# Toroidal orders and related phenomena
in nonmagnetic and magnetic materials

Hiroaki Kusunose1,2 & Satoru Hayami3 1Department of Physics, Meiji University,
Kawasaki 214-8571, Japan  2Quantum Research Center for Chirality, Institute
for Molecular Science, Okazaki 444-8585, Japan  3Department of Applied
Physics, the University of Tokyo, Tokyo 113-8656, Japan<EMAIL_ADDRESS>
###### Abstract

In this short article, we overview a concept of electronic toroidal
multipoles, and their ordering with associated physical properties in non-
magnetic and magnetic materials. The toroidal multipoles are introduced as
microscopic electronic variables in view of symmetry and connection to Dirac
theory. They are classified according to crystallographic and magnetic point
groups, which allows us to discuss various possible cross correlations in a
transparent and unified manner. The representative examples of toroidal orders
and related phenomena, and the mutual relationship between these orders are
given, with focusing on monopoles and dipoles. The concept of toroidal
multipoles would promote future studies toward observations and
identifications of unknown electronic phases and their related physical
phenomena.

##  1 Introduction

Symmetry of materials and their physical properties are closely related.
Representative examples that spontaneously break fundamental symmetries, i.e.,
the time-reversal and spatial inversion symmetries, are ferromagnetism and
ferroelectricity, respectively. They are long-range order of microscopic
variables, magnetic (M) and electric (E) dipoles. In recent years, it has been
extensively investigated alternative orders that break both time-reversal and
spatial inversion symmetries [1, 2, 3] or do not break both of them [4, 5].
They are called as toroidal or ferroaxial orders, respectively. These orders
are characterized by the alignment of a certain type of microscopic variables
as well, which are the main subject of the present paper.

In contrast to an electron in vacuum, electrons in materials have a rich
variety of degrees of freedom, i.e., charge, orbital, spin, and sublattice in
underlying molecular or crystal structures. By considering their combinations,
various microscopic variables other than ordinary electric and magnetic
dipoles can be constructed, which become order parameters of unconventional
orders and bring about associated exotic physical properties.

In the classification of electronic degrees of freedom in materials, it has
been recognized two types of multipoles, E and M multipoles [6, 7, 8].
Subsequently, additional two types of multipoles, the electric toroidal (ET)
and magnetic toroidal (MT) multipoles, that are indispensable for a unified
description of a state of matter, have been incorporated [9, 10, 11, 12, 13,
14, 15]. The ET and MT multipoles have the opposite spatial-inversion parity
to the conventional E and M multipoles, and these four types of multipoles (E,
M, ET, and MT) constitute a complete basis set to describe arbitrary
electronic degrees of freedom in materials in molecules and crystals [16]. For
example, ET and MT dipoles are the microscopic variables of ferroaxial and
toroidal orders, respectively [2, 3, 17]. On the other hand, the ET monopole
having time-reversal even pseudoscalar property is a practical entity of
chirality [12, 13, 18, 19], which was thoughtfully discussed by Barron [20,
21, 22]. Moreover, the MT monopole can be realized in certain types of
antiferromagnet (AFM) [23], which will be discussed in this paper.

This paper is organized as follows. In Sec. 2, we introduce the concept of
toroidal multipole from symmetry viewpoint at the classical and quantum-
mechanical level. In addition to this, we show the connection to the chirality
in the Dirac theory. Then, we discuss the toroidal orders and related
phenomena, with focusing on monopoles and dipoles in Sec. 3. The relation
between point groups and active multipoles, and possible cross correlations
under multipole orders are also clarified. In Sec. 4, we give prime examples
for ET and MT monopole and dipole orders and related phenomena. The final
section summarizes the paper.

##  2 Concept of toroidal multipole

###  2.1 Multipole and symmetry

Table 1:  Symmetry properties of the four types of monopoles and dipoles. $\mathcal{M}_{\perp}$ ($\mathcal{M}_{\parallel}$) represents the mirror operation with respect to the plane perpendicular (parallel) to dipole. $\bar{\bm{r}}=\bm{r}/|\bm{r}|$, $\bm{l}$, and $\bm{\sigma}$ represent dimensionless position vector, orbital angular momentum, and Pauli matrix, respectively. The definitions of operators, $\bm{Q}^{\prime}$, $\bm{G}^{\prime}$, and $\bm{T}^{\prime}$, are given in [13].  MP | $\mathcal{R}$ | $\mathcal{T}$ | $\mathcal{P}$ | $\mathcal{M}_{\perp}$ | $\mathcal{M}_{\parallel}$ | #PG | #MPG | Form (orbital) | Form (spin)
---|---|---|---|---|---|---|---|---|---
$Q_{0}$ | $l=0$ | $+$ | $+$ | $+$ | $+$ | $32$ | $122$ | $1$ | $\frac{1}{\sqrt{3}}(\bm{l}\cdot\bm{\sigma})$ |
$M_{0}$ | $l=0$ | $-$ | $-$ | $-$ | $-$ |  | $32$ | — | $\frac{1}{\sqrt{3}}(\bar{\bm{r}}\cdot\bm{\sigma})$ |
$G_{0}$ | $l=0$ | $+$ | $-$ | $-$ | $-$ | $11$ | $32$ | — | $\frac{1}{\sqrt{3}}(\bm{t}\cdot\bm{\sigma})$ |
$T_{0}$ | $l=0$ | $-$ | $+$ | $+$ | $+$ |  | $32$ |  $i$ (off-diagonal) | — |
$\bm{Q}$ | $l=1$ | $+$ | $-$ | $-$ | $+$ | $10$ | $31$ | $\bar{\bm{r}}$ | $\frac{1}{\sqrt{3}}(\bm{\sigma}\times\bm{t})$ | $\bm{Q}^{\prime}$
$\bm{M}$ | $l=1$ | $-$ | $+$ | $+$ | $-$ |  | $31$ | $\bm{l}$ | $\bm{\sigma}$ | $\frac{1}{\sqrt{10}}[3(\bar{\bm{r}}\cdot\bm{\sigma})\bar{\bm{r}}-\bar{\bm{r}}^{2}\bm{\sigma}]$
$\bm{G}$ | $l=1$ | $+$ | $+$ | $+$ | $-$ | $13$ | $43$ | — | $\frac{1}{\sqrt{2}}(\bm{\sigma}\times\bm{l})$ | $\bm{G}^{\prime}$
$\bm{T}$ | $l=1$ | $-$ | $-$ | $-$ | $+$ |  | $31$ | $\bm{t}\equiv\frac{1}{6}(\bar{\bm{r}}\times\bm{l})+{\rm h.c.}$ | $\frac{1}{\sqrt{2}}(\bm{\sigma}\times\bar{\bm{r}})$ | $\bm{T}^{\prime}$
|  |  |  |  |  |  |  |  |  |

It is well known that a charge distribution brings about a polar quantity such
as an electric dipole $\bm{Q}$, while an electric current distribution yields
an axial quantity such as a magnetic dipole $\bm{M}$. Once such dipoles exist,
a vortex-like alignment of them gives us another independent quantity
characterized by a dipole ($\bm{G}\propto\sum_{j}\bm{R}_{j}\times\bm{Q}_{j}$
or $\bm{T}\propto\sum_{j}\bm{R}_{j}\times\bm{M}_{j}$ where $\bm{R}_{j}$ is a
position of the dipole), which has an opposite spatial parity of $\bm{Q}$ or
$\bm{M}$ as the cross product contains the totally anti-symmetric (Levi-
Civita) tensor. This is the so-called electric-toroidal (ET) or magnetic-
toroidal (MT) dipole, representing the degree of vorticity of a distribution
of electric or magnetic dipoles. The concept of toroidal dipoles (rank $l=1$
with the component $m$) can be generalized straightforwardly to any ranks
(called a toroidal multipole $G_{lm}$ or $T_{lm}$) introduced as a classical
quantity in the middle of 1980s [9, 10].

The concept of multipole is closely related to the symmetry. The principal
symmetry of interest for materials science consists of the time-reversal
$\mathcal{T}$, spatial inversion $\mathcal{P}$, and rotation $\mathcal{R}$,
while the translational operation is less important in the present argument.
As similar to the multipole expansion for characterizing an anisotropic
distribution of electromagnetic fields, an anisotropic state of materials can
be characterized by “multipoles” with a set of properties of
$(\mathcal{T},\mathcal{P},\mathcal{R})$, i.e., “electric$(+)$/magnetic$(-)$”
of $\mathcal{T}$, “polar[$(-1)^{l}$]/axial[$(-1)^{l+1}$]” of $\mathcal{P}$,
and $(l,m)$ of $\mathcal{R}$. The enumeration of all combinations corresponds
either to $Q_{lm}$ (electric polar), $M_{lm}$ (magnetic axial), $G_{lm}$
(electric axial), or $T_{lm}$ (magnetic polar), and hence, these four-type of
multipoles constitute a symmetry-adapted complete basis set [11, 12, 13, 14,
15, 16]. In other words, a state of materials is described by a linear
combination of these four-type multipole bases as a building block with an
emphasis on the symmetry. For instance, the properties of the multipoles for
$l=0,1$ are summarized in Table 1. It is noted that the mirror operation
$\mathcal{M}(\bm{n})$ with respect to a plane perpendicular to a unit vector
$\bm{n}$ is equivalent to $\mathcal{P}$ in addition to $180^{\circ}$ rotation
around $\bm{n}$ [$C_{2}(\bm{n})$], i.e.,
$\mathcal{M}(\bm{n})=\mathcal{P}C_{2}(\bm{n})$. Therefore, $\mathcal{P}$ and
$\mathcal{M}(\bm{n})$ have no differences for monopole, while $\mathcal{P}$
and $\mathcal{M}(\bm{n})$ for dipole with $\bm{n}$ being perpendicular to it
($\mathcal{M}_{\parallel}$) differ since $C_{2}(\bm{n})$ gives an additional
sign for dipole.

###  2.2 Symmetry lowering and emergence of multipole

As long as a system is invariant under $(\mathcal{T},\mathcal{P},\mathcal{R})$
operations, an electric monopole [$l=m=0$, $\mathcal{T}=+$,
$(\mathcal{P},\mathcal{M}_{\perp},\mathcal{M}_{\parallel})=(+,+,+)$], i.e.,
electric charge, is only allowed to exist. However, since some of
$(\mathcal{T},\mathcal{P},\mathcal{R})$ is lost in realistic materials, the
corresponding multipoles other than $Q_{0}$ can appear. In other words, the
appearance of multipoles characterizes how the symmetry of a system is lost.
In the case of macroscopic systems, such symmetry-adapted multipole represents
a possible candidate of order parameter, while for an isolated molecule, it
characterizes its wave function in terms of symmetry. Note that $\mathcal{P}$
or $\mathcal{M}(\bm{n})$ alone can exist in the case without $C_{2}(\bm{n})$
in some point groups, in contrast to the rotation group where $C_{2}(\bm{n})$
and hence both $\mathcal{P}$ and $\mathcal{M}(\bm{n})$ always exist.

As shown above, the toroidal multipole was introduced as a classical quantity,
where the dipole moment, $\bm{Q}_{j}$ in $\bm{G}$ or $\bm{M}_{j}$ in $\bm{T}$,
is the expectation value of the corresponding quantum-mechanical electric or
magnetic dipole operator. Therefore, a certain type of antiferromagnets
accompany the ferroic ordering of $\bm{T}$ which breaks
$(\mathcal{T},\mathcal{P},\mathcal{M}_{\perp})$. Such antiferromagnets have
been investigated extensively, referred to as various terms such as magneto-
electric, (magnetic) toroidal, anapole, or MT-dipole ordering. Precisely
speaking, an MT state is characterized by a component of $\bm{T}$ belonging to
the identity representation in the ordered phase. UNi4B [24, 25, 26, 27],
$\alpha$-Cu2V2O7 [28, 29, 30, 31], and CuMnAs [32, 33, 34] are the prime
examples showing the MT-dipole ordering.

Meanwhile, since the MT monopole $T_{0}$ has properties
$(\mathcal{P},\mathcal{M}_{\perp},\mathcal{M}_{\parallel})=(+,+,+)$ and
$\mathcal{T}=-$, an emergence of $T_{0}$ breaks only $\mathcal{T}$ symmetry,
which mixes between time-reversal partners, i.e., $Q_{lm}$ and $T_{lm}$ or
$G_{lm}$ and $M_{lm}$ without changing other symmetry properties. This time-
reversal switching aspect due to $T_{0}$ leads to intriguing cross
correlations as shown later [23]. From a symmetry point of view, $T_{0}$ is
expressed as $T_{0}\propto\sum_{j}\bm{R}_{j}\cdot\bm{T}_{j}$ for example,
which indicates that a vortex-like magnetic structure has a divergent
property. The practical examples will be discussed later.

Similarly, a vortex-like alignment of electric dipole moments accompanies a
ferroic ordering of $\bm{G}$. Since it corresponds to a static structural
rotational distortion, it is called as rotational, ferroaxial, or ET-dipole
ordering. The gyrotropic vector, which plays a significant role in optical
phenomena in anisotropic media, belongs to the same category of $\bm{G}$ from
the symmetry of view. In this case, a component of $\bm{G}$ becomes the
identity representation in the symmetry-lowering state, which breaks the
vertical mirror $\mathcal{M}_{\parallel}$ with keeping $\mathcal{T}$,
$\mathcal{P}$, and $\mathcal{M}_{\perp}$ as $\bm{G}$ has the properties
$(\mathcal{T},\mathcal{P},\mathcal{M}_{\perp})=(+,+,+)$ and
$\mathcal{M}_{\parallel}=-$. RbFe(MoO4)2 [4, 35], NiTiO3 [5, 35, 36], and
Ca5Ir3O12 [37, 38, 39, 40, 41] are the prime examples showing the ET-dipole
ordering.

Moreover, chirality is characterized by pure rotation operations only with
keeping the time-reversal symmetry, and there are no mirror and spatial
inversion operations [21, 22, 42, 43]. This type of symmetry breaking
corresponds to the emergence of $G_{0}$ which has the properties
$\mathcal{T}=+$ and
$(\mathcal{P},\mathcal{M}_{\perp},\mathcal{M}_{\parallel})=(-,-,-)$ [12, 13,
18, 19]. As $G_{0}$ is $\mathcal{T}$-even axial monopole (pseudoscalar), one
possible realization of chirality is characterized by
$G_{0}\propto\sum_{j}\bm{R}_{j}\cdot\bm{G}_{j}$ for example, which implies a
three-dimensional structure containing a divergent character of a vortex-like
alignment of electric dipole moments corresponding to structural chirality.

As shown in the above arguments, a toroidal monopole represents a degree of
divergence of corresponding dipole distribution, while a toroidal dipole
represents a degree of vorticity (rotation) of opposite-parity dipole
distribution. They are comprehensive information of dipole distribution of
materials.

###  2.3 Quantum-mechanical operator for toroidal multipole

In contrast to the above classical view of toroidal multipoles, the quantum-
mechanical operators of them in atomic scale were found recently [11, 13]. The
expressions of monopoles and dipoles with or without the spin degrees of
freedom are summarized in Table 1, and the complete expressions of the atomic-
scale multipoles are found in literature [11, 13, 16].

The operator expression for $G_{0}$ naturally arises from the Dirac equation
[44, 45, 46]. It is well known that for a given four-component wave function
$\psi_{\mu}(x)$, the chiral density $\chi(x)$ at the space-time point
$x=(\bm{r},t)$ is expressed as
$\chi(x)=\sum_{\mu\nu}{\psi_{\mu}}^{\dagger}(x)\gamma_{5}^{\mu\nu}\psi_{\nu}(x)$,
where $\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ in terms of the
Dirac $\gamma$-matrices represents the operator of chirality acting on
$\psi_{\mu}(x)$. Note that there is the helicity operator,
$\bm{\Sigma}\cdot\bm{p}=-\gamma^{0}\gamma_{5}\bm{\gamma}\cdot\bm{p}$
($\bm{\Sigma}$ is $4\times 4$ Pauli matrices), which has the common
eigenstates of $\gamma_{5}$ for a massless fermion, although the helicity and
chirality are completely independent quantities under Lorentz invariance.

In the weak-relativistic regime by expanding the inverse of a mass $m$ (or
speed of light $c$), $\psi_{\mu}(x)$ is related to an ordinary two-spinor wave
function $\psi_{s}(x)$ ($s=\uparrow,\downarrow$) in the Schrödinger-Pauli
equation [47, 48, 49, 50]. The corresponding chiral density in this regime
becomes [51, 52, 53]

$\chi(x)=\sum_{ss^{\prime}}{\psi_{s}^{*}(x)\frac{\bm{\pi}\cdot{\bm{\sigma}}_{ss^{\prime}}}{mc}\psi_{s^{\prime}}(x)},\quad(\bm{\pi}=\bm{p}+\frac{e}{c}\bm{A}),$
(1)

where $\bm{p}$, $\bm{\sigma}$, $-e$, and $\bm{A}$ are canonical momentum
operator, Pauli matrices, electron charge, and vector potential, respectively,
and in the following we consider the case of no external electromagnetic
field, $\bm{A}=0$. It should be emphasized that there is no essential
distinction between the chirality and helicity in the weak-relativistic
regime.

The chiral density operator can be cast into the multipole language in the
spherical coordinate by using the mathematical identity [6] for $\bm{p}$ in
terms of the dimensionless orbital angular-momentum operator
$\bm{l}=(\bm{r}\times\bm{p})/\hbar$ as

$\bm{p}=-\frac{\hbar}{r}[\bar{\bm{r}}\times\bm{l}+i\bar{\bm{r}}r\frac{\partial}{\partial
r}],\quad\bar{\bm{r}}=\frac{\bm{r}}{r},\;\;\;r=|\bm{r}|.$ (2)

Then, we obtain the chiral density operator in the multipole language as

$\hat{\chi}=\frac{\bm{p}\cdot\bm{\sigma}}{mc}=-\frac{\hbar}{mcr}[\bar{\bm{r}}\cdot(\bm{l}\times\bm{\sigma})+i(\bar{\bm{r}}\cdot\bm{\sigma})r\frac{\partial}{\partial
r}].$ (3)

From this expression, we can confirm that $\hat{\chi}$ contains
$\hat{G}_{0}\propto\bar{\bm{r}}\cdot\hat{\bm{G}}$ with
$\hat{\bm{G}}\propto\bm{l}\times\bm{\sigma}$. Moreover, $\hat{\chi}$ also
contains $\hat{T}_{0}\hat{M}_{0}$ if the imaginary unit $i$ is regarded as the
MT-monopole operator, $\hat{T}_{0}$. Note that $\hat{T}_{0}=i$ alone can
appear only in the off-diagonal element in more than two-dimensional Hilbert
space, as $i$ itself cannot be an hermitian operator. Meanwhile, from Eq. (2),
$\bar{\bm{r}}\times\bm{l}$ is regarded as the MT-dipole type ($\bm{T}$) as
$\bm{p}$ is categorized to $\bm{T}$ as well. In this sense, the relativistic
SOC [$\propto(\bm{\sigma}\times\bm{E})\cdot\bm{p}$] is regarded as the
coupling between $\bm{p}$ (spin-independent MT-dipole operator) and the spin-
dependent MT-dipole operator,
$\hat{\bm{T}}\propto\bm{\sigma}\times\bar{\bm{r}}$ as the dipole moment
$\bm{r}$ may be proportional to the electric field $\bm{E}$.

For the simplest example of the atomic-scale toroidal multipoles, let us
consider the Hilbert space in the total angular momentum $j=1/2$ ($j_{z}=\pm
1/2$) arising from $s$ and $p$ orbitals. The angular dependence of the basis
functions in the two-component spinor form are given as

$\displaystyle\psi_{s,+\frac{1}{2}}(\theta,\phi)=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\quad\psi_{s,-\frac{1}{2}}(\theta,\phi)=\left(\begin{array}[]{c}0\\\
1\end{array}\right),$ (8)
$\displaystyle\psi_{p,+\frac{1}{2}}(\theta,\phi)=\left(\begin{array}[]{c}-\cos\theta\\\
-\sin\theta
e^{i\phi}\end{array}\right),\quad\psi_{p,-\frac{1}{2}}(\theta,\phi)=\left(\begin{array}[]{c}-\sin\theta
e^{-i\phi}\\\ \cos\theta\end{array}\right).$ (13)

In this Hilbert space, $\hat{G}_{0}$ and $\hat{\bm{T}}$ apart from the
normalization constant are expressed as

$\hat{G}_{0}=\left(\begin{array}[]{cc}0&-i\tilde{\sigma}_{0}\\\
i\tilde{\sigma}_{0}&0\end{array}\right),\quad\hat{\bm{T}}=\left(\begin{array}[]{cccc}0&-i\tilde{\bm{\sigma}}\\\
i\tilde{\bm{\sigma}}&0\end{array}\right)\quad\begin{array}[]{c}(s)\\\
(p)\end{array},$ (14)

where $\tilde{\sigma}_{0}$ and $\tilde{\bm{\sigma}}$ are the unit and Pauli
matrices acting on $j_{z}=\pm 1/2$ states, while $\hat{\bm{G}}$ and
$\hat{T}_{0}$ are inactive in the present Hilbert space.

The eigenstates of $\hat{G}_{0}$ are easily obtained as

$\psi_{a\pm}(\theta,\phi)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\mp
i\cos\theta\\\ \mp i\sin\theta
e^{i\phi}\end{array}\right),\quad\psi_{b\pm}(\theta,\phi)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\mp
i\sin\theta e^{-i\phi}\\\ 1\pm i\cos\theta\end{array}\right),$ (15)

where the states with the subscript $(a+,b+)$ or $(a-,b-)$ have degenerate
eigenvalue of $\hat{G}_{0}$ as they are $\mathcal{T}$ partner ($\mathcal{T}$
operation is $-i\tau_{y}$ and complex conjugation). On the other hand,
$(a+,a-)$ or $(b+,b-)$ are $\mathcal{P}$ partner having the eigenvalue of
$\hat{G}_{0}$ with opposite sign, which is confirmed by applying $\mathcal{P}$
operation, $\theta\to\pi-\theta$ and $\phi\to\phi+\pi$. Therefore, such
$\mathcal{P}$ partner constitutes a pair of enantiomers of chirality in the
atomic scale. The eigenstates of $\hat{G}_{0}$ are also those of
$\hat{\bm{T}}$ since $\hat{G}_{0}$ and $\hat{\bm{T}}$ are commutable. In this
case, $(a+,b-)$ or $(a-,b+)$ have degenerate eigenvalue of $\hat{T}_{z}$ as
they are $\mathcal{TP}$ partner, while $(a+,a-)$ or $(b+,b-)$ have the
eigenvalue of $\hat{T}_{z}$ with opposite sign.

Although the direct observation for the atomic-scale toroidal multipoles as
shown in the above has not been succeeded so far, such microscopic degrees of
freedom could be essential ingradients in understanding various cross
correlations at microscopic quantum-mechanical level. For clarity of
terminology, we use in the following “atomic” multipoles for describing
electronic degrees of freedom in terms of orbital and/or spin angular
momentum, “site-cluster” multipoles for composite ones consisting of atomic
multipoles at several sites in a cluster unit, and “bond-cluster” multipoles
for ones on a bond connecting between two sites.

##  3 Toroidal ordering and related phenomena

In this section, we introduce the relationship between the 32 crystallographic
and 122 magnetic point groups and active multipoles with $l=0,1$, and the
expected cross correlations in the presence of active multipoles.

###  3.1 Crystallographic point groups and multipoles

The 32 crystallographic point groups (C-PGs) are defined by the group of
symmetry operations $\mathcal{G}$ from $\mathcal{R}$, $\mathcal{P}$ and their
combinations, i.e., roto-inversion operation, $\mathcal{RP}$
[$\mathcal{G}\in(\mathcal{R},\mathcal{P},\mathcal{RP})$]. Note that the mirror
and roto-reflection operations can be expressed by $\mathcal{RP}$. Since
$\mathcal{T}$ is not included in the symmetry operations of C-PG, there is no
meaning to distinguish between electric and magnetic properties. Therefore, we
discuss the electric multipoles in this subsection by assuming non-magnetic
systems.

The relation between C-PGs and active E dipole ($\bm{Q}$) and ET monopole
($G_{0}$) and dipole ($\bm{G}$) is shown in Fig. 1 where the number in
parenthesis in the Venn diagram indicates that of C-PGs that activates each
multipole [12]. The total number of C-PGs for each multipole is listed in
Table 1, where $Q_{0}$ is always active in 32 C-PGs. Since $\bm{G}$ itself
does not break the inversion symmetry, the noncentrosymmetric PGs in the
$\bm{G}$ category (indicated by grey background) should accompany higher-rank
odd-parity multipoles. There are 23 C-PGs in Fig. 1, and the rest of nine
C-PGs have active multipoles with $l>1$.

Figure 1:  Relation between crystallographic point groups and active electric-
type multipoles. The background color of each point group represents the
presence or absence of $\mathcal{P}$ symmetry. The additional
$\mathcal{P}$-odd multipoles ($l>1$) exist in $\bar{4}$ and $\bar{6}$. The
correspondence between the international and Schoenflies symbols is also
shown.

###  3.2 Magnetic point groups and multipoles

The 122 magnetic point groups (M-PGs) are defined by the combination of
$\mathcal{T}$ and symmetry operations of C-PG. They are further classified
into three types depending on how $\mathcal{T}$ is included [54]. Type I (32
groups) is the same as C-PGs where $\mathcal{T}$ is completely lost, which is
indicated by no operations with prime. Type II (32 groups) called as grey
group has the symmetry operations $\mathcal{TG}$ in addition to $\mathcal{G}$.
This type of M-PG is expressed by attaching $1^{\prime}$ to the corresponding
C-PG. The rest of 58 M-PGs are in Type III called as black-and-white (BW) PG.

Since the presence or absence of $\mathcal{T}$ is taken into account in M-PGs,
there is a proper meaning of electric-type and magnetic-type multipoles. The
relation between M-PGs and active monopoles and dipoles are shown in Fig. 2
for (a) electric-type multipoles and (b) magnetic-type multipoles, where the
number in parenthesis in the Venn diagram indicates that of M-PGs that
activates each multipole [55]. The presence or absence of
$(\mathcal{P},\mathcal{T},\mathcal{PT})$ is indicated by the background color
of each M-PG. There are $77$ and $73$ M-PGs in Fig. 2(a) and (b),
respectively, and the rest of $45$ and $49$ M-PGs have active electric-type
and magnetic-type multipoles with $l>1$.

Figure 2:  Relation between magnetic point groups and active multipoles. (a)
the electric-type multipoles and (b) the magnetic-type multipoles. The
background color of each magnetic point group represents the presence or
absence of $(\mathcal{P},\mathcal{T},\mathcal{PT})$ symmetries. The additional
$\mathcal{P}$-odd multipoles ($l>1$) exist in the noncentrosymmetric point
groups of pure $\bm{G}$, $T_{0}$, and $\bm{M}$ categories.

The nonmagnetic states accompanying only electric-type multipoles should
belong to the Type II grey groups (grey or green backgrounds) which do not
appear in Fig. 2(b). For example, 11 chiral states in Type II (green
background) are nonmagnetic chiral states, while chiral states such as $222$
in Type I and $2^{\prime}2^{\prime}2$ in Type III coexist other magnetic-type
multipoles, like $M_{0}$ and $T_{0}$ for $222$ and $\bm{M}$ and $\bm{T}$ for
$2^{\prime}2^{\prime}2$ as shown in Fig. 2(b). Note that there are symmetry
correspondences, $T_{0}M_{0}$, $\bm{T}\cdot\bm{M}\leftrightarrow G_{0}$,
indicating that $G_{0}$ coexists with $\bm{T}$ and $\bm{M}$ that are parallel
to each other in $2^{\prime}2^{\prime}2$. Similarly, 13 axial states in Type
II (grey or green backgrounds) are nonmagnetic axial states, while axial
states such as $\bar{1}$ in Type I and $\bar{1}^{\prime}$ in Type III coexist
with ($M_{0},\bm{T}$) and ($T_{0},\bm{M}$), respectively, as the symmetry
correspondence $\bm{G}\leftrightarrow T_{0}\bm{M}\leftrightarrow M_{0}\bm{T}$
holds. Moreover, $2^{\prime}/m$ and $2^{\prime}/m^{\prime}$ in Type III
coexist $\bm{T}$ and $\bm{M}$, respectively, where the symmetry correspondence
$\bm{G}\leftrightarrow\bm{T}\times\bm{T}$ or
$\bm{G}\leftrightarrow\bm{M}\times\bm{M}$ indicates two distinct vectors
($\bm{T}_{1},\bm{T}_{2}$) or ($\bm{M}_{1},\bm{M}_{2}$) are orthogonal with
each other in $2^{\prime}/m$ and $2^{\prime}/m^{\prime}$, respectively. On the
contrary, there is the symmetry correspondences $\bm{Q}\leftrightarrow
M_{0}\bm{M}\leftrightarrow T_{0}\bm{T}$ and
$\bm{Q}\leftrightarrow\bm{T}\times\bm{M}$. Thus, the M-PGs in the
intersections of $(M_{0},\bm{M})$ and $(T_{0},\bm{T})$ in Fig. 2(b) appear in
$\bm{Q}$ in Fig. 2(a), and $\bm{T}$ and $\bm{M}$ are orthogonal to each other
in $m^{\prime}m2^{\prime}$ for example.

It is noted that since $(M_{0}$,$T_{0}$) or $(\bm{M}$,$\bm{T}$) have opposite
properties of ($\mathcal{P},\mathcal{M}_{\perp},\mathcal{M}_{\parallel}$), the
Venn diagram in Fig. 2(b) becomes symmetric between the left and right sides,
where the dual relation, $m\leftrightarrow m^{\prime}$ and
$\bar{n}\leftrightarrow\bar{n}^{\prime}$ in M-PGs, hold. On the other hand, as
the combinations of $\bm{M}$ and $\bm{T}$ appear differently in $\bm{G}$ and
$\bm{Q}$, the Venn diagram in Fig. 2(a) becomes asymmetric.

###  3.3 Cross correlations

Table 2:  Symmetry correspondences among four-type of multipoles ($l=0,1$), conjugated fields, and responses or ordered states.  MP | Conjugate field | Response/Order
---|---|---
$Q_{0}$ | $\bm{\nabla}\cdot\bm{E}$ |  $\delta T$ (temperature difference), charge
$M_{0}$ | $\bm{E}\cdot\bm{B}$ | monopole-type AFM
$G_{0}$ |  $\bm{E}\cdot(\bm{\nabla}\times\bm{E})$, $\bm{B}\cdot(\bm{\nabla}\times\bm{B})$ (optical chirality; Zilch) | chirality
$T_{0}$ |  $\bm{E}\cdot(\bm{\nabla}\times\bm{B})$, $\bm{B}\cdot(\bm{\nabla}\times\bm{E})$ | MT-monopole-type AFM
$\bm{Q}$ |  $\bm{E}$ (electric field), $-\bm{\nabla}T$ (temperature gradient) |  $\bm{P}$ (electric polarization), $\bm{u}$ (displacement), ferroelectricity
$\bm{M}$ |  $\bm{B}$ (magnetic field) |  $\bm{M}$ (magnetization), FM
$\bm{G}$ |  $\bm{\nabla}\times\bm{E}$, $\partial\bm{B}/\partial t$ |  $\bm{\nabla}\times\bm{u}$ (static rotational distortion), ferroaxial
$\bm{T}$ |  $\bm{\nabla}\times\bm{B}$, $\partial\bm{E}/\partial t$ |  $\bm{J}$ (electric current), $\bm{J}_{\rm T}$ (thermal current), vortex-type AFM
|  |

As mentioned above, the toroidal multipoles can coexist with other multipoles
according to the symmetry correspondence. Even in the high-symmetry PGs such
as nonmagnetic grey groups where the equilibrium value of magnetic-type
multipoles should vanish, the electric-type multipole can couple with
magnetic-type multipoles as a fluctuation. For example, there exists $G_{0}$
in $321^{\prime}$, which can couple with $\bm{T}\cdot\bm{M}$ as it has the
same symmetry properties of $G_{0}$. It also couples with $\bm{Q}\cdot\bm{G}$
as well. In this case, the Landau free energy can be written as

$F=\frac{\chi^{-1}_{\bm{T}}}{2}\bm{T}^{2}+\frac{\chi^{-1}_{\bm{M}}}{2}\bm{M}^{2}-\bm{M}\cdot\bm{H}-\bm{T}\cdot(\bm{\nabla}\times\bm{H})-\alpha
G_{0}(\bm{T}\cdot\bm{M})+\cdots,$ (16)

where $\chi_{A}$ is the susceptibility for $A=\bm{M},\bm{T}$ and $\alpha$ is
the trilinear coupling constant. The third and fourth terms represent the
interaction between $A$ and its conjugate field where $\bm{H}$ is an external
magnetic field. The equilibrium values for $\bm{M}$ and $\bm{T}$ in the
presence of $G_{0}$ can be obtained by minimizing $F$ with respect to them as

$\left(\begin{array}[]{c}\bm{M}\\\
\bm{T}\end{array}\right)=\frac{\chi_{\bm{M}}\chi_{\bm{T}}}{1-\chi_{\bm{M}}\chi_{\bm{T}}(\alpha
G_{0})^{2}}\left(\begin{array}[]{cc}\chi_{\bm{T}}^{-1}&\alpha G_{0}\\\ \alpha
G_{0}&\chi^{-1}_{\bm{M}}\end{array}\right)\left(\begin{array}[]{c}\bm{H}\\\
\bm{\nabla}\times\bm{H}\end{array}\right).$ (17)

From this expression, by applying the electric current, which leads to finite
$\bm{\nabla}\times\bm{H}$ along the applied current direction, the magnetic
dipole $\bm{M}$ is induced linearly through the trilinear coupling. This is
nothing but the Edelstein effect, i.e., linear electric-current induced
magnetization [56, 57, 58, 59, 60]. It should be noted however that since the
situation in the presence of steady current is not an equilibrium state, this
type of cross correlation should be treated by the Kubo formula or Boltzmann
transport equation, in a rigorous sense [61, 62, 63, 64, 65].

Here, there are a few comments. In the Landau theory, the free energy is
constructed under the guidance of symmetry in terms of macroscopic variables
irrespective of their microscopic origin. Thus, the argument holds for various
microscopic origins of macroscopic variables arising either from cluster-type
distributions of classical dipoles or pure quantum-mechanical degrees of
freedom. It is often considered the cluster-type toroidal orders where the
primary variables are ordinary dipoles. In this case, without using the
concept of toroidal moment, the coupling term to a toroidal moment, such as
$\bm{T}\cdot(\bm{\nabla}\times\bm{H})$ is often dropped. Moreover, a quantum-
mechanical type of toroidal order has a finite thermal average of the
corresponding toroidal operator itself as discussed in the previous section
without any accompanying dipole moments.

In the above discussion, the trilinear term determines possible linear cross
correlations. As was already stated, $G_{0}$ representing chiral state can
couple with $T_{0}M_{0}$, $\bm{Q}\cdot\bm{G}$, and $\bm{T}\cdot\bm{M}$. At the
same time, the corresponding composite fields of $\bm{Q}\cdot\bm{G}$ and
$\bm{T}\cdot\bm{M}$, i.e., $\bm{E}\cdot(\bm{\nabla}\times\bm{E})$ and
$\bm{B}\cdot(\bm{\nabla}\times\bm{B})$, whose sum is known as the optical
chirality (Zilch) [66, 67, 68] and is activated by circularly polarized light,
represent the conjugate field of $G_{0}$ [69]. These couplings imply various
cross correlations under chiral state, such as the electric-field induced
rotational distortion ($\bm{E}\to\bm{G}$) [18, 70], the current-induced
Magnetization (Edelstein effect) ($\bm{J}\to\bm{M}$) [56, 57, 58, 59, 60], and
the magnetic-field induced vortex-like spin alignment ($\bm{B}\to\bm{T}$).

Similarly, $\bm{G}$ representing ferroaxial state can couple with
$G_{0}\bm{Q}$, $T_{0}\bm{M}$, $M_{0}\bm{T}$, and $\bm{X}_{1}\times\bm{X}_{2}$
[71]. These couplings under ferroaxial state lead to the electric-field
induced chirality (electro-gyration) ($\bm{E}\to G_{0}$) [5, 35], the
magnetic-field induced MT monopole ($\bm{B}\to T_{0}$), and the transverse
responses by fields ($\bm{X}\to\bm{G}\times\bm{X}$) [17, 72, 73, 74]. The last
response will be discussed later.

The MT-monopole $T_{0}$ order can be realized in antiferromagnets (AFMs) with
proper structures as shown later [23]. In this case, $T_{0}$ can couple with
$G_{0}M_{0}$, $\bm{G}\cdot\bm{M}$, and $\bm{Q}\cdot\bm{T}$. Therefore, we
expect the time-reversal switching cross correlations, such as the static
electromagnetic-field induced chirality ($\bm{E}\cdot\bm{B}\to G_{0}$), the
magnetic-field induced static rotational distortion ($\bm{B}\to\bm{G}$), and
electric-field induced vortex-like AFM ($\bm{E}\to\bm{T}$). The MT-dipole
$\bm{T}$ orders in some AFMs and the associated cross correlations have
already been discussed extensively [2, 3, 26, 75, 76, 77, 78, 79, 80, 81, 82].
$\bm{T}$ couples with $\bm{G}\cdot\bm{T}$, $\bm{Q}\cdot\bm{M}$ and so on.
Among them, the magneto-electric effect ($\bm{E}\to\bm{M}$ or
$\bm{B}\to\bm{Q}$) is the representative cross correlation.

Figure 3:  Possible conversion among monopoles and dipoles by external fields:
electric field ($\bm{E}$), magnetic field ($\bm{B}$), electric current,
rotational stress, and circularly polarized light. The sign in the parenthesis
indicates the spatial parity of multipole.

The symmetry correspondence among multipoles ($l=0,1$), conjugate fields, and
responses or ordered states, and conversion among multipoles by external
fields are summarized in Table 2 and Fig. 3. When we consider such analysis
for multipoles with $l>1$, e.g., second-rank stress field ($Q_{2m}$), spin
current $J_{i}M_{j}$, etc., we expect a further variety of cross correlations
with toroidal states in a unified way.

##  4 Examples

###  4.1 Chirality as electric-toroidal monopole $G_{0}$

As was already mentioned, from a symmetry point of view the definition of
chirality matches an ET monopole, $G_{0}$ [12, 13, 18, 19]. For example, the
typical chiral crystal is elemental tellurium (Te), which belongs to the
trigonal space group $P3_{1}21$ (#152,$D_{3}^{4}$) or $P3_{2}21$
(#154,$D_{3}^{6}$) as a pair of right-handed and left-handed enantiomers. In
the unit cell, three sublattices constitute the threefold helical chain,
representing three-dimensional structural chirality.

Figure 4:  Schematic representations of chirality $G_{0}$, (a) site-cluster
type $G_{0}^{(1)}$ and (b) bond-cluster (itinerant) type $G_{0}^{(2)}$ [18].

The electronic states around H point in the hexagonal Brillouin zone near the
Fermi level consist of $p$ orbitals, and they are analyzed by the symmetry-
adapted multipole basis [18]. The results show that three types of $G_{0}$
give dominant contributions to the tight-binding model of Te,

$G_{0}^{(1)}=\frac{1}{\sqrt{2}}(Q_{x}^{\rm(c)}G_{x}^{\rm(a)}+Q_{y}^{\rm(c)}G_{y}^{\rm(a)}),\quad
G_{0}^{(2)}=T_{z}^{\rm(b)}M_{z}^{\rm(a)},\quad
G_{0}^{(3)}=\frac{1}{\sqrt{2}}(T_{x}^{\rm(b)}M_{x}^{\rm(a)}+T_{y}^{\rm(b)}M_{y}^{\rm(a)}),$
(18)

where the superscripts, c, b, and a, represent cluster, bond, and atomic
bases, respectively, and schematically shown in Fig. 4. Here,
$\bm{M}^{\rm(a)}$ mainly arises from spin degrees of freedom. $G_{0}^{(1)}$
represents divergent alignment of
$\bm{G}^{(a)}\propto\bm{l}\times\bm{\sigma}$, while $G_{0}^{(2)}$ and
$G_{0}^{(3)}$ describe the spin-dependent imaginary hopping processes along
and perpendicular to the helical chain, respectively. The latter $G_{0}^{(2)}$
and $G_{0}^{(3)}$ are the microscopic origin of Edelstein effect observed by
the spectrum shift of 125Te in NMR measurement [58, 60]. On the other hand,
$G_{0}^{(1)}$ can contribute another cross correlation, such as the electro-
rotation (electric-field induced rotational distortion) effect
($\bm{E}\to\bm{G}$) as discussed in the previous section, since the expression
consists of $\bm{Q}$ and $\bm{G}$ [18]. Once the existence of such microscopic
couplings was confirmed by the experiments, one could achieve absolute
enantioselection by using composite conjugate fields to access $G_{0}$ through
such couplings. Note that the ET quadrupole $G_{u}$ also belongs to the
identity representation of $D_{3}$, which gives the monoaxial anisotropy.

There is another attempt to quantify the chirality of molecule by using the ET
monopole. For such purpose, a twisted methane is analyzed as a prototype of
the method, and the results indeed show that the ET monopole becomes a
quantitative indicator for chirality [83]. In the twisted methane, it is
clarified that the handedness of chirality corresponds to the sign of the
expectation value of the electric toroidal monopole, and that the most
important ingredient is the modulation of the spin-dependent imaginary hopping
between the hydrogen atoms, while the relativistic spin-orbit coupling within
the carbon atom is irrelevant for chirality. The method can be
straightforwardly applied to more realistic chiral molecules, which would
unveil the nature of chirality quantitatively at the quantum-mechanical level.

###  4.2 Ferroaxial order ($\bm{G}$), and electro-gyration effect and spin
current generation

Figure 5:  Crystal structure of RbFe(MoO4)2 in (left) normal and (right)
ferroaxial phases (modified from [35]). The twin domains appear in the ordered
phase depending on the rotated direction of octahedra and tetrahedra units.
Figure 6:  Imaging of the ferroaxial domain structure of RbFe(MoO4)2 by the
optical rotation induced by electro-gyration effect, where $\Delta T/T$ is the
relative rotation angle of the light polarization [35].

The ferroaxial order is described by the uniform alignment of $\bm{G}$.
RbFe(MoO4)2 is one of the typical crystals showing the ferroaxial order. The
crystal structure of this compound is shown in the left panel of Fig. 5, where
FeO6 octahedra sharing vertices with MoO4 tetrahedra is stacked along $c$
axis. It exhibits the structural phase transition at $T_{c}\sim 195$ K, in
which octahedra and tetrahedra rotate in opposite direction around $c$ axis as
shown in the right panel of Fig. 5, and the point group changes from
$\bar{3}m$ ($D_{\rm 3d}$) to $\bar{3}$ ($C_{\rm 3i}$) with losing vertical
mirror operations. This structural transition has been detected by the high-
sensitive rotational second-harmonic generation method, in which the electric-
quadrupole transition is analyzed [4].

In the ET dipole order $\bm{G}$, the electric field can induce chirality
$G_{0}$ (electro-gyration effect: $\bm{E}\to G_{0}$), and the
enantioselection, i.e., the sign of $G_{0}$, can be realized by the direction
of the electric field as long as the single domain is assumed. In reality,
however, there are twin domains, and the electro-gyration effect is utilized
for domain detection as shown in Fig. 6 for RbFe(MoO4)2, because the optical
rotation angle of the linear polarized light depends on the domain of $\bm{G}$
with the fixed applied electric field [35]. The visualization of the
ferroaxial domain is particularly important as the achievement of the
enantioselection strongly depends on how the domain is distributed in the
sample.

A similar method is also applied to NiTiO3, which has elucidated that the
domain size depends on the cooling rate around $T_{c}$ [5, 35, 36]. Moreover,
state-of-the-art three-dimensional imaging by circularly polarized second
harmonic generation microscopy has also been developed to observe the ET
dipole in real space [84]. These experimental techniques further stimulate the
detection of $\bm{G}$ in other candidates, such as Co3Nb2O8 [85], CaMn7O12
[86], Ca5Ir3O12 [37, 38, 39, 40, 41], BaCoSiO4 [87], K2Zr(PO4)2 [88],
Na2Hf(BO3)2 [89], and the typical candidates are Ca(BH4)2 [90], PdS [91],
TaPO5 [92], VPO5 [93], and NbTe2 [94] belonging to $C_{\rm 4h}$ point group.

Figure 7:  (a) Concept of a nanorotator by $\bm{G}$ order. The inputs of
$\bm{H}$, $\bm{E}$, and $\bm{J}$ can be converted to the transverse output of
$\bm{M}$, $\bm{Q}$, and $\bm{T}$, respectively [17]. (b) The electric dipole
$\bm{Q}$ in the momentum space is equivalent to the spin current perpendicular
to $\bm{Q}$.

In addition to the above conversion from dipole to pseudoscalar, the
conversion among dipoles can also be expected in $\bm{G}$ order owing to the
coupling between $\bm{G}$ and $\bm{X}_{1}\times\bm{X}_{2}$ [17]. This cross
correlation is characterized by the conversion where the input and output
vectors are orthogonal with each other, and hence $\bm{G}$ order plays a role
of a nanorotator as shown in Fig. 7(a). Therefore, the inputs of $\bm{H}$,
$\bm{E}$, and $\bm{J}$ can be converted to the transverse output of $\bm{M}$,
$\bm{Q}$, and $\bm{T}$, respectively. In particular, as the electric dipole
$\bm{Q}$ in the momentum space is expressed as
$\bm{Q}\propto\bm{k}\times\bm{\sigma}$, the transverse output is equivalent to
the longitudinal spin current along the applied electric field as shown in
Fig. 7(b). The experimental confirmation of such cross correlation is highly
desired.

###  4.3 Magneto-electric gyration under magnetic-toroidal monopole order
($T_{0}$)

Table 3:  Classification for magnetic-toroidal monopole order ($T_{0}$) in Type I magnetic point groups and candidate materials [23]. Other multipoles indicated by open circles are accompanied with $T_{0}$ order.  PG | $T_{0}$ | $M_{z}$ | $T_{z}$ | $M_{0}$ | Materials
---|---|---|---|---|---
$O_{\rm h}$, $T_{\rm d}$, $T_{\rm h}$, $D_{\rm 2h}$, $D_{\rm 3h}$, $D_{\rm 4h}$, $D_{\rm 6h}$, $D_{\rm 2d}$, $D_{\rm 3d}$ | $\circ$ |  |  |  |  KMnF3, Ca2RuO4
$C_{\rm 2h}$, $C_{\rm 3h}$, $C_{\rm 4h}$, $C_{\rm 6h}$, $S_{4}$, $C_{\rm 3i}$, $C_{\rm i}$ | $\circ$ | $\circ$ |  |  |  MnV2O4, Mn3As2
$C_{\rm 2v}$, $C_{\rm 3v}$, $C_{\rm 4v}$, $C_{\rm 6v}$ | $\circ$ |  | $\circ$ |  |  YMnO3, Er2Cu2O5
$O$, $T$, $D_{2}$, $D_{3}$, $D_{4}$, $D_{6}$ | $\circ$ |  |  | $\circ$ |  Ho2Ge2O7, Mn3IrGe
$C_{\rm s}$ | $\circ$ | $\circ$ | $\circ$ |  |  Mn4Nb2O9
$C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $C_{6}$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ |  ScMnO3, Mn2FeMoO6
|  |  |  |  |
Figure 8:  Magnetic structure in Ca2RuO4 accompanying $T_{0}$, and expected optical rotation measurement in the applied electric and magnetic fields, $\bm{E}$ and $\bm{H}$ [23]. The rotation angle depends on the parallel or anti-parallel fields of $\bm{E}$ and $\bm{H}$.  Table 4:  Magnetic moments at four Ru sites in Ca2RuO4 [95]. $\bm{T}^{\rm(c)}_{u}$, $\bm{T}^{\rm(c)}_{v}$, and $\bm{M}^{\rm(c)}_{xyz}$ represent the cluster $3z^{2}-r^{2}$-type and $x^{2}-y^{2}$-type MT quadrupoles, and $xyz$-type magnetic octupole, respectively, which are all belonging to the identity representation.  Position | $\bm{M}$ | $\bm{T}^{\rm(c)}_{u}$ | $\bm{T}^{\rm(c)}_{v}$ | $\bm{M}^{\rm(c)}_{xyz}$
---|---|---|---|---
$(0,0,0)$ | $(0,+1.0,+0.1)$ | $\frac{1}{2\sqrt{2}}(1,-1,0)$ | $\frac{1}{2\sqrt{6}}(1,1,-2)$ | $\frac{1}{2\sqrt{3}}(1,1,1)$
$(1/2,0,1/2)$ | $(0,-1.0,-0.1)$ | $\frac{1}{2\sqrt{2}}(-1,1,0)$ | $\frac{1}{2\sqrt{6}}(-1,-1,-2)$ | $\frac{1}{2\sqrt{3}}(-1,-1,1)$
$(0,1/2,1/2)$ | $(0,+1.0,-0.1)$ | $\frac{1}{2\sqrt{2}}(-1,-1,0)$ | $\frac{1}{2\sqrt{6}}(-1,1,2)$ | $\frac{1}{2\sqrt{3}}(-1,1,-1)$
$(1/2,1/2,0)$ | $(0,-1.0,-0.1)$ | $\frac{1}{2\sqrt{2}}(1,1,0)$ | $\frac{1}{2\sqrt{6}}(1,-1,2)$ | $\frac{1}{2\sqrt{3}}(1,-1,-1)$
|  |  |  |

As discussed in the previous section, a divergent alignment of the vortex-like
AFM represents the cluster-type MT monopole, $T_{0}$ [23]. Indeed, AFMs
belonging to the same paramagnetic point group only without $\mathcal{T}$
operation, i.e., Type I magnetic point group, can exhibit the MT monopole
order, and they accompany additional multipoles in low-symmetry point groups.
The classification of active multipoles and the candidate AFMs are summarized
in Table 3.

Among them, Ca2RuO4 for example shows AFM order ($mmm1^{\prime}\to mmm$,
$D_{\rm 2h}$) as shown in Fig. 8, and the ordered moment at each position is
given by $\bm{M}$ in Table 4. There are nine cluster-type magnetic multipoles,
$\bm{T}_{u}^{\rm(c)}$, $\bm{T}_{v}^{\rm(c)}$, and $\bm{M}_{xyz}^{\rm(c)}$,
belonging to the identity representation, and the observed magnetic structure
can be expressed as the linear combination as

$\bm{M}=-1.414\bm{T}_{u}^{\rm(c)}+0.653\bm{T}_{v}^{\rm(c)}+1.270\bm{M}_{xyz}^{\rm(c)}.$
(19)

Therefore, we expect various cross correlations in the presence of $T_{0}$ as
discussed previously. For example, a possible experimental setup is shown in
Fig. 8, which is expected to show the optical rotation in the presence of
static electric and magnetic fields, and the rotation angle depends on the
relative directions between electric and magnetic fields. The direction of
incident light can be chosen arbitrarily since the order parameter has a
scalar nature. In this experiment, it would be also important to know how the
domain of $T_{0}$ is distributed in the sample. This type of experiment also
provides a way of absolute enantioselection by static electric and magnetic
fields as long as the single domain is assumed.

##  5 Summary

In summary, we have introduced the concept of toroidal multipoles both from
classical and quantum viewpoint. By using the concept, we have shown the
correspondence between active multipoles and crystallographic point groups in
non-magnetic materials, and magnetic point groups in magnetic materials, with
focusing on monopoles and dipoles. The electric-toroidal monopole is a
practical entity of chirality, while the electric-toroidal dipole is a
microscopic variable of ferroaxial order. The magnetic-toroidal monopole and
dipole are often realized in certain type of antiferromagnets. Once the
relevant symmetry-adapted multipoles are identified in materials, we can
expect various cross correlations in a unified way at the quantum-mechanical
level. The proper understanding of these toroidal moments would stimulate
further investigation of mutual conversions among peculiar states of matter.

## Acknowledgement

This paper is based on fruitful collaborations with Yuki Yanagi, Megumi
Yatsushiro, Rikuto Oiwa, Jun-ichiro Kishine, and Hiroshi H. Yamamoto. We would
also like to thank Makoto Naka, Michi-to Suzuki, Yukitoshi Motome, Hitoshi
Seo, Takuya Nomoto, Ryotaro Arita, Hiroshi Amitsuka, Tatsuya Yanagisawa,
Takahisa Arima, Hiraku Saito, for helpful discussions. This research was
supported by JSPS KAKENHI Grants Numbers JP21H01031, JP21H01037, JP22H04468,
JP22H00101, JP22H01183, JP23K03288, JP23H04869, JP23H00091 and by JST PRESTO
(JP- MJPR20L8) and JST CREST (JPMJCR23O4), and the grants of Special Project
(IMS program 23IMS1101), and OML Project (NINS program No. OML012301) by the
National Institutes of Natural Sciences.

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