Multipole Moments Under Square Vortex and Skyrmion Crystals

Abstract

Non-coplanar spin textures such as magnetic vortices and skyrmions manifest themselves in unusual physical phenomena owing to their topologically nontrivial properties. Here, we investigate emergent multipole moments under vortex and skyrmion crystals in the centrosymmetric tetragonal system. Depending on the vorticity and helicity of the vortex or skyrmion, various multipole moments, including magnetic toroidal and electric toroidal multipoles, are induced on the atomic scale. In particular, the vortex and skyrmion spin textures consisting of multiple spin density waves give rise to density waves in terms of other multipole moments. Our results reveal a close relationship between non-coplanar multiple-Q spin textures and multipole moments.

1. Introduction

The interplay among internal electronic degrees of freedom such as charge, spin, and orbit has attracted much attention in condensed matter physics. For example, coupling between the charge and spin degrees of freedom leads to magnetoresistance [1,2,3,4,5,6,7], multiferroic properties [8,9,10,11,12,13,14,15,16], and the topological Hall/Nernst effect [17,18,19,20,21,22,23,24,25,26,27,28], whereas coupling between the spin and orbital degrees of freedom leads to the spin Hall effect [29,30,31,32,33,34,35,36,37,38,39,40], the Edelstein effect [41,42,43,44,45,46,47,48,49], and noncentrosymmetric superconductivity [50,51,52,53,54,55,56]. Thus, understanding the relationship between microscopic electronic degrees of freedom and macroscopic physical phenomena has been a central issue for years.

The concept of multipoles can connect such microscopic and macroscopic properties, as the symmetry-adapted basis consisting of multipoles, referred to as the symmetry-adapted multipole basis (SAMB), corresponds to a complete basis set in physical space [57,58]. The SAMB is constituted of four multipoles: the electric multipole, characterized by a time-reversal-even polar tensor; the magnetic multipole, characterized by a time-reversal-odd axial tensor; the magnetic toroidal multipole, characterized by a time-reversal-odd polar tensor; and the electric toroidal multipole, characterized by a time-reversal-even axial tensor. Reflecting its completeness, the SAMB can describe various order parameters under complicated electronic, magnetic, and lattice structures, such as chiral charge ordering with the electric toroidal monopole [59,60,61], ferroaxial ordering with the electric toroidal dipole [62,63,64,65,66,67,68,69], and non-coplanar magnetic ordering with the magnetic toroidal octupole [70]. Furthermore, emergent multipole moments can account for various physical properties, such as band-dispersion modulation, cross-correlation, and nonlinear transport from the microscopic viewpoint [71].

In the present study, we focus on the multipole moments induced by multiple-Q spin textures. A multiple-Q state is defined as a superposition of multiple spin density waves [72,73,74,75,76,77]. These states describe various topologically nontrivial spin textures, including vortex and skyrmion crystals [78,79,80,81,82,83,84,85]. Because these multiple-Q states often exhibit non-coplanar spin textures, various types of symmetry reduction are expected; hence, various multipole moments can be induced. To demonstrate this, we consider the double-Q vortex and skyrmion spin textures in the s–p orbital model on a two-dimensional square lattice, where the s–p orbital basis includes multiple monopolar, dipolar, quadrupolar, and octupolar degrees of freedom in its Hilbert space. We show the multipole moments induced by the vortex and skyrmion spin textures with different vorticities and helicities. Our results provide rich multipole physics brought about by multiple-Q states.

The rest of this paper is organized as follows. In Section 2, we introduce a tight-binding model with s and p orbitals. We list all the multipole degrees of freedom in s-p orbital space. We also introduce the vortex and skyrmion spin textures consisting of a superposition of double-Q spiral waves. Then, we show the multipole moments induced under vortex and skyrmion crystals in Section 3, where both uniform and finite-q components of the multipole moments appear according to the type of vortex or skyrmion crystal. Finally, Section 4 concludes the paper.

2. Model

We start by introducing multipole moments. There are four types of multipoles, according to their spatial inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) parities: electric multipoles $Q_{lm}$, with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,+1]$; magnetic multipoles $M_{lm}$, with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},-1]$; magnetic toroidal multipoles $T_{lm}$, with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,-1]$; and electric toroidal multipoles $G_{lm}$, with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},+1]$, where l and m stand for the rank of the multipole and its component, respectively. Hereafter, we denote the monopole ($l=0$) as $X_0$, the dipole ($l=1$) as $(X_x,X_y,X_z)$, the quadrupole ($l=2$) as $(X_u,X_v,X_{yz},X_{zx},X_{xy})$, and the octupole ($l=3$) as $(X_{xyz},X_x^{\alpha },X_y^{\alpha },X_z^{\alpha },X_x^{\beta },X_y^{\beta },X_z^{\beta })$ for $X=Q,M,T,G$.

When we consider the four orbitals, consisting of one s orbital and three p orbitals, i.e., $p_x$, $p_y$, and $p_z$, the Hilbert space spanned by these four orbitals can be described by an $8\times 8$ matrix with the spin degree of freedom; in other words, there are 64 independent matrix elements in the s–p hybridized system. By using the multipole operators defined in [86], the correspondence between the multipoles and matrix elements is obtained; see Appendix A for the specific expressions. In the spinless Hilbert space, the 16 active multipoles are as follows: the electric monopoles $Q_{s,0}$ and $Q_{p,0}$, electric dipoles $\mathit{Q}=(Q_x,Q_y,Q_z)$, electric quadrupoles $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$, magnetic dipoles $\mathit{M}=(M_x,M_y,M_z)$, and magnetic toroidal dipoles $\mathit{T}=(T_x,T_y,T_z)$. In the spinful Hilbert space, the other 48 multipoles with spin dependence are the electric monopole $Q_0^{\left(\mathrm{s}\right)}$, the electric dipoles $(Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)},Q_z^{\left(\mathrm{s}\right)})$, the electric quadrupoles $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$, the magnetic monopole $M_0^{\left(\mathrm{s}\right)}$, the magnetic dipoles $(M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)},M_{s,z}^{\left(\mathrm{s}\right)})$, $(M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)},M_{p,z}^{\left(\mathrm{s}\right)})$, and $(M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)},M_{a,z}^{\left(\mathrm{s}\right)})$, the magnetic quadrupoles $(M_u^{\left(\mathrm{s}\right)},M_v^{\left(\mathrm{s}\right)},M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)},M_{xy}^{\left(\mathrm{s}\right)})$, the magnetic octupoles $(M_{xyz}^{\left(\mathrm{s}\right)},M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)},M_z^{\alpha \left(\mathrm{s}\right)},M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)},M_z^{\beta \left(\mathrm{s}\right)})$, the magnetic toroidal dipoles $(T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)},T_z^{\left(\mathrm{s}\right)})$, the magnetic toroidal quadrupoles $(T_u^{\left(\mathrm{s}\right)},T_v^{\left(\mathrm{s}\right)},T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)},T_{xy}^{\left(\mathrm{s}\right)})$, the electric toroidal monopole $G_0^{\left(\mathrm{s}\right)}$, the electric toroidal dipoles $(G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)},G_z^{\left(\mathrm{s}\right)})$, and the electric toroidal quadrupoles $(G_u^{\left(\mathrm{s}\right)},G_v^{\left(\mathrm{s}\right)},G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)},G_{xy}^{\left(\mathrm{s}\right)})$, where the superscript $\left(\mathrm{s}\right)$ denotes a spin-dependent multipole. The active multipoles in the s–p hybridized orbital system are summarized in

Table 1. Classification of multipole moments in the s–p hybridized orbital system: the upper panel shows the multipoles activated in spinless space, while the lower panel shows those activated in spinful space. Here, l means the rank of the multipole; see the main text for the details of the multipole notation.

- Without spin dependence
basis	$l=0$	$l=1$	$l=2$
s-s	$Q_{s,0}$
p-p	$Q_{p,0}$	$M_x,M_y,M_z$	$Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy}$
s-p		$Q_x,Q_y,Q_z$
		$T_x,T_y,T_z$
- With spin dependence
basis	$l=0$	$l=1$	$l=2$	$l=3$
s-s		$M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)},M_{s,z}^{\left(\mathrm{s}\right)}$
p-p	$Q_0^{\left(\mathrm{s}\right)}$	$M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)},M_{p,z}^{\left(\mathrm{s}\right)}$	$Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)}$	$M_{xyz}^{\left(\mathrm{s}\right)},M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)},M_z^{\alpha \left(\mathrm{s}\right)},M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)},M_z^{\beta \left(\mathrm{s}\right)}$
		$M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)},M_{a,z}^{\left(\mathrm{s}\right)}$	$T_u^{\left(\mathrm{s}\right)},T_v^{\left(\mathrm{s}\right)},T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)},T_{xy}^{\left(\mathrm{s}\right)}$
		$G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)},G_z^{\left(\mathrm{s}\right)}$
s-p	$G_0^{\left(\mathrm{s}\right)}$	$Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)},Q_z^{\left(\mathrm{s}\right)}$	$G_u^{\left(\mathrm{s}\right)},G_v^{\left(\mathrm{s}\right)},G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)},G_{xy}^{\left(\mathrm{s}\right)}$
	$M_0^{\left(\mathrm{s}\right)}$	$T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)},T_z^{\left(\mathrm{s}\right)}$	$M_u^{\left(\mathrm{s}\right)},M_v^{\left(\mathrm{s}\right)},M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)},M_{xy}^{\left(\mathrm{s}\right)}$

. It is noteworthy that similar multipoles are also active for different orbitals, such as the p–d and d–f orbitals; the following results for the s–p model can be straightforwardly applied to the other orbital cases, as the qualitative results are unaltered by of the choice of orbitals.

We consider a tight-binding model on a two-dimensional square lattice with spatial inversion symmetry under the $D_{4\mathrm{h}}$ symmetry, as there are no odd-parity multipoles or other unconventional multipole moments in the paramagnetic state, which enables us to investigate the relationship between multipole moments and magnetic structures. The Hamiltonian is provided by

$$\begin{array}{cc}\hfill \mathcal{H}& =-\sum _{ij\alpha {\alpha }^{\prime }\sigma }(t_{ij}^{\alpha {\alpha }^{\prime }}{c}_{i{\alpha }^{\prime }\sigma }^{†}{c}_{j\alpha \sigma }+\mathrm{H}.\mathrm{c}.)+{\displaystyle \frac{\lambda }{2}}\sum _{i\tilde{\alpha }{\tilde{\alpha }}^{\prime }\sigma {\sigma }^{\prime }}{c}_{i\tilde{\alpha }\sigma }^{†}{H}^{\mathrm{SOC}}{c}_{i{\tilde{\alpha }}^{\prime }{\sigma }^{\prime }}+J\sum _{i\alpha \sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{\mathbf{\sigma }}_{\sigma {\sigma }^{\prime }}{c}_{i\alpha {\sigma }^{\prime }}·{\mathit{M}}_i^{\prime },\hfill \end{array}$$

(1)

where $c_{i\alpha \sigma }^{†}$ and $c_{i\alpha \sigma }$ represent the creation and annihilation Fermion operators at site i, orbital $\alpha =s$, $p_x$, $p_y$, and $p_z$ (or $\tilde{\alpha }=p_x$, $p_y$, and $p_z$), and spin $\sigma$, respectively. The first term represents the kinetic energy of electrons, which includes nearest-neighbor hopping $t_s$ between the s orbitals, hopping $t_{xy}$ between the $p_x$ and $p_y$ orbitals, hopping $t_z$ between the $p_z$ orbitals, and hopping $t_{sp}$ between the s and $(p_x,p_y)$ orbitals, all of which are taken so as to preserve the symmetry of the tetragonal square-lattice structure. We ignore the orbital-dependent onsite potential for simplicity. The second term represents the atomic spin–orbit coupling for the p orbitals, where ${\mathcal{H}}^{\mathrm{SOC}}$ is represented by the $6\times 6$ matrix provided by

$$\begin{array}{c}\hfill H^{\mathrm{SOC}}=\left(\begin{array}{ccc}0& -i{\sigma }^z& i{\sigma }^y\\ i{\sigma }^z& 0& -i{\sigma }^x\\ -i{\sigma }^y& i{\sigma }^x& 0\end{array}\right),\end{array}$$

(2)

where ${\sigma }^{\mu }$ is the $\mu =x,y,z$ component of the Pauli matrix in spin space. In the following calculations, we set $t_s=-1$, $t_{xy}=0.7$, $t_z=0.4$, $t_{sp}=0.6$, and $\lambda =0.5$, where $t_s$ is set as the energy unit of the model.

The third term represents the effect of magnetic mean fields arising from the vortex and skyrmion spin textures; $J=1$ is the coupling constant, and the orbital component is neglected for simplicity. We suppose that the effective site-dependent magnetic field ${\mathit{M}}_i^{\prime }$ is represented in the following forms. For the Néel-type vortex and skyrmion spin textures, ${\mathit{M}}_i^{\prime }$ is represented by

$$\begin{array}{c}\hfill {\mathit{M}}_i^{\prime }={\mathcal{N}}_i{\left(\begin{array}{c}cos{\mathit{Q}}_1·{\mathit{r}}_i\\ cos{\mathit{Q}}_2·{\mathit{r}}_i\\ {\tilde{M}}_z-sin{\mathit{Q}}_1·{\mathit{r}}_i-sin{\mathit{Q}}_2·{\mathit{r}}_i\end{array}\right)}^{\mathrm{T}},\end{array}$$

(3)

where the double-Q ordering wave vectors are provided by ${\mathit{Q}}_1=(\pi /3,0)$ and ${\mathit{Q}}_2=(0,\pi /3)$; ${\mathit{r}}_i$ is the position vector, ${\mathcal{N}}_i$ represents the local constraint to satisfy $|{\mathit{M}}_i^{\prime }|=1$, and ${\tilde{M}}_z$ is the variational parameter corresponding to a net magnetization. For the vortex (skyrmion) crystal, ${\tilde{M}}_z=0$ (${\tilde{M}}_z=0.5$) in the following calculations. The real-space spin configurations of the Néel-type vortex and skyrmion crystals are shown in Figure 1a and Figure 1b, respectively. In the vortex crystal in Figure 1a, the skyrmion number is zero owing to the cancellation of the spin scalar chirality between the cores with $M_z^{\prime }=+1$ and $M_z^{\prime }=-1$, where the vortex cores are located at the center of the square plaquette [87]. This state is also referred to as a meron–antimeron crystal. Meanwhile, the skyrmion number is $-1$ in the skyrmion crystal in Figure 1b.

The different types of vortex and skyrmion crystals are obtained by changing the in-plane spin components in Equation (3). The Bloch-type vortex and skyrmion spin textures with different helicities from the Néel-type ones are provided by

$$\begin{array}{c}\hfill {\mathit{M}}_i^{\prime }={\mathcal{N}}_i{\left(\begin{array}{c}cos{\mathit{Q}}_2·{\mathit{r}}_i\\ -cos{\mathit{Q}}_1·{\mathit{r}}_i\\ {\tilde{M}}_z-sin{\mathit{Q}}_1·{\mathit{r}}_i-sin{\mathit{Q}}_2·{\mathit{r}}_i\end{array}\right)}^{\mathrm{T}},\end{array}$$

(4)

where the skyrmion number is also characterized by $-1$ owing to the same vorticity. The spin configurations of the Bloch-type vortex and skyrmion crystals are shown in Figure 2a and Figure 2b, respectively. Similarly, we can define the spin textures for two types of anti-type vortex and skyrmion spin textures with skyrmion number $+1$, i.e., with different vorticities from the Néel-type and Bloch-type ones. The spin ansatz of type-I anti-vortex and anti-skyrmion crystals is provided by

$$\begin{array}{c}\hfill {\mathit{M}}_i^{\prime }={\mathcal{N}}_i{\left(\begin{array}{c}cos{\mathit{Q}}_2·{\mathit{r}}_i\\ cos{\mathit{Q}}_1·{\mathit{r}}_i\\ {\tilde{M}}_z-sin{\mathit{Q}}_1·{\mathit{r}}_i-sin{\mathit{Q}}_2·{\mathit{r}}_i\end{array}\right)}^{\mathrm{T}},\end{array}$$

(5)

while that of type-II anti-vortex and anti-skyrmion crystals is provided by

$$\begin{array}{c}\hfill {\mathit{M}}_i^{\prime }={\mathcal{N}}_i{\left(\begin{array}{c}-cos{\mathit{Q}}_1·{\mathit{r}}_i\\ cos{\mathit{Q}}_2·{\mathit{r}}_i\\ {\tilde{M}}_z-sin{\mathit{Q}}_1·{\mathit{r}}_i-sin{\mathit{Q}}_2·{\mathit{r}}_i\end{array}\right)}^{\mathrm{T}}.\end{array}$$

(6)

The spin configurations corresponding to these anti-type vortex and skyrmion crystals are presented in Figure 2c–f. Furthermore, we can define hybrid vortex and skyrmion crystals which consist of a linear combination of the Néel-type and Bloch-type spin textures. The spin configurations of such hybrid vortex and skyrmion crystals are shown in Figure 2g and Figure 2h, respectively.

From the viewpoint of stability, the double-Q vortex and skyrmion crystals are stabilized when taking into account the effect of the Dzyaloshinskii–Moriya interaction in non-centrosymmetric lattice structures [88,89] or frustrated and multiple-spin interactions in centrosymmetric lattice structures [81,90,91,92]. Especially in the latter mechanism, the Néel-type, Bloch-type, and anti-type vortex and skyrmion crystals are energetically degenerated when the effect of the spin–orbit coupling is ignored [93]. This degeneracy is lifted by the magnetic anisotropy arising from the interplay between the spin–orbit coupling and the crystalline electric field [94]. For example, the magnetic anisotropy under $C_{4\mathrm{h}}$ symmetry favors hybrid vortex and skyrmion crystals [95]. Double-Q skyrmion crystals have been found in both non-centrosymmetric materials such as ${\mathrm{Co}}_{10-x/2}$${\mathrm{Zn}}_{10-x/2}$${\mathrm{Mn}}_x$ [96,97,98,99,100] and Cu₂OSeO₃ [101,102] and in centrosymmetric materials such as GdRu₂Si₂ [103,104,105,106,107].

3. Results

3.1. Néel-Type Vortex Crystal

First, we discuss the result in the case of the Néel-type vortex crystal by setting ${\tilde{M}}_z=0$ in Equation (3). We diagonalize the Hamiltonian in Equation (1) for the $N=6\times 6$ spins under the periodic boundary conditions; we also consider the $200\times 200$ supercells in order to reduce the finite-size effect. Figure 3 and Figure 4 show the electric-type and magnetic-type multipole structure factors at $\mu =0$ ($\mu$ represents the chemical potential), respectively. The multipole structure factor is defined as

$$\begin{array}{c}\hfill \tilde{X}\left(\mathit{q}\right)={\displaystyle \frac{1}{N}}\sum _{ij}{\langle X\rangle }_i{\langle X\rangle }_j{e}^{i\mathit{q}·({\mathit{r}}_i-{\mathit{r}}_j)},\end{array}$$

(7)

where X represents all the multipoles activated in the s–p hybridized system. Because the multipoles belonging to the same irreducible representation under the $D_{4\mathrm{h}}$ point group show qualitatively similar profiles in the multipole structure factor, we plot one of the multipole structure factors in each irreducible representation in Figure 3 and Figure 4: the data in Figure 3a–l correspond to those in the ${\mathrm{A}}_{1g}^{+}$, ${\mathrm{A}}_{2g}^{+}$, ${\mathrm{B}}_{1g}^{+}$, ${\mathrm{B}}_{2g}^{+}$, two ${\mathrm{E}}_g^{+}$, ${\mathrm{A}}_{1u}^{+}$, ${\mathrm{A}}_{2u}^{+}$, ${\mathrm{B}}_{1u}^{+}$, ${\mathrm{B}}_{2u}^{+}$, and two ${\mathrm{E}}_u^{+}$ irreducible representations, respectively, the data in Figure 4a–l correspond to those in the ${\mathrm{A}}_{1g}^{-}$, ${\mathrm{A}}_{2g}^{-}$, ${\mathrm{B}}_{1g}^{-}$, ${\mathrm{B}}_{2g}^{-}$, two ${\mathrm{E}}_g^{-}$, ${\mathrm{A}}_{1u}^{-}$, ${\mathrm{A}}_{2u}^{-}$, ${\mathrm{B}}_{1u}^{-}$, ${\mathrm{B}}_{2u}^{-}$, and two ${\mathrm{E}}_u^{-}$ irreducible representations, respectively, and the superscript of the irreducible representation stands for the parity in terms of the time-reversal operations. The correspondence between the irreducible representations and multipoles is summarized in

Table 2. Symmetry reduction from the magnetic point group $4/mmm{1}^{\prime }$ to $4mm{1}^{\prime }$ when the double-Q Néel-type vortex crystal occurs in Equation (3). The correspondence among the irreducible representation (Irrep.), multipole (MP), and main peak positions in the multipole structure factor is presented. The upper (lower) panel represents the electric-type (magnetic-type) multipoles. In the spinful basis, the irreducible representations $({\mathrm{A}}_{1g},{\mathrm{A}}_{2g},{\mathrm{B}}_{1g},{\mathrm{B}}_{2g},{\mathrm{E}}_g,{\mathrm{A}}_{1u},{\mathrm{A}}_{2u},{\mathrm{B}}_{1u},{\mathrm{B}}_{2u},{\mathrm{E}}_u)$ are replaced by $({\mathrm{E}}_{1/2g},{\mathrm{E}}_{1/2g},{\mathrm{E}}_{3/2g},{\mathrm{E}}_{3/2g},{\mathrm{E}}_{1/2g}\oplus {\mathrm{E}}_{3/2g},{\mathrm{E}}_{1/2u},{\mathrm{E}}_{1/2u},{\mathrm{E}}_{3/2u},{\mathrm{E}}_{3/2u},{\mathrm{E}}_{1/2u}\oplus {\mathrm{E}}_{3/2u})$.

Irrep.	MP	Peak Positions
${\mathrm{A}}_{1g}^{+}\to {\mathrm{A}}_1^{+}$	$Q_{s,0}$, $Q_{p,0}$, $Q_0^{\left(\mathrm{s}\right)}$, $Q_u$, $Q_u^{\left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_{1,2}$
${\mathrm{A}}_{2g}^{+}\to {\mathrm{A}}_2^{+}$	$G_z^{\left(\mathrm{s}\right)}$	$3{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $3{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{1g}^{+}\to {\mathrm{B}}_1^{+}$	$Q_v$, $Q_v^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_{1,2}$
${\mathrm{B}}_{2g}^{+}\to {\mathrm{B}}_2^{+}$	$Q_{xy}$, $Q_{xy}^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm 2{\mathit{Q}}_2$
${\mathrm{E}}_g^{+}\to {\mathrm{E}}^{+}$	$(Q_{yz},Q_{zx})$, $(Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)})$, $(G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_{1,2}$
${\mathrm{A}}_{1u}^{+}\to {\mathrm{A}}_2^{+}$	$G_0^{\left(\mathrm{s}\right)}$, $G_u^{\left(\mathrm{s}\right)}$	$3{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $3{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{A}}_{2u}^{+}\to {\mathrm{A}}_1^{+}$	$Q_z$, $Q_z^{\left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_{1,2}$
${\mathrm{B}}_{1u}^{+}\to {\mathrm{B}}_2^{+}$	$G_v^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm 2{\mathit{Q}}_2$
${\mathrm{B}}_{2u}^{+}\to {\mathrm{B}}_1^{+}$	$G_{xy}^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_{1,2}$
${\mathrm{E}}_u^{+}\to {\mathrm{E}}^{+}$	$(Q_x,Q_y)$, $(Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)})$, $(G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_{1,2}$
${\mathrm{A}}_{1g}^{-}\to {\mathrm{A}}_1^{-}$	$T_u^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{A}}_{2g}^{-}\to {\mathrm{A}}_2^{-}$	$M_z$, $M_{s,z}^{\left(\mathrm{s}\right)}$, $M_{p,z}^{\left(\mathrm{s}\right)}$, $M_{a,z}^{\left(\mathrm{s}\right)}$, $M_z^{\alpha \left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{1g}^{-}\to {\mathrm{B}}_1^{-}$	$T_v^{\left(\mathrm{s}\right)}$, $M_{xyz}^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{2g}^{-}\to {\mathrm{B}}_2^{-}$	$T_{xy}^{\left(\mathrm{s}\right)}$, $M_z^{\beta \left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{E}}_g^{-}\to {\mathrm{E}}^{-}$	$(M_x,M_y)$, $(M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)})$, $(M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)})$, $(M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)})$,	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
	$(T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)})$, $(M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)})$, $(M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)})$
${\mathrm{A}}_{1u}^{-}\to {\mathrm{A}}_2^{-}$	$M_0^{\left(\mathrm{s}\right)}$, $M_u^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{A}}_{2u}^{-}\to {\mathrm{A}}_1^{-}$	$T_z$, $T_z^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{1u}^{-}\to {\mathrm{B}}_2^{-}$	$M_v^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{2u}^{-}\to {\mathrm{B}}_1^{-}$	$M_{xy}^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{E}}_u^{-}\to {\mathrm{E}}^{-}$	$(T_x,T_y)$, $(T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)})$, $(M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$

. In addition, we show the real-space electric-type and magnetic-type multipole configurations in Figure A1 and Figure A2, respectively, in Appendix B for reference.

Under the Néel-type vortex crystal, the magnetic point group reduces from $4/mmm{1}^{\prime }$ to $4mm{1}^{\prime }$. It is notable that the time-reversal symmetry is not broken under the Néel-type vortex crystal, as the magnetic texture is invariant for the product of the time-reversal and translational operations. Thus, the uniform component of the magnetic-type multipoles is not induced in the Néel-type vortex crystal. Meanwhile, the symmetries with respect to the spatial inversion, the mirror in the horizontal plane, and the $C_2$ rotation around the x axis are broken, which indicates that the ${\mathrm{A}}_{2u}^{+}$ irreducible representation corresponds to the order parameter. Because the electric dipoles $Q_z$ and $Q_z^{\left(\mathrm{s}\right)}$ belong to the ${\mathrm{A}}_{2u}^{+}$ irreducible representation, they are regarded as the multipole order parameter of the Néel-type vortex crystal. According to symmetry reduction, electric dipole-related physical phenomena such as Rashba-type antisymmetric spin splitting in the band structure [108,109,110,111] and the transverse Edelstein effect [41] are expected.

In addition, we find that finite-q modulations, i.e., density waves, are induced in each multipole channel, as shown in Figure 3 and Figure 4. The nonzero modulations for the wave vector $\mathit{q}$ are understood from the effective coupling between multipoles and spin configurations [112]. For example, the $\mathit{q}$ component of the electric quadrupole $Q_v\propto x^2-y^2$ is coupled to the spins as $(S_{{\mathit{q}}_1}^x{S}_{{\mathit{q}}_2}^x-S_{{\mathit{q}}_1}^y{S}_{{\mathit{q}}_2}^y){\delta }_{{\mathit{q}}_1+{\mathit{q}}_2+\mathit{q},\mathbf{0}}$ in the lowest-order approximation, where $\delta$ is the Kronecker delta. From Equation (3), the above quantity becomes nonzero for ${\mathit{q}}_1={\mathit{Q}}_1$, ${\mathit{q}}_2={\mathit{Q}}_1$, and $\mathit{q}=-2{\mathit{Q}}_1$ (or ${\mathit{q}}_1={\mathit{Q}}_2$, ${\mathit{q}}_2={\mathit{Q}}_2$, and $\mathit{q}=-2{\mathit{Q}}_2$). This is why the intensities at $2{\mathit{Q}}_1$ and $2{\mathit{Q}}_2$ appear in ${\tilde{Q}}_v\left(\mathit{q}\right)$. Similarly, the $\mathit{q}$ component of the electric quadrupole $Q_{xy}\propto xy$ is coupled to $S_{{\mathit{q}}_1}^x{S}_{{\mathit{q}}_2}^y{\delta }_{{\mathit{q}}_1+{\mathit{q}}_2+\mathit{q},\mathbf{0}}$, which leads to the intensities at ${\mathit{Q}}_1\pm {\mathit{Q}}_2$ in ${\tilde{Q}}_{xy}\left(\mathit{q}\right)$. In this way, the finite-q peak positions in the multipole structure factor are accounted for by the two or higher-order spin-scattering processes.

3.2. Néel-Type Skyrmion Crystal

Next, we consider the Néel-type skyrmion crystal by setting ${\tilde{M}}_z=0.5$ in Equation (3). Figure 5 and Figure 6 represent the electric-type and magnetic-type multipole structure factors, respectively. Owing to the presence of uniform magnetization, the symmetry in the Néel-type skyrmion crystal is further reduced from $4mm{1}^{\prime }$ in the Néel-type vortex crystal to $4m^{\prime }{m}^{\prime }$. In addition to ${\mathrm{A}}_{2u}^{+}$, the ${\mathrm{A}}_{2g}^{-}$ and ${\mathrm{A}}_{1u}^{-}$ irreducible representations belong to the identity irreducible representation, as shown in

Table 3. Symmetry reduction from the magnetic point group $4/mmm{1}^{\prime }$ to $4m^{\prime }{m}^{\prime }$ when the double-Q Néel-type skyrmion crystal occurs in Equation (3). The correspondence among the irreducible representation (Irrep.), multipole (MP), and main peak positions in the multipole structure factor is presented. The upper (lower) panel represents the electric-type (magnetic-type) multipoles.

Irrep.	MP	Peak Positions
${\mathrm{A}}_{1g}^{+}\to {\mathrm{A}}^{+}$	$Q_{s,0}$, $Q_{p,0}$, $Q_0^{\left(\mathrm{s}\right)}$, $Q_u$, $Q_u^{\left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{A}}_{2g}^{+}\to {\mathrm{A}}^{-}$	$G_z^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{1g}^{+}\to {\mathrm{B}}^{+}$	$Q_v$, $Q_v^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_{1,2}$
${\mathrm{B}}_{2g}^{+}\to {\mathrm{B}}^{-}$	$Q_{xy}$, $Q_{xy}^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm 2{\mathit{Q}}_2$
${\mathrm{E}}_g^{+}\to {\mathrm{E}}^{(1,2)+}$	$(Q_{yz},Q_{zx})$, $(Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)})$, $(G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{A}}_{1u}^{+}\to {\mathrm{A}}^{-}$	$G_0^{\left(\mathrm{s}\right)}$, $G_u^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{A}}_{2u}^{+}\to {\mathrm{A}}^{+}$	$Q_z$, $Q_z^{\left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{B}}_{1u}^{+}\to {\mathrm{B}}^{-}$	$G_v^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm 2{\mathit{Q}}_2$
${\mathrm{B}}_{2u}^{+}\to {\mathrm{B}}^{+}$	$G_{xy}^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_{1,2}$
${\mathrm{E}}_u^{+}\to {\mathrm{E}}^{(1,2)-}$	$(Q_x,Q_y)$, $(Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)})$, $(G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{A}}_{1g}^{-}\to {\mathrm{A}}^{-}$	$T_u^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{A}}_{2g}^{-}\to {\mathrm{A}}^{+}$	$M_z$, $M_{s,z}^{\left(\mathrm{s}\right)}$, $M_{p,z}^{\left(\mathrm{s}\right)}$, $M_{a,z}^{\left(\mathrm{s}\right)}$, $M_z^{\alpha \left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{B}}_{1g}^{-}\to {\mathrm{B}}^{-}$	$T_v^{\left(\mathrm{s}\right)}$, $M_{xyz}^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm 2{\mathit{Q}}_1$
${\mathrm{B}}_{2g}^{-}\to {\mathrm{B}}^{+}$	$T_{xy}^{\left(\mathrm{s}\right)}$, $M_z^{\beta \left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_{1,2}$
${\mathrm{E}}_g^{-}\to {\mathrm{E}}^{(1,2)-}$	$(M_x,M_y)$, $(M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)})$, $(M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)})$, $(M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)})$,	${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
	$(T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)})$, $(M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)})$, $(M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)})$
${\mathrm{A}}_{1u}^{-}\to {\mathrm{A}}^{+}$	$M_0^{\left(\mathrm{s}\right)}$, $M_u^{\left(\mathrm{s}\right)}$	$\mathbf{0}$, ${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$
${\mathrm{A}}_{2u}^{-}\to {\mathrm{A}}^{-}$	$T_z$, $T_z^{\left(\mathrm{s}\right)}$	$2{\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_2\pm {\mathit{Q}}_1$
${\mathrm{B}}_{1u}^{-}\to {\mathrm{B}}^{+}$	$M_v^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_{1,2}$, $2{\mathit{Q}}_{1,2}$
${\mathrm{B}}_{2u}^{-}\to {\mathrm{B}}^{-}$	$M_{xy}^{\left(\mathrm{s}\right)}$	${\mathit{Q}}_1\pm {\mathit{Q}}_2$, $2{\mathit{Q}}_1\pm 2{\mathit{Q}}_2$
${\mathrm{E}}_u^{-}\to {\mathrm{E}}^{(1,2)+}$	$(T_x,T_y)$, $(T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)})$, $(M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)})$	${\mathit{Q}}_{1,2}$, ${\mathit{Q}}_1\pm {\mathit{Q}}_2$

. In other words, the uniform component of the magnetic monopole $M_0$ is induced, as shown in Figure 6g. Thus, the longitudinal magnetoelectric effect, which is caused by $M_0$, is expected in the Néel-type skyrmion crystal. Compared to the result in the Néel-type vortex crystal, the number of peak positions in the multipole structure factor increases in the Néel-type skyrmion crystal. This is because of the additional coupling between the multipoles and spin textures owing to the $\mathit{q}=\mathbf{0}$ component. For example, in the case of ${\tilde{Q}}_v\left(\mathit{q}\right)$, the ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ components are additionally induced, as $(S_{{\mathit{q}}_1}^x{S}_{{\mathit{q}}_2}^x-S_{{\mathit{q}}_1}^y{S}_{{\mathit{q}}_2}^y){\delta }_{{\mathit{q}}_1+{\mathit{q}}_2+\mathit{q},\mathbf{0}}$ becomes nonzero for ${\mathit{q}}_1={\mathit{Q}}_1+{\mathit{Q}}_2$, ${\mathit{q}}_2=-{\mathit{Q}}_2$, and $\mathit{q}=-{\mathit{Q}}_1$ (or ${\mathit{q}}_1={\mathit{Q}}_1+{\mathit{Q}}_2$, ${\mathit{q}}_2=-{\mathit{Q}}_1$, and $\mathit{q}=-{\mathit{Q}}_2$). It is notable that $S_{{\mathit{Q}}_1+{\mathit{Q}}_2}^x$ and $S_{{\mathit{Q}}_1+{\mathit{Q}}_2}^y$ become nonzero only for ${\tilde{M}}_z\ne 0$, as shown in Figure 4e,f and Figure 6e,f. The emergent multipole moments in the vortex and skyrmion crystal are consistent with the symmetry argument under magnetic point groups [71].

3.3. Other Vortex and Skyrmion Types

In this subsection, we discuss the multipole moments induced by other types of vortex and skyrmion crystals. For the Bloch-type vortex crystal in Figure 2a, the electric toroidal monopole $G_0$ and electric toroidal quadrupole $G_u$ are induced instead of $Q_z$ in the Néel-type vortex crystal. Because these multipoles are related to chirality, the Bloch-type vortex crystal exhibits the chirality-related physical phenomena, such as antisymmetric spin splitting in the form of $c_1(k_x{\sigma }_x+k_y{\sigma }_y)+c_2{k}_z{\sigma }_z$ ($c_1$ and $c_2$ are numerical coefficients) and the longitudinal Edelstein effect. When the effect of the magnetic field is taken into account, the Bloch-type vortex crystal turns into the Bloch-type skyrmion crystal in Figure 2b, where the magnetic toroidal dipole $T_z$ is additionally induced. Thus, the nonreciprocal transport along the z direction and the linear transverse magnetoelectric effect are expected to occur.

In the cases of the type-I and type-II anti-vortex crystals in Figure 2c,e, the rank-2 electric toroidal quadrupoles $G_v$ and $G_{xy}$ [113] are induced, which exhibit antisymmetric spin splittings in the functional forms of $k_x{\sigma }_x-k_y{\sigma }_y$ and $k_x{\sigma }_y+k_y{\sigma }_x$, respectively. When the type-I and type-II anti-skyrmion crystals in Figure 2d,f are realized under the external magnetic field, the magnetic quadrupoles $M_{xy}$ and $M_v$ are induced, both of which become the origin of the linear magnetoelectric effect, although the tensor components are different from those under $M_0$ and $T_z$.

Finally, further multipole moments can be induced when the hybrid vortex and skyrmion crystals in Figure 2g,h are considered. In this case, the electric toroidal dipole $G_z$ is additionally induced owing to the simultaneous appearance of $M_0$ and $T_z$, which becomes the origin of the rotational responses between conjugate fields and physical quantities with the same symmetry, such as the spin current generation [68]. A similar situation happens for anti-hybrid vortex (skyrmion) crystals consisting of a superposition of type-I and type-II anti-vortex (anti-skyrmion) crystals. We summarize the induced multipole moments under different types of vortex and skyrmion crystals in

Table 4. Uniform component of multipole moments induced by vortex and skyrmion crystals with different vorticities and helicities; $\mathit{H}$ represents the magnetic field and MPG stands for the magnetic point group. The multipoles are classified according to their spatial inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) parities.

			$(\mathcal{P},\mathcal{T})$
Type	$\mathit{H}$	MPG	$(+\mathbf{1},+\mathbf{1})$	$(-\mathbf{1},+\mathbf{1})$	$(+\mathbf{1},-\mathbf{1})$	$(-\mathbf{1},-\mathbf{1})$
para	$=0$	$4/mmm{1}^{\prime }$	$Q_0$, $Q_u$
Bloch	$=0$	${4221}^{\prime }$		$G_0$, $G_u$
	$\ne 0$	${42}^{\prime }{2}^{\prime }$		$G_0$, $G_u$	$M_z$, $M_z^{\alpha }$	$T_z$
Néel	$=0$	$4mm{1}^{\prime }$		$Q_z$
	$\ne 0$	$4m^{\prime }{m}^{\prime }$		$Q_z$	$M_z$, $M_z^{\alpha }$	$M_0$, $M_u$
anti-I	$=0$	$\overline{4}2m{1}^{\prime }$		$G_v$
	$\ne 0$	$\overline{4}{2}^{\prime }{m}^{\prime }$		$G_v$	$M_z$, $M_z^{\alpha }$	$M_{xy}$
anti-II	$=0$	$\overline{4}m{21}^{\prime }$		$G_{xy}$
	$\ne 0$	$\overline{4}{m}^{\prime }{2}^{\prime }$		$G_{xy}$	$M_z$, $M_z^{\alpha }$	$M_v$
hybrid	$=0$	${41}^{\prime }$	$G_z$	$G_0$, $Q_z$, $G_u$
	$\ne 0$	4	$G_z$	$G_0$, $Q_z$, $G_u$	$M_z$, $T_u$, $M_z^{\alpha }$	$M_0$, $T_z$, $M_u$
anti-hybrid	$=0$	$\overline{4}{1}^{\prime }$	$G_z$	$G_v$, $G_{xy}$
	$\ne 0$	$\overline{4}$	$G_z$	$G_v$, $G_{xy}$	$M_z$, $T_u$, $M_z^{\alpha }$	$M_v$, $T_{xy}$

.

4. Conclusions

In this paper, we have investigated the relationship between multipole moments and topologically nontrivial spin textures. By introducing four types of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal), we have shown that different types of multipoles are induced by various types of vortex and skyrmion crystals with different vorticities and helicities under tetragonal symmetry. In addition, we have classified the induced multipole moments in terms of the uniform and finite-q components. Because different multipoles give rise to different physical properties, rich physics can be expected in non-coplanar multiple-q states with various multipole moments, which can stimulate the future exploration of functional magnetic materials.