Cluster Odd-Parity Multipoles by Staggered Orbital Ordering in Locally Noncentrosymmetric Crystals

Abstract

Odd-parity multipoles in crystals manifest themselves not only in their peculiar electronic orderings but also in unconventional parity-violating physical phenomena. We here report the emergence of odd-parity multipoles by considering staggered orbital orderings in a locally noncentrosymmetric crystal system with the global inversion center but without the inversion center at each lattice site. We show that various odd-parity multipoles, such as the electric toroidal monopole, electric dipole, and electric toroidal quadrupole, are realized depending on the type of orbital orderings in the one-dimensional zigzag chain. Such odd-parity multipoles give rise to an antisymmetric spin splitting in the electronic band structure with the aid of the relativistic spin–orbit coupling. We also show that similar states with odd-parity multipoles are realized in other locally noncentrosymmetric crystals, such as the two-dimensional honeycomb and three-dimensional diamond structures.

1. Introduction

Spatial inversion symmetry is important in determining the physical properties of crystals. When the spatial inversion symmetry is lost, various parity-violating physical phenomena occur, such as spontaneous electric polarization and nonreciprocal transport [1,2,3,4,5]. The absence of spatial inversion symmetry also gives rise to an antisymmetric spin splitting in the electronic band structure in momentum space by combining the relativistic spin–orbit coupling (SOC), which is the so-called antisymmetric SOC (ASOC) [6,7]. Depending on the crystal symmetry, momentum dependence of the spin polarization is different [8,9,10]; the Rashba-type ASOC as $\mathit{k}\times \mathit{\sigma }$ under the polar point group [11,12,13,14], the hedgehog-type ASOC as $\mathit{k}·\mathit{\sigma }$ under the chiral point group [15,16], Ising-type ASOC as $k_x(k_x^2-3k_y^2){\sigma }_z$ under other noncentrosymmetric point groups [17,18,19,20], where $\mathit{k}$ represents the wave vector and $\mathit{\sigma }=2\mathit{s}$ represents the electron spin $\mathit{s}$. These ASOCs are the microscopic origin of further parity-violating phenomena [21,22,23,24,25,26], such as the spin Hall effect [27,28,29,30,31], the Edelstein effect [32,33,34,35,36], and noncentrosymmetric superconductivity [7,37,38,39].

The above parity-violating phenomena can be engineered in centrosymmetric crystals when the spatial inversion symmetry is spontaneously broken by an electronic-order-driven phase transition through the electron correlation [21,40,41]. The typical crystal structure to realize such a situation is a locally noncentrosymmetric crystal structure, where the global spatial inversion symmetry is present, whereas the inversion center at each lattice site is absent, as found in the one-dimensional zigzag chain [Figure 1a], two-dimensional honeycomb structure [Figure 1b], and three-dimensional diamond structure [Figure 1c]. In these situations, the staggered electronic orderings break the global spatial inversion symmetry, which results in the emergence of odd-parity multipoles [42]; the odd-parity multipoles characterize the degrees of freedom with spatial inversion odd, which is different from the conventional even-parity multipoles, such as the electric monopole and magnetic dipole. For example, the staggered electric charge (electric monopole) orderings lead to the odd-parity electric dipole leading to the uniform electric polarization, and the staggered spin (magnetic dipole) ordering with the magnetic moments along the out-of-chain direction leads to the odd-parity magnetic toroidal dipole [43,44]. In particular, the latter magnetic ordering induces fascinating physical phenomena, such as nonreciprocal magnon excitations [45], magnetoelectric effect [46], nonlinear longitudinal transports [47,48], nonlinear Hall effect [49,50], nonlinear spin Hall effect [51,52], and unconventional superconductivity [53].

In the present study, we investigate the spontaneous parity breaking by the orbital ordering rather than the magnetic ordering. The orbital ordering has been extensively studied in condensed matter physics [54,55,56,57], since it has been identified in various d-electron materials [58,59,60,61,62,63,64,65,66,67] and f-electron materials [68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96]. Moreover, such orbital degrees of freedom can be a source of further unconventional states, such as a CP² skyrmion [97,98,99,100] and higher-rank cluster multipole orderings [101]. By considering the staggered orbital ordering in the locally noncentrosymmetric crystals, such as the one-dimensional zigzag chain, two-dimensional honeycomb structure, and three-dimensional diamond structure, we show that the spontaneous antisymmetric spin-split band structure appears according to the breaking of the spatial inversion symmetry. Depending on the type of the orbital orderings, the odd-parity electric toroidal monopole and electric dipole are induced, which results in the chiral-type ASOC and Rashba-type ASOC, respectively. Our results indicate that further intriguing parity-violating physical phenomena are expected under the staggered orbital ordering in locally noncentrosymmetric crystals.

The rest of this paper is organized as follows. In Section 2, we discuss the correspondence between the staggered orbital ordering and odd-parity multipoles in locally noncentrosymmetric crystals based on symmetry analysis. Then, we introduce a minimal tight-binding model consisting of three p orbitals in the one-dimensional zigzag chain system in Section 3. We show the appearance of the antisymmetric spin splitting in the band dispersion under the staggered orbital ordering in Section 4. We describe the induced odd-parity multipoles in the two-dimensional honeycomb structure and the three-dimensional diamond structure in Section 5. We conclude the present result in Section 6.

2. Symmetry Analysis

In this section, we investigate the staggered orbital ordering on the one-dimensional zigzag chain consisting of two sublattices A and B, as shown in Figure 1a, based on the symmetry analysis. Specifically, we suppose the $2e$ site under the space group $Pmma$; the position vector of two sublattices are given by ${\mathit{r}}_{\mathrm{A}}=(1/4,0,1/4)$ and ${\mathit{r}}_{\mathrm{B}}=(3/4,0,-1/4)$. For the zigzag-chain structure, we consider the staggered alignment of the electric quadrupoles with five components $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$ for $u=3z^2-r^2$ and $v=x^2-y^2$. The irreducible representations of $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$ are given by $(A_g,A_g,B_{3g},B_{2g},B_{1g})$, respectively. Since the staggered potential for two sublattices belongs to the irreducible representation $B_{1u}$, the staggered alignment of $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$ belongs to the irreducible representation $(B_{1u},B_{1u},B_{2u},B_{3u},A_u)$, respectively; hereafter, we denote the staggered alignment of $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$ as $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$. The schematic orbital configurations of $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$ are shown in Figure 2. As shown in Section 3, the above electric quadrupole degrees of freedom with the orbital angular moment $l=2$ can be constructed by the p-orbital ($l=1$) physical space as the expectation value of the one-body order paramter [102].

The staggered orbital ordering breaks the global inversion symmetry in the one-dimensional zigzag chain, which results in the emergence of odd-parity multipoles in the cluster unit. By adopting symmetry-adapted cluster multipole theory combined with the virtual cluster method [103,104], each staggered ordering accompanies any of electric toroidal monopole $G_0$, electric dipole $(Q_x,Q_y,Q_z)$, and electric toroidal quadrupole $(G_u,G_v,G_{yz},G_{zx},G_{xy})$ according to their irreducible representation [105,106]. Here, the electric dipole corresponds to the rank-1 polar vector, while the electric toroidal monopole and quadrupole correspond to the rank-0 pseudoscalar and rank-2 axial tensor, respectively. In the case of $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)})$ belonging to the $B_{1u}$ representation, $Q_z$ and $G_{xy}$ are induced. Similarly, $Q_{yz}^{\left(\mathrm{s}\right)}$, $Q_{zx}^{\left(\mathrm{s}\right)}$, and $Q_{xy}^{\left(\mathrm{s}\right)}$ induce $(Q_y,G_{zx})$, $(Q_x,G_{yz})$, and $(G_0,G_u,G_v)$, respectively. Thus, the staggered orbital ordering for $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)})$ leads to polar-type parity-violating phenomena, while that for $Q_{xy}^{\left(\mathrm{s}\right)}$ leads to chiral-type parity-violating phenomena. For example, the former gives rise to the Rashba-type ASOC, while the latter gives rise to the chiral-type ASOC, where each odd-parity multipole shows the spin-momentum locking as follows:

$$\begin{array}{cc}\hfill G_0& \leftrightarrow \mathit{k}·\mathit{\sigma },\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathit{Q}& \leftrightarrow (\mathit{k}\times \mathit{\sigma }),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill G_u& \leftrightarrow 3k_z{\sigma }_z-\mathit{k}·\mathit{\sigma },\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill G_v& \leftrightarrow k_x{\sigma }_x-k_y{\sigma }_y,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill G_{yz}& \leftrightarrow k_y{\sigma }_z+k_z{\sigma }_y,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill G_{zx}& \leftrightarrow k_z{\sigma }_x+k_x{\sigma }_z,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill G_{xy}& \leftrightarrow k_x{\sigma }_y+k_y{\sigma }_x,\hfill \end{array}$$

(7)

where $\mathit{k}=(k_x,k_y,k_z)$ and $\mathit{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ are the wave vector and spin, respectively. When multiple odd-parity multipoles appear at the same time, the resultant ASOC is described by their linear combination. We summarize the correspondence among the staggered orbital orderings, odd-parity multipoles, and ASOCs in the zigzag chain in

Table 1. Classification of staggered orbital orderings in the one-dimensional zigzag chain, which is constructed by the $2e$ site under the space group $Pmma$. $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$ stand for the electric quadrupole ($u=3z^2-r^2$ and $v=x^2-y^2$), where the superscript $\left(\mathrm{s}\right)$ represents the staggered alignment. In each staggered quadrupole ordering, the irreducible representations (IR) for the orbital and cluster in the unit cell are shown. The fourth column represents the odd-parity multipoles up to rank 2; $G_0$ is the electric dipole, $(Q_x,Q_y,Q_z)$ are the electric dipole, and $(G_u,G_v,G_{yz},G_{zx},G_{xy})$ are the electric toroidal quadrupole. The fifth column represents the antisymmetric spin–orbit coupling (ASOC), where $c_1$, $c_2$, and $c_3$ are numerical coefficients.

OP	IR (Orbital)	IR (Cluster)	Odd-Parity Multipole	ASOC
$Q_u^{\left(\mathrm{s}\right)}$	$A_g$	$B_{1u}$	$Q_z$, $G_{xy}$	$c_1{k}_x{\sigma }_y+c_2{k}_y{\sigma }_x$
$Q_v^{\left(\mathrm{s}\right)}$	$A_g$	$B_{1u}$	$Q_z$, $G_{xy}$	$c_1{k}_x{\sigma }_y+c_2{k}_y{\sigma }_x$
$Q_{yz}^{\left(\mathrm{s}\right)}$	$B_{3g}$	$B_{2u}$	$Q_y$, $G_{zx}$	$c_1{k}_z{\sigma }_x+c_2{k}_x{\sigma }_z$
$Q_{zx}^{\left(\mathrm{s}\right)}$	$B_{2g}$	$B_{3u}$	$Q_x$, $G_{yz}$	$c_1{k}_y{\sigma }_z+c_2{k}_z{\sigma }_y$
$Q_{xy}^{\left(\mathrm{s}\right)}$	$B_{1g}$	$A_u$	$G_0$, $G_u$, $G_v$	$c_1{k}_x{\sigma }_x+c_2{k}_y{\sigma }_y+c_3{k}_z{\sigma }_z$

. It is noted that the above analyses can be straightforwardly applied to other orderings consisting of two or more orbitals.

3. Model

To show the emergence of the ASOC under the staggered orbital ordering in the zigzag chain along the x direction, we consider a minimal tight-binding model consisting of three p orbitals, ($p_x$, $p_y$, $p_z$), which is given by

$$\begin{array}{cc}\hfill \mathcal{H}& =-\sum _{ij\alpha {\alpha }^{\prime }\sigma }(t_{ij}^{\alpha {\alpha }^{\prime }}{c}_{i{\alpha }^{\prime }\sigma }^{†}{c}_{j\alpha \sigma }+\mathrm{H}.\mathrm{c}.)+\frac{\lambda }{2}\sum _{i\alpha {\alpha }^{\prime }\sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{\left(H^{\mathrm{SOC}}\right)}_{\alpha {\alpha }^{\prime }}^{\sigma {\sigma }^{\prime }}{c}_{i{\alpha }^{\prime }{\sigma }^{\prime }}-\sum _{i\alpha {\alpha }^{\prime }\sigma }{h}_i{c}_{i\alpha \sigma }^{†}{\left(H^{\mathrm{MF}}\right)}_{\alpha {\alpha }^{\prime }}{c}_{i{\alpha }^{\prime }\sigma },\hfill \end{array}$$

(8)

where $c_{i\alpha \sigma }^{†}$ and $c_{i\alpha \sigma }$ represent the creation and annihilation fermion operators at site i, orbital $\alpha =p_x$, $p_y$, and $p_z$, and spin $\sigma$, respectively. The position vector at sublattice A (B) is given by $(a/4,0,a/4)$ and $(3a/4,0,-a/4)$, where a is the lattice constant along the x direction and we set $a=1$ as the length unit. The first term represents the hopping term between sublattices A and B; we adopt two hopping parameters $t_{pp\sigma }$ and $t_{pp\pi }$ for $t_{ij}^{\alpha {\alpha }^{\prime }}$ based on the Slater-Koster parameter. The second term represents the atomic SOC that originates from the relativistic effect; $\lambda$ denotes the magnitude of the atomic SOC. The $6\times 6$ matrix $H^{\mathrm{SOC}}$ is given by

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

(9)

where ${\sigma }_{\mu }$ represents the $\mu$ component of the $2\times 2$ Pauli matrix in spin space. The third term represents the mean field to induce the staggered orbital ordered state, which originates from the two-body electron correlations; $h_i$ denotes the magnitude of the mean field at site i. We consider five types of orbital orderings, whose order parameters are described by the electric quadrupole $(Q_u,Q_v,Q_{yz},Q_{zx},Q_{xy})$. The $3\times 3$ matrix form in the single atom is given by

$$H^{\mathrm{MF},Q_u}=\left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 2\end{array}\right),H^{\mathrm{MF},Q_v}=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right),$$

(10)

$$H^{\mathrm{MF},Q_{yz}}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),H^{\mathrm{MF},Q_{zx}}=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right),H^{\mathrm{MF},Q_{xy}}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right).$$

(11)

Owing to the staggered orbital ordering, $h_i$ is taken as h for sublattice A and $-h$ for sublattice B. We take $(t_{pp\sigma },t_{pp\pi },\lambda ,h)=(-0.8,0.4,0.5,0.5)$ in the following calculations. We here do not consider the lattice distortions for simplicity.

4. Results

In this section, we discuss the effect of the staggered orbital ordering in the one-dimensional zigzag-chain model in Equation (8). Since the staggered orbital ordering breaks the spatial inversion symmetry, the ASOC emerges, where the functional form is expected from the symmetry argument in Table 1. For reference, we show the band structure without the mean-field term in Figure 3; there are six bands and each band is doubly degenerate owing to the presence of both the spatial inversion and time-reversal symmetries.

When the mean-field term for the $Q_u$ multipole, i.e., $H^{\mathrm{MF}}\to H^{\mathrm{MF},Q_u}$, is turned on, the antisymmetric spin-split band structure appears, as shown in Figure 4a; the spin polarization with the y component is antisymmetric in terms of the wave number $k_x$. Meanwhile, there is no spin polarization in the x and z components. This result indicates the emergence of the ASOC in the functional form of $k_x{\sigma }_y$, which is attributed to the appearance of the odd-parity multipoles, $Q_z\leftrightarrow k_x{\sigma }_y-k_y{\sigma }_x$ and $G_{xy}\leftrightarrow k_x{\sigma }_y+k_y{\sigma }_x$, as discussed in Section 2. Thus, the current-induced magnetization, where the y-component of the magnetization is induced by applying the electric current along the x direction, is expected. The ASOC vanishes when $\lambda$ is turned off, which indicates that the relativistic SOC plays an important role. In addition, the ASOC also vanishes for $t_{pp\sigma }=t_{pp\pi }$, which is attributed to the fact that the off-diagonal hopping between the $p_x$ and $p_z$ orbitals with the antisymmetric $k_x$ dependence vanishes. A similar antisymmetric spin-split band structure emerges when the $Q_v^{\left(\mathrm{s}\right)}$ ordering is considered, as shown in Figure 4b.

For the other staggered orderings, the ASOC is induced as expected by the symmetry. The $Q_{yz}^{\left(\mathrm{s}\right)}$ ordering leads to the antisymmetric spin splitting in the form of $k_x{\sigma }_z$, as shown in Figure 4c, and the $Q_{xy}^{\left(\mathrm{s}\right)}$ ordering leads to the antisymmetric spin splitting in the form of $k_x{\sigma }_x$, as shown in Figure 4d. The longitudinal coupling between $\mathit{k}$ and $\mathit{\sigma }$ is owing to the chiral nature without the mirror symmetry. Meanwhile, there is no antisymmetric spin splitting in the $Q_{zx}^{\left(\mathrm{s}\right)}$ order, since its induced odd-parity multipoles are related to the ASOC in the form of $c_1{k}_y{\sigma }_z+c_2{k}_z{\sigma }_y$, which does not include the $k_x$ dependence. In this case, the antisymmetric spin splitting can be found when the two-dimensional structure consisting of the zigzag chain is considered so that $k_y$ or $k_z$ appears. In this way, the staggered orbital ordered state breaking the spatial inversion symmetry gives rise to the antisymmetric spin splitting in the band structure, which becomes the origin of the parity-violating phenomena, such as the Edelstein effect.

In order to further understand the role of the relativistic SOC in the antisymmetric spin-split band structure, we decompose the SOC term in the model in Equation (8) as follows:

$$\begin{array}{c}\hfill \frac{{\lambda }_x}{2}\sum _{i\alpha {\alpha }^{\prime }\sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{H}_x^{\mathrm{SOC}}{c}_{i{\alpha }^{\prime }{\sigma }^{\prime }}+\frac{{\lambda }_y}{2}\sum _{i\alpha {\alpha }^{\prime }\sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{H}_y^{\mathrm{SOC}}{c}_{i{\alpha }^{\prime }{\sigma }^{\prime }}+\frac{{\lambda }_z}{2}\sum _{i\alpha {\alpha }^{\prime }\sigma {\sigma }^{\prime }}{c}_{i\alpha \sigma }^{†}{H}_z^{\mathrm{SOC}}{c}_{i{\alpha }^{\prime }{\sigma }^{\prime }}\end{array}$$

(12)

where

$$\begin{array}{cc}\hfill H_x^{\mathrm{SOC}}& =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i{\sigma }_x\\ 0& i{\sigma }_x& 0\end{array}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill H_y^{\mathrm{SOC}}& =\left(\begin{array}{ccc}0& 0& i{\sigma }_y\\ 0& 0& 0\\ -i{\sigma }_y& 0& 0\end{array}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill H_z^{\mathrm{SOC}}& =\left(\begin{array}{ccc}0& -i{\sigma }_z& 0\\ i{\sigma }_z& 0& 0\\ 0& 0& 0\end{array}\right),\hfill \end{array}$$

(15)

where ${\lambda }_x$, ${\lambda }_y$, and ${\lambda }_z$ represent the amplitude of the x, y, and z spin components of the SOC. This decomposition is performed in order to investigate which spin component plays an important role in inducing the ASOC.

We find that ${\lambda }_y$ plays an important role in inducing the ASOC in the $Q_u^{\left(\mathrm{s}\right)}$ and $Q_v^{\left(\mathrm{s}\right)}$ orderings, which means that ASOC vanishes for ${\lambda }_y=0$ but remains for ${\lambda }_x={\lambda }_z=0$. Similarly, ${\lambda }_x$ and ${\lambda }_z$ are essential for the ASOC in the $Q_{xy}^{\left(\mathrm{s}\right)}$ and $Q_{yz}^{\left(\mathrm{s}\right)}$ orderings, respectively. This result indicates that the $\mu$-spin component in the SOC contributes to the ASOC in terms of the $\mu$-spin component under the staggered orbital ordering. Thus, the three orbital degrees of freedom in the present model are minimal ingredients to induce the ASOC.

5. Discussion

The spontaneous parity breaking by staggered orbital orderings also occurs in other locally noncentrosymmetric lattice structures. We briefly discuss the cases of the two-dimensional honeycomb structure in Section 5.1 and the three-dimensional diamond structure in Section 5.2.

5.1. Honeycomb Structure

We consider the relationship between the staggered orbital ordering and odd-parity multipole in the two-dimensional honeycomb structure [41,107,108,109,110,111,112]. We suppose the $2c$ site under the space group $P6/mmm$. Similarly to the zigzag-chain case, the staggered orbital ordering leads to the odd-parity multipoles and the ASOCs. For the $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$ orderings, as shown in Figure 5, $[Q_{3b},G_v,G_{xy},(Q_y,G_{zx}),(Q_x,G_{yz})]$ are induced, where $Q_{3b}$ represents the electric octupole. Meanwhile, no chiral-type ASOC that originates from $G_0$ appears in the honeycomb structure. The correspondence among the staggered orbital ordering, odd-parity multipole, and ASOC is shown in

Table 2. Classification of staggered orbital orderings in the two-dimensional honeycomb structure, which is constructed by the $2c$ site under the space group $P6/mmm$. For the $Q_u^{\left(\mathrm{s}\right)}$ ordering with the odd-parity electric octupole $Q_{3b}$, the antisymmetric spin polarization also occurs in the x- and y-spin components, which is expressed as $k_z(k_x^2-k_y^2){\sigma }_x-2k_x{k}_y{k}_z{\sigma }_y-k_x(k_x^2-3k_y^2){\sigma }_z$. We take the x axis as the $C_2^{\prime }$ rotation axis. The notations in the table are the same as those in Table 1.

OP	IR (Orbital)	IR (Cluster)	Odd-Parity Multipole	ASOC
$Q_u^{\left(\mathrm{s}\right)}$	$A_g$	$B_{2u}$	$Q_{3b}$	$k_x(k_x^2-3k_y^2){\sigma }_z$
$Q_{zx}^{\left(\mathrm{s}\right)}$	$E_{1g}$	$E_{2u}$	$G_v$	$k_x{\sigma }_x-k_y{\sigma }_y$
$Q_{yz}^{\left(\mathrm{s}\right)}$	$E_{1g}$	$E_{2u}$	$G_{xy}$	$k_x{\sigma }_y+k_y{\sigma }_x$
$Q_v^{\left(\mathrm{s}\right)}$	$E_{2g}$	$E_{1u}$	$Q_y$, $G_{zx}$	$c_1{k}_y{\sigma }_z+c_2{k}_z{\sigma }_y$
$Q_{xy}^{\left(\mathrm{s}\right)}$	$E_{2g}$	$E_{1u}$	$Q_x$, $G_{yz}$	$c_1{k}_x{\sigma }_y+c_2{k}_y{\sigma }_x$

.

5.2. Diamond Structure

The three-dimensional diamond structure is another lattice structure in locally noncentrosymmetric structure [64,113,114,115,116,117]. By taking the $8b$ site under the space group $Fd\overline{3}m$, we consider the staggered orbital ordering in terms of the electric quadrupole in Figure 6. The $(Q_u^{\left(\mathrm{s}\right)},Q_v^{\left(\mathrm{s}\right)},Q_{yz}^{\left(\mathrm{s}\right)},Q_{zx}^{\left(\mathrm{s}\right)},Q_{xy}^{\left(\mathrm{s}\right)})$ orderings accompany the odd-parity multipoles as $(G_u,G_v,Q_x,Q_y,Q_z)$, respectively. Similar to the honeycomb structure, the chiral-type ASOC does not appear in the diamond structure. The correspondence among the staggered orbital ordering, odd-parity multipole, and ASOC is shown in

Table 3. Classification of staggered orbital orderings in the three-dimensional diamond structure, which is constructed by the $8b$ site under the space group $Fd\overline{3}m$. The notations in the table are the same as those in Table 1.

OP	IR (Orbital)	IR (Cluster)	Odd-Parity Multipole	ASOC
$Q_u^{\left(\mathrm{s}\right)}$	$E_g$	$E_u$	$G_u$	$2k_z{\sigma }_z-k_x{\sigma }_x-k_y{\sigma }_y$
$Q_v^{\left(\mathrm{s}\right)}$	$E_g$	$E_u$	$G_v$	$k_x{\sigma }_x-k_y{\sigma }_y$
$Q_{yz}^{\left(\mathrm{s}\right)}$	$T_{2g}$	$T_{1u}$	$Q_x$	$k_y{\sigma }_z-k_z{\sigma }_y$
$Q_{zx}^{\left(\mathrm{s}\right)}$	$T_{2g}$	$T_{1u}$	$Q_y$	$k_z{\sigma }_x-k_x{\sigma }_z$
$Q_{xy}^{\left(\mathrm{s}\right)}$	$T_{2g}$	$T_{1u}$	$Q_z$	$k_x{\sigma }_y-k_y{\sigma }_x$

.

6. Conclusions

We have investigated the spontaneous parity breaking under the staggered orbital ordering in locally noncentrosymmetric crystals. Similarly to staggered antiferromagnetic ordering, staggered orbital ordering leads to odd-parity multipoles, which become the microscopic origin of the ASOC and its related physical phenomena. We demonstrate the appearance of the ASOC by taking the one-dimensional zigzag chain as an example. We also classify the odd-parity multipoles in the two-dimensional honeycomb and three-dimensional diamond structures. The present results indicate that staggered-type electronic orderings in locally noncentrosymmetric crystals provide a rich playground for parity-violating physical phenomena.