Revealing Spin and Spatial Symmetry Decoupling: New Insights into Magnetic Systems with Dzyaloshinskii-Moriya Interaction

Symmetry, as formulated by group theory, is fundamental to the laws of nature. In condensed-matter physics, symmetry has long been central, as it dictates the physical properties, phase transitions, and behavior of various materials[1]. It governs the conservation laws [2], determines selection rules [3], and plays a particularly crucial role in fields of Landau theory of phase transition [4], topological phases [5, 6, 7, 8, 9] and quasiparticle excitations [10, 11, 12]. The symmetries of three-dimensional (3D) solids are comprehensively described by crystallographic group theory, with spatial operations, including rotations, reflections, translations, and their combinations, forming the 32 point groups (PGs) and 230 space groups (SGs) for nonmagnetic systems [13]. When spin-orbit coupling (SOC) effect is considered, the double SGs are applicable. On the other hand, for magnetic materials, the SOC generally locks spin and spatial degrees of freedom, preventing spin from rotating independently, and the symmetry of magnetic systems should be described by 122 magnetic point groups (MPGs) and 1651 magnetic space groups (MSGs) [13].

In 1960s, it is found that the Heisenberg model neglecting the Dzyaloshinskii-Moriya interaction (DMI), which is the leading order of SOC effect [14, 15], enjoys higher symmetries than MSGs, and the concept of spin space groups (SSGs) is introduced [16, 17, 18, 19]. It has attracted significant attention in areas such as Landau theory of phase transitions [20, 21], neutron scattering [22], and magnon band topology [23, 24, 25, 26]. Very recently, the full classification of collinear SSGs and the associated magnon band topology for collinear magnets [27] are developed, offering new pathways to exploring exotic phenomena of magnonic systems.

In SSGs, the spin and spatial operations become partially decoupled, giving rise to an expanded set of symmetries expressed as [18, 19, 28],

where $G_{SO}$ stands for the spin-only group that only contains spin operations, $G_{NSS}$ stands for the nontrivial spin space group that contains no pure spin operations [28].

In recent years, SSGs have been extended to electronic systems, and drawn growing attention due to their relevance in the study of magnetic materials with negligible SOC [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. In field of magnetic topological electronic states, the classifications and the irreducible corepresentations of spin groups for coplanar and collinear magnetic structures are derived [28], and the Z₂ topological phases [28] and exotic nodal-line or -point semimetals [29, 30] protected by SSG symmetry are predicted. Meanwhile, SSGs are also crucial for understanding the unconventional spin-momentum locking without SOC[31, 32, 33, 34, 35, 36, 37, 38], which was later recognized as a characteristic feature of altermagnetism[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. Very recently, an exhaustive analysis of SSGs symmetries provides a full classification and investigates the enriched features of electronic bands within SSGs framework [52, 53, 54], which paves the way for deeper understanding and further exploration of emergent phenomena in magnetic materials.

The symmetry of SSGs is generally an approximation, holding strictly only when SOC is absent. However, as a relativistic effect, SOC always exists and plays a crucial role in various fields of condensed matter physics, such as permanent magnets [55, 56], spintronics [57], spin-orbitronics [58], and topological physics [59, 60]. The DMI, which has been widely regarded as the “SOC” in spin models and magnon systems [16, 61, 62, 63, 64, 65], is also always present even in the centrosymmetric compound, since not all bond centers are inversion centers. In topological magnons, DMI generates the Berry curvature and plays a central role in the study of magnon band topology [66, 67, 68, 69, 70, 68, 71]. Meanwhile, in the field of magnon transport, DMI is essential in many phenomena such as the magnon Hall effect[61, 72, 73, 74, 75, 76, 77, 78] and the magnon spin Nernst effect[63, 62, 79, 80, 81, 82], where DMI takes on a role analogous to that of SOC in the spin Hall effect.

The DMI and SSG symmetries have both played significant roles in the exploration of exotic phenomena in magnetic materials. However, as mentioned above, they are inherently in conflict. Could there be scenarios where SSG symmetries remain strictly applicable even when huge DMI is taken into account? The coexistence of DMI and SSG symmetries in one magnetic system will undoubtedly provide new avenues for study novel physical phenomena. In this work, we prove that the inclusion of DMI does not necessarily lead to a complete locking of spin and spatial rotations, and identify a novel symmetry to describe the spin model and magnon system with even considerable DMIs.

Group Theory Analysis

First we consider the coplanar spin configurations with the spin moments lying in the $xy$ plane. The symmetry of spin-coplanar SSGs could be written as Eq. (1), where the spin-only group for spin-coplanar SSGs has the expresssion as [18, 19, 28],

where $T$ and $U_{z}(\pi)$ represent time-reversal symmetry and the two-fold spin rotation operation along the z-axis, respectively.

In the spin Hamiltonian, the antisymmetric DMI is expressed as $H_{DMI}=\sum_{i,j}\bm{D_{ij}}\cdot(\bm{S_{i}}\times\bm{S_{j}})$ [14, 15], where $\bm{S_{i}}$ and $\bm{S_{j}}$ are the spin operators at the lattice sites $i$ and $j$, while $\bm{D_{ij}}$ is a vector arising from SOC and satisfies the antisymmetry condition $\bm{D_{ij}}=-\bm{D_{ji}}$. The spin Hamiltonian may also contain other spin interactions like anisotropic exchange or dipole-dipole couplings, but these are typically orders of magnitude weaker than the DM interaction [14, 15, 83] and thus are not considered in this work. In general, the inclusion of DMI locks spin and spatial rotations, and reduces the Hamiltonian symmetry to MSGs. However, we find that under two following cases of magnetic configurations, the spin and spatial rotations are still partially decoupled:

(i) The first case occurs in 2D systems, when the magnetic atoms lie within the $xy$ plane, which possesses horizontal mirror symmetry $\sigma_{h}$. In this case, only the component of DMI perpendicular to the mirror plane, i.e. $D^{z}$, is left [15], which makes that the spin Hamiltonian retains the symmetry $\{TU_{z}(\pi)||E\}$. Therefore, the spin-only group $G_{SO}=\{E,TU_{z}(\pi)\}$ in spin-coplanar SSG is strictly preserved. Below we provide a criterion to determine whether the operations in $G_{NSS}$ are preserved:

An operation $R\in G_{NSS}$ in 2D systems could be written as $R=\{A||B\}$, where $A$ and $B$ represents the spin and spatial operations, respectively. The generators for the spin operation $A$ include $U_{z}(\frac{2\pi}{n})$ and $U_{x}(\pi)$,where $n=1,2,3,4,6$[28], while the generators for the spatial operation $B$ include $C_{nz}$, $C_{2x}$, $S_{4}$, $P$, $\sigma_{h}$ and $\sigma_{v}$[84], where $S_{4}$ presents the four-fold rotation inversion operation and $P$ presents inversion operation, while $\sigma_{h}$ and $\sigma_{v}$ correspond to mirror operactions along the horizontal and vertical planes, respectively.

The DMI with only $D^{z}$ can be expressed as $H_{DMI}=\sum_{i,j}D_{ij}^{z}(S_{i}^{x}S_{j}^{y}-S_{i}^{y}S_{j}^{x})$. By applying the generators to the DMI term, the spatial operations $B$ act on the site $i$ and $j$, while the spin operations $A$ act on the directions of spin operations $S^{x}$ and $S^{y}$, respectively. The spin Hamiltonian is constrained by the space group symmetry and should remain invariant under the space group operation [15]. Note that in the spin group framework, this symmetry operation of space group is expressed as $R_{SG}=\{B^{\prime}||B\}$, where $B^{\prime}$ denotes the proper rotation part of the spatial operation $B$. Therefore, whether a coplanar SSG $R=\{A||B\}$ remains valid depends on whether the two spin-space rotations $A$ and $B^{\prime}$ alter the sign of $S_{i}^{x}S_{j}^{y}-S_{i}^{y}S_{j}^{x}$ in a consistent manner. Considering the influence of spin part on the DMI term, we categorize these generators into two types: $S_{i}^{x}S_{j}^{y}-S_{i}^{y}S_{j}^{x}$ changes sign under $U_{x}(\pi)$ (the corresponding spatial operations $B$ being $C_{2x}$ and $\sigma_{v}$), while $U_{z}(\frac{2\pi}{n})$ leaves $S_{i}^{x}S_{j}^{y}-S_{i}^{y}S_{j}^{x}$ invariant (the corresponding spatial operactions $B$ are $C_{nz}$, $S_{4}$, $P$ and $\sigma_{h}$). For a given operation $R=\{A||B\}$, when the operations $A$ and $B$ belong to the same type described above, the operation $R$ is preserved even in the presence of DMI. Otherwise, the operation $R$ would be broken.

(ii) The second case arises in 1D systems, when the magnetic atoms are aligned along $z$-axis chain, and the chain has double rotational symmetry $C_{2z}$, which also constrain the direction of the DMI to $z$ direction[15]. Therefore, the DMI with only $z$-components $D_{z}$ ensures that the SSG symmetry $\{TU_{z}(\pi)||E\}$ is preserved. Analogous to case (i), the symmetry of spin Hamiltonian maintain the form of spin-coplanar SSG with the spin-only group $G_{SO}=\{E,TU_{z}(\pi)\}$. For any given SSG operation $R=\{A||B\}$, one can also determine whether it remains valid using the criterion discussed in case (i).

Furthermore, for collinear spin configuration within the framework of the linear spin wave theory (LSWT), we also find interesting symmetries here. The spin-only group $G_{SO}$ for the collinear spin configurations could be written as [18, 19, 28],

where the direction of spin moments is defined along the $z$-direction, and $U_{z}(\phi),$ $\phi\in[0,2\pi)$ means full spin rotation along $z$-axis, $\{E,TU_{n}(\pi)\}$ contains the identity operation as well as the combined operation of time reversal $T$ and two-fold spin rotation about $n$-axis perpendicular to the spin $z$-axis. Within the framework of LSWT, the Holstein-Primakoff (HP) transformation results in that only $D_{ij}^{z}$ term contains quadratic contribution of magnons. Through symmetry analysis, we demonstrate that the $D_{ij}^{z}$ term preserves $\{U_{z}(\phi)||E\}$ symmetry but breaking the operation $\{TU_{n}(\pi)||E\}$. Therefore, the magnons in spin-collinear systems has the novel symmetry expressed as $G_{SS}=G_{NSS}\times G_{SO}^{\prime}$, where the spin-only group could be written as

while the nontrivial spin space group $G_{NSS}$ is a subgroup of the one for original SSG without DMI. It should be noted that, the above expression of the spin group $G_{SS}$ has spin-only symmetry beyond MSG, but does not belong to the conventional definitions of collinear or coplanar spin groups, whose spin-only groups are given by Eq. (2) and (3) respectively.

The coexistence of DMI interactions with SSG symmetries directly gives rise to emergent physical phenomena. In the following we present a concrete application in magnon transport. In magnon spintronics, the ability to generate a pure transverse spin current without a thermal current is of great interest [62, 63]. Generally, the DM interactions play an important role in the magnon spin Hall effect [61, 62, 63, 72, 73, 74, 75], thus researchers typically employ MSG symmetries to analyze/discuss the system with magnon spin Hall but without magnon thermal Hall. For example, in Refs [62, 63], the suppression of the magnon thermal Hall effect is achieved in collinear antiferromagnets via $PT$ symmetry, where $P$ and $T$ represent the inversion and time-reversal symmetry. The using of MSG significantly narrows the range of materials capable of generating pure magnon spin currents.

In contrast, in the two spin-coplanar cases we proposed, the SSG symmetry $\{TU_{z}(\pi)||E\}$ could ensure that the magnon Berry curvature is an odd function with respect to the quasimomentum $\bm{k}$, $\Omega_{n}(\bm{k})=-\Omega_{n}(-\bm{k})$, thereby completely suppressing the magnon thermal Hall response. Notably, the DMI of $D^{z}$ in these cases induces the magnon spin conductivity $\sigma^{z}_{\alpha\beta}$, where the in-plane indices $\alpha$ and $\beta$ denote the directions of the magnon spin current and the temperature gradient, respectively. Thus, the pure magnon spin current driven by a temperature gradient can be achieved in systems with the coexistence of DMI interactions and SSG symmetries. This is different from previous theoretical frameworks [62, 63]. In the following, we would also present the potential material realization.

Candidate Materials

Based on the above symmtry analysis, we will extract the criteria to identify the magnetic materials whether the spin and spatial operations are still partially decoupled even with large DMI. The 2D systems are described by wallpaper groups or layer groups [87]. In the case of wallpaper groups, only the $xy$-direction is defined, thus the $xy$-plane is inherently mirror symmetric. For all the 17 wallpaper groups, when the spins lie in the $xy$-plane, the spin-only group $G_{SO}$ in spin-coplanar SSG is preserved.

On the other hand, since most 2D materials typically consist of multiple atomic layers or have finite extension in the $z$-direction, the layer group usually provides a more appropriate description [87]. There are 17 layer groups having the mirror symmetry $\sigma_{h}$, which are summarized in TABLE 1. If a magnetic material belongs to the layer groups in TABLE 1, and its magnetic atoms are also located at the given Wyckoff positions in TABLE 1, i.e. $z=0$, while their spins lie in the $xy$-plane, the presence of DMI does not degrade the spin Hamiltonian symmetry to MSGs, but instead preserves the spin-coplanar SSGs symmetries. We predict that the coexistence of DMI and SSG symmetries exists in 17 out of 80 layer groups ($21\%$), demonstrating that this phenomenon is not limited to isolated cases but rather occurs widely across diverse magnetic systems.

For any 2D magnetic material, one can easily identify their symmetries using our TABLE 1. To illustrate the result, we apply our theory to a popular database Computational 2D Materials Database (C2DB)[85, 86] and identify 33 materials as summarized in TABLE 2. The spin Hamiltonian for these materials would maintain the SSG symmetry even with large DMI. As an illustrative example, we consider the layered material $\mathrm{VSe_{2}}$ in TABLE 2, whose layer group is $p-6m2$, with magnetic V atoms residing on the mirror plane. It was suggested that $\mathrm{VSe_{2}}$ adopts an in-plane collinear ferromagnetic ground state [88]. Due to the strong SOC in selenium [89], this material would exhibit significant DM interactions. Conventionally, such materials should be described by MSG theory, thus naively one will expect both magnon thermal Hall and the magnon spin conductivity. However, our symmetry analysis above reveals that the DMI and the SSG symmetry $\{TU_{z}(\pi)||E\}$ in this compound induces magnon spin Hall and forbids the magnon thermal Hall, respectively. Thus, the coexistence of DMI and SSG symmetry results in the pure magnon spin current ($\sigma^{z}_{\alpha\beta}$) in this material.

In addition to the classifications in 2D groups in TABLE 1, we can also expand it to 3D systems through the corresponding mappings from layer groups to 3D space groups [84]. Each layer group is isomorphic to the factor group $G/T(1)$, where $G$ represent a 3D space group, and $T(1)$ is a 1D translation subgroup of $G$. It allows each layer group to correspond to a specific 3D space group, which represents the structure obtained when the layer group is extended along the $z$ direction. The corresponding space groups and required Wyckoff positions of the magnetic atoms for 3D materials are summarized in TABLE 3. For 3D materials with sufficiently large spacing between their mirror planes, where interlayer anisotropic interactions can be neglected as an additional approximation, one can also determine whether they possess decoupled spin and spatial symmetry using TABLE 3.

For 1D materials, their symmetry is typically described using rod groups, which possess translational symmetry along a single axis (marked as $z$-direction) with 3D space[87]. Based on the symmetry analysis above, we provide the TABLE 4, when magnetic materials belongs to the given rod groups and the magnetic atoms occupy the required Wyckoff positions, while their magnetic moments lie in the $xy$ plane, these materials could preserve spin-coplanar SSG symmetries described by Eq. (1) and Eq. (2). The TABLE 4 can also be expanded to identify the quasi-1D magnetic candidate materials. Since many magnetic materials exhibit quasi-1D characteristics, such as Sr₃MIrO₆(M = Li, Na, Zn, Cd) [90], thereby the exploration of SSG symmetry in these materials is worthy of further investigation.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grants No. 12188101, No. 12334007, and No. 12474233), the National Key R&D Program of China (Grant No. 2022YFA1403601), Innovation Program for Quantum Science and Technology (Grant No.2021ZD0301902, and No. 2024ZD0300101), Natural Science Foundation of Jiangsu Province (No. BK20233001, BK20243011).