Locally Odd-Parity Hybridization Induced by Spiral Magnetic Textures

Abstract

We study unconventional multipole moments arising from noncollinear magnetic structures within an augmented framework encompassing electric, magnetic, magnetic toroidal, and electric toroidal multipoles. Employing a tight-binding model for an s-p hybridized orbital system, we analyze two spiral magnetic textures and classify the resulting multipoles according to magnetic point group symmetry. Different spiral wave types, such as cycloidal and proper-screw forms, activate distinct multipole components, with odd-parity multipoles emerging from local s-p parity mixing induced by magnetically driven inversion-symmetry breaking. Calculated multipole structure factors reveal finite-q peaks originating from higher-order magnetic-dipole-scattering processes and their characteristic couplings between Fourier components of the magnetic dipole texture. Our results demonstrate that magnetic ordering can generate parity-mixed states without intrinsic structural inversion asymmetry, offering new pathways to realize cross-correlation phenomena in functional magnetic materials.

1. Introduction

Magnetic structures in solids arise from the cooperative ordering of localized or itinerant electronic degrees of freedom, most notably the spin and orbital angular momenta of electrons. In their simplest form, such structures can be described in terms of magnetic dipole moments, whose spatial arrangements give rise to the canonical ferromagnetic, antiferromagnetic, and ferrimagnetic orders, depending on the interplay among strong spin–orbit coupling (SOC), crystalline electric fields, and electron–electron correlations. When the effect of the SOC is strong, the spin and orbital moments are no longer independent but become entangled into total-angular-momentum states that can host higher-rank multipole moments, such as rank-2 quadrupole, rank-3 octupole, and rank-4 hexadecapole moments, which can describe anisotropic charge and spin distributions in an atomic scale [1,2,3]. Such high-rank multipole orderings have been found in f-electron materials, such as CeB₆ [4,5,6,7,8,9,10,11,12,13], PrPb₃ [14,15,16,17,18,19], Pr$T_2{X}_{20}$ ($T=\mathrm{Ir},\mathrm{Rh},X=\mathrm{Zn};T=\mathrm{V},X=\mathrm{Al}$) [20,21,22,23,24,25,26,27], UPd₃ [28,29,30,31,32,33], CeTe [34,35], and RB₂C₂($R=\mathrm{Dy},\mathrm{Ho}$) [36,37,38,39]. Recently, atomic-scale multipole orderings have been found in d-electron materials, such as Ba₂MgReO₆ [40,41,42].

Recently, antiferromagnetic structures over multiple sites have also been described within the framework of multipoles, which is referred to as cluster multipoles [3,43,44]. This description enables us to express the complex arrangement of local dipole moments within a crystallographic unit cell as the ferroic alignment of higher-rank multipoles, which provides a useful perspective for understanding the cross-correlation phenomena exhibited by antiferromagnets. The typical example is the noncollinear antiferromagnet Mn₃Sn, exhibiting the large anomalous Hall and Nernst effects even with a negligible small magnetization [45,46,47,48]; the noncollinear antiferromagnetic structure is related to the magnetic octupole [44] or anisotropic magnetic dipole [49,50,51,52], both of which can lead to the ferromagnetic-related physical phenomena. Similarly, the concept of cluster multipoles offers insights into understanding cross-correlation phenomena arising from antiferromagnetic structures in various materials. Examples include the linear magnetoelectric effect associated with the magnetic quadrupole in Cr₂O₃ [53,54,55,56,57], Co₄Nb₂O₉ [58,59,60,61], and KOsO₄ [62,63], and the linear magnetoelectric effect associated with the magnetic toroidal dipole in UNi₄B [64], nonreciprocal magnon excitations in $\alpha$-Cu₂V₂O₇ [65,66,67,68], and the magnetopiezo effect associated with the magnetic quadrupole (hexadecapole) ordering in Ba$T_2$As₂ ($T=$ Mn, Fe) and EuTBi₂ ($T=$ Mn, Zn) [69,70]. Thus, magnetic structures and multipoles are closely intertwined, and understanding their correspondence is expected to pave the way for more efficient exploration of functional materials in the future.

In the present study, we explore the relationship between spiral magnetic structures and multipoles using an augmented multipole framework that encompasses four types of multipoles with distinct spatial inversion and time-reversal parities: electric, magnetic, magnetic toroidal, and electric toroidal multipoles [71,72]. Based on group-theoretical and model analyses, we demonstrate that varying the type of spiral wave leads to the induction of different classes of multipoles; the cycloidal spiral magnetic texture leads to density waves in terms of the electric dipole, electric toroidal dipole, magnetic monopole, magnetic toroidal dipole, and so on, where the proper-screw spiral magnetic texture leads to density waves in terms of the electric toroidal monopole, electric toroidal dipole, electric dipole, magnetic toroidal dipole, and so on. In particular, we show that odd-parity multipoles in an atomic scale arise from parity mixing between different orbitals, reflecting the breaking of local inversion symmetry by the spiral magnetic texture. The present results indicate that complex magnetic structures can activate unconventional multipole moments, giving rise to a variety of emergent cross-correlation phenomena and thereby opening new avenues for the design of functional magnetic materials.

The rest of this paper is organized as follows. In Section 2, we introduce the s-p orbital model in the one-dimensional chain system. In addition, we show the magnetic dipole structure under the spiral type of the magnetic dipole density wave. Then, we briefly discuss the active multipole moments in the s-p hybridized Hilbert space in Section 3, which includes both odd-parity and even-parity multipoles. In Section 4, we show the real-space and momentum-space distributions of multipoles under the cycloidal and proper-screw spiral states. We also discuss the effect of the relativistic SOC. Finally, we conclude the present results in Section 5.

2. Model

We consider a tight-binding model on a one-dimensional lattice that possesses spatial inversion symmetry and belongs to the magnetic point group $mmm{1}^{\prime }$; the chain direction is taken as the x direction. By considering one s and three p orbitals at each lattice site, the model Hamiltonian is given by

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

(1)

where $c_{i\alpha \sigma }^{†}$ and $c_{i\alpha \sigma }$ denote the fermionic creation and annihilation operators, respectively, at site i with orbital index $\alpha =s$, $p_x$, $p_y$, and $p_z$ (or $\tilde{\alpha }=p_x$, $p_y$, and $p_z$) and spin $\sigma$.

The first term describes the electronic kinetic energy, incorporating nearest-neighbor hoppings of several types: $t_s$ between s orbitals, $t_x$ between $p_x$ orbitals, $t_z$ between $p_z$ orbitals, and $t_{sp}$ between s and $p_x$ orbitals. These hopping amplitudes are chosen to be consistent with the symmetry requirements of the one-dimensional chain structure under the $mmm{1}^{\prime }$ symmetry. We set $t_s=-1$, $t_x=0.7$, $t_z=0.4$, and $t_{sp}=0.6$ without loss of generality; $t_s$ is taken as the energy unit of the model. For simplicity, we neglect any onsite potential differences between orbitals.

The second term accounts for the atomic SOC acting on the p orbitals, with $H^{\mathrm{SOC}}$ expressed as the $6\times 6$ matrix given by

$$\begin{array}{c}{H}^{\mathrm{SOC}}=\left(\begin{array}{ccc}0& -i{\sigma }_z& i{\sigma }_y\\ i{\sigma }_z& 0& -i{\sigma }_x\\ -i{\sigma }_y& i{\sigma }_x& 0\end{array}\right),\end{array}$$

(2)

where ${\sigma }_{\mu }$ ($\mu =x,y,z$) denotes the Pauli matrix acting in spin space. We set $\lambda =0.5$ or $\lambda =0$ in the following calculations.

The third term describes the magnetic mean-field contribution originating from the spiral magnetic textures, with the coupling constant set to $J=1$. We assume that the effective site-dependent magnetic field ${\mathit{M}}_i^{\prime }$ takes the following form. For the cycloidal spiral magnetic texture, ${\mathit{M}}_i^{\prime }$ is given by

$$\begin{array}{c}{\mathit{M}}_i^{\prime }=\left(cosQ{x}_i,0,sinQ{x}_i\right),\end{array}$$

(3)

where the ordering wave vector Q is defined as $Q=\pi /4$ and $x_i$ denotes the position of site i. The real-space magnetic dipole configuration of the cycloidal spiral wave is shown in Figure 1a. The spiral plane of the cycloidal spiral state is parallel to the spiral axis. We also consider another spiral wave, with its spiral plane lying in the $yz$ plane and oriented perpendicular to the spiral axis. This state is referred to as the proper-screw spiral wave, whose magnetic dipole configuration is given by

$$\begin{array}{c}{\mathit{M}}_i^{\prime }=\left(0,cosQ{x}_i,sinQ{x}_i\right).\end{array}$$

(4)

The real-space magnetic dipole configuration of the proper-screw spiral wave is shown in Figure 1b.

Let us comment on the stabilization mechanisms of spiral waves in centrosymmetric lattice structures. One possible origin is the competition between ferromagnetic exchange interactions and further-neighbor antiferromagnetic exchange interactions [73,74,75,76,77]. Another is the long-range interaction arising from the nesting of the Fermi surface in itinerant magnets [78,79,80], commonly known as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [81,82,83]. Since both mechanisms are independent of the magnetic dipole orientation, the spiral plane can be arbitrary. Its direction is determined by additional effects such as the SOC. We implicitly assume that the present spiral state is stabilized by such mechanisms. The spiral magnetic state often leads to fascinating physical phenomena, such as magnetic-order-driven electric polarization [84,85,86,87,88,89] and antisymmetric spin-split band structures without relying on the relativistic SOC [90,91,92].

3. Active Multipole

We discuss the active atomic-scale multipoles in the s-p hybridized orbital system. Multipoles can be classified into four types according to their parities under spatial inversion ($\mathcal{P}$) and time reversal ($\mathcal{T}$) [71,72]: the electric multipole $Q_{lm}$ with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,+1]$; the magnetic multipole $M_{lm}$ with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},-1]$; the magnetic toroidal multipole $T_{lm}$ with $(\mathcal{P},\mathcal{T})=[{(-1)}^l,-1]$; and the electric toroidal multipole $G_{lm}$ with $(\mathcal{P},\mathcal{T})=[{(-1)}^{l+1},+1]$. Here, l denotes the multipole rank and m denotes its component. For convenience, we label 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 $(X_{xyz},X_x^{\alpha },X_y^{\alpha },X_z^{\alpha },X_x^{\beta },X_y^{\beta },X_z^{\beta })$, with $X=Q,M,T,G$.

For a basis consisting of four orbitals, one s orbital and three p orbitals ($p_x$, $p_y$, $p_z$), the Hilbert space, including spin, is described by an $8\times 8$ matrix. This corresponds to 64 independent matrix elements in the s-p hybridized orbital system. Among them, according to the different spatial inversion parities of s and p orbitals, the active 40 multipoles in the s-s and p-p space correspond to even-parity multipoles with $\mathcal{P}=+1$, whereas 24 multipoles in the s-p space correspond to odd-parity multipoles with $\mathcal{P}=-1$. The active multipoles in the s-s orbital space are given by electric monopole $Q_{s,0}$ and magnetic dipoles $(M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)},M_{s,z}^{\left(\mathrm{s}\right)})$; the active multipoles in the p-p orbital space are given by two electric monopoles $Q_{p,0}$ and $Q_0^{\left(\mathrm{s}\right)}$, two electric quadrupoles $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$ and $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$, three magnetic dipoles $\mathit{M}=(M_x,M_y,M_z)$, $(M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)},M_{p,z}^{\left(\mathrm{s}\right)})$, and $(M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)},M_{a,z}^{\left(\mathrm{s}\right)})$, magnetic octupoles $(M_{xyz}^{\left(\mathrm{s}\right)},M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)},M_z^{\alpha \left(\mathrm{s}\right)},M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)},M_z^{\beta \left(\mathrm{s}\right)})$, magnetic toroidal quadrupoles $(T_u^{\left(\mathrm{s}\right)},T_v^{\left(\mathrm{s}\right)},T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)},T_{xy}^{\left(\mathrm{s}\right)})$, and electric toroidal dipoles $(G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)},G_z^{\left(\mathrm{s}\right)})$; the active multipoles in the s-p orbital space are given by two electric dipoles $\mathit{Q}=(Q_x,Q_y,Q_z)$ and $(Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)},Q_z^{\left(\mathrm{s}\right)})$, magnetic monopole $M_0^{\left(\mathrm{s}\right)}$, magnetic quadrupoles $(M_u^{\left(\mathrm{s}\right)},M_v^{\left(\mathrm{s}\right)},M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)},M_{xy}^{\left(\mathrm{s}\right)})$, two magnetic toroidal dipoles $\mathit{T}=(T_x,T_y,T_z)$ and $(T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)},T_z^{\left(\mathrm{s}\right)})$, electric toroidal monopole $G_0^{\left(\mathrm{s}\right)}$, and electric toroidal quadrupoles $(G_u^{\left(\mathrm{s}\right)},G_v^{\left(\mathrm{s}\right)},G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)},G_{xy}^{\left(\mathrm{s}\right)})$. Here, the superscript $\left(\mathrm{s}\right)$ stands for the multipoles activated in spinful space. A complete list of the active multipoles for the s-p hybridized orbital system is given in

Table 1. Multipole moment classification in an s-p hybridized orbital system. l stands for the rank of the multipole. The upper, middle, and lower panels show the multipoles activated in s-s, p-p, and s-p Hilbert space, respectively. The abbreviation as $M_3^{\left(\mathrm{s}\right)}=\{M_{xyz}^{\left(\mathrm{s}\right)},M_x^{\alpha \left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)},M_z^{\alpha \left(\mathrm{s}\right)},M_x^{\beta \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)},M_z^{\beta \left(\mathrm{s}\right)}\}$ is used. See the main text and Appendix A for the notation and the specific expressions of the multipole matrix elements.

Basis	$(\mathcal{P},\mathcal{T})$	$\mathit{l}=0$	$\mathit{l}=1$	$\mathit{l}=2$	$\mathit{l}=3$
s-s	$(+1,+1)$	$Q_{s,0}$
	$(+1,-1)$		$M_{s,x}^{\left(\mathrm{s}\right)},M_{s,y}^{\left(\mathrm{s}\right)},M_{s,z}^{\left(\mathrm{s}\right)}$
p-p	$(+1,+1)$	$Q_{p,0}$	$G_x^{\left(\mathrm{s}\right)},G_y^{\left(\mathrm{s}\right)},G_z^{\left(\mathrm{s}\right)}$	$Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy}$
		$Q_0^{\left(\mathrm{s}\right)}$		$Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)}$
	$(+1,-1)$		$M_x,M_y,M_z$	$T_u^{\left(\mathrm{s}\right)},T_v^{\left(\mathrm{s}\right)},T_{yz}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)},T_{xy}^{\left(\mathrm{s}\right)}$	$M_3^{\left(\mathrm{s}\right)}$
			$M_{p,x}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)},M_{p,z}^{\left(\mathrm{s}\right)}$
			$M_{a,x}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)},M_{a,z}^{\left(\mathrm{s}\right)}$
s-p	$(-1,+1)$	$G_0^{\left(\mathrm{s}\right)}$	$Q_x,Q_y,Q_z$	$G_u^{\left(\mathrm{s}\right)},G_v^{\left(\mathrm{s}\right)},G_{yz}^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)},G_{xy}^{\left(\mathrm{s}\right)}$
			$Q_x^{\left(\mathrm{s}\right)},Q_y^{\left(\mathrm{s}\right)},Q_z^{\left(\mathrm{s}\right)}$
	$(-1,-1)$	$M_0^{\left(\mathrm{s}\right)}$	$T_x,T_y,T_z$	$M_u^{\left(\mathrm{s}\right)},M_v^{\left(\mathrm{s}\right)},M_{yz}^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)},M_{xy}^{\left(\mathrm{s}\right)}$
			$T_x^{\left(\mathrm{s}\right)},T_y^{\left(\mathrm{s}\right)},T_z^{\left(\mathrm{s}\right)}$

. The matrix elements for each multipole are derived following the method in [93]; we show their explicit expressions in Appendix A.

Under the magnetic point group $mmm{1}^{\prime }$ without magnetic ordering, the irreducible representations of each multipole as follows: $(Q_{s,0},Q_{p,0},Q_0^{\left(\mathrm{s}\right)},Q_u,Q_u^{\left(\mathrm{s}\right)},Q_v,Q_v^{\left(\mathrm{s}\right)})\in {\mathrm{A}}_g^{+}$, $(G_z^{\left(\mathrm{s}\right)},Q_{xy},Q_{xy}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{1g}^{+}$, $(G_y^{\left(\mathrm{s}\right)},Q_{zx},Q_{zx}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{2g}^{+}$, $(G_x^{\left(\mathrm{s}\right)},Q_{yz},Q_{yz}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{3g}^{+}$, $(G_0^{\left(\mathrm{s}\right)},G_u^{\left(\mathrm{s}\right)},G_v^{\left(\mathrm{s}\right)})\in {\mathrm{A}}_u^{+}$, $(Q_z,Q_z^{\left(\mathrm{s}\right)},G_{xy}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{1u}^{+}$, $(Q_y,Q_y^{\left(\mathrm{s}\right)},G_{zx}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{2u}^{+}$, $(Q_x,Q_x^{\left(\mathrm{s}\right)},G_{yz}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{3u}^{+}$, $(T_u^{\left(\mathrm{s}\right)},T_v^{\left(\mathrm{s}\right)},M_{xyz}^{\left(\mathrm{s}\right)})\in {\mathrm{A}}_g^{-}$, $(M_z,M_{s,z}^{\left(\mathrm{s}\right)},M_{p,z}^{\left(\mathrm{s}\right)},M_{a,z}^{\left(\mathrm{s}\right)},T_{xy}^{\left(\mathrm{s}\right)},M_z^{\alpha \left(\mathrm{s}\right)},M_z^{\beta \left(\mathrm{s}\right)})\in {\mathrm{B}}_{1g}^{-}$, $(M_y,M_{s,y}^{\left(\mathrm{s}\right)},M_{p,y}^{\left(\mathrm{s}\right)},M_{a,y}^{\left(\mathrm{s}\right)},T_{zx}^{\left(\mathrm{s}\right)},M_y^{\alpha \left(\mathrm{s}\right)},M_y^{\beta \left(\mathrm{s}\right)})\in {\mathrm{B}}_{2g}^{-}$, $(M_x,M_{s,x}^{\left(\mathrm{s}\right)},M_{p,x}^{\left(\mathrm{s}\right)},M_{a,x}^{\left(\mathrm{s}\right)},T_{yz}^{\left(\mathrm{s}\right)},M_x^{\alpha \left(\mathrm{s}\right)},M_x^{\beta \left(\mathrm{s}\right)})\in {\mathrm{B}}_{3g}^{-}$, $(M_0^{\left(\mathrm{s}\right)},M_u^{\left(\mathrm{s}\right)},M_v^{\left(\mathrm{s}\right)})\in {\mathrm{A}}_u^{-}$, $(T_z,T_z^{\left(\mathrm{s}\right)},M_{xy}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{1u}^{-}$, $(T_y,T_y^{\left(\mathrm{s}\right)},M_{zx}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{2u}^{-}$, and $(T_x,T_x^{\left(\mathrm{s}\right)},M_{yz}^{\left(\mathrm{s}\right)})\in {\mathrm{B}}_{3u}^{-}$ [94]. Here, the sign for the superscript in the irreducible representation represents the parity in terms of the time-reversal operation.

4. Density Waves

We evaluate the expectation values of the active multipoles in the presence of spiral magnetic textures by diagonalizing the Hamiltonian in Equation (1) while the chemical potential is kept at $\mu =0$. Calculations are performed for a system of $N=8$ sites under periodic boundary conditions, and finite-size effects are suppressed by averaging over a supercell comprising 20,000 repetitions of the unit cell. We define the multipole structure factor as

$$\begin{array}{c}\tilde{X}\left(q\right)={\displaystyle \frac{1}{N}}\sum _{ij}{\langle X\rangle }_i{\langle X\rangle }_j{e}^{iq(x_i-x_j)},\end{array}$$

(5)

where X denotes all multipoles active in the s-p hybridized system and q is the wave number. We set the lattice constant as unity.

In the case of the absence of magnetic ordering, i.e., $J=0$, nonzero components of $\tilde{X}\left(q\right)$ are ${\tilde{Q}}_{s,0}\left(0\right)$, ${\tilde{Q}}_{p,0}\left(0\right)$, ${\tilde{Q}}_0^{\left(\mathrm{s}\right)}\left(0\right)$, ${\tilde{Q}}_u\left(0\right)$, ${\tilde{Q}}_u^{\left(\mathrm{s}\right)}\left(0\right)$, ${\tilde{Q}}_v\left(0\right)$, and ${\tilde{Q}}_v^{\left(\mathrm{s}\right)}\left(0\right)$; only the uniform component ($q=0$) appears and no density waves occur. This is understood from the fact that only the multipoles belonging to the identity irreducible representation under the $mmm{1}^{\prime }$ symmetry are symmetry-allowed. Finite-q components are also prohibited owing to the translational symmetry of the one-dimensional chain.

By considering the effect of the site-dependent magnetic field arising from the spiral magnetic dipole texture, additional contributions appear depending on the symmetry reduction. We first discuss the case of the cycloidal spiral state, where the magnetic point group changes from $mmm{1}^{\prime }$ to $2mm{1}^{\prime }$. The cycloidal spiral state preserves the time-reversal symmetry, as the magnetic texture is invariant under the combined operation of time reversal and translation. This indicates that there is no uniform ($q=0$) component in magnetic and magnetic toroidal multipoles. In contrast, the cycloidal spiral state breaks both the spatial inversion symmetry and the mirror symmetry with respect to the $xy$ plane, identifying the ${\mathrm{B}}_{1u}^{+}$ irreducible representation as the corresponding order parameter. This means that the electric dipoles $Q_z$ and $Q_z^{\left(\mathrm{s}\right)}$ and the electric toroidal quadrupole $G_{xy}^{\left(\mathrm{s}\right)}$ accompany the uniform component of the multipole structure factor. In other words, they are regarded as the multipole order parameter of the cycloidal spiral state. This symmetry reduction allows for electric-dipole-driven phenomena, including Rashba-type antisymmetric spin splitting in the band structure [95,96,97,98] and the transverse Edelstein effect [99]. We show the correspondence between the irreducible representations of $mmm{1}^{\prime }$ and $2mm{1}^{\prime }$, as well as that between the multipoles and the irreducible representations, in

Table 2. The symmetry reduction from the magnetic point group $mmm{1}^{\prime }$ to $2mm{1}^{\prime }$ once the cycloidal spiral state in Equation (3) occurs. The correspondence among the irreducible representation (Irrep.), multipole (MP), and main peak positions of the multipole structure factor with and without the relativistic spin–orbit coupling (SOC) at $\mu =0$ is presented from left to right. The upper (lower) panel represents the electric-type (magnetic-type) multipoles.

Irrep.	MP	Peak Positions w/SOC	w/o SOC
${\mathrm{A}}_g^{+}\to {\mathrm{A}}_1^{+}$	$Q_{s,0}$, $Q_{p,0}$, $Q_0^{\left(\mathrm{s}\right)}$, $Q_u$, $Q_u^{\left(\mathrm{s}\right)}$, $Q_v$, $Q_v^{\left(\mathrm{s}\right)}$	0, $2Q$, $4Q$	0
${\mathrm{B}}_{1g}^{+}\to {\mathrm{A}}_2^{+}$	$G_z^{\left(\mathrm{s}\right)}$, $Q_{xy}$, $Q_{xy}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{2g}^{+}\to {\mathrm{B}}_1^{+}$	$G_y^{\left(\mathrm{s}\right)}$, $Q_{zx}$, $Q_{zx}^{\left(\mathrm{s}\right)}$	$2Q$	—
${\mathrm{B}}_{3g}^{+}\to {\mathrm{B}}_2^{+}$	$G_x^{\left(\mathrm{s}\right)}$, $Q_{yz}$, $Q_{yz}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{A}}_u^{+}\to {\mathrm{A}}_2^{+}$	$G_0^{\left(\mathrm{s}\right)}$, $G_u^{\left(\mathrm{s}\right)}$, $G_v^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{1u}^{+}\to {\mathrm{A}}_1^{+}$	$Q_z$, $Q_z^{\left(\mathrm{s}\right)}$, $G_{xy}^{\left(\mathrm{s}\right)}$	0, $2Q$, $4Q$	—
${\mathrm{B}}_{2u}^{+}\to {\mathrm{B}}_2^{+}$	$Q_y$, $Q_y^{\left(\mathrm{s}\right)}$, $G_{zx}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{3u}^{+}\to {\mathrm{B}}_1^{+}$	$Q_x$, $Q_x^{\left(\mathrm{s}\right)}$, $G_{yz}^{\left(\mathrm{s}\right)}$	$2Q$	—
${\mathrm{A}}_g^{-}\to {\mathrm{A}}_1^{-}$	$T_u^{\left(\mathrm{s}\right)}$, $T_v^{\left(\mathrm{s}\right)}$, $M_{xyz}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{1g}^{-}\to {\mathrm{A}}_2^{-}$	$M_z$, $M_{s,z}^{\left(\mathrm{s}\right)}$, $M_{p,z}^{\left(\mathrm{s}\right)}$, $M_{a,z}^{\left(\mathrm{s}\right)}$, $T_{xy}^{\left(\mathrm{s}\right)}$, $M_z^{\alpha \left(\mathrm{s}\right)}$, $M_z^{\beta \left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{B}}_{2g}^{-}\to {\mathrm{B}}_1^{-}$	$M_y$, $M_{s,y}^{\left(\mathrm{s}\right)}$, $M_{p,y}^{\left(\mathrm{s}\right)}$, $M_{a,y}^{\left(\mathrm{s}\right)}$, $T_{zx}^{\left(\mathrm{s}\right)}$, $M_y^{\alpha \left(\mathrm{s}\right)}$, $M_y^{\beta \left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{3g}^{-}\to {\mathrm{B}}_2^{-}$	$M_x$, $M_{s,x}^{\left(\mathrm{s}\right)}$, $M_{p,x}^{\left(\mathrm{s}\right)}$, $M_{a,x}^{\left(\mathrm{s}\right)}$, $T_{yz}^{\left(\mathrm{s}\right)}$, $M_x^{\alpha \left(\mathrm{s}\right)}$, $M_x^{\beta \left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{A}}_u^{-}\to {\mathrm{A}}_2^{-}$	$M_0^{\left(\mathrm{s}\right)}$, $M_u^{\left(\mathrm{s}\right)}$, $M_v^{\left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{B}}_{1u}^{-}\to {\mathrm{A}}_1^{-}$	$T_z$, $T_z^{\left(\mathrm{s}\right)}$, $M_{xy}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{2u}^{-}\to {\mathrm{B}}_2^{-}$	$T_y$, $T_y^{\left(\mathrm{s}\right)}$, $M_{zx}^{\left(\mathrm{s}\right)}$	$1Q$, $3Q$	1Q
${\mathrm{B}}_{3u}^{-}\to {\mathrm{B}}_1^{-}$	$T_x$, $T_x^{\left(\mathrm{s}\right)}$, $M_{yz}^{\left(\mathrm{s}\right)}$	—	—

.

Such a symmetry argument is confirmed by the multipole structure factor shown in Figure 2. As multipoles belonging to the same irreducible representation of the $mmm{1}^{\prime }$ magnetic point group exhibit qualitatively similar behavior in their structure factors, we present only a representative multipole structure factor for each irreducible representation. Figure 2a,b present the results for the electric monopole $Q_0^{\left(\mathrm{s}\right)}$ and the electric dipole $Q_z$, respectively. Since both multipoles belong to the identity irreducible representation under the $2mm{1}^{\prime }$ symmetry, the $q=0$ component becomes nonzero. Additionally, these multipoles can exhibit finite-q modulations (density waves) at $q=2Q$ and $q=4Q$, whose behavior is also found by the sublattice-dependent form of real-space multipole configurations shown in Appendix B. In addition, one finds that the electric toroidal dipole $G_y^{\left(\mathrm{s}\right)}$ and the electric dipole $Q_x$, which belong to the ${\mathrm{B}}_1^{+}$ representation, exhibit the density wave at $q=2Q$, as shown in Figure 2c and Figure 2d, respectively. These finite-q modulations occur as a secondary effect of the onset of the magnetic dipole density wave under the spiral magnetic texture, as detailed below.

The density wave also occurs in magnetic and magnetic toroidal multipoles, although no $q=0$ component is induced. Figure 2e,f show the structure factors of the different components of the magnetic dipoles, $M_z$ and $M_x$, respectively, which exhibit the intensities at $q=Q$ and $3Q$. The presence of the 3Q component is owing to the interplay between the SOC and hopping anisotropy for three p orbitals, which deforms the spiral plane from the circular shape to the elliptical shape, as found by the inequivalent intensities of ${\tilde{M}}_z\left(Q\right)$ and ${\tilde{M}}_x\left(Q\right)$. In addition to $M_z$ and $M_x$, the odd-parity magnetic monopole $M_0^{\left(\mathrm{s}\right)}$ and the magnetic toroidal dipole $T_y$ exhibit nonzero intensities at the same wave numbers, as shown in Figure 2g,h, since they belong to the same irreducible representations of $M_z$ and $M_x$ under the magnetic point group $2mm{1}^{\prime }$, respectively.

The finite modulations at wave number q under the spiral magnetic texture can be interpreted in terms of the effective coupling between the multipoles and the underlying magnetic dipole configurations [100]. As an example, in the lowest-order approximation, the $2Q$ component of the electric monopole, ${\tilde{Q}}_0^{\left(\mathrm{s}\right)}\left(2Q\right)$, couples to the magnetic dipole (spin) degrees of freedom through the combination $M_x^{\prime }\left(Q\right)M_x^{\prime }\left(Q\right)+M_y^{\prime }\left(Q\right)M_y^{\prime }\left(Q\right)+M_z^{\prime }\left(Q\right)M_z^{\prime }\left(Q\right)$, where ${\mathit{M}}^{\prime }\left(Q\right)$ means the Fourier transform of ${\mathit{M}}_i^{\prime }$ in Equation (3). From Equation (3), one finds that $M_x^{\prime }\left(Q\right)\ne 0$ and $M_z^{\prime }\left(Q\right)\ne 0$, which results in a nonzero intensity of ${\tilde{Q}}_0^{\left(\mathrm{s}\right)}\left(2Q\right)$, as shown in Figure 2a. In a similar manner, the 2Q component of the electric quadrupole $Q_{zx}$, which belongs to the same irreducible representation as $G_y^{\left(\mathrm{s}\right)}$ in Figure 2c, is induced when $M_z^{\prime }\left(Q\right)M_x^{\prime }\left(Q\right)\ne 0$. Furthermore, the nonzero ${\tilde{Q}}_z\left(0\right)$ is also attributed to the coupling $M_z^{\prime }\left(Q\right)M_x^{\prime }(-Q)-M_x^{\prime }\left(Q\right)M_z^{\prime }(-Q)$ so that the vector chirality becomes nonzero [90]. Thus, the finite-q peaks observed in the multipole structure factor can be attributed to magnetic-dipole-scattering processes of second or higher order.

As shown in Figure 2b,d,g,h, the odd-parity multipoles are induced in the presence of the cycloidal spiral state. The emergence of such atomic-scale odd-parity multipoles is owing to the local parity mixing between s and p orbitals through the inversion symmetry breaking at each lattice site [101]. Although the local inversion symmetry breaking is found in the lattice structures with the sublattice degrees of freedom, such as the zigzag chain [102,103,104,105] and honeycomb structure [106,107,108], the present results indicate that magnetic orderings are another route to induce the local parity mixing. The results of the multipole structure factor under the cycloidal spiral magnetic texture are summarized in Table 2.

Finally, we discuss the effect of the SOC. The results at $\lambda =0$ are summarized in the rightmost column of Table 2. In contrast to the result with the SOC, electric-type multipoles show no intensities except for the $q=0$ component of the electric monopoles $Q_{s,0}$ and $Q_{p,0}$, which remains nonzero even for $J=0$. Thus, the SOC plays an important role in inducing the odd-parity mixing between s and p orbitals. In addition, there are no intensities at higher-harmonic wave vectors in magnetic-type multipoles, which is attributed to the spin rotational symmetry in the absence of the SOC.

Next, we discuss the behavior of the multipole moments induced by the proper-screw spiral wave in Equation (4). In this case, the magnetic point group is lowered from $mmm{1}^{\prime }$ to ${2221}^{\prime }$, where the electric toroidal monopole $G_0$ and the electric toroidal quadrupoles $G_u$ and $G_v$ are induced instead of $Q_z$ in the cycloidal spiral state. Since these multipoles are related to structural chirality [109,110], the proper-screw spiral state exhibits the chirality-related physical phenomena, such as the longitudinal Edelstein effect and nonreciprocal transport under a magnetic field [111,112,113,114]. The nonzero components of the multipole structure factor with and without the SOC are understood as in the case of the cycloidal spiral state, which are summarized in

Table 3. The symmetry reduction from the magnetic point group $mmm{1}^{\prime }$ to ${2221}^{\prime }$ once the proper-screw spiral state in Equation (4) occurs. The correspondence among the irreducible representation (Irrep.), multipole (MP), and main peak positions of the multipole structure factor with and without the relativistic spin–orbit coupling (SOC) at $\mu =0$ is presented from left to right. The upper (lower) panel represents the electric-type (magnetic-type) multipoles.

Irrep.	MP	Peak Positions w/SOC	w/o SOC
${\mathrm{A}}_g^{+}\to {\mathrm{A}}^{+}$	$Q_{s,0}$, $Q_{p,0}$, $Q_0^{\left(\mathrm{s}\right)}$, $Q_u$, $Q_u^{\left(\mathrm{s}\right)}$, $Q_v$, $Q_v^{\left(\mathrm{s}\right)}$	0, $2Q$	0
${\mathrm{B}}_{1g}^{+}\to {\mathrm{B}}_1^{+}$	$G_z^{\left(\mathrm{s}\right)}$, $Q_{xy}$, $Q_{xy}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{2g}^{+}\to {\mathrm{B}}_2^{+}$	$G_y^{\left(\mathrm{s}\right)}$, $Q_{zx}$, $Q_{zx}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{3g}^{+}\to {\mathrm{B}}_3^{+}$	$G_x^{\left(\mathrm{s}\right)}$, $Q_{yz}$, $Q_{yz}^{\left(\mathrm{s}\right)}$	$2Q$	—
${\mathrm{A}}_u^{+}\to {\mathrm{A}}^{+}$	$G_0^{\left(\mathrm{s}\right)}$, $G_u^{\left(\mathrm{s}\right)}$, $G_v^{\left(\mathrm{s}\right)}$	0, $2Q$	—
${\mathrm{B}}_{1u}^{+}\to {\mathrm{B}}_1^{+}$	$Q_z$, $Q_z^{\left(\mathrm{s}\right)}$, $G_{xy}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{2u}^{+}\to {\mathrm{B}}_2^{+}$	$Q_y$, $Q_y^{\left(\mathrm{s}\right)}$, $G_{zx}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{3u}^{+}\to {\mathrm{B}}_3^{+}$	$Q_x$, $Q_x^{\left(\mathrm{s}\right)}$, $G_{yz}^{\left(\mathrm{s}\right)}$	$2Q$	—
${\mathrm{A}}_g^{-}\to {\mathrm{A}}^{-}$	$T_u^{\left(\mathrm{s}\right)}$, $T_v^{\left(\mathrm{s}\right)}$, $M_{xyz}^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{1g}^{-}\to {\mathrm{B}}_1^{-}$	$M_z$, $M_{s,z}^{\left(\mathrm{s}\right)}$, $M_{p,z}^{\left(\mathrm{s}\right)}$, $M_{a,z}^{\left(\mathrm{s}\right)}$, $T_{xy}^{\left(\mathrm{s}\right)}$, $M_z^{\alpha \left(\mathrm{s}\right)}$, $M_z^{\beta \left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{B}}_{2g}^{-}\to {\mathrm{B}}_2^{-}$	$M_y$, $M_{s,y}^{\left(\mathrm{s}\right)}$, $M_{p,y}^{\left(\mathrm{s}\right)}$, $M_{a,y}^{\left(\mathrm{s}\right)}$, $T_{zx}^{\left(\mathrm{s}\right)}$, $M_y^{\alpha \left(\mathrm{s}\right)}$, $M_y^{\beta \left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{B}}_{3g}^{-}\to {\mathrm{B}}_3^{-}$	$M_x$, $M_{s,x}^{\left(\mathrm{s}\right)}$, $M_{p,x}^{\left(\mathrm{s}\right)}$, $M_{a,x}^{\left(\mathrm{s}\right)}$, $T_{yz}^{\left(\mathrm{s}\right)}$, $M_x^{\alpha \left(\mathrm{s}\right)}$, $M_x^{\beta \left(\mathrm{s}\right)}$	—	—
${\mathrm{A}}_u^{-}\to {\mathrm{A}}^{-}$	$M_0^{\left(\mathrm{s}\right)}$, $M_u^{\left(\mathrm{s}\right)}$, $M_v^{\left(\mathrm{s}\right)}$	—	—
${\mathrm{B}}_{1u}^{-}\to {\mathrm{B}}_1^{-}$	$T_z$, $T_z^{\left(\mathrm{s}\right)}$, $M_{xy}^{\left(\mathrm{s}\right)}$	$1Q$, $3Q$	$1Q$
${\mathrm{B}}_{2u}^{-}\to {\mathrm{B}}_2^{-}$	$T_y$, $T_y^{\left(\mathrm{s}\right)}$, $M_{zx}^{\left(\mathrm{s}\right)}$	$1Q$, $3Q$	1Q
${\mathrm{B}}_{3u}^{-}\to {\mathrm{B}}_3^{-}$	$T_x$, $T_x^{\left(\mathrm{s}\right)}$, $M_{yz}^{\left(\mathrm{s}\right)}$	—	—

.

5. Conclusions

We have explored the interplay between noncollinear magnetic structures and unconventional multipole moments within an augmented multipole framework consisting of electric, magnetic, magnetic toroidal, and electric toroidal types. Using a tight-binding model for an s-p hybridized system under various spiral magnetic textures, we identified the emergence of distinct multipole orders dictated by the underlying magnetic point group symmetry. Group-theoretical and model analyses reveal that changing the spiral wave type, such as from cycloidal to proper screw, activates different multipoles. In particular, odd-parity multipoles arise from local parity mixing between s and p orbitals, induced by inversion-symmetry breaking from the magnetic order itself. The computed multipole structure factors indicate that finite-q peaks stem from higher-order magnetic-dipole-scattering processes, with specific multipole components directly linked to characteristic couplings between Fourier components of the magnetic dipole texture. These results demonstrate that magnetic ordering can generate parity-mixed states and unconventional multipoles even in centrosymmetric crystals, offering a route to engineer emergent cross-correlation phenomena, such as magnetoelectric and magnetopiezoelectric effects, and guiding the design of functional magnetic materials.

Our symmetry analysis and computed structure factors point to concrete material platforms where magnetically induced local odd-parity hybridization could be realized. Prominent candidates include centrosymmetric spiral magnets such as layered halides NiBr₂ [73] and its Zn-doped derivative, as well as RKKY-driven helical states in rare-earth metals and intermetallic compounds [78,79,80], which are expected to host the predicted odd-parity multipoles.