Magnetic Toroidal Monopole in a Single-Site System

Abstract

A magnetic toroidal monopole, which characterizes time-reversal-odd polar-charge quantity, manifests itself not only in antiferromagnetism but also in time-reversal switching physical responses. We theoretically investigate an atomic-scale description of the magnetic toroidal monopole based on multipole representation theory, which consists of four types of multipoles. We show that the magnetic toroidal monopole degree of freedom is activated as the off-diagonal imaginary hybridization between the single-site orbitals with the same orbital angular momentum but different principal quantum numbers. We demonstrate that the expectation value of the magnetic toroidal monopole becomes nonzero when both electric and magnetic fields are applied to the system.

1. Introduction

Time-reversal symmetry is one of the most fundamental concepts in physics. When the time-reversal symmetry is present, the laws of physics are invariant when the direction of time is reversed. In condensed matter physics, the presence and absence of time-reversal symmetry affects not only microscopic electronic band structures but also macroscopic physical phenomena. For example, energy levels are at least twofold degenerate, which is the so-called Kramers degeneracy, in a time-reversal symmetric nonmagnetic system with half-integer spin. Another example is the robustness of gapless surface states in topological insulators against nonmagnetic disorder when the time-reversal symmetry is preserved [1,2,3,4,5,6].

On the other hand, when the time-reversal symmetry is broken, magnetic properties and phenomena emerge. The most typical example is a ferromagnetic state with a net magnetization, which becomes a source of the anomalous Hall effect owing to a finite Berry curvature in momentum space [7,8,9,10,11,12,13,14,15,16]. Recently, it was recognized that such an anomalous Hall effect can be induced even in antiferromagnets without the net magnetization when a time-reversal symmetry breaking occurs in antiferromagnets with collinear [17,18,19,20,21,22,23,24], coplanar [25,26,27,28,29], and noncoplanar spin textures [30,31,32,33] as found in ferromagnets. When the spatial inversion symmetry is further broken in addition to the time-reversal symmetry, the system exhibits multiferroic phenomena, such as the magnetoelectric effect [34,35,36,37]. Furthermore, the breaking of the time-reversal symmetry leads to unconventional superconductivity, such as chiral p-wave superconductivity [38,39,40,41,42,43,44].

In crystals, the systems usually have other point-group symmetries that preserve crystal structures, such as rotational and mirror symmetries [45]. When the time-reversal symmetry is broken under magnetic phase transitions, some point-group symmetries are often lost simultaneously. For example, in the case of the ferromagnets, the mirror symmetries parallel to the magnetic-moment direction are broken as well as the time-reversal symmetry. With such symmetry breaking in mind, there is a natural question arising as follows: what happens when only the time-reversal symmetry is broken while the other symmetries are preserved? The recent theoretical study has revealed that such a situation occurs when magnetic orderings in the system are characterized by magnetic toroidal monopole ordering, which corresponds to a time-reversal-odd polar-charge quantity [46]. It is noted that the present magnetic toroidal monopole ordering is qualitatively different from the concept of altermagnetism [47], which has been recently recognized as the time-reversal symmetry breaking antiferromagnets. Altermagnetism mainly covers the following two topics: one is the emergence of the anomalous Hall effect in antiferromagnets without a net magnetization, and the other is the emergence of the momentum-dependent spin-split band structure without the relativistic spin–orbit coupling. From the multipole viewpoint, the former is explained by the activation of the magnetic dipole (axial vector quantity), while the latter is explained by the activation of the magnetic toroidal quadrupole (rank-2 polar tensor quantity), magnetic octupole (rank-3 axial tensor quantity), and so on. Meanwhile, the present magnetic toroidal monopole (rank-0 polar scalar) itself exhibits neither the anomalous Hall effect nor momentum-dependent spin-split band structure; the magnetic toroidal monopole does not couple to the momentum with the spin degree of freedom, which means that the magnetic toroidal monopole is not related to the spin splitting. However, under the magnetic toroidal monopole ordering, unconventional time-reversal switching responses are expected, such as magnetic field-induced rotational distortion, electric field-induced spin vortex, and electromagnetic field-induced chirality, which do not occur in conventional ferromagnets. Recently, such an antiferromagnet with a magnetic toroidal monopole has been identified in Co₂SiO₄, where the electric field-induced nonreciprocal dichroism [48,49] was observed [50]. Although physical phenomena driven by the magnetic toroidal monopole have been observed, their microscopic description is still lacking. In particular, it is unclear when and how such a magnetic toroidal monopole degree of freedom emerges in an atomic-scale wave function.

In the present study, we theoretically investigate the atomic-scale description of the magnetic toroidal monopole based on the multipole representation theory [51,52]. We show that the magnetic toroidal monopole degree of freedom corresponds to the off-diagonal imaginary component between the orbitals with the same orbital angular momentum but different principal quantum numbers. In other words, the magnetic toroidal monopole can be activated as the imaginary hybridization between the 1s–2s orbitals, 2p–3p orbitals, 3d–4d orbitals, 4f–5f orbitals, and so on. We demonstrate that such an imaginary hybridization can occur by applying both electric and magnetic fields in appropriate directions. The present results indicate that the time-reversal-odd polar-charge degree of freedom can be described by the atomic-scale electronic degrees of freedom, which enables us to express all the multipole degrees of freedom by atomic wave functions.

The rest of this paper is organized as follows. In Section 2, we introduce the magnetic toroidal monopole degree of freedom that characterizes the time-reversal-odd polar-charge quantity. Then, after presenting a cluster description of the magnetic toroidal monopole when the system exhibits antiferromagnetic orderings in Section 3, we show a symmetry relationship between the magnetic toroidal monopole and atomic-scale electronic degrees of freedom in Section 4. In Section 5, we demonstrate that the expectation value of the magnetic toroidal monopole becomes nonzero when the symmetry of the system satisfies its emergence condition. Finally, Section 6 is devoted to the conclusion of the present paper.

2. Magnetic Toroidal Monopole

Let us start by introducing the magnetic toroidal dipole, which has been extensively studied in the context of multiferroics [53,54,55,56,57,58,59,60,61]. Since the magnetic toroidal dipole is represented by a vector product between the position vector ${\mathit{r}}_i$ and the spin vector ${\mathit{S}}_i$ (the subscript i stands for the site index), i.e., ${\mathit{r}}_i\times {\mathit{S}}_i$, so as to form the vortex-type spin configuration [Figure 1a], it breaks both the spatial inversion and time-reversal symmetries. Accordingly, intriguing physical phenomena have been explored in both theory and experiments, such as the linear magnetoelectric effect [48,62,63,64,65,66,67,68], asymmetric magnon dispersions [69,70,71,72,73], directional-dependent nonlinear transport [74,75,76,77,78,79], nonlinear spin Hall effect [80,81], and intrinsic nonlinear Hall effect [82,83,84,85]. Various antiferromagnetic materials hosting the magnetic toroidal dipole have been observed in collinear antiferromagnetic structures like Cr₂O₃ [62,86,87], GaFeO₃ [88,89,90,91], LiCoPO₄ [92,93,94,95], Ba₂CoGe₂O₇ [96,97,98,99], LiFeSi₂O₆ [100,101,102], CuMnAs [82,103,104,105,106], BaCoSiO₄ [107,108], and PbMn₂Ni₆Te₃O₁₈ [109] and noncollinear antiferromagnetic structures like UNi₄B [110,111,112].

The monopole object is generally constructed by taking the inner product of the dipole moment and the position vector. In other words, the magnetic toroidal monopole can be constructed by $\mathit{T}·\mathit{r}$, where $\mathit{T}$ stands for the magnetic toroidal dipole. The schematic picture of the magnetic toroidal monopole is presented in Figure 1b. Since $\mathit{T}$ and $\mathit{r}$ correspond to the time-reversal-odd and time-reversal-even polar vectors, respectively, the magnetic toroidal monopole is characterized by the time-reversal-odd polar charge. Thus, the emergence of the magnetic toroidal monopole itself breaks the time-reversal symmetry but does not break the point-group symmetry. This indicates that the magnetic toroidal monopole belongs to the identity irreducible representation of the 32 crystal point groups without the time-reversal operation, such as $m\overline{3}m$ and $4/mmm$ [113]. In addition to Co₂SiO₄, which was experimentally identified as the magnetic toroidal monopole ordering, there are several candidate antiferromagnetic materials hosting the magnetic toroidal monopole, such as KMnF₃ [114], Ca₂RuO₄ [115], MnV₂O₄ [116], Mn₃As₂ [117], MnO₃ [118], Er₂Cu₂O₅ [119], Ho₂Ge₂O₇ [120], Mn₃IrGe [121], ScMnO₃ [118], and Mn₂FeMoO₆ [122]. In these materials, unconventional cross correlations between the physical quantities with different time-reversal parities can be expected, such as magnetic field-induced rotational distortion, electric field-induced spin vortex, and electromagnetic field-induced chirality.

3. Cluster Description

In this section, we show the relationship between antiferromagnetic structures and magnetic toroidal monopole based on cluster multipole representation [28,123,124,125,126]. As discussed in Section 2, there are various antiferromagnetic structures accompanying the magnetic toroidal monopole irrespective of cluster or periodic crystal structures [46]. Although we focus on the atomic-scale magnetic toroidal monopole without antiferromagnetic ordering in the present study, we here present two examples in order to gain a deeper insight into the magnetic toroidal monopole degree of freedom intuitively.

Figure 2a shows a noncollinear spin configuration under the magnetic point group $4/m$ with the 8c Wyckoff sites. The vortex-type noncollinear spin configuration gives rise to the local magnetic toroidal dipole; the positive (negative) z component of the magnetic toroidal dipole is induced in the plane HGCD (EFBA), which is denoted by the red arrows in Figure 2a. Since the magnitude of the induced magnetic toroidal dipole for two planes is equivalent, there is no net magnetic toroidal dipole in the whole cluster. Instead, one can easily find the monopole component of the magnetic toroidal dipole distribution, i.e., $\mathit{T}·\mathit{r}\ne 0$, where the origin of $\mathit{r}$ is taken at the center of the square prism. Thus, the noncollinear antiferromagnetic structure in Figure 2a is regarded as the magnetic toroidal monopole ordering.

Figure 2b represents another example of the antiferromagnetic structure hosting the magnetic toroidal monopole. The collinear-type magnetic structure under the magnetic point group $mmm$ with the 8g Wyckoff sites satisfies the symmetry of the magnetic toroidal monopole. Indeed, when we calculate the magnetic toroidal dipole on each plaquette in the rectangular parallelepiped, one can find the two-in and two-out structure of the magnetic toroidal dipole, as shown in Figure 2b. It is noted that the distribution of the magnetic toroidal dipole has the monopole component, since the magnitude of the magnetic toroidal dipole on the plane ABCD or EFGH is not equivalent to that on the plane BCGF or ADHE, owing to the rectangular symmetry. In other words, the distribution of the magnetic toroidal dipole is decomposed into the monopole and quadrupole components. Thus, the collinear magnetic structure in Figure 2b is also regarded as the magnetic toroidal monopole ordering, which exhibits physical phenomena driven by the magnetic toroidal monopole. A similar discussion holds for periodic systems as long as the same magnetic symmetry is considered. In this way, cluster antiferromagnetic structures consisting of multiple sites can host the magnetic toroidal monopole.

4. Atomic-Scale Description

In order to discuss the possibility of the atomic-scale magnetic toroidal monopole, we introduce the atomic-scale multipole operators consisting of electric, magnetic, magnetic toroidal, and electric toroidal multipoles, where each multipole is characterized by two quantum numbers as follows: the orbital angular momentum l and its z component m ($-l\le m\le l$) [127]. The four types of multipoles are characterized by different spatial inversion and time-reversal parities; electric (electric toroidal) multipoles are characterized by the time-reversal-even polar (axial) tensor, while magnetic toroidal (magnetic) multipoles are characterized by the time-reversal-odd polar (axial) tensor.

From the microscopic viewpoint, whether multipoles are activated in the physical space is determined by the addition rule of the angular momenta in terms of the wave function. When the active multipoles are considered for the physical space spanned by two wave functions with the orbital angular momenta $L_1$ and $L_2$, only the rank-l multipoles satisfying $|L_1-L_2|\le l\le L_1+L_2$ are active. By taking the real wave function as the basis for s, p, d, and f orbitals, here and hereafter, the electric and electric toroidal multipoles correspond to the real-number matrix, while the magnetic and magnetic toroidal multipoles correspond to the imaginary-number matrix. For example, in the s-p orbital system without the spin degree of freedom, the active multipoles are a rank-1 electric dipole and a rank-1 magnetic toroidal dipole. Such a correspondence between the active multipoles and atomic s, p, d, and f wave functions is summarized in Ref. [52], where it was shown that all the multipoles except for the magnetic toroidal monopole can become active in the atomic scale.

In the present study, we focus on the multipole degrees of freedom between the wave functions with the same orbital angular momentum but different principal quantum numbers n. In other words, we consider the multipole degrees of freedom in the physical space spanned by 1s–2s orbitals, 2p–3p orbitals, 3d–4d orbitals, 4f–5f orbitals, and so on. By adopting two orbitals with different principal quantum numbers, one can find another electronic degree of freedom, which corresponds to the imaginary off-diagonal matrix elements. Indeed, such a degree of freedom can express the magnetic toroidal monopole degree of freedom, as detailed below.

Let us consider the active multipole degrees of freedom in the 1s–2s orbitals. Since there is only one degree of freedom in the s orbital, there are one real and one imaginary matrix elements in the physical space spanned by the $1s$ and $2s$ orbitals. When we adopt the real wave function, the real (imaginary) matrix component corresponds to the time-reversal-even (time-reversal-odd) multipoles from the time-reversal operation. In addition, the rank of the active multipoles must be zero because both $1s$ and $2s$ orbitals are characterized by zero orbital angular momentum from the addition rule of the angular momenta. Thus, one identifies that the off-diagonal real matrix element between the 1s and 2s orbitals corresponds to the electric monopole, while the off-diagonal imaginary matrix element corresponds to the magnetic toroidal monopole. These results indicate that the magnetic toroidal monopole can be described by the atomic-scale wave functions invoking the imaginary hybridization between the s orbitals with different principal quantum numbers.

Similarly, the magnetic toroidal monopole can be active for the other orbitals with different principal quantum numbers, such as p, d, and f orbitals. In these orbitals with nonzero orbital angular momenta, unconventional higher-rank multipole degrees of freedom are also active, which include the rank-1 electric toroidal dipole related to the ferroaxial (ferrorotational) ordering [128,129,130,131,132,133,134] and rank-2 magnetic toroidal quadrupole. All the active multipoles in these atomic wave functions are summarized in

Table 1. Active multipoles in the atomic s, p, d, and f wave functions with different principal quantum numbers. E, M, MT, and ET represent electric, magnetic, magnetic toroidal, and electric toroidal multipoles, respectively. # represents the number of independent multipoles.

L	Orbital	#	$\mathit{l}=0$	1	2	3	4	5	6
0	s-s	2	E/MT	–	–	–	–	–	–
1	p-p	18	E/MT	M/ET	E/MT	–	–	–	–
2	d-d	50	E/MT	M/ET	E/MT	M/ET	E/MT	–	–
3	f-f	98	E/MT	M/ET	E/MT	M/ET	E/MT	M/ET	E/MT

.

5. Model Analysis

We demonstrate that the expectation values of the magnetic toroidal monopole become nonzero by performing the microscopic model calculations. We introduce the lattice model, where each atom is loosely connected by effective exchange interactions. Each site consists of $2s$- and $3s$-orbital wave functions $({\psi }_{s1},{\psi }_{s2})$ and three $2p$-orbital wave functions $({\psi }_x,{\psi }_y,{\psi }_z)$. The model Hamiltonian within the single-site unit cell is given by

$$\begin{array}{c}\hfill \mathcal{H}={\mathrm{\Delta }}_1{Q}_{0p}+{\mathrm{\Delta }}_2{Q}_{0s2}+\lambda \mathit{L}·\mathit{S}-h_{Q_z}{Q}_z-h_{T_z}{T}_z-h_{Q_z}^{\prime }{Q}_z^{\prime }-h_{T_z}^{\prime }{T}_z^{\prime },\end{array}$$

(1)

where $(Q_{0p},Q_{0s2},\mathit{L},\mathit{S},Q_z,T_z,Q_z^{\prime },T_z^{\prime })$ are $10\times 10$ matrices, which are obtained by the formula in Ref. [127]. The first and second terms represent the atomic energy levels, which arise from the different atomic energies of s and p orbitals. The first term stands for the atomic energy level difference between the 2s and 2p orbitals by ${\mathrm{\Delta }}_1$, while the second term stands for that between the 2s and 3s orbitals by ${\mathrm{\Delta }}_2$. The expressions of $Q_{0p}$ and $Q_{0s2}$ are explicitly given by

$$\begin{array}{c}\hfill Q_{0p}=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(2)

$$\begin{array}{c}\hfill Q_{0s2}=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right),\end{array}$$

(3)

where the basis is represented by $\{{\psi }_{s1\uparrow },{\psi }_{x\uparrow },{\psi }_{y\uparrow },{\psi }_{z\uparrow },{\psi }_{s2\uparrow },{\psi }_{s1\downarrow },{\psi }_{x\downarrow },{\psi }_{y\downarrow },{\psi }_{z\downarrow },{\psi }_{s2\downarrow }\}$. We set ${\mathrm{\Delta }}_1=1$ and ${\mathrm{\Delta }}_2=2$. The second term represents the relativistic spin–orbit coupling, which works on the p orbital; $\mathit{L}$ and $\mathit{S}$ stand for the orbital and spin angular momentum operators of the p orbital, respectively. We set $\lambda =0.3$.

The fourth to seventh terms are introduced to induce the magnetic toroidal monopole. Since the magnetic toroidal monopole is related to $\mathit{T}·\mathit{r}$, as discussed in Section 2, we consider the molecular fields corresponding to $\mathit{T}$ and $\mathit{r}$ in an atomic scale. One is the electric dipole field corresponding to $\mathit{r}$ in the fourth and sixth terms, where $Q_z$ and $Q_z^{\prime }$ are given by

$$\begin{array}{c}\hfill Q_z=\left(\begin{array}{cccccccccc}0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(4)

$$\begin{array}{c}\hfill Q_z^{\prime }=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}\\ 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0\end{array}\right).\end{array}$$

(5)

Thus, the fourth and sixth terms describe the real hybridization between $2s$–$2p_z$ and $3s$–$2p_z$ orbitals, respectively. We set the different molecular fields for the $2s$–$2p_z$ orbital space and $3s$–$2p_z$ orbital space, $h_{Q_z}$ and $h_{Q_z}^{\prime }$, respectively. Such an electric dipole degree of freedom is activated by the polar crystalline electric field or the electric field. Meanwhile, the fifth and seventh terms mimic the effect of the magnetic toroidal dipole field corresponding to $\mathit{T}$, whose matrices are given by

$$\begin{array}{c}\hfill T_z=\left(\begin{array}{cccccccccc}0& 0& 0& {\displaystyle \frac{i}{3\sqrt{3}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -{\displaystyle \frac{i}{3\sqrt{3}}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{i}{3\sqrt{3}}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{\displaystyle \frac{i}{3\sqrt{3}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(6)

$$\begin{array}{c}\hfill T_z^{\prime }=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\displaystyle \frac{i}{3\sqrt{3}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{i}{3\sqrt{3}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{i}{3\sqrt{3}}}\\ 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{i}{3\sqrt{3}}}& 0\end{array}\right).\end{array}$$

(7)

These matrices indicate that the fifth and seventh terms describe the imaginary hybridization between $2s$–$2p_z$ and $3s$–$2p_z$ orbitals, respectively. We set the different molecular fields for the $2s$–$2p_z$ orbital space and $3s$–$2p_z$ orbital space, $h_{T_z}$ and $h_{T_z}^{\prime }$, respectively. The model in Equation (1) possesses neither time-reversal symmetry nor product symmetry of spatial inversion and time-reversal symmetries so that the magnetic toroidal monopole can be active. Since the Hamiltonian is hermitian, all the eigenvalues of the Hamiltonian are real.

Figure 3a represents the $h_{T_z}^{\prime }$ dependence of the expectation values in terms of relevant multipoles as follows: the magnetic toroidal monopole $T_0$ and two magnetic toroidal dipoles $T_z$ and $T_z^{\prime }$. Here, the magnetic toroidal monopole is defined as the imaginary hybridization between 2s and $3s$ orbitals, whose matrix is given by

$$\begin{array}{c}\hfill T_0=\left(\begin{array}{cccccccccc}0& 0& 0& 0& i& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -i& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& i\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -i& 0& 0& 0& 0\end{array}\right).\end{array}$$

(8)

The symbol $\langle \dots \rangle$ stands for the expectation values at the temperature $T=0.1$. The other model parameters are taken at $h_{Q_z}=h_{Q_z}^{\prime }=0.2$ and $h_{T_z}=0.1$.

The results in Figure 3a show that the magnetic toroidal monopole is induced irrespective of $h_{T_z}^{\prime }$ for nonzero $h_{Q_z}$, $h_{Q_z}^{\prime }$, and $h_{T_z}$, which means that the atomic-scale magnetic toroidal monopole becomes the atomic-scale order parameter once the irreducible representation of the magnetic toroidal monopole corresponds to the identity irreducible representation. In order to understand the microscopic essence to induce the magnetic toroidal monopole, we perform an expansion method to $\langle T_0\rangle$ [135,136]. As a result, we find that $\langle T_0\rangle$ is proportional to $(h_{Q_z}^{\prime }{h}_{T_z}-h_{Q_z}{h}_{T_z}^{\prime })$, which means that the spin–orbit coupling $\lambda$ and atomic energy levels ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ play no significant role in inducing the magnetic toroidal monopole, whereas both the electric dipole and magnetic toroidal dipole fields are important.

Next, we show that the magnetic toroidal monopole can be induced by applying both external electric and magnetic fields without the magnetic toroidal dipole field. We consider the model Hamiltonian as follows:

$$\begin{array}{c}\hfill \mathcal{H}={\mathrm{\Delta }}_1{Q}_{0p}+{\mathrm{\Delta }}_2{Q}_{0s2}+\lambda \mathit{L}·\mathit{S}-h_{Q_z}{Q}_z-h_{Q_x}{Q}_x-h_{Q_z}^{\prime }{Q}_z^{\prime }-h_{Q_x}^{\prime }{Q}_x^{\prime }-h_{M_y}{M}_y,\end{array}$$

(9)

where the first to third terms are the same as those in the model in Equation (1). Instead of the magnetic toroidal dipole field in the fifth and seventh terms in Equation (1), we introduce two different dipole fields. One is the x component of the electric dipole field appearing in the fifth and seventh terms, where the matrix forms of $Q_x$ and $Q_x^{\prime }$ are given by

$$\begin{array}{c}\hfill Q_x=\left(\begin{array}{cccccccccc}0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill Q_x^{\prime }=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{\sqrt{3}}}& 0& 0& 0\end{array}\right).\end{array}$$

(11)

This indicates the hybridization between s and $p_x$ orbitals. The other is the magnetic dipole field appearing in the last term, where $M_y$ is given by

$$\begin{array}{c}\hfill M_y=\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& i& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -i& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& i& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -i& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

(12)

where we neglect the effect of the magnetic field on the spin moments for simplicity. Thus, we consider the situation where the electric field is applied in the $xz$ plane and the magnetic field is applied along the y direction. In the following, we parameterize $h_{Q_z}=0.2cos\theta$, $h_{Q_x}=0.1sin\theta$, $h_{Q_z}^{\prime }=0.2cos\theta$, and $h_{Q_x}^{\prime }=0.05sin\theta$.

Figure 3b represents the $\theta$ dependence of $\langle T_0\rangle$, $\langle T_z\rangle$, and $\langle T_z^{\prime }\rangle$, where $\langle T_0\rangle$ becomes nonzero except for $\theta =0$, $\pi /2$, and $\pi$. The disappearance of the magnetic toroidal monopole at $\theta =0$, $\pi /2$, and $\pi$ is due to the absence of x or z components of the electric dipole field. This means that the application of the electric field along the high-symmetry directions, such as the x and z directions, is not enough to induce the magnetic toroidal monopole. Such behavior is qualitatively understood from the fact that the magnetic toroidal monopole is induced when both the electric dipole and magnetic toroidal dipole are induced, as demonstrated in Figure 3a; the z component of the magnetic toroidal dipole in Equation (1) corresponds to the product of the x component of the electric dipole and the y component of the magnetic dipole in Equation (9). Indeed, the essential model parameters for $\langle T_0\rangle$ in the present model, which are obtained by the expansion method [135,136], are given by $h_{M_y}(h_{Q_z}^{\prime }{h}_{Q_x}-h_{Q_z}{h}_{Q_x}^{\prime })$; the two components of the electric field as well as the magnetic field perpendicular to the electric field are necessary.

6. Conclusions

In conclusion, we have investigated the possibility of the atomic-scale magnetic toroidal monopole. Based on the symmetry, multipole representation, and model analyses, we have shown that the atomic-scale magnetic toroidal monopole corresponds to the off-diagonal imaginary hybridization between the orbitals with the same orbital angular momentum but different principal quantum numbers. The present results indicate that the atomic-scale magnetic toroidal monopole can be a potential order parameter in antiferromagnets for the magnetic phase transitions that break only the time-reversal symmetry.

Let us comment on the active magnetic toroidal monopole for other degrees of freedom. The magnetic toroidal monopole can also be activated through the bond degree of freedom as the imaginary hybridization between the different sites with the same orbital, depending on the spatial distribution of the imaginary hybridizations among multiple sites. In such a situation, since the imaginary hybridization occurs within the same orbital, e.g., 1s–1s orbital, the larger hybridization effect compared to the atomic-scale case might be expected.