Ferroaxial Property Under Hybrid Skyrmion Crystals

Abstract

The ferroaxial moment, a time-reversal-even axial dipole degree of freedom, plays a key role not only in conventional quantum states of matter but also in anomalous off-diagonal cross-correlated responses. Here, we theoretically demonstrate that a skyrmion crystal with a swirling non-coplanar spin texture can exhibit ferroaxiality. In particular, we show that a hybrid skyrmion crystal, formed by the superposition of Néel- and Bloch-type vortices, hosts a finite ferroaxial moment. The degree of ferroaxiality is quantified by calculating the expectation value of the electric toroidal dipole within a multi-orbital tight-binding model based on multipole representation theory. We further reveal characteristic off-diagonal responses associated with magnetic and magnetic toroidal multipoles under external magnetic fields. These results establish the hybrid skyrmion crystal as a promising platform for exploring the fundamental nature of ferroaxial moments.

1. Introduction

Multipole degrees of freedom provide a unifying framework for classifying complex electronic orders and cross-correlation responses in solids [1,2,3,4,5,6,7,8]. Beyond conventional electric and magnetic multipoles, electric toroidal and magnetic toroidal multipoles have emerged as key factors of anisotropic spin and orbital degrees of freedom in an atomic scale, together with their unconventional couplings to external fields [8]. Among them, the magnetic toroidal dipole, which is characterized as a time-reversal-odd polar vector, has been extensively studied from both atomic and thermodynamic perspectives [9,10,11], since it often gives rise to unconventional cross-correlation phenomena [12,13], including the linear magnetoelectric effect [14,15,16,17,18,19,20], asymmetric magnon excitations [21,22,23], and the nonlinear spin Hall effect [24,25].

In contrast, the ferroaxial moment represents a distinct axial degree of freedom, characterized as a time-reversal-even axial dipole [26,27]. Unlike magnetic toroidal dipoles, which embody parity-breaking magnetic responses, the ferroaxial moment encodes the sense of static rotation or twisting of a system and thus serves as an axial-order parameter associated with structural asymmetries [28,29,30]. From a symmetry standpoint, ferroaxiality can be regarded as the ferroic alignment of electric toroidal dipoles. It has been classified as one of the fundamental ferroic orders, on equal footing with ferroelectricity, ferromagnetism, and ferrotoroidicity, and it is allowed in crystallographic point groups lacking mirror symmetry. This degree of freedom couples to anomalous off-diagonal cross-correlation responses [31], such as the antisymmetric thermopolarization [32], intrinsic longitudinal spin current generation [31,33], and transverse nonlinear magnetic responses [34]. Recent experimental efforts have identified ferroaxial moments in a wide range of materials, which includes Co₃Nb₂O₈ [35], CaMn₇O₁₂ [36], RbFe(MoO₄)₂ [28,37], NiTiO₃ [37,38,39,40], Ca₅Ir₃O₁₂ [41,42,43,44], BaCoSiO₄ [45], K₂Zr(PO₄)₂ [40,46], Na₂Hf(BO₃)₂ [47], Na-superionic conductors [48], Na₂BaM(PO₄)₂ ($M=$ Mg, Mn, Co, and Ni) [49], MnTiO₃ [50], NiCo₂TeO₆ [51], and rare-earth tritellurides [52].

In the present study, we investigate an alternative route to induce the ferroaxial moment by focusing on magnetic phase transitions [53]. We demonstrate that a hybrid skyrmion crystal, which is defined as a spin texture formed by the superposition of Néel- and Bloch-type skyrmions, can host a finite ferroaxial moment even in the absence of structural rotational distortions. By analyzing a multi-orbital tight-binding model on a two-dimensional square lattice, we evaluate the expectation value of the electric toroidal dipole as a quantitative measure of ferroaxility and reveal that spin–orbit coupling plays a crucial role in inducing the atomic-scale ferroaxial moment. Furthermore, the hybrid skyrmion crystal exhibits magnetic field-induced cross-correlation responses characteristic of ferroaxiality and relevant odd-parity multipoles, such as transverse magnetization and magnetic toroidal dipoles. These findings establish the hybrid skyrmion crystal as a fertile platform for exploring ferroaxial physics in magnetism, thereby extending the scope of ferroaxiality beyond crystallographic distortions.

The remainder of this paper is organized as follows: in Section 2, we introduce the multi-orbital tight-binding model consisting of s and p orbitals on a two-dimensional square lattice, together with the spin texture of the hybrid skyrmion crystal, and discuss the associated symmetry reduction under magnetic ordering. In Section 3, we present our results on the emergence of ferroaxiality in the hybrid skyrmion crystal and clarify the role of relativistic spin–orbit coupling. Then, in Section 4, we discuss magnetic field-induced off-diagonal responses, including transverse magnetization and magnetic toroidal dipoles. Finally, Section 5 provides a conclusion of the present paper.

2. Model

We consider a multi-orbital tight-binding model on a two-dimensional square lattice. To explicitly incorporate the ferroaxial moment in the low-energy Hilbert space, we study an s-p hybridized system, in which the electric toroidal dipole degree of freedom, corresponding to the atomic-scale ferroaxial moment, is naturally included as described below. The tight-binding Hamiltonian is given by

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

(1)

We define the fermionic operators $c_{i\alpha \sigma }^{†}$ and $c_{i\alpha \sigma }$ as creating and annihilating an electron with spin $\sigma$ on orbital $\alpha$ ($\alpha =s$, $p_x$, $p_y$, $p_z$ or $\tilde{\alpha }=p_x$, $p_y$, $p_z$) at square lattice site i, respectively; throughout this paper, the lattice constant is set to unity.

The kinetic energy part of the Hamiltonian function, given in the first term, includes several nearest-neighbor hopping processes, including $t_s$ between s orbitals, $t_{xy}$ between $p_x$ and $p_y$ orbitals, $t_z$ between $p_z$ orbitals, and $t_{sp}$ between the s and $(p_x,p_y)$ orbitals. These transfer integrals are chosen to ensure that the Hamiltonian function respects the tetragonal $D_{4\mathrm{h}}$ symmetry of the square lattice. To reduce complexity, orbital-dependent onsite potentials are omitted. The second part of the Hamiltonian function accounts for the atomic spin–orbit coupling within the p-orbital manifold, represented by the $6\times 6$ matrix $H^{\mathrm{SOC}}$. The specific expression of $H^{\mathrm{SOC}}$ is given 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)

with basis $(p_{x\sigma },p_{y\sigma },p_{z\sigma })$. ${\sigma }_{\mu }$ ($\mu =x,y,z$) denotes the Pauli matrices acting in the spin space. Unless otherwise noted, the following parameter values are employed: $t_s=-1$, $t_{xy}=0.7$, $t_z=0.4$, $t_{sp}=0.6$, and $\lambda =0.5$. The adopted parameter set is designed to represent a generic situation where orbital hybridization and spin–orbit coupling occur at comparable energy scales. The hopping amplitudes correspond to typical energy scales of $0.01$–1 eV in transition metal systems, while the strength of the spin–orbit coupling $\lambda$ corresponds to tens to hundreds of meV depending on the d-orbital character. We note that the qualitative conclusions of this study are not sensitive to the fine details of these parameters: the ferroaxial moment robustly appears in the hybrid skyrmion crystal whenever the spin–orbit coupling is finite, whereas it vanishes in the pure Néel- and Bloch-type cases. Changing the relative magnitudes of the hopping integrals modifies quantitative values but does not alter these essential symmetry-based results. The overall energy scale of the model is measured in units of $t_s$. The third part of the Hamiltonian function describes the coupling to magnetic mean fields originating from skyrmion spin textures, with the coupling constant fixed at $J=1$. Here, the coupling constant J represents the effective interaction between the itinerant electron spins and the localized moments forming the skyrmion crystal texture, as described below. Physically, J corresponds to a local exchange field that enforces the alignment of the conduction electron spin with the background magnetic texture, analogous to Hund’s coupling or Kondo coupling in itinerant electron models [54,55,56]. In typical transition metal compounds, J is of the order of 0.01–1 eV depending on the localization of the magnetic moments.

We model the effective site-dependent field $\mathit{M}{’}_i$ of the skyrmion crystal, which corresponds to the fixed localized spins ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$, as follows:

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

(3)

where the double-Q structure is defined by the two modulation wave vectors ${\mathit{Q}}_1=(\pi /3,0)$ and ${\mathit{Q}}_2=(0,\pi /3)$; we schematically illustrate the momentum space positions of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the square lattice Brillouin zone in Figure 1d, where these two vectors are related by the $C_4$ rotational symmetry of the lattice and their superposition gives rise to the double-Q structure that corresponds to the hybrid skyrmion crystal. In this representation, ${\mathit{r}}_i$ is the position vector of site i, ${\tilde{M}}_z=0.5$ serves as a variational parameter associated with a uniform magnetization, and ${\mathcal{N}}_i$ enforces the normalization condition $|{\mathit{M}}_i^{\prime }|=1$. The parameter $\theta$ controls the helicity of the skyrmion spin texture; the state with $\theta =0,\pi$ denotes the Néel-type skyrmion crystal, that with $\theta =\pm \pi /2$ denotes the Bloch-type skyrmion crystal, and that with $\theta \ne 0,\pm \pi /2,\pi$ denotes the hybrid skyrmion crystal. The resulting spin configurations for $\theta =0$, $\theta =\pi /2$ and $\theta =\pi /3$ are illustrated in Figure 1a, Figure 1b, and Figure 1c, respectively.

The skyrmion spin texture described above has been observed in various materials since its discovery in chiral and polar magnets [57,58,59,60,61,62,63,64]; Néel-type skyrmion crystals are typically realized in polar magnets [65,66,67,68], whereas Bloch-type skyrmion crystals are stabilized in chiral magnets [57,58,59,60,62], consistent with the Lifshitz invariants in free energy [69,70,71,72]. In particular, square lattice skyrmion crystals have been reported in noncentrosymmetric compounds such as ${\mathrm{Co}}_{10-x/2}{\mathrm{Zn}}_{10-x/2}{\mathrm{Mn}}_x$ [73,74,75,76,77] and Cu₂OSeO₃ [78,79], as well as in centrosymmetric compounds such as GdRu₂Si₂ [80,81,82,83,84] and GdRu₂Ge₂ [85]. Several stabilization mechanisms for square skyrmion crystals have been proposed theoretically [86]. In noncentrosymmetric cases, the Dzyaloshinskii–Moriya interaction [87,88] plays a central role in inducing the skyrmion crystal [89,90,91,92], whereas in centrosymmetric cases, frustrated exchange interactions combined with bond-dependent anisotropic interactions and/or dipolar interactions are essential [93,94,95]. Although most skyrmion crystals are classified as either Néel- or Bloch-type, recent theoretical studies have suggested the possibility of hybrid skyrmion crystals arising from competing interactions in the momentum space [96] and crystalline-dependent magnetic anisotropy [97]. Experimentally, hybrid skyrmion crystals have been identified in synthetic multilayers [98,99,100] and in bulk magnets such as EuNiGe₃ [101,102], which can exhibit peculiar helicity-dependent dynamics [103,104].

The resulting magnetic point group symmetry depends on the helicity of the skyrmion. For the Néel-type skyrmion crystal in Figure 1a, the magnetic point group symmetry is reduced from $4/mmm{1}^{\prime }$ to $4m^{\prime }{m}^{\prime }$. Since the presence of $m^{\prime }$ symmetry forbids a ferroaxial moment parallel to the mirror plane, the Néel-type skyrmion crystal does not host a ferroaxial moment. Similarly, in the Bloch-type skyrmion crystal in Figure 1b, the ferroaxial moment is prohibited due to the combined symmetry of the twofold rotational operation about the x and y axes with the time-reversal operation. Indeed, the time-reversal-even axial vector oriented along the z axis does not transform as the identity irreducible representation of the magnetic point group ${42}^{\prime }{2}^{\prime }$ in the Bloch-type skyrmion crystal. In contrast, the hybrid skyrmion crystal belongs to the magnetic point group 4, which breaks both $m^{\prime }$ and $2^{\prime }$ symmetries. Consequently, the z component of the ferroaxial moment is symmetry-allowed in the hybrid skyrmion crystal. We summarize the relation between the magnetic point groups of the three skyrmion textures and the symmetry-allowed multipoles relevant to the present study in

Table 1. Magnetic point groups (MPGs) of Néel-, Bloch-, and hybrid-type skyrmion crystals (SkX) and the corresponding allowed and forbidden multipoles. $G_z^{\left(s\right)}$ denotes the ferroaxial moment (electric toroidal dipole), $M_0^{\left(s\right)}$ the magnetic monopole, and $T_z^{\left(s\right)}$ the magnetic toroidal dipole. The irreducible representation (Irrep.) under $4/mmm{1}^{\prime }$ is also shown in the rightmost column.

SkX	MPG	Allowed Multipoles	Forbidden Multipoles	Irrep.
Néel-type	$4m^{\prime }{m}^{\prime }$	$M_0^{\left(s\right)}$	$G_z^{\left(s\right)}$, $T_z^{\left(s\right)}$	${\mathrm{A}}_{2g}^{-}$, ${\mathrm{A}}_{1u}^{-}$
Bloch-type	${42}^{\prime }{2}^{\prime }$	$T_z^{\left(s\right)}$	$G_z^{\left(s\right)}$, $M_0^{\left(s\right)}$	${\mathrm{A}}_{2g}^{-}$, ${\mathrm{A}}_{2u}^{-}$
Hybrid-type	4	$G_z^{\left(s\right)}$, $M_0^{\left(s\right)}$, $T_z^{\left(s\right)}$	–	${\mathrm{A}}_{2g}^{-}$, ${\mathrm{A}}_{1u}^{-}$, ${\mathrm{A}}_{2u}^{-}$

. This overview highlights that the ferroaxial moment is uniquely allowed in the hybrid skyrmion crystal, while the Néel-type and Bloch-type skyrmion crystals only support the magnetic monopole and magnetic toroidal dipole, respectively.

3. Ferroaxiality Under Skyrmion Spin Textures

We begin by examining the ferroaxiality under skyrmion spin textures. Figure 2a shows the $\theta$ dependence of the expectation value with respect to the z component of the electric toroidal dipole $G_z^{\left(\mathrm{s}\right)}$ at the chemical potential $\mu =-2$. The operator expression of the z-directional electric toroidal dipole is given by [105]

$$\begin{array}{c}\hfill G_z^{\left(\mathrm{s}\right)}={\displaystyle \frac{1}{\sqrt{2}}}({\sigma }_x\otimes M_y-{\sigma }_y\otimes M_x),\end{array}$$

(4)

where $M_x$ and $M_y$ represent the orbital components of the magnetic dipole, whose matrix forms for the basis $(s_{\sigma },p_{x\sigma },p_{y\sigma },p_{z\sigma })$ are represented by

$$M_x=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -i\\ 0& 0& i& 0\end{array}\right),M_y=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& i\\ 0& 0& 0& 0\\ 0& -i& 0& 0\end{array}\right).$$

(5)

Equation (4) corresponds to the definition of the electric toroidal dipole introduced in Ref. [105], where the operator is defined at the single-ion level. In the present study, multipole operators are defined on each lattice site in terms of the spin and orbital degrees of freedom, following the conventional single-atom definitions. The system-averaged multipoles shown here are obtained by taking the expectation value of these operators in the mean-field electronic state and summing them over all lattice sites, normalized by the total number of sites, where we take $6\times 6=36$ sites. Equivalently, this corresponds to evaluating the $\mathit{q}=0$ component of the corresponding multipole operator in reciprocal space.

The atomic-scale electric toroidal dipole moment is defined as the vector product of spin and orbital angular momenta. As shown in Figure 2a, $\langle G_z^{\left(\mathrm{s}\right)}\rangle$ becomes nonzero except for $\theta =0$ and $\theta =\pi /2$; the corresponding real-space configurations of $\langle G_z^{\left(s\right)}\rangle$ are shown in Figure 3d–f, where density waves in terms of $\langle G_z^{\left(s\right)}\rangle$ can be found in all cases. In other words, the ferroaxial moment with the same symmetry as the electric toroidal dipole is induced in the hybrid skyrmion crystal but not in the Néel- and Bloch-type skyrmion crystals, which is in agreement with the symmetry analysis in Table 1. These results demonstrate that the ferroaxial moment can be activated by a magnetic phase transition through symmetry lowering, even in the absence of rotational distortions of ligand ions [106]. The expectation value $\langle G_z^{\left(\mathrm{s}\right)}\rangle$ reaches its maximum around $\theta =0.25\pi$, highlighting the importance of the coupling between the Néel- and Bloch-type helicities. With respect to the model parameters, $\langle G_z^{\left(\mathrm{s}\right)}\rangle$ remains nonzero when $t_{sp}=0$, but it vanishes for $\lambda =0$. This behavior highlights the crucial role of spin–orbit coupling in the present model. Although a non-coplanar spin texture can induce orbital hybridization, the emergence of a ferroaxial moment requires an additional microscopic channel that couples spin and orbital sectors. The spin–orbit interaction provides this channel by mixing orbitals of different symmetries, thereby converting the non-coplanar spin texture into an electric toroidal dipole moment. In the absence of SOC ($\lambda =0$), this spin–orbital entanglement is lost, and the ferroaxial moment $\langle G_z^{\left(s\right)}\rangle$ consequently vanishes.

To highlight the distinct behavior of $\langle G_z^{\left(\mathrm{s}\right)}\rangle$ for different $\theta$ values, we evaluate its structure factor defined as

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

(6)

where $\mathit{q}$ is the wave vector and ${\langle \dots \rangle }_i$ means the expectation value at site i. Physically, the structure factor in Equation (6) quantifies the spatial correlations of the electric toroidal dipole moments across the lattice, analogous to how the spin structure factor measures correlations of spins. A peak at $\mathit{q}=\mathbf{0}$ corresponds to a ferroaxial alignment, i.e., a uniform ordering of electric toroidal dipoles, whereas peaks at finite $\mathit{q}$ vectors indicate modulated or antiferroaxial arrangements. This quantity allows us to identify whether the skyrmion textures host uniform or spatially modulated ferroaxial moments.

Figure 3a–c represents the structure factors of $\langle G_z^{\left(\mathrm{s}\right)}\rangle$ at $\theta =0$, $\theta =\pi /6$, and $\theta =\pi /2$, respectively. The data for the hybrid skyrmion crystal in Figure 3b exhibits the intensity at $\mathit{q}=\mathbf{0}$, while those for the Néel-type skyrmion crystal in Figure 3a and the Bloch-type skyrmion crystal in Figure 3c do not. In all cases, however, finite-$\mathit{q}$ components such as ${\mathit{Q}}_1+2{\mathit{Q}}_2$ remain visible. These results indicate that the Néel- and Bloch-type skyrmion crystals accompany the antiferroic alignment of the axial moment without a net uniform component. The corresponding real-space configurations of $\langle G_z^{\left(\mathrm{s}\right)}\rangle$, i.e., the spatial distribution of the ferroaxial moment, are shown in Figure 3d–f. These maps visualize how the ferroaxial moment forms density wave-like patterns depending on the helicity, thereby providing a complementary picture to the spin textures in Figure 1 and facilitating a comparison with experimental imaging techniques such as second-harmonic generation microscopy.

In addition to the electric toroidal dipole, the skyrmion spin texture also induces other odd-parity multipoles. As examples, we present the $\theta$ dependence of the expectation value of the magnetic monopole $M_0^{\left(\mathrm{s}\right)}$ in Figure 2b and that of the z component of the magnetic toroidal dipole $T_z^{\left(\mathrm{s}\right)}$ in Figure 2c. Their operator expressions are given by

$$\begin{array}{cc}\hfill M_0^{\left(\mathrm{s}\right)}& ={\displaystyle \frac{1}{\sqrt{3}}}(Q_x\otimes {\sigma }_x+Q_y\otimes {\sigma }_y+Q_z\otimes {\sigma }_z),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill T_z^{\left(\mathrm{s}\right)}& ={\displaystyle \frac{1}{\sqrt{2}}}({\sigma }_x\otimes Q_y-{\sigma }_y\otimes Q_x),\hfill \end{array}$$

(8)

where $(Q_x,Q_y,Q_z)$ represents the atomic-scale electric dipole operators, defined as

$$\begin{array}{c}\begin{array}{c}{Q}_x={\displaystyle \frac{1}{\sqrt{3}}}\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\hfill \end{array}\\ \begin{array}{c}{Q}_y={\displaystyle \frac{1}{\sqrt{3}}}\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\hfill \end{array}\\ \begin{array}{c}{Q}_z={\displaystyle \frac{1}{\sqrt{3}}}\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right).\hfill \end{array}\end{array}$$

(9)

Equations (7) and (8) represent the magnetic monopole operator and magnetic toroidal dipole operator, respectively, in the convention of Ref. [105].

The data show that the magnetic monopole is activated in the Néel-type skyrmion crystal, while the magnetic toroidal dipole is activated in the Bloch-type skyrmion crystal. This trend is consistent with the real-space spin textures shown in Figure 1a,b: in the case of the Néel-type skyrmion crystal, the in-plane spins are arranged radially around the core, reflecting the symmetry of a magnetic monopole. In contrast, for the Bloch-type skyrmion crystal, the in-plane spins form a vortex-like configuration around the core, which corresponds to the symmetry of a magnetic toroidal dipole. The appearance of the magnetic monopole (magnetic toroidal dipole) implies the longitudinal (transverse) magnetoelectric effect [97,107]. Since the hybrid skyrmion crystal hosts both odd-parity multipoles, multiple components of the magnetoelectric tensor become finite.

4. Transverse Responses Under an External Magnetic Field

Next, we examine the cross-correlation responses associated with the ferroaxial moment under an external magnetic field. To this end, we add a Zeeman term to the Hamiltonian function in Equation (1):

$$\begin{array}{cc}\hfill {\mathcal{H}}^Z& =\sum _{i\alpha \sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{\mathit{\sigma }}_{\sigma {\sigma }^{\prime }}{c}_{i\alpha {\sigma }^{\prime }}·\mathit{H},\hfill \end{array}$$

(10)

where $\mathit{H}=(H_x,H_y,0)$ represents the in-plane external magnetic field. We consider the hybrid skyrmion spin texture with $\theta =\pi /6$.

Figure 4a presents the x component of the spin magnetization $M_x$ under a magnetic field applied along the x direction, i.e., $H_x\ne 0$ and $H_y=0$. Here, the $\mu$ component of the spin magnetization is defined as

$$\begin{array}{c}\hfill M_{\mu }=Q_0\otimes {\sigma }_{\mu },\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill & Q_0=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).\hfill \end{array}$$

(12)

The expectation value of the magnetization $\langle M_x\rangle$ increases linearly with $H_x$, consistent with the standard response of a physical quantity to its conjugate external field. A similar response is obtained when the magnetic field is applied along the y direction ($H_x=0$ and $H_y\ne 0$), where $\langle M_y\rangle$ exhibits a linear dependence on $H_y$, as shown in Figure 4d. The equivalence between the x and y directional responses reflects the fourfold rotational symmetry of the hybrid skyrmion crystal.

Meanwhile, the presence of the ferroaxial moment leads to additional off-diagonal transverse responses. In this case, the y component of the spin magnetization is induced by the x-directional magnetic field, as shown in Figure 4b, and the x component is induced by the y-directional magnetic field, as shown in Figure 4c [34]. It is noted that the sign of the induced spin magnetization is opposite for different magnetic field directions. Such a transverse response vanishes for the Néel- and Bloch-type skyrmion crystals, since the ferroaxial moment is not present. In other words, such a transverse response is a hallmark of the hybrid skyrmion crystal.

Moreover, additional cross-correlation responses arise from other active odd-parity multipole moments in the skrmion spin texture [108]. As an illustrative example, we focus on the magnetic field-induced toroidal dipole in the hybrid skyrmion spin texture. Figure 5a–d displays the magnetic field dependence of the induced magnetic toroidal dipole moment under the hybrid skyrmion spin texture. Here, the magnetic toroidal dipoles $T_x^{\left(\mathrm{s}\right)}$ and $T_y^{\left(\mathrm{s}\right)}$ are defined as

$$\begin{array}{cc}\hfill T_x^{\left(\mathrm{s}\right)}& ={\displaystyle \frac{1}{\sqrt{2}}}({\sigma }_y\otimes Q_z-{\sigma }_z\otimes Q_y),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill T_y^{\left(\mathrm{s}\right)}& ={\displaystyle \frac{1}{\sqrt{2}}}({\sigma }_z\otimes Q_x-{\sigma }_x\otimes Q_z).\hfill \end{array}$$

(14)

The behavior of the induced magnetic toroidal dipole follows that of the induced magnetization shown in Figure 4. In the longitudinal channel, where ${\mathit{T}}^{\left(\mathrm{s}\right)}\Vert \mathit{H}$, the induced magnetic toroidal dipole has the same sign and magnitude, as shown in Figure 5a,d, while in the transverse channel, where ${\mathit{T}}^{\left(\mathrm{s}\right)}\perp \mathit{H}$, the induced magnetic toroidal dipole has the same magnitude but opposite sign, as shown in Figure 5b,c. Since the magnetic toroidal dipole becomes the microscopic origin of the nonlinear nonreciprocal transport [109,110,111,112,113], one may anticipate the nonreciprocal conductivity along both x and y directions in the hybrid skyrmion spin texture under the external in-plane magnetic field.

The emergence of the longitudinal magnetic toroidal dipole parallel to the applied magnetic field can be ascribed to the activation of the electric toroidal monopole, which mediates an inner-product coupling between polar and axial vector quantities as $\mathit{H}·\mathit{T}$. In contrast, the emergence of the transeverse magnetic toroidal dipole perpendicular to the magnetic field originates from the activation of the electric dipole, giving rise to an outer-product coupling between the polar and axial vector quantities as $\mathit{H}\times \mathit{T}$. Since the electric toroidal monopole and electric dipole are activated in the Bloch-type and Néel-type skyrmion crystals, respectively, the hybrid skyrmion crystal exhibits both contributions. Thus, the hybrid skyrmion crystal serves as a unique platform in which longitudinal and transverse magnetic toroidal dipole responses coexist.

From an experimental perspective, promising candidate systems for realizing such hybrid skyrmion crystals include synthetic multilayers [98,99,100] and bulk polar magnets such as EuNiGe₃ [101,102], where hybrid helicities have already been reported. In these materials, the coexistence of hybrid skyrmions and ferroaxial moments is anticipated. Detecting ferroaxial responses may be possible via nonlinear optical techniques, such as second-harmonic generation microscopy that can directly image ferroaxial domains [50]. This probe offers complementary approaches to test the theoretical predictions presented in this study.

It is worth noting that the activation of the ferroaxial moment through the electric toroidal dipole is not limited to skyrmion textures [53]. In principle, other non-coplanar multiple-Q spin states that break the same set of mirror and twofold symmetries [114] can also induce ferroaxiality, even if they do not carry a quantized skyrmion number. The hybrid skyrmion crystal serves as a representative case in which these symmetry conditions are naturally realized.

5. Conclusions

We have theoretically demonstrated that the ferroaxial moment, a time-reversal-even axial dipole, can be induced in a hybrid skyrmion crystal without relying on structural rotational distortions. By employing a multi-orbital tight-binding model with s and p orbitals on a square lattice, we quantified ferroaxiality through the expectation value of the electric toroidal dipole. Our analysis revealed that the hybrid skyrmion texture, formed by the superposition of Néel- and Bloch-type vortices, breaks the relevant symmetries and gives rise to a finite ferroaxial moment, in contrast to the pure Néel- or Bloch-type cases. We further clarified that relativistic spin–orbit coupling is essential for inducing this ferroaxiality at the atomic scale.

In addition to establishing the ferroaxial moment, we identified characteristic magnetic field-induced cross-correlation responses in the hybrid skyrmion crystal. These include a transverse spin magnetization originating from the electric toroidal dipole, as well as longitudinal and transverse magnetic toroidal dipoles induced by electric toroidal monopoles and electric dipoles, respectively. Such responses arise from the dual Néel- and Bloch-type characteristics inherent in the hybrid spin texture. Taken together, our results establish the hybrid skyrmion crystal as a promising platform for exploring magnetic-order-driven ferroaxial physics, opening avenues for future applications in spintronics and multiferroics.