Field-Induced Ferroaxiality in Antiferromagnets with Magnetic Toroidal Quadrupole

Abstract

Magnetic toroidal multipoles have recently emerged as key descriptors of unconventional cross-correlation phenomena in antiferromagnetic systems. Among them, the rank-2 magnetic toroidal quadrupole, which is characterized as a time-reversal-odd polar tensor, has been theoretically associated with a variety of cross-correlation phenomena arising from the time-reversal symmetry breaking. In this study, we investigate the interplay between magnetic toroidal quadrupoles and electric toroidal dipoles in antiferromagnets, with a particular focus on magnetic field-induced ferroaxiality. Through symmetry analysis and microscopic model calculations, we demonstrate that ferroaxiality can be induced by an external magnetic field, depending on both the field direction and the type of the magnetic toroidal quadrupole. We classify all magnetic point groups that possess magnetic toroidal quadrupoles and identify various candidate materials based on the MAGNDATA database. Our findings reveal a route to coupling spin and lattice degrees of freedom via toroidal multipoles.

1. Introduction

Magnetic toroidal multipoles have recently attracted considerable attention in condensed matter physics [1,2,3,4,5]. These magnetic toroidal multipoles differ from conventional magnetic multipoles in that they are time-reversal-odd polar tensors, rather than time-reversal-odd axial tensors. As such, several intriguing physical phenomena characteristic of magnetic toroidal multipoles have been recognized. A representative example is the rank-1 magnetic toroidal dipole, which is typically realized as a vortex-like spin configuration that simultaneously breaks spatial inversion and time-reversal symmetries [4,5,6,7]. This symmetry breaking leads to a host of parity-violating responses, including the linear magnetoelectric effect [8,9,10,11,12,13,14], asymmetric magnon dispersions [15,16,17,18,19], directional-dependent nonlinear transport [20,21,22,23,24,25], nonlinear spin Hall effect [26,27], and intrinsic nonlinear Hall effect [28,29,30]. Some of these phenomena have been experimentally observed in insulating and metallic multiferroic materials such as Cr₂O₃ [8,31,32], GaFeO₃ [33,34,35,36], LiCoPO₄ [37,38,39,40], CuMnAs [28,41,42,43,44], UNi₄B [45,46,47,48], and HoAgGe [49,50,51,52]. Beyond the dipole component, recent studies have shown that the other-rank magnetic toroidal multipoles also lead to exotic responses. For instance, the rank-0 magnetic toroidal monopole has been linked to time-reversal-switching responses [9,53,54], while the rank-2 magnetic toroidal quadrupole has been connected to magnetically driven spin current generation [55,56] and quadrupole anomalous Hall effect [57]. These magnetic toroidal multipoles naturally appear in antiferromagnetic materials irrespective of whether their spin configurations are collinear or noncollinear. A number of candidate materials exhibiting such magnetic toroidal multipoles have been theoretically proposed and experimentally examined, including Co₂SiO₄ [58], KMnF₃ [59], Ca₂RuO₄ [60], Mn₃As₂ [61], Mn₃IrGe [62], Mn₂FeMoO₆ [63], CuFeS₂ [64], ErGe_1.83 [65], and Ce₄Sb₃ [66].

In addition to magnetic toroidal multipoles, electrons in solids possess another toroidal multipole degree of freedom known as electric toroidal multipoles [1,2,6,7]. The electric toroidal multipoles are characterized as time-reversal-even axial tensors, exhibiting parity properties opposite to those of magnetic toroidal multipoles in terms of both spatial inversion and time-reversal operations; the electric toroidal dipole is related to the vortex structure of the electric dipole, as shown in Figure 1 [67,68]. Recent spectroscopic studies have observed ferroic ordering of electric toroidal dipoles, which is so-called ferroaxial or ferrorotational ordering, in a variety of materials, including RbFe(MoO₄)₂ [69,70], NiTiO₃ [70,71,72], Ca₅Ir₃O₁₂ [73,74,75,76], and BaCoSiO₄ [77]. The electric toroidal dipoles can be the origin of various unconventional physical phenomena [78,79,80], such as the antisymmetric thermopolarization [81], intrinsic longitudinal spin current generation [79,82], and transverse nonlinear magnetic response [83]. In this way, both magnetic and electric toroidal multipoles offer promising pathways to realizing exotic physical functionalities. However, the interplay between these two types of toroidal multipoles remains largely unexplored.

In the present study, we show an effective coupling between magnetic toroidal and electric toroidal degrees of freedom based on symmetry considerations and microscopic model analyses. We specifically focus on ferroaxiality in antiferromagnets hosting the magnetic toroidal quadrupole. Our results show that applying an external magnetic field in the presence of such a magnetic toroidal quadrupole order can induce ferroaxiality, with the response strongly dependent on the field direction. To illustrate this mechanism, we analyze fundamental antiferromagnetic systems with both collinear and noncollinear spin configurations for a tight-binding model. We further classify all magnetic point groups to possess the magnetic toroidal quadrupoles in antiferromagnets, and provide a list of candidate materials extracted from the MAGNDATA database [84]. These findings reveal a route to functionalities arising from the interplay between magnetic toroidal and electric toroidal multipole degrees of freedom in antiferromagnetic systems.

The remainder of this paper is organized as follows. In Section 2, we briefly introduce the concept of the magnetic toroidal quadrupole and discuss relevant antiferromagnetic spin structures. We also present a classification of all magnetic point groups possessing active magnetic toroidal quadrupoles. In Section 3, we explore the emergence of magnetic field-induced ferroaxiality under the magnetic toroidal quadrupole ordering based on the symmetry analysis. Section 4 provides microscopic model calculations that demonstrate the cross-correlation effects discussed in Section 3. Finally, Section 5 summarizes the main conclusions of this work.

2. Magnetic Toroidal Quadrupole

The magnetic toroidal quadrupole is a rank-2 magnetic toroidal multipole, consisting of five components. Its operator expressions are given by [85]

$$\begin{array}{cc}\hfill T_u& =3z_j{t}_2^z-{\mathit{r}}_j·{\mathit{t}}_2,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill T_v& =\sqrt{3}(x_j{t}_2^x-y_j{t}_2^y),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill T_{yz}& =\sqrt{3}(y_j{t}_2^z+z_j{t}_2^y),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill T_{zx}& =\sqrt{3}(z_j{t}_2^x+x_j{t}_2^z),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill T_{xy}& =\sqrt{3}(x_j{t}_2^y+y_j{t}_2^x).\hfill \end{array}$$

(5)

Here, ${\mathit{t}}_2\left({\mathit{r}}_j\right)$ denotes the rank-2 magnetic toroidal moment defined as

$$\begin{array}{cc}\hfill {\mathit{t}}_2\left({\mathit{r}}_j\right)& ={\displaystyle \frac{1}{6}}{\mathit{r}}_j\times \left({\mathit{l}}_j+4{\mathit{s}}_j\right),\hfill \end{array}$$

(6)

where ${\mathit{r}}_j$, ${\mathit{l}}_j$, and ${\mathit{s}}_j$ stand for the position, orbital angular momentum, and spin angular momentum operators of the j-th electron, respectively. The magnetic toroidal moment ${\mathit{t}}_2\left({\mathit{r}}_j\right)$ is odd under both spatial inversion and time-reversal operations. As a result, the magnetic toroidal quadrupoles with five components, $(T_u,T_v,T_{yz},T_{zx},T_{xy})$, are even under the spatial inversion operation and odd under the time-reversal operation. This symmetry property indicates that the breaking of the time-reversal symmetry is essential for activating magnetic toroidal quadrupole moments.

A variety of magnetic structures can accommodate magnetic toroidal quadrupoles. We present several examples of antiferromagnetic spin configurations that host the magnetic toroidal quadrupoles in a tetragonal cluster with $D_{4\mathrm{h}}$ symmetry. These spin configurations are used for the microscopic model analyses discussed in Section 4. Under the $D_{4\mathrm{h}}$ point group, the irreducible representations of the magnetic toroidal quadrupoles are classified as follows: $T_u\in A_{1g}$, $T_v\in B_{1g}$, $T_{yz},T_{zx}\in E_g$, and $T_{xy}\in B_{2g}$ [86]. Considering the 8e Wyckoff position under $D_{4\mathrm{h}}$ symmetry, we obtain a total of 24 independent magnetic structures, whose symmetry representations are decomposed as $A_{1g}\oplus 2A_{2g}\oplus 2B_{1g}\oplus B_{2g}\oplus 3E_g\oplus 2A_{1u}\oplus A_{2u}\oplus B_{1u}\oplus 2B_{2u}\oplus 3E_u$. Among these, the irreducible representations $3E_g\oplus 2A_{1u}\oplus A_{2u}\oplus B_{1u}\oplus 2B_{2u}\oplus 3E_u$ correspond to odd-parity magnetic and magnetic toroidal multipoles, while $A_{1g}\oplus 2A_{2g}\oplus 2B_{1g}\oplus B_{2g}\oplus 3E_g$ corresponds to even-parity ones. The latter category includes magnetic dipole, magnetic toroidal quadrupole, magnetic octupole, and so on. In particular, the magnetic structures belonging to $A_{1g}$, $B_{1g}$, $B_{2g}$, and $E_g$ indicate that they can be regarded as hosting uniform magnetic toroidal quadrupoles from the viewpoint of cluster multipole theory [87,88,89,90]. We present antiferromagnetic structures with $T_u$ in Figure 2a, $T_v$ in Figure 2b,c, and $T_{xy}$ in Figure 2d; the magnetic structures in Figure 2a,b,d are noncollinear, while that in Figure 2c is collinear.

The emergence of magnetic toroidal quadrupoles, as illustrated in Figure 2, can be intuitively understood by examining the spatial distribution of magnetic toroidal dipole moment defined as ${\sum }_i{\mathit{r}}_i\times {\mathit{s}}_i$ without the orbital degree of freedom. By setting the origin of ${\mathit{r}}_i$ at the center of each face of the cuboid in Figure 2a, one finds that the magnetic toroidal dipole moments point outward on the side faces, whereas they point inward on the top and bottom surfaces. This distribution clearly corresponds to the magnetic toroidal quadrupole $T_u$ defined in Equation (1). Similarly, a spatial distribution of magnetic toroidal dipole moments exhibiting the same symmetry as $T_v$ can be realized, independent of whether the magnetic structure is collinear and noncollinear, as shown in Figure 2b,c. For the case of $T_{xy}$ shown in Figure 2d, the corresponding quadrupole distribution can be obtained by defining the origin of ${\mathit{r}}_i$ at the midpoint of the bond connecting the top and bottom faces of the cuboid.

The above examples illustrate that a wide variety of antiferromagnetic structures can host the magnetic toroidal quadrupoles. Indeed, from the viewpoint of the magnetic point group symmetry, the magnetic toroidal quadrupoles are symmetry-allowed in 42 out of the 122 magnetic point groups [86]; among them, 19 magnetic point groups possess the ferroaxiality (electric toroidal dipoles). We provide a comprehensive list of all magnetic point groups that permit magnetic toroidal quadrupoles or electric toroidal dipole under magnetic orderings, along with representative candidate materials extracted from MAGNDATA [84], in

Table 1. List of magnetic point groups (MPGs) to possess the magnetic toroidal quadrupole (MTQ) and electric toroidal dipole (ETD). # in the leftmost column shows the index of the magnetic point group. Representative materials extracted from MAGNDATA [84] are also listed in the rightmost column. The materials extracted from Ref. [56] are denoted by †.

	MPG	MTQ	ETD	Materials
#1	1	$T_u$, $T_v$, $T_{yz}$, $T_{zx}$, $T_{xy}$	$G_x$, $G_y$, $G_z$	Mn2ScSbO6 [92], CuB2O4 [93]
#3	$\overline{1}$	$T_u$, $T_v$, $T_{yz}$, $T_{zx}$, $T_{xy}$	$G_x$, $G_y$, $G_z$	RbMnF4 [94], YVO3 [95]
#6	2	$T_u$, $T_v$, $T_{zx}$	$G_y$	LiFeP2O7 [96], HoNiO3 [97]
#8	$2^{\prime }$	$T_{xy}$, $T_{yz}$	$G_y$	BaDy2O4 [98], BaFe2Se4 [99]
#9	m	$T_u$, $T_v$, $T_{zx}$	$G_y$	Mn4Nb2O9 [100]
#11	$m^{\prime }$	$T_{xy}$, $T_{yz}$	$G_y$	ScFeO3 [101], MnSiN2 [102]
#12	$2/m$	$T_u$, $T_v$, $T_{zx}$	$G_y$	Cu2OSO4 [103], NiCO3 [104]
#16	$2^{\prime }/m^{\prime }$	$T_{xy}$, $T_{yz}$	$G_y$	Mn3Ti2Te6 [105], CoV2O6 [106]
#17	222	$T_u$, $T_v$		FePO4 [107]
#19	$2^{\prime }{2}^{\prime }2$	$T_{xy}$		VNb3S6 [108], CsNiCl3 [109]
#20	$mm2$	$T_u$, $T_v$		FeSb2O4 [110], Er2Cu2O5 [111]
#22	$m^{\prime }m{2}^{\prime }$	$T_{zx}$		CaBaCo4O7 [112], Ba2CoGe2O7 [113]
#23	$m^{\prime }{m}^{\prime }2$	$T_{xy}$		$\alpha$-Cu2V2O7 [114], BaCuF4 [115]
#24	$mmm$	$T_u$, $T_v$		$\alpha$-Mn2O3 [116], NiTe2O5 [117], Fe2PO5 [118]
#27	$m^{\prime }{m}^{\prime }m$	$T_{xy}$		NiF2 [119], NaOsO3 [120], YFeO3 ($Y=$ Ce, Nd, Dy) [121,122,123]
#29	4	$T_u$	$G_z$	Ce5TeO8 [124]
#31	$4^{\prime }$	$T_v$, $T_{xy}$	$G_z$
#32	$\overline{4}$	$T_u$	$G_z$
#34	${\overline{4}}^{\prime }$	$T_v$, $T_{xy}$	$G_z$	CsCoF4 † [125]
#35	$4/m$	$T_u$	$G_z$	Mn3CuN [126], MnV2O4 [127]
#37	$4^{\prime }/m$	$T_v$, $T_{xy}$	$G_z$
#40	422	$T_u$		Ho2Ge2O7 † [128]
#42	$4^{\prime }{22}^{\prime }$	$T_v$		Er2Ge2O7 [129,130]
#44	$4mm$	$T_u$
#46	$4^{\prime }m{m}^{\prime }$	$T_v$		PbNi1.76Mg0.24V2O8 [131]
#48	$\overline{4}2m$	$T_u$		Ba2MnSi2O7 [132], CuFeS2 † [64]
#50	${\overline{4}}^{\prime }m{2}^{\prime }$	$T_v$
#51	${\overline{4}}^{\prime }2m^{\prime }$	$T_v$		Ce4Sb3 † [66]
#53	$4/mmm$	$T_u$		CdYb2S4 † [133], KMnF3 [59]
#56	$4^{\prime }/mm{m}^{\prime }$	$T_v$		CoF2 † [134], Er2Sn2O7 [135]
#60	3	$T_u$	$G_z$	Cu2OSeO3 [136], Na2MnTeO6 [137]
#62	$\overline{3}$	$T_u$	$G_z$	NiN2O6 [138], CaFe3Ti4O12 [139]
#65	32	$T_u$		La0.33Sr0.67FeO3 [140]
#68	$3m$	$T_u$		PbNiO3 [141], Ba3MnNb2O9 † [142]
#71	$\overline{3}m$	$T_u$		Li2MnTeO6 [143], LaCrO3 [144]
#76	6	$T_u$	$G_z$	BaCoSiO4 [145], ScMnO3 [146]
#79	$\overline{6}$	$T_u$	$G_z$
#82	$6/m$	$T_u$	$G_z$	FeF3 [147]
#87	622	$T_u$
#91	$6mm$	$T_u$		HoMnO3 [148], YMnO3 [146]
#95	$\overline{6}m2$	$T_u$		Ba3CoSb2O9 [149]
#100	$6/mmm$	$T_u$

. These candidate antiferromagnetic compounds represent promising platforms for realizing magnetic field-induced ferroaxiality, which will be discussed in the next section.

3. Field-Induced Axiality

Since the magnetic toroidal quadrupole is a rank-2 time-reversal-odd polar tensor, it mediates coupling between two rank-1 vectors: a time-reversal-even one, corresponding to the electric toroidal dipole, i.e., ferroaxiality related to static rotational lattice distortion, and a time-reversal-odd one, corresponding to the magnetic dipole. Considering the magnetic dipole shares the same symmetry as the external magnetic field, the electric toroidal dipole $\mathit{G}=(G_x,G_y,G_z)$ can be induced by the magnetic field $\mathit{H}=(H_x,H_y,H_z)$ in the presence of the magnetic toroidal quadrupole.

From the tensor product [150], their relationship is clearly expressed as follows:

$$\begin{array}{cc}\hfill G_x& =\left(-{\displaystyle \frac{1}{\sqrt{3}}}{T}_u+T_v\right)H_x+T_{xy}{H}_y+T_{zx}{H}_z,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill G_y& =T_{xy}{H}_x-\left({\displaystyle \frac{1}{\sqrt{3}}}{T}_u+T_v\right)H_y+T_{yz}{H}_z,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill G_z& =T_{zx}{H}_x+T_{yz}{H}_y+{\displaystyle \frac{2}{\sqrt{3}}}{T}_u{H}_z.\hfill \end{array}$$

(9)

These expressions show that two components of the magnetic toroidal quadrupoles, $T_u$ and $T_v$, can give rise to longitudinal couplings between $\mathit{G}$ and $\mathit{H}$; $G_x$, $G_y$, and $G_z$ are induced by $H_x$, $H_y$, and $H_z$ under $T_u$, while $G_x$ and $G_y$ are induced by $H_x$ and $H_y$ under $T_v$. In contrast, $T_{yz}$, $T_{zx}$, and $T_{xy}$, lead to transverse couplings between $\mathit{G}$ and $\mathit{H}$; for example, $G_x$ and $G_y$ are induced by $H_y$ and $H_x$ under $T_{xy}$, respectively. Thus, ferroaxiality can be induced by the external magnetic field whenever magnetic toroidal quadrupoles are activated in antiferromagnetic systems. The relationship between the ferroaxial moment and magnetic field is summarized in

Table 2. Ferroaxiality induced by the magnetic field under magnetic toroidal quadrupoles (MTQs). $(H_x,H_y,H_z)$ stand for the components of the applied magnetic field, and $(G_x,G_y,G_z)$ stand for the components of the electric toroidal dipole corresponding to the axial moment.

MTQ	${\mathit{H}}_{\mathit{x}}$	${\mathit{H}}_{\mathit{y}}$	${\mathit{H}}_{\mathit{z}}$
$T_u$	$G_x$	$G_y$	$G_z$
$T_v$	$G_x$	$G_y$	–
$T_{yz}$	–	$G_z$	$G_y$
$T_{zx}$	$G_z$	–	$G_x$
$T_{xy}$	$G_y$	$G_x$	–

. Similarly, the magnetic toroidal monopole $T_0$ also induces the coupling between $\mathit{G}$ and $\mathit{H}$ in the form of $\mathit{G}·\mathit{H}$, indicating the emergence of isotropic longitudinal field-induced ferroaxiality, as discussed in Ref. [54].

It is also noteworthy that, from a symmetry viewpoint, the magnetic toroidal quadrupole enables coupling between a rank-1 time-reversal-even polar vector and a rank-1 time-reversal-odd polar vector. The former corresponds to an electric dipole, which shares symmetry with physical quantities such as the spin current and electric field, while the latter corresponds to a magnetic toroidal dipole, which shows the same transformation as the electric current. Consequently, this coupling leads to phenomena, such as electric-field-induced magnetic toroidal dipoles and spin current generation driven by electric currents [55,56].

4. Model Analyses

In order to demonstrate the magnetic-field-induced ferroaxiality in antiferromagnets hosting the magnetic toroidal quadrupole, we analyze a fundamental tight-binding model. We consider a bilayer lattice structure consisting of eight sublattices under tetragonal $D_{4\mathrm{h}}$ symmetry, as illustrated in Figure 3. The bond lengths are set as $a=a^{\prime }=0.5$ and $c=1$. Each lattice site hosts one s orbital and three p orbitals, enabling the inclusion of electric toroidal degrees of freedom within the Hilbert space. The tight-binding model is given by

$$\begin{array}{c}\hfill \mathcal{H}=\sum _{\begin{array}{c}\mathit{k}\gamma \alpha \sigma \\ {\gamma }^{\prime }{\alpha }^{\prime }{\sigma }^{\prime }\end{array}}{c}_{\mathit{k}\gamma \alpha \sigma }^{†}({\delta }_{\sigma {\sigma }^{\prime }}{H}^t+{\delta }_{\gamma {\gamma }^{\prime }}{H}^{\mathrm{SOC}}-h{\delta }_{\gamma {\gamma }^{\prime }}{\delta }_{\alpha {\alpha }^{\prime }}{H}^{\mathrm{M}}-{\delta }_{\gamma {\gamma }^{\prime }}{\delta }_{\alpha {\alpha }^{\prime }}{H}^{\mathrm{Z}})c_{\mathit{k}{\gamma }^{\prime }{\alpha }^{\prime }{\sigma }^{\prime }},\end{array}$$

(10)

where $c_{\mathit{k}\gamma \alpha \sigma }^{(†)}$ represents the annihilation (creation) operator of electrons, where the subscripts $\mathit{k}$, $\gamma$, $\alpha$, and $\sigma$ stands for the wave vector, sublattices A–H, orbitals s, $p_x$, $p_y$, and $p_z$, and spin, respectively. The first term, $H^t$, represents the kinetic energy contributions, including both intra- and inter-unit cuboid hoppings. The intra-cuboid hoppings are parameterized as follows: along the in-plane bond direction, t denotes the hopping between s orbitals ($\alpha ,{\alpha }^{\prime }=s$), $t_p$ denotes the hopping between $(p_x,p_y)$ orbitals ($\alpha ,{\alpha }^{\prime }=p_x,p_y$), $t_z$ denotes the hopping between $p_z$ orbitals ($\alpha ,{\alpha }^{\prime }=p_z$), and $t_{sp}$ denotes the hopping between different s-$(p_x,p_y)$ orbitals ($\alpha =s$ and ${\alpha }^{\prime }=p_x,p_y$ and vice versa). We also set the hoppings along the z direction as $t_s^z$ between s orbitals, $t_p^z$ between $p_z$ orbitals, and $t_{sp}^z$ between s and $p_z$ orbitals. We set $t=-1$ as the energy unit of the model, and the other hopping parameters are chosen as $t_p=0.7$, $t_z=0.2$, $t_{sp}=0.3$, $t_s^z=0.5$, $t_p^z=0.1$, and $t_{sp}^z=0.15$. For the inter-cuboid hopping parameters, each parameter is scaled by a factor of 0.8 relative to the corresponding intra-cuboid value. The following results are not qualitatively altered for different choices of the hopping parameters. The second term in Equation (10), $H^{\mathrm{SOC}}$, represents the atomic spin–orbit coupling, acting on the three p orbitals; $\lambda$ denotes the magnitude of the spin–orbit coupling and is set as $0.5$.

The third term in Equation (10), $H^{\mathrm{M}}$, describes the antiferromagnetic mean-field term with the magnitude h; we set $h=2$. We consider four types of antiferromagnetic spin configurations hosting the magnetic toroidal quadrupole, as shown in Figure 2a–d. The fourth term in Equation (10), $H^{\mathrm{Z}}$, represents the Zeeman-coupling term with an external magnetic field $\mathit{H}=(H_x,H_y,H_z)$. Here, we neglect orbital effects of the magnetic field, as they do not affect the following results qualitatively. In all calculations, we consider a low-electron filling per site $n_{\mathrm{e}}=(1/N)\langle {\sum }_{\mathit{k}\gamma \alpha \sigma }{c}_{\mathit{k}\gamma \alpha \sigma }^{†}{c}_{\mathit{k}\gamma \alpha \sigma }\rangle =0.2$ ($n_{\mathrm{e}}=8$ corresponds to the full filling); the total system size is taken to be $N=8\times {1600}^2$, where ${1600}^2$ represents the number of supercell.

Figure 4 shows the magnetic field dependence of ferroaxiality for four types of antiferromagnetic spin configurations. We here evaluate the degree of ferroaxiality by calculating the expectation value of the atomic-scale electric toroidal dipole operator, which is defined as

$$\begin{array}{c}\hfill \mathit{G}=\mathit{l}\times \mathit{\sigma },\end{array}$$

(11)

where $\mathit{l}$ represents the orbital angular momentum operator for the p orbitals and $\mathit{\sigma }=2\mathit{s}$. It is noteworthy that this outer-product form of coupling between $\mathit{l}$ and $\mathit{\sigma }$, characteristic of the electric toroidal dipole, stands in stark constrast to the inner-product form of coupling $\mathit{l}·\mathit{\sigma }$ that appears in conventional spin–orbit coupling terms. While such an outer-product spin–orbit interaction has been linked to static rotational lattice distortion in nonmagnetic systems [151], our results demonstrate that it can also be induced by an external magnetic field in magnetic systems with the magnetic toroidal quadrupole.

In Figure 4a, we present the results for the antiferromagnetic ordering associated with $T_u$. As expected from the symmetry argument in Section 3, both $\langle G_z\rangle$ and $\langle G_x\rangle$ become finite upon applying the magnetic field along the z and x directions, respectively. Moreover, these responses exhibit an odd-function behavior with respect to the magnetic field, which indicates that the direction of ferroaxiality can be reversed by reversing the field direction. Similarly, the other antiferromagnetic spin configurations with $T_v$ and $T_{xy}$ display the magnetic field-induced ferroaxiality consistent with the symmetry expectations; $\langle G_x\rangle$ is induced by $H_x$ in the presence of $T_v$, as shown in Figure 4b,c, while $\langle G_y\rangle$ is induced by $H_x$ in the presence of $T_{xy}$, as shown in Figure 4d. We confirmed that similar results satisfying the relationship in Table 2 are also obtained for the magnetic field along the y direction. We also emphasize the crucial role of the spin–orbit coupling $\lambda$ in inducing ferroaxiality: in the absence of the spin–orbit coupling ($\lambda =0$), the expectation value $\langle \mathit{G}\rangle$ vanishes entirely in the present model.

5. Conclusions

In conclusion, we have investigated cross-correlation phenomena mediated by the magnetic toroidal quadrupoles in antiferromagnetic systems. Our analysis reveals that ferroaxiality, which is a structural degree of freedom associated with rotational lattice distortions, can be induced by an external magnetic field in systems that host the magnetic toroidal quadrupole. We have demonstrated this magnetic field-induced ferroaxiality by means of both symmetry arguments and explicit microscopic model calculations. This result highlights a novel route for coupling spin and lattice degrees of freedom via magnetic toroidal quadrupoles, which becomes a source of a new type of magnetoelastic interactions. Given that a wide variety of antiferromagnetic materials are predicted to exhibit magnetic toroidal quadrupoles, our findings provide a guiding principle for identifying materials that enable such coupling phenomena. Our results are expected to stimulate experimental investigations of ferroaxial responses driven by the magnetic toroidal quadrupole. From an experimental perspective, second-harmonic generation, which covers processes including higher-order multipole contributions like magnetic-dipole and electric-quadrupole transitions even under centrosymmetric lattice structures, is a promising technique for detecting field-induced ferroaxiality, as it is highly sensitive to subtle symmetry changes in the crystal structure [152].