Symmetry Classification of Antiferromagnets with Four Types of Multipoles

Abstract

A plethora of antiferromagnetic structures have been so far found in condensed matter physics, where the antiferromagnetic phase transition is characterized by symmetry lowering under the magnetic point group. Depending on the types of symmetry lowering, various cross-correlation phenomena, such as the anomalous Hall effect, magneto-electric effect, and magneto-piezoelectric effect, emerge below the critical temperature. We revisit a close relationship between the symmetry of the antiferromagnetic structures and cross-correlations based on the augmented multipoles consisting of electric, magnetic, magnetic toroidal, and electric toroidal multipoles with different spatial inversion and time-reversal parities. The symmetry classification will be useful for further exploration of functional antiferromagnetic materials.

1. Introduction

Magnetic ordering is one of the most fundamental electronic orderings in solids, where the spin degree of freedom in electrons is ordered through the electron correlation. Owing to the breaking of time-reversal symmetry, various physical phenomena emerge below the magnetic phase transition temperature; ferromagnetic ordering gives rise to the anomalous Hall effect [1,2,3,4,5,6,7] and antiferromagnetic ordering breaking the spatial inversion symmetry gives rise to the linear magnetoelectric effect [8,9,10,11,12,13,14]. Meanwhile, recent studies have revealed that the same physical phenomena are often caused under totally different magnetic structures when the symmetry of the magnetic structures is the same. For example, the anomalous Hall effect is induced by various types of antiferromagnetic structures including collinear [15,16,17,18,19,20,21,22], noncollinear [23,24,25,26,27], and noncoplanar ones [28,29,30,31] when the symmetry of the antiferromagnetic state is the same as that of the ferromagnetic state. This example suggests that antiferromagnetic ordering has the potential to exhibit further intriguing physical phenomena, which would be important for the realization of functional antiferromagnetic materials applied to spintronic devices.

The concept of augmented multipoles has been introduced to smoothly make the antiferromagnetic structures correspond to physical phenomena [32,33]. There are four types of multipoles with different spatial inversion and time-reversal parities: electric multipole with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,+1]$, magnetic multipole with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},-1]$, magnetic toroidal multipole with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,-1]$, and electric toroidal multipole with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},+1]$, where $\mathcal{P}$ and $\mathcal{T}$ stand for the spatial inversion and time-reversal parities, respectively, and l is the rank of the multipole. Since the four types of multipoles constitute a complete basis set in physical Hilbert space, any antiferromagnetic structures can be described by a ferroic alignment of any multipoles [34,35]. Figure 1 represents the examples of symmetry lowering by magnetic phase transitions under the tetragonal symmetry $4/mmm{1}^{\prime }$; the antiferromagnetic structures are characterized by the magnetic monopole $M_0$ and the magnetic toroidal dipole $T_z$ with $(\mathcal{P},\mathcal{T})=(-1,-1)$. Furthermore, the systematic classification of antiferromagnetic structures has been performed under the 122 magnetic point groups based on magnetic representation analysis [36].

In the present study, we revisit such a correspondence between antiferromagnetic structures and multipoles under 122 magnetic point groups in order to demonstrate which magnetic materials exhibit functional properties. By classifying the 122 magnetic point groups in terms of the four types of multipoles, we show that almost all of the magnetic point groups accompany the multipole moments up to rank 3, and hence, we expect rich cross-correlation phenomena in antiferromagnetic materials. In order to stimulate the experimental observation, we list candidate materials and expected cross-correlations in each multipole. The present results provide useful information not only in exploring functional antiferromagnetic materials but also in reexamining the well-known materials from the multipole viewpoint.

The rest of this paper is organized as follows: In Section 2, we first introduce the four types of multipoles. Then, we show the correspondence between antiferromagnetic structures and multipoles by taking an example in the tetragonal system in Section 3. In Section 4, we discuss the expected cross-correlations when the multipole is induced. We conclude the present results in Section 5.

2. Four Types of Multipoles

First, we briefly review the four types of multipoles, consisting of electric multipole $Q_{lm}$, magnetic multipole $M_{lm}$, magnetic toroidal multipole $T_{lm}$, and electric toroidal multipole $G_{lm}$, where the subscripts l and m represent the orbital angular momentum (rank of multipole) and its z component, respectively; $m=-l,-l+1,\cdots ,l$. The operator expressions of the four multipoles in spinless space are given by [37,38]

$$\begin{array}{cc}\hfill Q_{lm}=& O_{lm},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill M_{lm}=& {\displaystyle \frac{1}{2}}\left[\left(\mathsf{\nabla }{O}_{lm}\right)·\widehat{{\mathit{m}}_l}+\widehat{{\mathit{m}}_l}·(\mathsf{\nabla }{O}_{lm})\right],\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill T_{lm}=& {\displaystyle \frac{1}{2}}\left[(\mathsf{\nabla }{O}_{lm})·\widehat{\mathit{t}}-\widehat{\mathit{t}}·(\mathsf{\nabla }{O}_{lm})\right],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill G_{lm}=& {\displaystyle \frac{1}{2}}\left[\left({\nabla }_{\alpha }{\nabla }_{\beta }{O}_{lm}\right){\widehat{g}}_l^{\alpha \beta }-{\left({\widehat{g}}_l^{\alpha \beta }\right)}^{†}\left({\nabla }_{\alpha }{\nabla }_{\beta }{O}_{lm}\right)\right],\hfill \end{array}$$

(4)

where $O_{lm}\left(\mathit{r}\right)$ is proportional to the spherical harmonics as follows:

$$\begin{array}{c}\hfill O_{lm}\left(\mathit{r}\right)=\sqrt{{\displaystyle \frac{4\pi }{2l+1}}}{r}^l{Y}_{lm}\left(\widehat{\mathit{r}}\right),\end{array}$$

(5)

with $\widehat{\mathit{r}}=\mathit{r}/r$. ${\widehat{\mathit{m}}}_l$, ${\widehat{\mathit{t}}}_l$, and ${\widehat{g}}_l^{\alpha \beta }$ are represented by

$$\begin{array}{cc}\hfill {\widehat{\mathit{m}}}_l& ={\displaystyle \frac{2\widehat{\mathit{l}}}{l+1}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\widehat{\mathit{t}}}_l& ={\displaystyle \frac{2}{(l+1)(l+2)}}(\mathit{r}\times \widehat{\mathit{l}}),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\widehat{g}}_l^{\alpha \beta }& ={\widehat{t}}_l^{\alpha }{\widehat{m}}_l^{\beta },\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \widehat{\mathit{l}}& =-i(\mathit{r}\times \mathsf{\nabla }).\hfill \end{array}$$

(9)

The expression of multipoles in spinful space is obtained by the addition rule of the above multipole operator and Pauli matrix in spin space [38].

We denote the monopole as $X_0$, dipole as $X_{1m}=(X_x,X_y,X_z)$, quadrupole as $X_{2m}=(X_u,X_v,X_{yz},X_{zx},X_{xy})$, and octupole as $X_{3m}=(X_{xyz},X_x^{\alpha },X_y^{\alpha },X_z^{\alpha },X_x^{\beta },X_y^{\beta },X_z^{\beta })$ for $X=Q,M,T,G$. The specific expressions of $O_{lm}$ up to rank 3 are given by

$$\begin{array}{cc}\hfill O_0& =1,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill (O_x,O_y,O_z)& =(x,y,z),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill O_u& ={\displaystyle \frac{1}{2}}(3z^2-r^2),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill O_v& ={\displaystyle \frac{\sqrt{3}}{2}}(x^2-y^2),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill (O_{yz},O_{zx},O_{xy})& =\sqrt{3}(yz,zx,xy),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill O_{xyz}& =\sqrt{15}xyz,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \left(O_x^{\alpha },O_y^{\alpha },O_z^{\alpha }\right)& ={\displaystyle \frac{1}{2}}\left(x(5x^2-3r^2),y(5y^2-3r^2),z(5z^2-3r^2)\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \left(O_x^{\beta },O_y^{\beta },O_z^{\beta }\right)& ={\displaystyle \frac{\sqrt{15}}{2}}\left(x(y^2-z^2),y(z^2-x^2),z(x^2-y^2)\right).\hfill \end{array}$$

(17)

The spatial inversion and time-reversal parities in each multipole are summarized in

Table 1. Four types of multipoles (MPs) up to rank 3. “#" in the sixth column represents the number of magnetic point groups to possess multipoles in the totally symmetric irreducible representation. The seventh column represents the band dispersion in the presence of multipoles. The eighth, ninth, and tenth columns denote the induced multipoles under the electric field $\mathit{E}$, magnetic field $\mathit{H}$, and electric current $\mathit{J}$.

MP	Rank	Notation	$\mathcal{P}$	$\mathcal{T}$	#	Band Dispersion	E	H	J
E	0	$Q_0$	$+1$	$+1$	122	1	$Q_{1m}$	$M_{1m}$	$T_{1m}$
E	1	$Q_{1m}$	$-1$	$+1$	31	$\mathit{k}\times \mathit{\sigma }$	$Q_0,G_{1m},Q_{2m}$	$M_0,T_{1m},M_{2m}$	$T_0,M_{1m},T_{2m}$
E	2	$Q_{2m}$	$+1$	$+1$	106	$O_{lm}\left(\mathit{k}\right)$	$Q_{1m},G_{2m},Q_{3m}$	$M_{1m},T_{2m},M_{3m}$	$T_{1m},M_{2m},T_{3m}$
E	3	$Q_{3m}$	$-1$	$+1$	58	$(\mathit{k}\times \mathit{\sigma })·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)$	$Q_{2m},G_{3m},Q_{4m}$	$M_{2m},T_{3m},M_{4m}$	$T_{2m},M_{3m},T_{4m}$
M	0	$M_0$	$-1$	$-1$	32	–	$M_{1m}$	$Q_{1m}$	$G_{1m}$
M	1	$M_{1m}$	$+1$	$-1$	31	$\mathit{\sigma }$	$M_0,T_{1m},M_{2m}$	$Q_0,G_{1m},Q_{2m}$	$G_0,Q_{1m},G_{2m}$
M	2	$M_{2m}$	$-1$	$-1$	42	–	$M_{1m},T_{2m},M_{3m}$	$Q_{1m},G_{2m},Q_{3m}$	$G_{1m},Q_{2m},G_{3m}$
M	3	$M_{3m}$	$+1$	$-1$	58	$\mathit{\sigma }·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)$	$M_{2m},T_{3m},M_{4m}$	$Q_{2m},G_{3m},Q_{4m}$	$G_{2m},Q_{3m},G_{4m}$
MT	0	$T_0$	$+1$	$-1$	32	–	$T_{1m}$	$G_{1m}$	$Q_{1m}$
MT	1	$T_{1m}$	$-1$	$-1$	31	$\mathit{k}$	$T_0,M_{1m},T_{2m}$	$G_0,Q_{1m},G_{2m}$	$Q_0,G_{1m},Q_{2m}$
MT	2	$T_{2m}$	$+1$	$-1$	42	$(\mathit{k}\times \mathit{\sigma })·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)$	$T_{1m},M_{2m},T_{3m}$	$G_{1m},Q_{2m},G_{3m}$	$Q_{1m},G_{2m},Q_{3m}$
MT	3	$T_{3m}$	$-1$	$-1$	58	$O_{lm}\left(\mathit{k}\right)$	$T_{2m},M_{3m},T_{4m}$	$G_{2m},Q_{3m},G_{4m}$	$Q_{2m},G_{3m},Q_{4m}$
ET	0	$G_0$	$-1$	$+1$	32	$\mathit{k}·\mathit{\sigma }$	$G_{1m}$	$T_{1m}$	$M_{1m}$
ET	1	$G_{1m}$	$+1$	$+1$	43	–	$G_0,Q_{1m},G_{2m}$	$T_0,M_{1m},T_{2m}$	$M_0,T_{1m},M_{2m}$
ET	2	$G_{2m}$	$-1$	$+1$	42	$\mathit{\sigma }·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)$	$G_{1m},Q_{2m},G_{3m}$	$T_{1m},M_{2m},T_{3m}$	$M_{1m},T_{2m},M_{3m}$
ET	3	$G_{3m}$	$+1$	$+1$	71	–	$G_{2m},Q_{3m},G_{4m}$	$T_{2m},M_{3m},T_{4m}$	$M_{2m},T_{3m},M_{4m}$

.

The four types of multipoles also describe the spin-split band structures and band deformations in momentum space. By reading $\mathit{r}$ with the wave vector $\mathit{k}$ in $O_{lm}\left(\mathit{r}\right)$, the momentum-space description of multipoles is given by [39]

$$\begin{array}{cc}\hfill & Q_{lm}\left(\mathit{k}\right)\equiv \left\{\begin{array}{cc}{\sigma }_0{O}_{lm}\left(\mathit{k}\right)\hfill & (l=0,2,4,6,\cdots )\hfill \\ (\mathit{k}\times \mathit{\sigma })·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)\hfill & (l=1,3,5,\cdots )\hfill \end{array}\right.\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & M_{lm}\left(\mathit{k}\right)\equiv \left\{\begin{array}{cc}0\hfill & (l=0,2,4,6,\cdots )\hfill \\ \mathit{\sigma }·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)\hfill & (l=1,3,5,\cdots )\hfill \end{array}\right.\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & T_{lm}\left(\mathit{k}\right)\equiv \left\{\begin{array}{cc}0\hfill & (l=0)\hfill \\ (\mathit{k}\times \mathit{\sigma })·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)\hfill & (l=2,4,6,\cdots )\hfill \\ {\sigma }_0{O}_{lm}\left(\mathit{k}\right)\hfill & (l=1,3,5,\cdots )\hfill \end{array}\right.\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & G_{lm}\left(\mathit{k}\right)\equiv \left\{\begin{array}{cc}\mathit{k}·\mathit{\sigma }\hfill & (l=0)\hfill \\ \mathit{\sigma }·{\mathsf{\nabla }}_{\mathit{k}}{O}_{lm}\left(\mathit{k}\right)\hfill & (l=2,4,6,\cdots )\hfill \\ 0\hfill & (l=1,3,5,\cdots )\hfill \end{array}\right.\hfill \end{array}$$

(21)

where ${\sigma }_0$ and $\mathit{\sigma }$ denote the identity and Pauli matrices in spin space, respectively. Thus, the even-rank (odd-rank) electric (magnetic toroidal) multipoles describe the symmetric (antisymmetric) band deformation without spin dependence. Meanwhile, the odd-rank (even-rank) electric (magnetic toroidal) multipoles and the even-rank (odd-rank) electric toroidal (magnetic) multipoles describe the antisymmetric (symmetric) spin splitting in the band structure.

The multipole description is also useful for understanding the cross-correlation phenomena when the external field or current is applied [39,40]. When the applied field/current is a rank-1 quantity, the rank of the induced multipoles is characterized by $l-1$, l, and $l+1$ for the rank-l multipole in antiferromagnets. For example, when the electric field $\mathit{E}$, which corresponds to $Q_{1m}$, is applied to the antiferromagnet with the magnetic toroidal dipole $T_{1m}$, any of the magnetic toroidal monopole $T_0$, magnetic dipole $M_{1m}$, or magnetic toroidal quadrupole $T_{2m}$ are induced depending on the field direction owing to the tensor product $Q_{1m}\otimes T_{1m}\to T_0\oplus M_{1m}\oplus T_{2m}$. Thus, the linear magneto-electric effect as a consequence of the cross-coupling between $M_{1m}$ and $Q_{1m}$ is expected under the $T_{1m}$ order. In addition, one finds that cross-correlations between $Q_{1m}$ and $T_0$ ($T_{2m}$) occur, which means that $T_0$ ($T_{2m}$) is induced by the electric field in antiferromagnets with $T_{1m}$. Similar cross-correlation phenomena are straightforwardly investigated for different fields/currents, such as the magnetic field $\mathit{H}$ corresponding to $M_{1m}$ and the electric current $\mathit{J}$ corresponding to $T_{1m}$. We summarize the correspondence among the multipoles, band dispersions, and couplings to $\mathit{E}$, $\mathit{H}$, and $\mathit{J}$ in Table 1.

3. Multipoles in Antiferromagnets

By using the four types of multipoles, we describe any complicated magnetic structures by the ferroic alignment of the multipoles. In order to exemplify this, we consider the magnetic point group $4/mmm{1}^{\prime }$. As shown in

Table 2. Irreducible representations (irreps) of four types of multipoles under the magnetic point group (MPG) $4/mmm{1}^{\prime }$ [36]. The superscript “−” of irrep indicates the parity with respect to the antiunitary time-reversal operation. MP in the second column represents the multipole, which corresponds to the order parameter under magnetic orderings. From the fourth to seventh columns, possible $\mathit{q}=\mathit{0}$ magnetic orderings for the Wyckoff positions under the space group $P4/mmm$ are shown [35].

Irrep	MP	Subgroup	$1\mathit{a}$ (0 0 0)	$2\mathit{f}$ (0 $1/2$ 0)	$4\mathit{l}$ (x 0 0)	$8\mathit{p}$ (x y 0)
${\mathrm{A}}_{1\mathrm{g}}^{-}$	$T_0,T_u$	$4/mmm$	–	–	–	✓
${\mathrm{A}}_{2\mathrm{g}}^{-}$	$M_z,M_z^{\alpha }$	$4/m{m}^{\prime }{m}^{\prime }$	✓	✓	✓	✓
${\mathrm{B}}_{1\mathrm{g}}^{-}$	$T_v$, $M_{xyz}$	$4^{\prime }/mm{m}^{\prime }$	–	–	–	✓
${\mathrm{B}}_{2\mathrm{g}}^{-}$	$T_{xy}$, $M_z^{\beta }$	$4^{\prime }/m{m}^{\prime }m$	–	✓	✓	✓
${\mathrm{E}}_{\mathrm{g}}^{-}$	$T_{yz},M_x,M_x^{\alpha },M_x^{\beta }$	$m{m}^{\prime }{m}^{\prime }$	✓	✓	✓	✓
	$T_{zx},M_y,M_y^{\alpha },M_y^{\beta }$	$m^{\prime }m{m}^{\prime }$	✓	✓	✓	✓
${\mathrm{A}}_{1\mathrm{u}}^{-}$	$M_0,M_u$	$4/m^{\prime }{m}^{\prime }{m}^{\prime }$	–	–	✓	✓
${\mathrm{A}}_{2\mathrm{u}}^{-}$	$T_z,T_z^{\alpha }$	$4/m^{\prime }mm$	–	–	✓	✓
${\mathrm{B}}_{1\mathrm{u}}^{-}$	$T_{xyz}$, $M_v$	$4^{\prime }/m^{\prime }{m}^{\prime }m$	–	–	✓	✓
${\mathrm{B}}_{2\mathrm{u}}^{-}$	$T_z^{\beta }$, $M_{xy}$	$4^{\prime }/m^{\prime }m{m}^{\prime }$	–	–	✓	✓
${\mathrm{E}}_{\mathrm{u}}^{-}$	$T_x,T_x^{\alpha },T_x^{\beta }$, $M_{yz}$	$m^{\prime }mm$	–	–	✓	✓
	$T_y,T_y^{\alpha },T_y^{\beta }$, $M_{zx}$	$m{m}^{\prime }m$	–	–	✓	✓

, all the irreducible representations without the time-reversal symmetry are characterized by either magnetic or magnetic toroidal multipoles [36]. For example, when the magnetic structure is characterized by the irreducible representation ${\mathrm{B}}_{1g}^{-}$, the magnetic toroidal quadrupole $T_v$ and magnetic octupole $M_{xyz}$ are induced. In such a situation, one expects the appearance of the symmetric spin-split band structure and electric-field-induced magnetic quadrupole from Table 1.

The correspondence between the irreducible representations and magnetic patterns is investigated for the space group $P4/mmm$. We show the possible multipole orderings when the ions with the magnetic moments are located at the $1a$, $2f$, $4l$, and $8p$ sites in Table 2 [35]. For the $1a$ site, the possible magnetic structure is a ferromagnetic one, which indicates that only the irreducible representations corresponding to the magnetic dipole $(M_x,M_y,M_z)$ are possible. When the site symmetry is lowered, other multipoles belonging to different irreducible representations are induced; the magnetic toroidal quadrupole $T_{xy}$ belonging to the irreducible representation ${\mathrm{B}}_{2g}^{-}$ is possible for the $2f$ site and odd-parity multipoles like $M_0$ and $T_z$ are possible for the $4l$ site. Furthermore, in the case of the $8p$ site, all the irreducible representations are possible. The real-space spin configurations belonging to the different irreducible representations in the $8p$ site are shown in Figure 2. One finds that unconventional multipole orderings, such as the magnetic toroidal monopole $T_0$ belonging to ${\mathrm{A}}_{1g}^{-}$ and magnetic toroidal quadrupole $T_v$ belonging to ${\mathrm{B}}_{1g}^{-}$, are constructed various collinear and noncollinear spin configurations.

The above example indicates that various multipoles are activated in the antiferromagnetic structures according to their magnetic point groups. Indeed, 32, 31, 42, and 58 out of 122 magnetic point groups possess $M_0$, $M_{1m}$, $M_{2m}$, and $M_{3m}$, respectively. Similarly, 122, 31, 106, and 58 magnetic point groups possess $Q_0$, $Q_{1m}$, $Q_{2m}$, and $Q_{3m}$, respectively, 32, 31, 42, and 58 magnetic point groups possess $T_0$, $T_{1m}$, $T_{2m}$, and $T_{3m}$, respectively, and 32, 43, 42, and 71 magnetic point groups possess $G_0$, $G_{1m}$, $G_{2m}$, and $G_{3m}$, respectively. In order to show the active multipoles under 122 magnetic point groups, we classify them in each magnetic point group, as shown in

Table 3. Classification of four types of multipoles up to rank 3 under the 122 magnetic point groups. The number in the column for the multipoles $X_{lm}$ ($X=Q,M,T,G$) represents the number of active multipoles belonging to the totally symmetric irreducible representation. The materials taken from MAGNDATA [41] are also listed in the rightmost column.

		$(\mathcal{P},\mathcal{T})=(+1,+1)$					$(-1,+1)$					$(+1,-1)$					$(-1,-1)$
	MPG	${\mathit{Q}}_{\mathit{0}}$	${\mathit{Q}}_{\mathit{2}\mathit{m}}$	${\mathit{G}}_{\mathit{1}\mathit{m}}$	${\mathit{G}}_{\mathit{3}\mathit{m}}$		${\mathit{G}}_{\mathit{0}}$	${\mathit{G}}_{\mathit{2}\mathit{m}}$	${\mathit{Q}}_{\mathit{1}\mathit{m}}$	${\mathit{Q}}_{\mathit{3}\mathit{m}}$		${\mathit{T}}_{\mathit{0}}$	${\mathit{T}}_{\mathit{2}\mathit{m}}$	${\mathit{M}}_{\mathit{1}\mathit{m}}$	${\mathit{M}}_{\mathit{3}\mathit{m}}$		${\mathit{M}}_{\mathit{0}}$	${\mathit{M}}_{\mathit{2}\mathit{m}}$	${\mathit{T}}_{\mathit{1}\mathit{m}}$	${\mathit{T}}_{\mathit{3}\mathit{m}}$		Material
#1	1	1	5	3	7		1	5	3	7		1	5	3	7		1	5	3	7		Mn2ScSbO6 [42]
#2	${11}^{\prime }$	1	5	3	7		1	5	3	7												LiFeAs2O7 [43]
#3	$\overline{1}$	1	5	3	7							1	5	3	7							RbMnF4 [44]
#4	$\overline{1}{1}^{\prime }$	1	5	3	7																	CuMnO2 [45]
#5	${\overline{1}}^{\prime }$	1	5	3	7												1	5	3	7		MnPSe3 [46]
#6	2	1	3	1	3		1	3	1	3		1	3	1	3		1	3	1	3		LiFeP2O7 [47]
#7	${21}^{\prime }$	1	3	1	3		1	3	1	3												Yb2CoMnO6 [48]
#8	$2^{\prime }$	1	3	1	3		1	3	1	3			2	2	4			2	2	4		BaDy2O4 [49]
#9	m	1	3	1	3			2	2	4		1	3	1	3			2	2	4		Mn4Nb2O9 [50]
#10	$m{1}^{\prime }$	1	3	1	3			2	2	4												DyFeWO6 [51]
#11	$m^{\prime }$	1	3	1	3			2	2	4			2	2	4		1	3	1	3		ScFeO3 [52]
#12	$2/m$	1	3	1	3							1	3	1	3							Cu2OSO4 [53]
#13	$2/m{1}^{\prime }$	1	3	1	3																	CuSe2O5 [54]
#14	$2^{\prime }/m$	1	3	1	3													2	2	4		YbCl3 [55]
#15	$2/m^{\prime }$	1	3	1	3												1	3	1	3		Co2V2O7 [56]
#16	$2^{\prime }/m^{\prime }$	1	3	1	3								2	2	4							Mn3Ti2Te6 [57]
#17	222	1	2		1		1	2		1		1	2		1		1	2		1		FePO4 [58]
#18	${2221}^{\prime }$	1	2		1		1	2		1												AgNiO2 [59]
#19	$2^{\prime }{2}^{\prime }2$	1	2		1		1	2		1			1	1	2			1	1	2		VNb3S6 [60]
#20	$mm2$	1	2		1			1	1	2		1	2		1			1	1	2		FeSb2O4 [61]
#21	$mm{21}^{\prime }$	1	2		1			1	1	2												EuNiO3 [62]
#22	$m^{\prime }m{2}^{\prime }$	1	2		1			1	1	2			1	1	2			1	1	2		CaBaCo4O7 [63]
#23	$m^{\prime }{m}^{\prime }2$	1	2		1			1	1	2			1	1	2		1	2		1		$\alpha$-Cu2V2O7 [64]
#24	$mmm$	1	2		1							1	2		1							$\alpha$-Mn2O3 [65]
#25	$mmm{1}^{\prime }$	1	2		1																	BaFe2As2 [66]
#26	$mm{m}^{\prime }$	1	2		1													1	1	2		U3Ru4Al12 [67]
#27	$m^{\prime }{m}^{\prime }m$	1	2		1								1	1	2							NiF2 [68]
#28	$m^{\prime }{m}^{\prime }{m}^{\prime }$	1	2		1												1	2		1		TbB4 [69]
#29	4	1	1	1	1		1	1	1	1		1	1	1	1		1	1	1	1		Ce5TeO8 [70]
#30	${41}^{\prime }$	1	1	1	1		1	1	1	1												CeAuAl3 [71]
#31	$4^{\prime }$	1	1	1	1		1	1	1	1			2		2			2		2
#32	$\overline{4}$	1	1	1	1			2		2		1	1	1	1			2		2
#33	$\overline{4}{1}^{\prime }$	1	1	1	1			2		2
#34	${\overline{4}}^{\prime }$	1	1	1	1			2		2			2		2		1	1	1	1		CsCoF4 [72]
#35	$4/m$	1	1	1	1							1	1	1	1							Mn3CuN [73]
#36	$4/m{1}^{\prime }$	1	1	1	1																	Sr2FeOsO6 [74]
#37	$4^{\prime }/m$	1	1	1	1								2		2
#38	$4/m^{\prime }$	1	1	1	1												1	1	1	1		TlFe1.6Se2 [75]
#39	$4^{\prime }/m^{\prime }$	1	1	1	1													2		2		KOsO4 [76]
#40	422	1	1				1	1				1	1				1	1				Ho2Ge2O7 [77]
#41	${4221}^{\prime }$	1	1				1	1														Ba(TiO)Cu4(PO4)4 [78]
#42	$4^{\prime }{22}^{\prime }$	1	1				1	1					1		1			1		1		Er2Ge2O7 [79]
#43	${42}^{\prime }{2}^{\prime }$	1	1				1	1						1	1				1	1		Nd5Si4 [80]
#44	$4mm$	1	1						1	1		1	1						1	1
#45	$4mm{1}^{\prime }$	1	1						1	1												CeRhGe3 [81]
#46	$4^{\prime }m{m}^{\prime }$	1	1						1	1			1		1			1		1
#47	$4m^{\prime }{m}^{\prime }$	1	1						1	1				1	1		1	1				CeIrGe3 [82]
#48	$\overline{4}2m$	1	1					1		1		1	1					1		1		Ba2MnSi2O7 [83]
#49	$\overline{4}2m{1}^{\prime }$	1	1					1		1												GeCu2O4 [84]
#50	${\overline{4}}^{\prime }m{2}^{\prime }$	1	1					1		1			1		1				1	1
#51	${\overline{4}}^{\prime }2m^{\prime }$	1	1					1		1			1		1		1	1				Ce4Sb3 [85]
#52	$\overline{4}{2}^{\prime }{m}^{\prime }$	1	1					1		1				1	1			1		1		EuCr2As2 [86]
#53	$4/mmm$	1	1									1	1									CdYb2S4 [87]
#54	$4/mmm{1}^{\prime }$	1	1																			EuMn2Si2 [88]
#55	$4/m^{\prime }mm$	1	1																1	1		Co3Al2Si3O12 [89]
#56	$4^{\prime }/mm{m}^{\prime }$	1	1										1		1							CoF2 [90]
#57	$4^{\prime }/m^{\prime }{m}^{\prime }m$	1	1															1		1		BaMn2Bi2 [91]
#58	$4/m{m}^{\prime }{m}^{\prime }$	1	1											1	1							Ho2Ru2O7 [92]
#59	$4/m^{\prime }{m}^{\prime }{m}^{\prime }$	1	1														1	1				GdB4 [93]
#60	3	1	1	1	3		1	1	1	3		1	1	1	3		1	1	1	3		Cu2OSeO3 [94]
#61	${31}^{\prime }$	1	1	1	3		1	1	1	3												RbFe(MoO4)2 [95]
#62	$\overline{3}$	1	1	1	3							1	1	1	3							NiN2O6 [96]
#63	$\overline{3}{1}^{\prime }$	1	1	1	3																	LaMn3V4O12 [97]
#64	${\overline{3}}^{\prime }$	1	1	1	3												1	1	1	3		MgMnO3 [98]
#65	32	1	1		1		1	1		1		1	1		1		1	1		1		La0.33Sr0.67FeO3 [99]
#66	${321}^{\prime }$	1	1		1		1	1		1												DyFe3(B3)4 [100]
#67	${32}^{\prime }$	1	1		1		1	1		1				1	1				1	1		BaCu3V2O8(OD)2 [101]
#68	$3m$	1	1		1				1	1		1	1		1				1	1		PbNiO3 [102]
#69	$3m{1}^{\prime }$	1	1		1				1	1												Ba3Nb2NiO9 [103]
#70	$3m^{\prime }$	1	1		1				1	1				1	1		1	1		1		CrSe [104]
#71	$\overline{3}m$	1	1		1							1	1		1							Li2MnTeO6 [105]
#72	$\overline{3}m{1}^{\prime }$	1	1		1																	SrRu2O6 [106]
#73	${\overline{3}}^{\prime }m$	1	1		1														1	2		Ca2YZr2Fe3O12 [107]
#74	${\overline{3}}^{\prime }{m}^{\prime }$	1	1		1												1	1		1		Na2MnTeO6 [108]
#75	$\overline{3}{m}^{\prime }$	1	1		1									1	2							Co3Sn2S2 [109]
#76	6	1	1	1	1		1	1	1	1		1	1	1	1		1	1	1	1		BaCoSiO4 [110]
#77	${61}^{\prime }$	1	1	1	1		1	1	1	1
#78	$6^{\prime }$	1	1	1	1		1	1	1	1					1					1		YMnO3 [111]
#79	$\overline{6}$	1	1	1	1					1		1	1	1	1					1
#80	$\overline{6}{1}^{\prime }$	1	1	1	1					1												ErAuIn [112]
#81	${\overline{6}}^{\prime }$	1	1	1	1					1					1		1	1	1	1		Tb14Ag51 [113]
#82	$6/m$	1	1	1	1							1	1	1	1							FeF3 [114]
#83	$6/m{1}^{\prime }$	1	1	1	1
#84	$6^{\prime }/m$	1	1	1	1															2		K2Mn3(VO4)2CO3 [115]
#85	$6/m^{\prime }$	1	1	1	1												1	1	1	1		U14Au51 [116]
#86	$6^{\prime }/m^{\prime }$	1	1	1	1										2
#87	622	1	1				1	1				1	1				1	1
#88	${6221}^{\prime }$	1	1				1	1														ScMn6Ge6 [117]
#89	$6^{\prime }{22}^{\prime }$	1	1				1	1							1					1
#90	${62}^{\prime }{2}^{\prime }$	1	1				1	1						1	1				1	1		EuIn2As2 [118]
#91	$6mm$	1	1						1	1		1	1						1	1		HoMnO3 [119]
#92	$6mm{1}^{\prime }$	1	1						1	1
#93	$6^{\prime }m{m}^{\prime }$	1	1						1	1					1					1		Co2Mo3O8 [120]
#94	$6m^{\prime }{m}^{\prime }$	1	1						1	1				1	1		1	1				LuFeO3 [121]
#95	$\overline{6}m2$	1	1							1		1	1							1		Ba3CoSb2O9 [122]
#96	$\overline{6}m{21}^{\prime }$	1	1							1												CsCr0.94Fe0.06F4 [123]
#97	${\overline{6}}^{\prime }{m}^{\prime }2$	1	1							1					1		1	1				UNiGa [124]
#98	${\overline{6}}^{\prime }m{2}^{\prime }$	1	1							1					1				1	1		CsFeCl3 [125]
#99	$\overline{6}{m}^{\prime }{2}^{\prime }$	1	1							1				1	1					1		HoPdIn [126]
#100	$6/mmm$	1	1									1	1
#101	$6/mmm{1}^{\prime }$	1	1																			FeGe [127]
#102	$6/m^{\prime }mm$	1	1																1	1
#103	$6^{\prime }/mm{m}^{\prime }$	1	1																	1
#104	$6^{\prime }/m^{\prime }m{m}^{\prime }$	1	1												1							CsCoCl3 [128]
#105	$6/m{m}^{\prime }{m}^{\prime }$	1	1											1	1							Fe2.71GeTe2 [129]
#106	$6/m^{\prime }{m}^{\prime }{m}^{\prime }$	1	1														1	1
#107	23	1			1		1			1		1			1		1			1		Mn3IrSi [130]
#108	${231}^{\prime }$	1			1		1			1												MnGe [131]
#109	$m\overline{3}$	1			1							1			1							MnTe2 [132]
#110	$m\overline{3}{1}^{\prime }$	1			1																	Au72Al14Tb14 [133]
#111	$m^{\prime }{\overline{3}}^{\prime }$	1			1												1			1
#112	432	1					1					1					1					SrCuTe2O6 [134]
#113	${4321}^{\prime }$	1					1
#114	$4^{\prime }{32}^{\prime }$	1					1								1					1		BaCuTe2O6 [135]
#115	$\overline{4}3m$	1								1		1								1
#116	$\overline{4}3m{1}^{\prime }$	1								1												Gd2Ti2O7 [136]
#117	${\overline{4}}^{\prime }3m^{\prime }$	1								1					1		1
#118	$m\overline{3}m$	1										1
#119	$m\overline{3}m{1}^{\prime }$	1																				NdZn [137]
#120	$m^{\prime }{\overline{3}}^{\prime }m$	1																		1
#121	$m\overline{3}{m}^{\prime }$	1													1							Tb3Ga5O12 [138]
#122	$m^{\prime }{\overline{3}}^{\prime }{m}^{\prime }$	1															1

. It is noted that magnetic structures without the breaking of the time-reversal symmetry are possible when the $\mathit{q}\ne \mathit{0}$ state is considered. The typical magnetic materials from MAGNDATA [41] are also listed. This table is useful for seeing which types of multipoles are present in magnetic materials.

4. Cross-Correlations in Antiferromagnets with Multipoles

In this section, we briefly show the expected cross-correlations and transports in magnetic materials with multipoles: the electric dipole $Q_{1m}$ in Section 4.1, the electric quadrupole $Q_{2m}$ in Section 4.2, the electric octupole $Q_{3m}$ in Section 4.3, the magnetic monopole $M_0$ in Section 4.4, the magnetic dipole $M_{1m}$ in Section 4.5, the magnetic quadrupole $M_{2m}$ in Section 4.6, the magnetic octupole $M_{3m}$ in Section 4.7, the magnetic toroidal monopole $T_0$ in Section 4.8, the magnetic toroidal dipole $T_{1m}$ in Section 4.9, the magnetic toroidal quadrupole $T_{2m}$ in Section 4.10, the magnetic toroidal octupole $T_{3m}$ in Section 4.11, the electric toroidal monopole $G_0$ in Section 4.12, the electric toroidal dipole $G_{1m}$ in Section 4.13, the electric toroidal quadrupole $G_{2m}$ in Section 4.14, and the electric toroidal octupole $G_{3m}$ in Section 4.15.

4.1. Electric Dipole

There are 31 magnetic point groups accompanying the electric dipole $Q_{1m}$, as shown in

Table 4. Classification of magnetic point groups (MPGs) with the electric dipole $Q_{1m}$ according to the presence and absence of the magnetic dipole $M_{1m}$ and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓
$mm2$, $4mm$, $3m$, $6mm$		✓
${11}^{\prime }$, ${21}^{\prime }$, $m{1}^{\prime }$, $mm{21}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, $4mm{1}^{\prime }$
$4^{\prime }m{m}^{\prime }$, ${31}^{\prime }$, $3m{1}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$, $6mm{1}^{\prime }$, $6^{\prime }m{m}^{\prime }$

. We classify 31 magnetic point groups in terms of the presence and absence of $M_{1m}$ and $T_{1m}$. Since $M_{1m}$ and $T_{1m}$ are coupled to the magnetic field and electric current, respectively, they can be used to control the domain of the antiferromagnetic state with $Q_{1m}$.

The magnetic materials with $Q_{1m}$ exhibit the antisymmetric spin polarization in the band structure, which corresponds to the Rashba-type spin–orbit coupling in the form of $\mathit{k}\times \mathit{\sigma }$. Accordingly, the transverse Edelstein effect, where the magnetization is induced by the electric current in a perpendicular way, occurs. The materials with $Q_{1m}$ also show the nonlinear Hall effect based on the Berry curvature dipole mechanism [139]. Moreover, the system with $Q_{1m}$ shows the magnetic toroidal moment in the external magnetic field. One of the candidate materials is CrSe, where the symmetry reduces from $P{6}_3/mmc$ to $P31m^{\prime }$ (magnetic point group $3m^{\prime }$) by the magnetic phase transition [104]. In this material, the above physical phenomena are induced only below the critical temperature. In addition, the single domain can be obtained by performing magnetic field cooling.

4.2. Electric Quadrupole

There are 106 magnetic point groups accompanying the electric quadrupole $Q_{2m}$. We classify them in terms of the presence and absence of $Q_{1m}$, $M_{1m}$, and $T_{1m}$, as shown in

Table 5. Classification of magnetic point groups (MPGs) with the electric quadrupole $Q_{2m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓		✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓	✓
${11}^{\prime }$, ${21}^{\prime }$, $m{1}^{\prime }$, $mm{21}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, $4mm{1}^{\prime }$	✓
$4^{\prime }m{m}^{\prime }$, ${31}^{\prime }$, $3m{1}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$, $6mm{1}^{\prime }$, $6^{\prime }m{m}^{\prime }$	✓
$\overline{1}$, $2/m$, $2^{\prime }/m^{\prime }$, $m^{\prime }{m}^{\prime }m$, $\overline{4}$, $4/m$, $\overline{4}{2}^{\prime }{m}^{\prime }$		✓
$4/m{m}^{\prime }{m}^{\prime }$, $\overline{3}$, $\overline{3}{m}^{\prime }$, $\overline{6}$, $6/m$, $\overline{6}{m}^{\prime }{2}^{\prime }$, $6/m{m}^{\prime }{m}^{\prime }$		✓
${\overline{1}}^{\prime }$, $2^{\prime }/m$, $2/m^{\prime }$, $mm{m}^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$			✓
$4/m^{\prime }mm$, ${\overline{3}}^{\prime }$, ${\overline{3}}^{\prime }m$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$, ${\overline{6}}^{\prime }m{2}^{\prime }$, $6/m^{\prime }mm$			✓
$\overline{1}{1}^{\prime }$, $2/m{1}^{\prime }$, 222, ${2221}^{\prime }$, $mmm$, $mmm{1}^{\prime }$, $m^{\prime }{m}^{\prime }{m}^{\prime }$
$\overline{4}{1}^{\prime }$, $4/m{1}^{\prime }$, $4^{\prime }/m$, $4^{\prime }/m^{\prime }$, 422, ${4221}^{\prime }$, $4^{\prime }{22}^{\prime }$, $\overline{4}2m$
$\overline{4}2m{1}^{\prime }$, ${\overline{4}}^{\prime }2m^{\prime }$, $4/mmm$, $4/mmm{1}^{\prime }$, $4^{\prime }/mm{m}^{\prime }$
$4^{\prime }/m^{\prime }{m}^{\prime }m$, $4/m^{\prime }{m}^{\prime }{m}^{\prime }$, $\overline{3}{1}^{\prime }$, 32, ${321}^{\prime }$, $\overline{3}m$
$\overline{3}m{1}^{\prime }$, ${\overline{3}}^{\prime }{m}^{\prime }$, $\overline{6}{1}^{\prime }$, $6/m{1}^{\prime }$, $6^{\prime }/m$, $6^{\prime }/m^{\prime }$, 622
${6221}^{\prime }$, $6^{\prime }{22}^{\prime }$, $\overline{6}m2$, $\overline{6}m{21}^{\prime }$, ${\overline{6}}^{\prime }{m}^{\prime }2$, $6/mmm$
$6/mmm{1}^{\prime }$, $6^{\prime }/mm{m}^{\prime }$, $6^{\prime }/m^{\prime }m{m}^{\prime }$, $6/m^{\prime }{m}^{\prime }{m}^{\prime }$

, where the presence of $Q_{1m}$ means that the single-domain formation can be controlled by applying the electric field when $Q_{1m}$ is simultaneously induced below the critical temperature. The nematic ordering belongs to this category.

Although $Q_{2m}$ has the same spatial inversion and time-reversal properties as the electric monopole $Q_0$, it exhibits intriguing electromagnetic responses according to its spatial anisotropy. In addition to the conjugate physical quantities, additional higher-rank multipoles are induced under external fields and currents. In other words, the responses against the fields and currents are different from the case with $Q_0$. For example, the electric toroidal quadrupole $G_{2m}$, the magnetic toroidal quadrupole $T_{2m}$, and the magnetic quadrupole $M_{2m}$ are induced by the electric field, magnetic field, and electric current, respectively. In particular, the momentum-dependent spin splitting, which arises from $T_{2m}$, in addition to Zeeman-type uniform spin splitting, occurs in the magnetic field.

4.3. Electric Octupole

The electric octupole $Q_{3m}$ is included in 58 magnetic point groups, where 27 magnetic point groups do not possess $Q_{1m}$, as shown in

Table 6. Classification of magnetic point groups (MPGs) with the electric octupole $Q_{3m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$. The upper (lower) columns represent the magnetic point groups with (without) $Q_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓		✓
${11}^{\prime }$, ${21}^{\prime }$, $m{1}^{\prime }$, $mm{21}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, $4mm{1}^{\prime }$	✓
$4^{\prime }m{m}^{\prime }$, ${31}^{\prime }$, $3m{1}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$, $6mm{1}^{\prime }$, $6^{\prime }m{m}^{\prime }$	✓
$2^{\prime }{2}^{\prime }2$, ${32}^{\prime }$		✓	✓
$\overline{4}$, $\overline{4}{2}^{\prime }{m}^{\prime }$, $\overline{6}$, $\overline{6}{m}^{\prime }{2}^{\prime }$		✓
${\overline{4}}^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$, ${\overline{6}}^{\prime }$, ${\overline{6}}^{\prime }m{2}^{\prime }$			✓
222, ${2221}^{\prime }$, $\overline{4}{1}^{\prime }$, $\overline{4}2m$, $\overline{4}2m{1}^{\prime }$, ${\overline{4}}^{\prime }2m^{\prime }$
32, ${321}^{\prime }$, $\overline{6}{1}^{\prime }$, $\overline{6}m2$, $\overline{6}m{21}^{\prime }$, ${\overline{6}}^{\prime }{m}^{\prime }2$
23, ${231}^{\prime }$, $\overline{4}3m$, $\overline{4}3m{1}^{\prime }$, ${\overline{4}}^{\prime }3m^{\prime }$

. Similarly to $Q_{1m}$, the magnetic materials with $Q_{3m}$ also exhibit the antisymmetric spin-split band structure including the Dresselhaus-type one in the form of $k_x(k_y^2-k_z^2){\sigma }_x+k_y(k_z^2-k_x^2){\sigma }_y+k_z(k_x^2-k_y^2){\sigma }_z$. Meanwhile, the magnetic octupole instead of the magnetic dipole (magnetization) is induced by the electric current in contrast to $Q_{1m}$. The candidate material is VNb₃S₆, where the symmetry reduction occurs from $P{6}_{3}22$ to $C{2}^{\prime }{2}^{\prime }{2}_1$ (magnetic point group $2^{\prime }{2}^{\prime }2$) [60]. Since $M_{1m}$ and $T_{1m}$ are simultaneously induced under $2^{\prime }{2}^{\prime }2$, either the magnetic field or electric current cooling can lead to the single domain of $Q_{3m}$.

4.4. Magnetic Monopole

Among 122 magnetic point groups, 32 magnetic point groups include the magnetic monopole $M_0$, as shown in

Table 7. Classification of magnetic point groups (MPGs) with the magnetic monopole $M_0$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $m^{\prime }$, 4, 3, 6	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓	✓
${\overline{1}}^{\prime }$, $2/m^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{3}}^{\prime }$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$			✓
222, $m^{\prime }{m}^{\prime }{m}^{\prime }$, 422, ${\overline{4}}^{\prime }2m^{\prime }$, $4/m^{\prime }{m}^{\prime }{m}^{\prime }$, 32, ${\overline{3}}^{\prime }{m}^{\prime }$
622, ${\overline{6}}^{\prime }{m}^{\prime }2$, $6/m^{\prime }{m}^{\prime }{m}^{\prime }$, 23, $m^{\prime }{\overline{3}}^{\prime }$, 432, ${\overline{4}}^{\prime }3m^{\prime }$, $m^{\prime }{\overline{3}}^{\prime }{m}^{\prime }$

. This becomes the origin of the linear longitudinal magneto-electric effect, where the magnetization $M_{1m}$ (electric polarization $Q_{1m}$) is induced in the electric-field (magnetic-field) direction [140,141,142,143]. TbB₄ (magnetic point group $m^{\prime }{m}^{\prime }{m}^{\prime }$) [69,144,145] and FePO₄ (magnetic point group 222) [58] are typical examples to exhibit physical phenomena related to magnetic monopoles.

4.5. Magnetic Dipole

The magnetic dipole $M_{1m}$ is included in 31 magnetic point groups, as shown in

Table 8. Classification of magnetic point groups (MPGs) with the magnetic dipole $M_{1m}$ according to the presence and absence of the electric dipole $Q_{1m}$ and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓
$\overline{1}$, $2/m$, $2^{\prime }/m^{\prime }$, $m^{\prime }{m}^{\prime }m$, $\overline{4}$, $4/m$, $\overline{4}{2}^{\prime }{m}^{\prime }$
$4/m{m}^{\prime }{m}^{\prime }$, $\overline{3}$, $\overline{3}{m}^{\prime }$, $\overline{6}$, $6/m$, $\overline{6}{m}^{\prime }{2}^{\prime }$, $6/m{m}^{\prime }{m}^{\prime }$

[146,147,148]. Since $M_{1m}$ corresponds to the ferroic alignment of spin, i.e., the ferromagnetic state, the Berry curvature occurs in the band structure, which becomes the origin of the physical phenomena under magnetization or magnetic field occur, such as the Hall effect, Nernst effect, and magneto-optical Kerr effect [149,150]. Meanwhile, the magnetic point groups with $M_{1m}$ are also realized in the antiferromagnetic structure even without (or with a negligibly small) net magnetization, as found in the noncollinear antiferromagnets like Mn₃Sn [25,26,151,152,153] and collinear antiferromagnets [19,154] like LaMO₃ ($M=$ Cr, Mn, and Fe) [15], the bilayer MnPSe₃ [16], $\kappa$-type organic conductors [18], and so on. Recently, different types of magnetic dipole have been introduced such as the “anisotropic magnetic dipole” $M_{1m}^{\prime }$, which consists of the product of the electric quadrupole $Q_{2m}$ and conventional magnetic dipole (spin) $M_{1m}$, and they are regarded as a microscopic indicator of the presence of the anomalous Hall effect in antiferromagnets [21,155]. Since the symmetry conditions for $M_{1m}$ and $M_{1m}^{\prime }$ are the same as each other, the magnetic materials belonging to 31 magnetic point groups in Table 8 result in “ferromagnetic” physical phenomena irrespective of the ferromagnetic and antiferromagnetic structures.

In addition, the materials with $M_{1m}$ exhibit further cross-correlations. One of the examples is the switching response between the magnetic dipole and magnetic toroidal dipole ($T_{1m}$) of circularly polarized light. Another example is the electric-current-induced chirality (electric toroidal monopole $G_0$).

4.6. Magnetic Quadrupole

A total of 42 out of 122 magnetic point groups possess the magnetic quadrupole $M_{2m}$, as shown in

Table 9. Classification of magnetic point groups (MPGs) with the magnetic quadrupole $M_{2m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, the magnetic toroidal dipole $T_{1m}$, and the magnetic monopole $M_0$. The upper (lower) columns represent the magnetic point groups with (without) $M_0$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$	${\mathit{M}}_0$
1, 2, $m^{\prime }$, 4, 3, 6	✓	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓	✓		✓
${\overline{1}}^{\prime }$, $2/m^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{3}}^{\prime }$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$			✓	✓
222, $m^{\prime }{m}^{\prime }{m}^{\prime }$, 422, ${\overline{4}}^{\prime }2m^{\prime }$, $4/m^{\prime }{m}^{\prime }{m}^{\prime }$				✓
32, ${\overline{3}}^{\prime }{m}^{\prime }$, 622, ${\overline{6}}^{\prime }{m}^{\prime }2$, $6/m^{\prime }{m}^{\prime }{m}^{\prime }$				✓
$2^{\prime }$, m, $m^{\prime }m{2}^{\prime }$	✓	✓	✓
$mm2$	✓		✓
$2^{\prime }{2}^{\prime }2$		✓	✓
$4^{\prime }$, $4^{\prime }m{m}^{\prime }$	✓
$\overline{4}$, $\overline{4}{2}^{\prime }{m}^{\prime }$		✓
$2^{\prime }/m$, $mm{m}^{\prime }$			✓
$4^{\prime }/m^{\prime }$, $4^{\prime }{22}^{\prime }$, $\overline{4}2m$, $4^{\prime }/m^{\prime }{m}^{\prime }m$

; of these, 27 magnetic point groups accompany $M_0$, while the remaining 15 magnetic point groups do not have $M_0$. Since the physical properties of $M_{2m}$ are similar to those of $M_0$ owing to the same spatial inversion and time-reversal parities, it is desired to focus on the materials listed in the lower columns when the pure nature of the magnetic quadrupole is investigated. Similarly to the materials with $M_0$, the materials with $M_{2m}$ show the linear magneto-electric effect [156,157,158], as found in Cr₂O₃ [159,160,161,162,163]. In addition, $M_{2m}$ becomes the microscopic origin of the magneto-piezoelectric effect found in EuMnBi [164,165], which arises from the spin-orbital-momentum locking in the band structure [166], and the intrinsic nonlinear Hall effect [167,168,169]. KOsO₄ (magnetic point group $4^{\prime }/m^{\prime }$) [76,170] and Er₂Ge₂O₇ (magnetic point group $4^{\prime }{22}^{\prime }$) [79] are other candidate materials with $M_{2m}$ but without $Q_{1m}$, $M_{1m}$, or $T_{1m}$.

4.7. Magnetic Octupole

There are 58 magnetic point groups to possess the magnetic octupole $M_{3m}$, as shown in

Table 10. Classification of magnetic point groups (MPGs) with the magnetic octupole $M_{3m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$. The upper (lower) columns represent the magnetic point groups with (without) $M_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $4m^{\prime }{m}^{\prime }$, $3m^{\prime }$, $6m^{\prime }{m}^{\prime }$	✓	✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓	✓
$\overline{1}$, $2/m$, $2^{\prime }/m^{\prime }$, $m^{\prime }{m}^{\prime }m$, $\overline{4}$, $4/m$, $\overline{4}{2}^{\prime }{m}^{\prime }$		✓
$4/m{m}^{\prime }{m}^{\prime }$, $\overline{3}$, $\overline{3}{m}^{\prime }$, $\overline{6}$, $6/m$, $\overline{6}{m}^{\prime }{2}^{\prime }$, $6/m{m}^{\prime }{m}^{\prime }$		✓
$mm2$, $3m$	✓		✓
$4^{\prime }$, $4^{\prime }m{m}^{\prime }$, $6^{\prime }$, $6^{\prime }m{m}^{\prime }$	✓
${\overline{4}}^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$, ${\overline{6}}^{\prime }$, ${\overline{6}}^{\prime }m{2}^{\prime }$			✓
222, $mmm$, $4^{\prime }/m$, $4^{\prime }{22}^{\prime }$, ${\overline{4}}^{\prime }2m^{\prime }$, $4^{\prime }/mm{m}^{\prime }$
32, $\overline{3}m$, $6^{\prime }/m^{\prime }$, $6^{\prime }{22}^{\prime }$, ${\overline{6}}^{\prime }{m}^{\prime }2$, $6^{\prime }/m^{\prime }m{m}^{\prime }$
23, $m\overline{3}$, $4^{\prime }{32}^{\prime }$, ${\overline{4}}^{\prime }3m^{\prime }$, $m\overline{3}{m}^{\prime }$

. Among them, 27 magnetic point groups are characterized by the magnetic point groups without $M_{1m}$; the pure effect of $M_{3m}$ is expected. The materials with magnetic octupole $M_{3m}$ induce rank-2, rank-3, and rank-4 multipoles when external fields and currents are applied. Since $T_{3m}$ ($Q_{2m}$) are induced for the applied electric field (magnetic field), the band structure is modulated in an antisymmetric (symmetric) way. In addition, $M_{3m}$ becomes the microscopic origin of the magnetic-field-induced striction, i.e., magnetostriction [171]. Er₂Ge₂O₇ (magnetic point group $4^{\prime }{22}^{\prime }$) [79] is one of the candidates to have $M_{3m}$.

4.8. Magnetic Toroidal Monopole

Magnetic toroidal monopole $T_0$ is included in the 32 magnetic point groups, as shown in

Table 11. Classification of magnetic point groups (MPGs) with the magnetic toroidal monopole $T_0$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, m, 4, 3, 6	✓	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓		✓
$\overline{1}$, $2/m$, $\overline{4}$, $4/m$, $\overline{3}$, $\overline{6}$, $6/m$		✓
222, $mmm$, 422, $\overline{4}2m$, $4/mmm$, 32, $\overline{3}m$, 622
$\overline{6}m2$, $6/mmm$, 23, $m\overline{3}$, 432, $\overline{4}3m$, $m\overline{3}m$

. Since $T_0$ corresponds to a time-reversal-odd scalar quantity, it leads to the time-reversal switching responses, such as the electric-field-induced magnetic toroidal dipole $T_{1m}$ and the magnetic-field-induced electric toroidal dipole $G_{1m}$ [172,173]. Such cross-correlations have recently been observed in Co₂SiO₄ [174]. Another candidate is Ho₂Ge₂O₇ under magnetic point group 422 [77], for example.

4.9. Magnetic Toroidal Dipole

Magnetic toroidal dipole $T_{1m}$ is allowed for 31 magnetic point groups, as shown in

Table 12. Classification of magnetic point groups (MPGs) with the magnetic toroidal dipole $T_{1m}$ according to the presence and absence of the electric dipole $Q_{1m}$ and the magnetic dipole $M_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓
${\overline{1}}^{\prime }$, $2^{\prime }/m$, $2/m^{\prime }$, $mm{m}^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$
$4/m^{\prime }mm$, ${\overline{3}}^{\prime }$, ${\overline{3}}^{\prime }m$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$, ${\overline{6}}^{\prime }m{2}^{\prime }$, $6/m^{\prime }mm$

[175]. It is the most typical multipole degree of freedom to exhibit the cross-correlation phenomena in magnetic materials. The representative phenomenon is the linear transverse magneto-electric effect in magnetic insulators [176,177,178,179,180], although a similar phenomenon has been observed in magnetic metals like UNi₄B [181,182,183,184,185]. In addition, as $T_{1m}$ corresponds to the time-reversal-odd polar vector, it gives rise to asymmetric band deformation without the spin dependence. Accordingly, $T_{1m}$ induces further cross-correlations and transports, such as nonreciprocal transport [186,187,188,189,190], asymmetric magnon excitations [191,192,193], and nonlinear spin Hall/Nernst effect [194,195].

4.10. Magnetic Toroidal Quadrupole

There are 42 magnetic point groups to possess the magnetic toroidal quadrupole $T_{2m}$, as shown in

Table 13. Classification of magnetic point groups (MPGs) with the magnetic toroidal quadrupole $T_{2m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, the magnetic toroidal dipole $T_{1m}$, and the magnetic toroidal monopole $T_0$. The upper (lower) columns represent the magnetic point groups with (without) $T_0$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$	${\mathit{T}}_0$
1, 2, m, 4, 3, 6	✓	✓	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓		✓	✓
$\overline{1}$, $2/m$, $\overline{4}$, $4/m$, $\overline{3}$, $\overline{6}$, $6/m$		✓		✓
222, $mmm$, 422, $\overline{4}2m$, $4/mmm$, 32				✓
$\overline{3}m$, 622, $\overline{6}m2$, $6/mmm$				✓
$2^{\prime }$, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$	✓	✓	✓
$m^{\prime }{m}^{\prime }2$	✓	✓
$2^{\prime }{2}^{\prime }2$		✓	✓
$4^{\prime }$, $4^{\prime }m{m}^{\prime }$	✓
$2^{\prime }/m^{\prime }$, $m^{\prime }{m}^{\prime }m$		✓
${\overline{4}}^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$			✓
$4^{\prime }/m$, $4^{\prime }{22}^{\prime }$, ${\overline{4}}^{\prime }2m^{\prime }$, $4^{\prime }/mm{m}^{\prime }$

; 27 out of 42 magnetic point groups possess $T_0$, whereas 15 magnetic point groups do not. The materials with $T_{2m}$ show the momentum-dependent symmetric spin-split band structure, which results in the directional-dependent spin current generation [196,197]. Moreover, in the materials with $T_{2m}$, the ferroaxial nature is induced when the magnetic field is applied. One of the candidate materials is CoF₂ under magnetic point group $4^{\prime }/mm{m}^{\prime }$ [90], although the domain control might be difficult in this material owing to no simple conjugate fields to $T_{2m}$. In this context, Mn₃Ti₂Te₆ under magnetic point group $2^{\prime }/m^{\prime }$ [57] is another candidate material, where the magnetic field cooling enables us to select a single domain.

4.11. Magnetic Toroidal Octupole

A total of 58 magnetic point groups possess the magnetic toroidal octupole $T_{3m}$, where 31 magnetic point groups also possess $T_{1m}$, as shown in

Table 14. Classification of magnetic point groups (MPGs) with the magnetic toroidal octupole $T_{3m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$. The upper (lower) columns represent the magnetic point groups with (without) $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$, 4, 3, 6	✓	✓	✓
$mm2$, $4mm$, $3m$, $6mm$	✓		✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓	✓
${\overline{1}}^{\prime }$, $2^{\prime }/m$, $2/m^{\prime }$, $mm{m}^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$			✓
$4/m^{\prime }mm$, ${\overline{3}}^{\prime }$, ${\overline{3}}^{\prime }m$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$, ${\overline{6}}^{\prime }m{2}^{\prime }$, $6/m^{\prime }mm$			✓
$m^{\prime }{m}^{\prime }2$, $3m^{\prime }$	✓	✓
$4^{\prime }$, $4^{\prime }m{m}^{\prime }$, $6^{\prime }$, $6^{\prime }m{m}^{\prime }$	✓
$\overline{4}$, $\overline{4}{2}^{\prime }{m}^{\prime }$, $\overline{6}$, $\overline{6}{m}^{\prime }{2}^{\prime }$		✓
222, $m^{\prime }{m}^{\prime }{m}^{\prime }$, $4^{\prime }/m^{\prime }$, $4^{\prime }{22}^{\prime }$, $\overline{4}2m$, $4^{\prime }/m^{\prime }{m}^{\prime }m$
32, ${\overline{3}}^{\prime }{m}^{\prime }$, $6^{\prime }/m$, $6^{\prime }{22}^{\prime }$, $\overline{6}m2$
$6^{\prime }/mm{m}^{\prime }$, 23, $m^{\prime }{\overline{3}}^{\prime }$, $4^{\prime }{32}^{\prime }$, $\overline{4}3m$, $m^{\prime }{\overline{3}}^{\prime }m$

. Since $T_{3m}$ induces asymmetric band modulation similar to $T_{1m}$, the materials with $T_{3m}$ show nonreciprocal transport against the electric field and thermal gradient [198,199,200]. One of the potential candidates is CrSe under the magnetic point group $3m^{\prime }$ [104], where physical phenomena related to the MT octupole can be controlled by both electric and magnetic fields.

4.12. Electric Toroidal Monopole

A total of 32 magnetic point groups possess the electric toroidal monopole $G_0$, as shown in

Table 15. Classification of magnetic point groups (MPGs) with the electric toroidal monopole $G_0$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, 4, 3, 6	✓	✓	✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓	✓
${11}^{\prime }$, ${21}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, ${31}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$	✓
222, ${2221}^{\prime }$, 422, ${4221}^{\prime }$, $4^{\prime }{22}^{\prime }$, 32, ${321}^{\prime }$, 622
${6221}^{\prime }$, $6^{\prime }{22}^{\prime }$, 23, ${231}^{\prime }$, 432, ${4321}^{\prime }$, $4^{\prime }{32}^{\prime }$

. As $G_0$ is characterized as the time-reversal-even pseudoscalar quantity, it corresponds to the chirality [201,202,203], which becomes the microscopic origin of the hedgehog-type antisymmetric spin–orbit coupling $\mathit{k}·\mathit{\sigma }$, the longitudinal Edelstein effect [204,205,206], and the electrical magnetochiral effect [207]. Although such a chirality is usually accompanied by lattice structures without the mirror and spatial inversion symmetries, it can be generated by the magnetic phase transition when $G_0$ additionally belongs to the totally symmetric irreducible representation below the critical temperature. For example, FePO₄ under magnetic point group 222 [58], ScMn₆Ge₆ under magnetic point group ${6221}^{\prime }$ [117], and La_0.33Sr_0.67FeO₃ under magnetic point group 32 [99] belong to the materials showing the $G_0$ property driven by the magnetic phase transition.

4.13. Electric Toroidal Dipole

The electric toroidal dipole $G_{1m}$ is included in 43 magnetic point groups, as shown in

Table 16. Classification of magnetic point groups (MPGs) with the electric toroidal dipole $G_{1m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, and the magnetic toroidal dipole $T_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, 4, 3, 6	✓	✓	✓
${11}^{\prime }$, ${21}^{\prime }$, $m{1}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, ${31}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$	✓
$\overline{1}$, $2/m$, $2^{\prime }/m^{\prime }$, $\overline{4}$, $4/m$, $\overline{3}$, $\overline{6}$, $6/m$		✓
${\overline{1}}^{\prime }$, $2^{\prime }/m$, $2/m^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{3}}^{\prime }$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$			✓
$\overline{1}{1}^{\prime }$, $2/m{1}^{\prime }$, $\overline{4}{1}^{\prime }$, $4/m{1}^{\prime }$, $4^{\prime }/m$, $4^{\prime }/m^{\prime }$
$\overline{3}{1}^{\prime }$, $\overline{6}{1}^{\prime }$, $6/m{1}^{\prime }$, $6^{\prime }/m$, $6^{\prime }/m^{\prime }$

. The electric toroidal dipole has recently been attracted since it brings about an unconventional electronic state, invariant under both spatial inversion and time-reversal operations; its ordering is referred to as ferro-rotational order or ferro-axial order [208,209,210,211,212], which have been observed in RbFe(MoO₄)₂ [210,213] and NiTiO₃ [213,214,215,216]. The materials with $G_{1m}$ exhibit transverse responses of the conjugate physical quantities, such as antisymmetric thermopolarization [217], longitudinal spin current generation [218,219], nonlinear transverse magnetization [220], and second-order nonlinear magnetostriction [221]. Such physical properties driven by $G_{1m}$ can be induced by the magnetic phase transition. FeF₃, which undergoes the phase transition from $P6/mmm$ to $P{6}_3/m$ (magnetic point group $6/m$), is one of the candidate materials to exhibit the property of $G_{1m}$ by magnetic phase transition [114]. Another candidate is a magnetic vortex accompanying both magnetic monopole and magnetic toroidal dipole [222]. In magnetic materials, $G_{1m}$ contributes to the magnetic anisotropy to tilt the spin moments from the crystal axis by combining the relativistic spin–orbit coupling [223].

4.14. Electric Toroidal Quadrupole

The electric toroidal quadrupole $G_{2m}$ can be found in 42 magnetic point groups, as shown in

Table 17. Classification of magnetic point groups (MPGs) with the electric toroidal quadrupole $G_{2m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, the magnetic toroidal dipole $T_{1m}$, and the electric toroidal monopole $G_0$. The upper (lower) columns represent the magnetic point groups with (without) $G_0$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$	${\mathit{G}}_0$
1, 2, $2^{\prime }$, 4, 3, 6	✓	✓	✓	✓
$2^{\prime }{2}^{\prime }2$, ${42}^{\prime }{2}^{\prime }$, ${32}^{\prime }$, ${62}^{\prime }{2}^{\prime }$		✓	✓	✓
${11}^{\prime }$, ${21}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, ${31}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$	✓			✓
222, ${2221}^{\prime }$, 422, ${4221}^{\prime }$, $4^{\prime }{22}^{\prime }$				✓
32, ${321}^{\prime }$, 622, ${6221}^{\prime }$, $6^{\prime }{22}^{\prime }$				✓
m, $m^{\prime }$, $m^{\prime }m{2}^{\prime }$	✓	✓	✓
$m^{\prime }{m}^{\prime }2$	✓	✓
$mm2$	✓		✓
$m{1}^{\prime }$, $mm{21}^{\prime }$	✓
$\overline{4}$, $\overline{4}{2}^{\prime }{m}^{\prime }$		✓
${\overline{4}}^{\prime }$, ${\overline{4}}^{\prime }m{2}^{\prime }$			✓
$\overline{4}{1}^{\prime }$, $\overline{4}2m$, $\overline{4}2m{1}^{\prime }$, ${\overline{4}}^{\prime }2m^{\prime }$

. Among them, 27 magnetic point groups also possess $G_0$. The materials with $G_{2m}$ show parity-violating physical phenomena, such as the nonlinear Hall effect based on the Berry curvature dipole mechanism [139] and Edelstein effect [224,225,226]. GeCu₂O₄ under the magnetic point group $\overline{4}2m{1}^{\prime }$ is one of the candidate materials to exhibit the physical phenomena of $G_{2m}$ below the critical temperature [84].

4.15. Electric Toroidal Octupole

There are 71 magnetic point groups that include the electric toroidal octupole $G_{3m}$, as shown in

Table 18. Classification of magnetic point groups (MPGs) with the electric toroidal octupole $G_{3m}$ according to the presence and absence of the electric dipole $Q_{1m}$, the magnetic dipole $M_{1m}$, the magnetic toroidal dipole $T_{1m}$, and the electric toroidal dipole $G_{1m}$. The upper (lower) columns represent the magnetic point groups with (without) $G_{1m}$.

MPG	${\mathit{Q}}_{1\mathit{m}}$	${\mathit{M}}_{1\mathit{m}}$	${\mathit{T}}_{1\mathit{m}}$	${\mathit{G}}_{1\mathit{m}}$
1, 2, $2^{\prime }$, m, $m^{\prime }$, 4, 3, 6	✓	✓	✓	✓
${11}^{\prime }$, ${21}^{\prime }$, $m{1}^{\prime }$, ${41}^{\prime }$, $4^{\prime }$, ${31}^{\prime }$, ${61}^{\prime }$, $6^{\prime }$	✓			✓
$\overline{1}$, $2/m$, $2^{\prime }/m^{\prime }$, $\overline{4}$, $4/m$, $\overline{3}$, $\overline{6}$, $6/m$		✓		✓
${\overline{1}}^{\prime }$, $2^{\prime }/m$, $2/m^{\prime }$, ${\overline{4}}^{\prime }$, $4/m^{\prime }$, ${\overline{3}}^{\prime }$, ${\overline{6}}^{\prime }$, $6/m^{\prime }$			✓	✓
$\overline{1}{1}^{\prime }$, $2/m{1}^{\prime }$, $\overline{4}{1}^{\prime }$, $4/m{1}^{\prime }$, $4^{\prime }/m$, $4^{\prime }/m^{\prime }$				✓
$\overline{3}{1}^{\prime }$, $\overline{6}{1}^{\prime }$, $6/m{1}^{\prime }$, $6^{\prime }/m$, $6^{\prime }/m^{\prime }$				✓
$m^{\prime }m{2}^{\prime }$	✓	✓	✓
$m^{\prime }{m}^{\prime }2$, $3m^{\prime }$	✓	✓
$mm2$, $3m$	✓		✓
$2^{\prime }{2}^{\prime }2$, ${32}^{\prime }$		✓	✓
$mm{21}^{\prime }$, $3m{1}^{\prime }$	✓
$m^{\prime }{m}^{\prime }m$, $\overline{3}{m}^{\prime }$		✓
$mm{m}^{\prime }$, ${\overline{3}}^{\prime }m$			✓
222, ${2221}^{\prime }$, $mmm$, $mmm{1}^{\prime }$, $m^{\prime }{m}^{\prime }{m}^{\prime }$, 32, ${321}^{\prime }$
$\overline{3}m$, $\overline{3}m{1}^{\prime }$, ${\overline{3}}^{\prime }{m}^{\prime }$, 23, ${231}^{\prime }$, $m\overline{3}$, $m\overline{3}{1}^{\prime }$, $m^{\prime }{\overline{3}}^{\prime }$

. Of these, 28 out of the 71 do not possess $G_{1m}$. Although the physical phenomena of $G_{3m}$ have not been investigated compared to other multipoles, the materials with $G_{3m}$ show similar physical phenomena to $G_{1m}$ owing to the same spatial inversion and time-reversal parities. For example, the longitudinal spin current generation is possible in materials with $G_{3m}$ [219]. FePO₄ (magnetic point group 222) [58] is one of the candidates to exhibit physical phenomena by $G_{3m}$ through the magnetic phase transition.

5. Conclusions

To summarize, we have revisited the multipole classification under 122 magnetic point groups, with an emphasis on magnetic materials. We have shown that four types of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal multipoles) emerge in magnetic materials irrespective of their rank. Since each multipole gives rise to different cross-correlations and transports, the systematic correspondence between multipoles and magnetic point groups enables us to design and engineer functional magnetic materials, including topological spin textures like magnetic skyrmions [227,228,229], which will be useful for future spintronic applications.