{4})$ | $4m^{\prime}m^{\prime}$ | JJJ | $4$ | $1$ | $1$ | $0$ | $1$ | $0$ | $5$ | $1$ | $0$ | $1$ | $0$
| $D_{4}\\{D_{2}\\}$ | $=D_{2d}(D_{2})$ | $\bar{4}^{\prime}2m^{\prime}$ | JJJ | $2$ | $3$ | $0$ | $1$ | $1$ | $0$ | $5$ | $1$ | $0$ | $1$ | $0$
| $D_{4}[D_{2}](D_{2}[C_{2}])$ | $=D_{2d}(C_{2v})$ | $\bar{4}^{\prime}m2^{\prime}$ | JJJ’ | $2$ | $3$ | $0$ | $1$ | $1$ | $4$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{4}[D_{2}](C_{4}[C_{2}])$ | $=D_{2d}(S_{4})$ | $\bar{4}2^{\prime}m^{\prime}$ | iii-III’ | $4$ | $1$ | $1$ | $0$ | $1$ | $2$ | $3$ | $0$ | $0$ | $1$ | $1$
| $D_{4}(D_{2})\times C_{i}$ | $=D_{4h}(D_{2h})$ | $4^{\prime}/mmm^{\prime}$ | ii-III | $2$ | $3$ | $0$ | $1$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{4}(C_{4})\times C_{i}$ | $=D_{4h}(C_{4h})$ | $4/mm^{\prime}m^{\prime}$ | ii-III | $4$ | $1$ | $1$ | $0$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{4}\times C_{i\theta}$ | $=D_{4h}(D_{4})$ | $4/m^{\prime}m^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $5$ | $1$ | $0$ | $1$ | $0$
| $D_{4}(C_{4})\times C_{i\theta}$ | $=D_{4h}(C_{4v})$ | $4/m^{\prime}mm$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $4$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{4}(D_{2})\times C_{i\theta}$ | $=D_{4h}(D_{2d})$ | $4^{\prime}/m^{\prime}m^{\prime}m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $2$ | $3$ | $0$ | $0$ | $1$ | $1$
Trigonal | $C_{3}\times C_{i\theta}$ | $=C_{3i}(C_{3})$ | $\bar{3}^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $3$
| $D_{3}(C_{3})$ | $=D_{3}(C_{3})$ | $32^{\prime}$ | i-III | $4$ | $1$ | $1$ | $0$ | $2$ | $4$ | $1$ | $0$ | $1$ | $0$ | $2$
| $D_{3}\\{C_{3}\\}$ | $=C_{3v}(C_{3})$ | $3m^{\prime}$ | JJJ | $4$ | $1$ | $1$ | $0$ | $2$ | $0$ | $3$ | $1$ | $0$ | $1$ | $1$
| $D_{3}(C_{3})\times C_{i}$ | $=D_{3d}(C_{3i})$ | $\bar{3}m^{\prime}$ | ii-III | $4$ | $1$ | $1$ | $0$ | $2$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{3}\times C_{i\theta}$ | $=D_{3d}(D_{3})$ | $\bar{3}^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $3$ | $1$ | $0$ | $1$ | $1$
| $D_{3}(C_{3})\times C_{i\theta}$ | $=D_{3d}(C_{3v})$ | $\bar{3}^{\prime}m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $4$ | $1$ | $0$ | $1$ | $0$ | $2$
Hexagonal | $C_{6}(C_{3})$ | $=C_{6}(C_{3})$ | $6^{\prime}$ | i-III | $4$ | $3$ | $0$ | $0$ | $2$ | $4$ | $3$ | $0$ | $0$ | $0$ | $2$
| $C_{6}\\{C_{3}\\}$ | $=C_{3h}(C_{3})$ | $\bar{6}^{\prime}$ | JJJ | $4$ | $3$ | $0$ | $0$ | $2$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{6}(C_{3})\times C_{i}$ | $=C_{6h}(C_{3i})$ | $6^{\prime}/m^{\prime}$ | ii-III | $4$ | $3$ | $0$ | $0$ | $2$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $C_{6}\times C_{i\theta}$ | $=C_{6h}(C_{6})$ | $6/m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $C_{6}(C_{3})\times C_{i\theta}$ | $=C_{6h}(C_{3h})$ | $6^{\prime}/m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $4$ | $3$ | $0$ | $0$ | $0$ | $2$
| $D_{6}(D_{3})$ | $=D_{6}(D_{3})$ | $6^{\prime}22^{\prime}$ | i-III | $4$ | $3$ | $0$ | $0$ | $1$ | $4$ | $3$ | $0$ | $0$ | $0$ | $1$
| $D_{6}(C_{6})$ | $=D_{6}(C_{6})$ | $62^{\prime}2^{\prime}$ | i-III | $6$ | $1$ | $1$ | $0$ | $1$ | $6$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{6}[C_{6}](D_{3}[C_{3}])$ | $=C_{6v}(C_{3v})$ | $6^{\prime}mm^{\prime}$ | iii’-III | $4$ | $3$ | $0$ | $0$ | $1$ | $4$ | $3$ | $0$ | $0$ | $0$ | $1$
| $D_{6}\\{C_{6}\\}$ | $=C_{6v}(C_{6})$ | $6m^{\prime}m^{\prime}$ | JJJ | $6$ | $1$ | $1$ | $0$ | $1$ | $0$ | $7$ | $1$ | $0$ | $1$ | $0$
| $D_{6}\\{D_{3}\\}$ | $=D_{3h}(D_{3})$ | $\bar{6}^{\prime}2m^{\prime}$ | JJJ | $4$ | $3$ | $0$ | $0$ | $1$ | $0$ | $7$ | $1$ | $0$ | $1$ | $0$
| $D_{6}[D_{3}](D_{3}[C_{3}])$ | $=D_{3h}(C_{3v})$ | $\bar{6}^{\prime}m2^{\prime}$ | JJJ’ | $4$ | $3$ | $0$ | $0$ | $1$ | $6$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{6}[D_{3}](C_{6}[C_{3}])$ | $=D_{3h}(C_{3h})$ | $\bar{6}m^{\prime}2^{\prime}$ | iii-III’ | $6$ | $1$ | $1$ | $0$ | $1$ | $4$ | $3$ | $0$ | $0$ | $0$ | $1$
| $D_{6}(D_{3})\times C_{i}$ | $=D_{6h}(D_{3d})$ | $6^{\prime}/m^{\prime}mm^{\prime}$ | ii-III | $4$ | $3$ | $0$ | $0$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{6}(C_{6})\times C_{i}$ | $=D_{6h}(C_{6h})$ | $6/mm^{\prime}m^{\prime}$ | ii-III | $6$ | $1$ | $1$ | $0$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{6}\times C_{i\theta}$ | $=D_{6h}(D_{6})$ | $6/m^{\prime}m^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $7$ | $1$ | $0$ | $1$ | $0$
| $D_{6}(C_{6})\times C_{i\theta}$ | $=D_{6h}(C_{6v})$ | $6/m^{\prime}mm$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $6$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{6}(D_{3})\times C_{i\theta}$ | $=D_{6h}(D_{3h})$ | $6^{\prime}/mmm^{\prime}$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $4$ | $3$ | $0$ | $0$ | $0$ | $1$
Table 19:  (_Continued._) |  |  |  |  | magnetopolar | antimagnetopolar
---|---|---|---|---|---|---
Crystal system | Schönflies |  Hermann- Mauguin | type | $\ell_{\min}^{(mt,g)}$ | $\ell_{\min}^{(m,u)}$ | $T_{1}^{+-}$ | $T_{2}^{+-}$ | $T_{3}^{+-}$ | $\ell_{\min}^{(m,g)}$ | $\ell_{\min}^{(mt,u)}$ | $T_{0}^{--}$ | $T_{1}^{--}$ | $T_{2}^{--}$ | $T_{3}^{--}$
Cubic | $T\times C_{i\theta}$ | $=T_{h}(T)$ | $m^{\prime}3^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $3$ | $1$ | $0$ | $0$ | $1$
| $O(T)$ | $=O(T)$ | $4^{\prime}32^{\prime}$ | i-III | $6$ | $3$ | $0$ | $0$ | $1$ | $6$ | $3$ | $0$ | $0$ | $0$ | $1$
| $O\\{T\\}$ | $=T_{d}(T)$ | $\bar{4}^{\prime}3m^{\prime}$ | JJJ | $6$ | $3$ | $0$ | $0$ | $1$ | $0$ | $9$ | $1$ | $0$ | $0$ | $0$
| $O(T)\times C_{i}$ | $=O_{h}(T_{h})$ | $m3m^{\prime}$ | ii-III | $6$ | $3$ | $0$ | $0$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $O\times C_{i\theta}$ | $=O_{h}(O)$ | $m^{\prime}3^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $9$ | $1$ | $0$ | $0$ | $0$
| $O(T)\times C_{i\theta}$ | $=O_{h}(T_{d})$ | $m^{\prime}3^{\prime}m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $6$ | $3$ | $0$ | $0$ | $0$ | $1$
Number of crystallographic groups allowing tensor  | $18$ | $15$ | $37$ |  |  | $21$ | $21$ | $29$ | $40$
Axial | $C_{\infty}\times C_{i\theta}$ | $=C_{\infty h}(C_{\infty})$ | $\infty/m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
| $D_{\infty}(C_{\infty})$ | $=D_{\infty}(C_{\infty})$ | $\infty 2^{\prime}$ | i-III | $\infty$ | $1$ | $1$ | $0$ | $1$ | $\infty$ | $1$ | $0$ | $1$ | $0$ | $1$
| $D_{\infty}\\{C_{\infty}\\}$ | $=C_{\infty v}(C_{\infty})$ | $\infty m^{\prime}$ | JJJ | $\infty$ | $1$ | $1$ | $0$ | $1$ | $0$ | $\infty$ | $1$ | $0$ | $1$ | $0$
| $D_{\infty}(C_{\infty})\times C_{i}$ | $=D_{\infty h}(C_{\infty h})$ | $\infty/mm^{\prime}$ | ii-III | $\infty$ | $1$ | $1$ | $0$ | $1$ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$
| $D_{\infty}\times C_{i\theta}$ | $=D_{\infty h}(D_{\infty})$ | $\infty/m^{\prime}m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $\infty$ | $1$ | $0$ | $1$ | $0$
| $D_{\infty}(C_{\infty})\times C_{i\theta}$ | $=D_{\infty h}(C_{\infty v})$ | $\infty/m^{\prime}m$ | JJ’ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $\infty$ | $1$ | $0$ | $1$ | $0$ | $1$
Spherical | $R\times C_{i\theta}$ | $=R_{\theta}$ | $\infty\infty m^{\prime}$ | JJ | $\infty$ | $\infty$ | $0$ | $0$ | $0$ | $0$ | $\infty$ | $1$ | $0$ | $0$ | $0$

Table 19 pertains to $\theta$/̄odd tensors $T_{\ell}^{s-}$. It lists the
number of independent components for such tensors with rank $\ell\leq 3$ under
the type/̄III magnetic point groups (58 crystallographic, 7 continuous).
According to the classification introduced in Sec. II.3, these comprise the
groups of type LABEL:gcat:i-III, LABEL:gcat:ii-III, LABEL:gcat:iiip-III,
LABEL:gcat:iii-IIIp, LABEL:gcat:JJJ, LABEL:gcat:JJJp, LABEL:gcat:JJJpp,
LABEL:gcat:JJ and LABEL:gcat:JJp. In the other types, $\theta$/̄odd tensors
are either equivalent to the $\theta$/̄even ones listed in Table 18 (types
LABEL:gcat:J, LABEL:gcat:ii-I and LABEL:gcat:iii-I), or they are forbidden by
symmetry (types LABEL:gcat:i-II, LABEL:gcat:ii-II and LABEL:gcat:iii-II). More
generally, the entries in Table 19 are related to those given in Table 18 as
per the relationships illustrated by small $2\times 2$ matrices in Tables 5,
8, 9, and further elucidated in Appendix E. As the magnetotoroidal
monopolarization $T_{0}^{+-}$ is forbidden by symmetry under all type/̄III
groups, it has been omitted to simplify the presentation. Table 19 also lists
the lowest even- and odd/̄$\ell$ magnetic and magnetotoroidal multipole
densities permitted under a given type/̄III magnetic point group.

A variety of magnetism-related physical properties can be inferred from Table
19. For example, ferromagnetism occurs for groups allowing the tensor
$T_{1}^{+-}$ [15, 16]. In groups where $T_{1}^{+-}$ is forbidden, the presence
of $T_{3}^{+-}$ signals altermagnetism [71]. Axion electrodynamics arises from
a finite $T_{0}^{--}$ [112].

Tables 18 and 19 list the number of independent components of spherical
tensors $T_{\ell}^{ss^{\prime}}$. These tables can likewise be used to infer
the number of independent components of Cartesian rank/̄$N$ tensors
$\mathcal{T}_{N}^{ss^{\prime}}$ representing materials properties [5, 8, 31,
32, 34, 36, 35, 33]. Cartesian rank/̄0 and rank/̄1 tensors are equivalent to
spherical rank/̄0 and rank/̄1 tensors, respectively. Using Eqs. (15) and (16),
a Cartesian rank/̄2 tensor $\mathcal{T}_{2}^{ss^{\prime}}$ can be decomposed
into irreducible spherical tensors

$\mathcal{T}_{2}^{ss^{\prime}}=T_{0}^{ss^{\prime}}+T_{1}^{ss^{\prime}}+T_{2}^{ss^{\prime}}\;,$
(48)

where $T_{0}^{ss^{\prime}}$ represents the trace of
$\mathcal{T}_{2}^{ss^{\prime}}$, $T_{1}^{ss^{\prime}}$ represents the
antisymmetric part of $\mathcal{T}_{2}^{ss^{\prime}}$, and
$T_{2}^{ss^{\prime}}$ represents the symmetric, traceless part of
$\mathcal{T}_{2}^{ss^{\prime}}$. Similarly, a Cartesian rank/̄3 tensor
$\mathcal{T}_{3}^{ss^{\prime}}$ can be decomposed into irreducible spherical
tensors

$\mathcal{T}_{3}^{ss^{\prime}}=T_{0}^{ss^{\prime}}+(T_{1}^{ss^{\prime}}+T_{3}^{ss^{\prime}})+2(T_{1}^{ss^{\prime}}+T_{2}^{ss^{\prime}})\;,$
(49)

where $T_{0}^{ss^{\prime}}$ represents the totally antisymmetric part of
$\mathcal{T}_{3}^{ss^{\prime}}$, $T_{1}^{ss^{\prime}}+T_{3}^{ss^{\prime}}$
represents the totally symmetric part of $\mathcal{T}_{3}^{ss^{\prime}}$, and
$2(T_{1}^{ss^{\prime}}+T_{2}^{ss^{\prime}})$ represents terms with mixed
permutation symmetry [113]. Current-induced spin polarization in nonmagnetic
crystals [89], also known as Edelstein effect [114], is represented by a
Cartesian rank/̄2 tensor $\mathcal{T}_{2}^{-+}$. This effect is allowed to
exist for any of the 18 groups permitting (at least) one of the tensors
$T_{0}^{-+}$, $T_{1}^{-+}$, or $T_{2}^{-+}$, compare Table 18. Similarly, one
can read off from Table 19 the 58 magnetic groups permitting
magnetoelectricity described by a tensor $\mathcal{T}_{2}^{--}$ [45]. Note
that neither the Edelstein effect nor magnetoelectricity impose constraints
for the respective material tensors $\mathcal{T}_{2}^{ss^{\prime}}$ under
permutation of the indices representing the Cartesian components of these
tensors.

##  Appendix E Rationalizing symmetry-allowed tensor components for magnetic
point groups

This Appendix discusses commonalities regarding the shape of tensors
describing systems belonging to the same Laue class $L(G_{\mathrm{p}})$.
Combining rotations in a proper point group $G_{\mathrm{p}}$ with space
inversion $i$ and time inversion $\theta$ yields the magnetic point groups of
the class $L(G_{\mathrm{p}})$. The intricate relation between symmetry
transformations for these groups manifests in commonalities regarding the form
of symmetry-allowed tensors. Similarities in the form of crystal-property
matrices are apparent from their tabulations (see, e.g., Refs. [5, 34, 100]
and Appendix D above), and they also underpin the observation of Opechowski’s
magic relations [33].

We visualize the commonalities between tensor configurations for groups within
a class via small $2\times 2$ matrices such as
$(\raisebox{0.86108pt}[0.0pt][0.0pt]{$\begin{array}[]{@{}>{\scriptstyle}cs{0.08em}>{\scriptstyle}c@{}}\mcirc\hfil\hskip
0.56001pt&\mfilledstar\scriptstyle\hfil\\\\[-6.88889pt] \mdiamond\hfil\hskip
0.56001pt&\mdiamond\end{array}$})$ shown in Tables 2, 3, 5, 8 and 9. Symbols
in these matrices mark congruences between tensors with signature $++$ (upper
left), $+-$ (upper right), $-+$ (lower left) and $--$ (lower right) for a
given group and between different groups within a given class. Filled symbols
indicate that a rank/̄1 tensor (i.e., a vector) is allowed (compare Tables 19
and 19). A missing symbol indicates that tensors with the respective signature
are forbidden for any $\ell$. For example, the matrix
$(\raisebox{0.86108pt}[0.0pt][0.0pt]{$\begin{array}[]{@{}>{\scriptstyle}cs{0.08em}>{\scriptstyle}c@{}}\mcirc\hfil\hskip
0.56001pt&\mfilledstar\scriptstyle\hfil\\\\[-6.88889pt] \mdiamond\hfil\hskip
0.56001pt&\mdiamond\end{array}$})$ listed in Table 9 for the group
$D_{2d}(S_{4})=D_{4}[D_{2}]\,(C_{4}[C_{2}])$ [class $L(D_{4})$] indicates that
this group allows tensors $T_{\ell}^{ss^{\prime}}$ for all four signatures
$ss^{\prime}$. In particular, this group allows rank/̄1 tensors with signature
$+-$, and rank/̄$\ell$ tensors with signature $-+$ have the same structure as
rank/̄$\ell$ tensors with signature $--$ (for $\ell\geq\ell_{\mathrm{min}}$ as
listed in Tables 19 and 19). Finally, repeated appearances of the symbol
$\mcirc$ in the matrices listed for other groups in the class $L(D_{4})$
indicate that the respective tensors $T_{\ell}^{ss^{\prime}}$ allowed for
these groups have the same structure as the tensors $T_{\ell}^{++}$ allowed
for the group $D_{2d}(S_{4})$ [and similarly for the other symbols in
$(\raisebox{0.86108pt}[0.0pt][0.0pt]{$\begin{array}[]{@{}>{\scriptstyle}cs{0.08em}>{\scriptstyle}c@{}}\mcirc\hfil\hskip
0.56001pt&\mfilledstar\scriptstyle\hfil\\\\[-6.88889pt] \mdiamond\hfil\hskip
0.56001pt&\mdiamond\end{array}$})$]. These congruences of tensor
configurations for groups within a given class $L(G_{\mathrm{p}})$ imply that
knowing the structure of rank/̄$\ell$ tensors for groups of type LABEL:gcat:J
and LABEL:gcat:iii-I is sufficient to infer the structure of rank/̄$\ell$
tensor for all groups in $L(G_{\mathrm{p}})$.

In the following, we use the concepts of duality (7) and triadic relationships
(10) between magnetic point groups to formally derive, for each signature
$ss^{\prime}$, the congruences of tensors $T_{\ell}^{ss^{\prime}}$ among the
groups in a given class $L(G_{\mathrm{p}})$. In particular, we show how it is
possible to relate the structure of tensors $T_{\ell}^{s-}$ allowed for
magnetic groups in $L(G_{\mathrm{p}})$ to the structure of the tensors
$T_{\ell}^{s+}$ allowed for nonmagnetic groups in $L(G_{\mathrm{p}})$. While
our discussion focuses on spherical tensors $T_{\ell}^{ss^{\prime}}$ [56, 57]
that are relevant, e.g., for describing multipole order in solids (see
discussion in Sec. IV), the congruences apply also to Cartesian tensors
representing materials properties [5, 8, 31, 32, 34, 36, 35, 33], see Appendix
D.

###  E.1 Tensors $\boldsymbol{T_{\ell}^{++}}$: Parapolarizations

Parapolarizations, having signature $++$, are agnostic w.r.t. both SIS and
TIS, hence their structure is determined by symmetry under proper rotations
only. Tensors $T_{\ell}^{++}$ thus have the same structure for all groups in a
given class $L(G_{\mathrm{p}})$, and this structure depends only on the
symmetries present in the class root $G_{\mathrm{p}}$.

###  E.2 Tensors $\boldsymbol{T_{\ell}^{-+}}$: Electropolarizations

Tensors that are odd under SIS and even under TIS are unaffected by whether
TIS may be broken. Tensors $T_{\ell}^{-+}$ thus have the same structure for
all groups in a subclass $L(G_{0}^{\theta},\theta)\subset L(G_{\mathrm{p}})$ ,
coinciding with the structure of $T_{\ell}^{-+}$ for the subclass root
$G_{0}^{\theta}$. For the groups in the subclass $L(G_{\mathrm{p}},\theta)$,
the structure of tensors $T_{\ell}^{-+}$ is the same as the structure of
tensors $T_{\ell}^{++}$, because these tensors have the same structure for the
subclass root $G_{\mathrm{p}}$.

In contrast, the structure of tensors $T_{\ell}^{-+}$ for the groups in
$L(G_{\mathrm{p}}[\tilde{G}_{\mathrm{p}}],\theta)$ is, in general, different
from the structure of the tensors $T_{\ell}^{++}$. No tensors of the form
$T_{\ell}^{-+}$ are allowed for any groups in a subclass
$L(G_{\mathrm{p}}\times C_{i},\theta)$. Table 3 illustrates the congruence of
tensors $T_{\ell}^{-+}$ between groups in a given subclass
$L(G_{0}^{\theta},\theta)$ (corresponding to columns in the upper part of the
table).

###  E.3 Tensors $\boldsymbol{T_{\ell}^{+-}}$: Magnetopolarizations

Tensors with signature $+-$ are unaffected by whether SIS may be broken. For
all groups in a subclass $L(G_{0}^{i},i)\subset L(G_{\mathrm{p}})$
(corresponding to rows in the upper part of Table 3) tensors $T_{\ell}^{+-}$
thus have the same structure, coinciding with the structure of $T_{\ell}^{+-}$
for the subclass root $G_{0}^{i}$. In particular, for the groups in
$L(G_{\mathrm{p}},i)$ tensors $T_{\ell}^{+-}$ have the same structure as
tensors $T_{\ell}^{++}$, while tensors of the form $T_{\ell}^{+-}$ are
forbidden by symmetry for all groups in $L(G_{\mathrm{p}}\times