Symmetry analysis with spin crystallographic groups: Disentangling effects free of spin-orbit coupling in emergent electromagnetism

Recent studies identified spin-order-driven phenomena such as spin-charge interconversion without relying on the relativistic spin-orbit interaction. Those physical properties can be prominent in systems containing light magnetic atoms due to sizable exchange splitting and may pave the way for realization of giant responses correlated with the spin degree of freedom. In this paper, we present a systematic symmetry analysis based on the spin crystallographic groups and identify the physical property of a vast number of magnetic materials up to 1500 in total. By decoupling the spin and orbital degrees of freedom, our analysis enables us to take a closer look into the relation between the dimensionality of spin structures and the resultant physical properties and to identify the spin and orbital contributions separately. In stark contrast to the established analysis with magnetic space groups, the spin crystallographic group manifests richer symmetry including spin-translation symmetry and leads to emergent responses. For representative examples, we discuss the geometrical nature of the anomalous Hall effect and magnetoelectric effect and classify the spin Hall effect arising from the nonrelativistic spin-charge coupling. Using the power of computational analysis, we apply our symmetry analysis to a wide range of magnets, encompassing complex magnets such as those with noncoplanar spin structures as well as collinear and coplanar magnets. We identify emergent multipoles relevant to physical responses and argue that our method provides a systematic tool for exploring sizable electromagnetic responses driven by spin order.

Figure 1

Procedure of symmetry analysis. (a) Input of crystal and spin structures. Dozens of data imported from magndata. (b) Spin space group $\mathcal{G}$ and magnetic space group $\mathbf{G}$ computationally identified. The effect of spin-orbit coupling (SOC) is not considered in $\mathcal{G}$ but is taken into account in $\mathbf{G}$. The spin space-group symmetry is given by the spin rotation ($W$) and by orbital-space operations such as rotation ($R$) and translation ($\mathit{t}$) operations. The magnetic-space group is comprised of the orbital space operations $(R,\mathit{t})$ with or without the time-reversal operation $\theta$ such as $R^{\prime }=\theta R$. For the magnetic-space group $\mathbf{G}$, the orbital-space operations $R,R^{\prime }$ also act on spins due to the SOC constraint. (c) The space group is reduced to its point group by omitting the translation operation $\mathit{t}$. Spin ($\mathcal{P}$) and magnetic ($\mathbf{P}$) point groups are obtained from $\mathcal{G}$ and $\mathbf{G}$, respectively. The spin point group is further divided into (${\mathcal{P}}_{\text{sp}},{\mathcal{P}}_{\text{orb}}$) consisting of either spin or orbital space symmetry. The spin crystallographic groups ($\mathcal{G},\mathcal{P}$) are reduced to the spin-orbital-coupled groups ($\mathbf{G},\mathbf{P}$) by SOC. (d) The symmetry-adapted form of a given tensor $\widehat{\chi }$ obtained by the spin or magnetic point-group symmetry. The red-colored components (${\chi }_{\text{sp}},{\chi }_{\text{orb}}$) originate from the spin-order-induced symmetry breaking without SOC, and the origin of each component is further attributed to the spin (${\chi }_{\text{sp}}$) and orbital (${\chi }_{\text{orb}}$) degrees of freedom. SOC entangles the spin contribution with the orbital (${\chi }_1,{\chi }_4$) and induces additional components (${\chi }_2,{\chi }_3$).

Figure 2

Spin-group transformation of electron depicted by position $\mathit{r}$, momentum $\mathit{p}$, and spin $\mathit{s}$. The spin-group operation $g=(R,W)$ ($R$ is orbital space rotation, $W$ is spin space rotation). (a) Orbital space operation acting on position and momentum while leading to no action on spins ($W=1$). (b) Spin space inversion operation same as the time-reversal operation ($R=1,W=-1$) flipping the time-reversal-odd quantities such as $\mathit{r}$ and $\mathit{s}$. (c) Spin space operation satisfying $R=1$ and the proper rotation condition (det $W=+1$). It gives no transformation related to the orbital space objects.

Figure 3

(a) Crystal (left panel) and spin (right panel) structures of ${\mathrm{CoTa}}_3{\mathrm{S}}_6$. (b) The tetrahedron spanned by four spins in the magnetic structure. The black lines denote the twofold rotation axes relevant to the spin-translation group.

Figure 4

(a) Crystal and spin structures of ${\mathrm{Cr}}_2{\mathrm{O}}_3$ where spins are collinear along the [001] direction. (b), (c) Chirality of the crystal structure of ${\mathrm{CoTa}}_3{\mathrm{S}}_6$ implied by ${\mathrm{TaS}}_6$ prism with the twisted coordination of Co atoms. The twisting arrangement determines the chirality $\chi =\pm 1$. When orbital magnetization (blue-colored vector) is formed along the [001] direction due to its noncoplanar spin ordering, (b) the toroidal moment (red-colored vector) is antiparallel to it, (c) while it is parallel with the opposite chirality.

Figure 5

(a) Spin structure of ${\mathrm{Mn}}_3\mathrm{Sn}$. (b) Transverse conversion between the charge and spin currents denoted by the rotator $R_{ij}$. The Hall plane perpendicular to the $x_i$ direction is for the current whose spin is polarized along the $x_j$ direction in the spin space.

Figure 6

Distribution of magnetic materials parametrized by the order of the magnetic space group ($\left|\mathbf{G}\right|$) and by the ratio between the order of the nontrivial spin space group ($|\overline{\mathcal{G}}|$) and $\left|\mathbf{G}\right|$. The distributions are categorized based on whether the spin-structure dimension is collinear, coplanar (but noncollinear), or noncoplanar.

Figure 7

Spin configurations of (a) ${\mathrm{Ba}}_3{\mathrm{MnSb}}_2{\mathrm{O}}_9$ with only Mn atoms and of (b), (c) ${\mathrm{Mn}}_3\mathrm{CuN}$. In (c), the spin space operation $((m_{\left(010\right)},\mathit{0}))m_{\alpha }$ is depicted. The mirror operations $W=m_{\beta },m_{\gamma }$ can be similarly obtained.