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Yutong Yu 1 , Xiaobing Chen 2,1,† Yanzhou Zhu 3 , Yuhui Li 1 , Renzheng Xiong 1 , Jiayu
Li 1,4 , Yuntian Liu 1 , and Qihang Liu 1,2,5,* 1 State Key Laboratory of Quantum Functional Materials, Department of Physics, and
Guangdong Basic Research Center of Excellence for Quantum Science, Southern
University of Science and Technology (SUSTech), Shenzhen 518055, China
2 Quantum Science Center of Guangdong–Hong Kong–Macao Greater Bay Area
(Guangdong), Shenzhen 518045, China
3 School of Mathematics Science, Xiamen University, Xiamen 361005, China 4 New Cornerstone Science Lab, Department of Physics, The University of Hong Kong,
Hong Kong, China 5 Guangdong Provincial Key Laboratory for Computational Science and Material
Design, Southern University of Science and Technology, Shenzhen 518055, China
† Email: [email protected] * Email: [email protected] 1
Unconventional magnets that combine antiferromagnetic structures with
ferromagnetic-like responses are essential for the development of next-generation
spintronics. Their emergent properties are fundamentally dictated by the interplay
between exchange-driven magnetic geometry and spin-orbit coupling, which are
described by spin space group (SSG) and magnetic space group (MSG) frameworks,
respectively. However, the lack of direct correspondence between these frameworks,
developed in different eras, hinders the systematic tracking of symmetry evolution of
these intertwined physical contributions. In this work, we introduce FINDSPINGROUP,
a computational architecture that implements the recently emerged, oriented spin space
group framework to unify SSG and MSG descriptions. By automating the tracking of
symmetry-breaking pathways from the non-relativistic to the relativistic limit, this
online platform enables the classification of magnetic phases and the derivation of
symmetry-constrained tensors for phenomena such as momentum-dependent spin
splitting and the anomalous Hall effect. By standardizing data exchange through the
spin crystallographic information file, this architecture establishes a computational
infrastructure for the high-throughput discovery and design of unconventional magnets.
Keywords: unconventional magnets, oriented spin space group, online platform, spin–
orbit coupling, spin polarization, anomalous Hall effect 2
The exploration of magnetic materials with high storage density, ultrafast dynamics,
and electric controllability is central to next-generation spintronic devices [1, 2]. In
recent years, unconventional magnets that combine ferromagnetic-like responses with
antiferromagnetic structures have emerged as promising candidates [3]. These systems
host diverse emergent phenomena arising from two distinct physical sources: the non-
relativistic magnetic geometry governed by exchange interactions and the relativistic
spin-orbit coupling (SOC) effect. However, the textbook magnetic space group (MSG)
framework assumes complete locking between lattice and spin spaces [4], which
naturally conflates effects driven by magnetic geometry from those solely induced by
SOC. Recently, the spin space group (SSG) theory has emerged as an extended
framework to provide a comprehensice description of magnetic geometry by
decoupling symmetry operations in lattice and spin spaces [5-16]. This has led to a new
symmetry paradigm for understanding the physical effects of unconventional magnets,
such as spin splitting, anomalous Hall effect, and magnetoelectric coupling [12, 17-28],
i.e., employing SSG to describe the effects induced by magnetic order and MSG to
characterize the SOC effects.
Despite this conceptual framework, its practical implementation for real materials
remains hindered by the lack of a unified platform. In practice, different conventions
between SSG and MSG frameworks, such as cell choices and coordinate systems, often
hinder direct comparisons of physical responses across the non-relativistic and
relativistic limits. Without a unified convention, reconciling these two symmetry
descriptions requires manual realignment, which is inefficient and error-prone. While
existing tools can identify SSG operations [29-31], they lack an automated workflow
to bridge the SSG and MSG descriptions. Consequently, tracing the symmetry-breaking
pathways of physical properties under SOC has remained largely manual, limiting
systematic and high-throughput discovery of unconventional magnetism.
In this work, we address these challenges by introducing FINDSPINGROUP, an
open-source platform for the automated identification of magnetic symmetry, magnetic 3 phases, and various symmetry-constrained physical properties. Specifically, the
platform is built on the recently developed oriented spin space group (OSSG)
framework [32], which unifies the SSG and MSG frameworks and explicitly elucidates
the symmetry-breaking pathways from SSG to MSG. FINDSPINGROUP generates
standardized outputs, including magnetic cells, spin Wyckoff positions, spin site-
symmetry groups, the spin Brillouin zone, and spin little co-groups. This automated
workflow enables a systematic investigation of how various physical properties evolve
as magnetic order and SOC are sequentially introduced, covering fundamental
properties such as magnetizations, electric polarization, and spin polarization, as well
as transport tensors, including the anomalous and nonlinear Hall effects. Beyond these
capabilities, the practical utility of this workflow is demonstrated by its integration with
other tools such as STensor [33] and Tensorsymmetry [34]. This interoperability, along
with its widespread adoption [20, 21, 23, 32-37], establishes FINDSPINGROUP as a
computational infrastructure for investigating unconventional magnets.
The symmetry operations of the SSG can be expressed as {𝑔 ! ||𝑔 " |𝜏}, where 𝑔 ! =
simultaneously reverses spin and momentum, while 𝑔 " = {𝐶 $ (𝜃), 𝑃𝐶 $ (𝜃)} comprises lattice rotations 𝐶 $ (𝜃) and the spatial inversion 𝑃 . A magnetic structure is characterized by specifying the atomic positions (r) and magnetic moments (m). The action of an SSG operation on the tuple (𝑟, 𝑚) is defined as:
and the product of two SSG operations is defined as:
The general form of an SSG, denoted by 𝐺 '' , can be expressed as the direct product of a nontrivial SSG 𝐺 (' and the spin-only group 𝐺 ') [8]:
4
where 𝐺 (' denotes the nontrivial SSG that every spin operation is combined with a spatial operation, except for the identity element, 𝐺 ') consists of pure spin operations {𝑔 ! ||𝐸|0} that describe the dimensionality of magnetic geometry. Based on the nomenclature of SSGs developed in Ref. [14], we extend the four-index nomenclature
the parent space group, sublattice space group, cell-expansion factor, and mapping
sequence in the database, respectively. The additional fifth index L or P represents a
collinear or coplanar spin-only group, respectively, and is omitted for non-coplanar
configurations. The complete database of all SSGs, with their standard international
notations, is available on the FINDSPINGROUP website.
When considering SOC effects, the operations in lattice and spin spaces become completely locked. The operations in the SSG that satisfy 𝑈 # (𝜑) = 𝐶 $ (𝜃) constitute the corresponding MSG. Consequently, spin rotations are absorbed by the lattice
rotations and are thus not explicitly expressed in MSGs. Although the SSG and MSG
provide comprehensive descriptions of magnetic geometry and SOC effects,
respectively, the arbitrary relative orientation between the spin and lattice spaces in the
SSG [Fig. 1(a)] prevents a direct mapping to the MSG. This ambiguity hinders the
comparative analysis of physical properties with and without SOC.
Historically, the characterization of phase transitions has relied on group-subgroup
relationships, in which the symmetry-breaking path from a parent to a daughter phase
is typically labeled by the irreducible representations of the parent group. For example,
magnetic representations [38, 39] are adopted to characterize the symmetry descent
from a paramagnetic space group to an MSG. However, the SSG possesses an intrinsic
O(3) auto-equivalence in spin space, and its magnetic subgroups are obtained by
selecting specific orientations. This unique property allows for a simultaneous
description of the SSG and its MSG subgroup through the oriented SSG [32] by fixing
the relative orientation between lattice and spin spaces [Fig. 1(b)], rather than
introducing additional SSG representations.
5 Besides the alignment of spin-space operations of the nontrivial SSG with respect
to the lattice, the OSSG introduces additional specification for spin-only groups in
collinear and coplanar systems. The crystallographic direction [αβγ] is used to label the principal axis of the magnetic moment in collinear systems, - #$% # 1 , and the normal direction of the spin plane in coplanar systems # #$% 1 , respectively. Based on these, the
connection between OSSG and the corresponding MSG can be established directly by locking the spin and lattice operations via the constraint 𝑈 # (𝜑) = 𝐶 $ (𝜃) [Fig. 1(c)].
Take the OSSG of two altermagnets, CrSb and MnTe, as examples; they both share
corresponds to [001] and [210], respectively. The corresponding MSG can be identified
from completely locked operations. For CrSb, the OSSG corresponds to the MSG
Figure 1. Evolution from SSG to MSG via the oriented framework. (a) In the SSG,
the solid line in the middle separates two spaces, indicating that lattice and spin spaces
are independent and have an arbitrary relative orientation. The multiple colored arrows
and the dashed circle represent the orientational degeneracy in spin space. (b) In the
OSSG, the dashed line indicates that the two spaces are aligned to define a material-
6 specific orientation of magnetic orders, but the operations remain decoupled. (c) Upon
introducing SOC, the disappearance of the central line indicates complete coupling
between the two spaces, and the magnetic moments are constrained along specific
crystallographic axes. Only the subset of OSSG operations compatible with spin-lattice
locking is preserved, yielding the corresponding MSG. The orientational degeneracy is
removed, and the dashed circle therefore disappears. This construction integrates non-
relativistic and relativistic descriptions, along with their symmetry evolution pathways,
into a unified framework.
Systematic identification of the OSSG requires a standard numerical framework
to ensure consistency across the non-relativistic and relativistic regimes. In
FINDSPINGROUP, the OSSG of a magnetic structure is determined by evaluating the
symmetry operations within the direct product of the parent space group and the point
group in spin space [14]. Throughout this work, the crystallographic part of the spin-
group setting, including the basis vectors, cell setting, and origin choice, is expressed
in the standard space-group convention of the Bilbao Crystallographic Server [40].
Besides, we introduce the .scif (spin crystallographic information file) format.
Analogous to the .mcif format, the .scif format encodes the OSSG information as an
extension to the underlying crystallographic structure. This approach preserves
symmetry-resolved information across different settings and supports both tensor-
symmetry analysis with STensor [33] and direct visualization of SSG symmetry in Jmol
[41]. Detailed information on identification and visualization is provided in the
Methods section.
Following the space groups and MSGs, the spin Wyckoff positions and spin site-
symmetry groups are introduced to evaluate the Wyckoff-position splitting induced by
magnetic order. Notably, the spin site-symmetry groups of an OSSG are always
isomorphic to a subset of the 941 spin point groups. This isomorphism ensures that site-
dependent effects can be exhaustively classified; for instance, only 375 spin point
7 groups allow local magnetic moments. In reciprocal space, the OSSG dictates the
symmetry of the spin Brillouin zone. While collinear and coplanar systems are
characterized by the 24 centrosymmetric arithmetic crystal classes, noncoplanar SSGs
correspond to all 73 arithmetic classes [42]. The construction of spin Wyckoff positions,
the associated spin site-symmetry groups, and the little co-groups of wave vectors
within the OSSG convention are discussed in Method. This ensures the precise
identification of high-symmetry wavevectors, providing a comprehensive description
of momentum-dependent spin physics.
According to Neumann’s principle, the symmetry of any physical property of a
crystal must include the symmetry operations of its point group [43]. Consequently,
crystallographic and magnetic point groups are often employed to impose symmetry
constraints on physical properties such as ferroelectricity, magnetic moments, and
various transport effects [44, 45]. However, unlike 32 point groups and 122 magnetic
point groups, the 598 nontrivial spin point groups [46] and 941 spin point groups [10]
are insufficient to provide an exhaustive description of SSGs. This is because SSGs permit arbitrary spin propagation {𝑔 ! ||𝐸|𝜏} , where 𝑔 ! could be a noncrystallographic spin rotation. To bridge this gap, we directly employ the OSSG to
impose symmetry constraints and trace the symmetry-breaking pathways as the
magnetic geometry descends into the relativistic limit, where the locking of spin and
lattice spaces can permit previously forbidden physical responses.
For a tensor T, the transformation under a symmetry operation of OSSG can be
expressed as:
where the indices 𝑖, 𝑗, 𝑘, 𝑙, 𝑚, 𝑛, … refer to Cartesian components x, y, and z. Following
standard conventions [14], the x- and z-axes are aligned with the crystallographic aand c-vectors to form a right-handed Cartesian system. The coefficients 𝑎 1" , 𝑎 2# , 𝑎 +$ … correspond to elements of the representation matrix derived from the symmetry 8 operation {𝑔 ! ||𝑔 " |𝜏}. The factor 𝜔 accounts for the parity under spatial inversion and time reversal.
Based on whether they change sign under spatial inversion and time reversal,
tensors are classified as either polar or axial, and as time-dependent and timeindependent, respectively. Therefore, 𝜔 is determined by the determinants of the
symmetry operations in lattice and spin spaces:
time-dependent polar vector
lattice and spin spaces, respectively. Improper rotations yield a determinant of -1 and
correspond to rotations combined with spatial inversion in lattice space or time reversal
in spin space.
The overall workflow of FINDSPINGROUP is illustrated in Fig. 2. Starting from structure files (.cif, .mcif, .scif), the program identifies the OSSG, assigns the corresponding standardized database index, and derives outputs such as conventional and primitive magnetic cells, spin Wyckoff positions, and the spin Brillouin zone. By applying the tensor constraints defined in Eq. (4), FINDSPINGROUP evaluates a range of physical properties, including magnetization, electric polarization, spin splitting, and various transport effects. Representative applications to these classes of physical
structure input in CIF, MCIF, or SCIF format, FINDSPINGROUP identifies the OSSG
and uses it as the working convention for downstream analysis. Within this unified
framework, the platform determines the magnetic cell, spin Wyckoff positions, and spin
Brillouin zone, and further evaluates symmetry constraints on magnetic phase, spin
splitting, electric polarization, and quantum transport in the presence or absence of spin-
orbit coupling.
As discussed before, the OSSG framework, which reconciles the SSG description
of the phenomenological dichotomy between ferromagnetism and antiferromagnetism
[32] with the MSG description of SOC effects, is employed to design time-reversal-odd
physical responses in unconventional magnets. Below, we begin with the net
magnetization M, a time-dependent axial vector, to show how the OSSG establishes a
symmetry classification of magnetism.
Symmetry analysis can be applied separately to the spin magnetization 𝑀 ' and the orbital magnetization 𝑀 ) in the nonrelativistic limit. Although both are timedependent axial vectors, 𝑀 ' is governed by the spin-space operation 𝑔 ! while 𝑀 ) is constrained by the lattice-space operation 𝑔 " and the time reversal component in 𝑔 ! . Specifically, the spin-space point group, formed by all spin-space operations in OSSG, constrains 𝑀 ' . In contrast, 𝑀 ) is constrained by the effective magnetic point group, which is obtained by mapping the operation {𝑔 ! ||𝑔 " |𝜏} to 𝑔 " and 𝑇𝑔 " for proper and improper spin-space operation 𝑔 ! , respectively. The respective transformation relations are expressed as:
This distinction leads to different physical consequences. The symmetry-protected zero 𝑀 ' defines the boundary between ferromagnets/ferrimagnets (FMs/FiMs) and 10 antiferromagnets (AFMs) [32], serving as a prerequisite for exploring unconventional
magnets with time-reversal-odd physical responses. On the other side, the spin-mirror operation in collinear and coplanar magnets enforces a zero 𝑀 ) in the absence of SOC, while non-coplanar magnets can support a nonzero 𝑀 ) originating from the nonrelativistic magnetic geometry. Furthermore, the symmetry requirements for 𝑀 ) are
identical to those of the anomalous Hall conductivity, which corresponds to the
antisymmetric part of the conductivity tensor and thus an axial pseudovector [47]. This
implies that a magnetic-geometry-induced AHE can only emerge in non-coplanar
magnets [48, 49]. In contrast, for collinear and coplanar systems, the AHE originates
solely from relativistic SOC effects.
The classification procedure of magnetic phases under OSSGs is summarized in
Fig. 3. A polar spin-space point group, defined as a subgroup of ∞m, indicates a nonzero 𝑀 ' . Systems with a nonzero 𝑀 ' are further distinguished by their spin Wyckoff
positions (SWPs), of which a single type identifies an FM and multiple inequivalent
types indicate a FiM. In the case of FiM, when the magnetization vanishes, the system
is further classified as a compensated ferrimagnet (cFiM). Conversely, nonpolar spin-
space point groups correspond to AFMs. Unlike FMs, which maintain a net
magnetization in both non-relativistic and relativistic limits, AFMs are further
distinguished by their response to SOC. If there exists a net magnetization induced by
SOC effects in an AFM, i.e., allowed by the fully coupled operations in its OSSG, it is
further classified as spin-orbit magnetism (SOM) [32]. Such a unique class of AFM
manifests FM-like properties related to finite magnetization, such as the anomalous
Hall effect, anomalous Nernst effect, and magneto-optical Kerr effect, etc.
11
12
Besides macroscopic properties, the OSSG evaluates how magnetic geometry
governs the spin-polarized band structure in reciprocal space, where the wave vector k
is treated as a time-dependent axial vector and remains invariant under proper spin
rotations. In FMs, FiMs, and cFiMs, the spin-space point group is polar, leading to
Zeeman-type splitting in such systems. In AFMs, the SSG protects a vanishing net
magnetization, corresponding to the absence of spin polarization at the Γ point in
momentum space. Away from Γ, when the spin-space part of the k-little co-group is
polar, the momentum-dependent spin splitting is allowed. This condition requires
breaking the combination of spatial inversion and time reversal (PT), which is also an
MSG operation. In addition, such spin splitting requires the absence of the D ∞ or D 2
group in spin space, which could occur in collinear and noncollinear magnetic
configurations, respectively [14]. This analysis also distinguishes between spin-
polarization components at any k-point induced by magnetic geometry and the
additional relativistic contributions that emerge only upon the introduction of SOC. For
collinear AFMs that support momentum-dependent spin splitting without SOC,
FINDSPINGROUP explicitly assigns the label "altermagnet" [12].
This method can also be applied to two-dimensional spin-split AFMs. Compared
with three-dimensional cases, a prerequisite step is to identify the orientation of the
non-periodic direction of the structural input and then to apply constraints on the spin
polarization. In practice, the three-dimensional OSSG is first determined. By removing
the k-point components along the non-periodic direction, the two-dimensional spin
Brillouin zone, high-symmetry k-points, and spin little co-groups for any k-points are
identified. The reduction in dimensionality can induce additional symmetry-protected
degeneracies, i.e., the spin polarization at general k points can be further constrained by
improper spin rotation coupled with lattice rotation C 2z , or proper spin rotations
associated with the mirror operation M z [50, 51].
13 To illustrate the functionality of our algorithm and program, we present several
examples in which FINDSPINGROUP is used to identify OSSGs and analyze their
associated physical properties, both in the SOC-free and SOC-included cases.
The full symmetry of two-dimensional (2D) altermagnet V 2 Se 2 O (Fig. 4(a)) [19, 52, 53] is described by the OSSG 𝑃 /% 4 / % 𝑚 % 𝑚 /% 𝑚 - !"# # 1 (123.47.1.1.L), where
[αβγ] denotes the easy axis under SOC. This system is classified as an AFM due to its
spin-space point group ∞/mm (Fig. 4(c)). In the absence of SOC, this OSSG belongs to
the arithmetic crystal class 4/mmmP, the corresponding reduced spin Brillouin zone is
constructed with all high-symmetric k-points identified by the red triangle in Fig. 4(b).
Symmetry analysis reveals that this collinear AFM is an altermagnet in the SOC-free
limit, characterized by momentum-dependent spin splitting. The spin polarization
consists of only the S x component, while the electronic bands along the Γ-M direction remain spin degenerate due to the {2 % ‖𝑚 %/%* } symmetry operation. Furthermore, the {2 % ‖4 **% } operation enforces a sign reversal of the spin texture under a fourfold lattice rotation in momentum space, thereby dictating the d-wave spin texture and
the spin-polarized Fermi surfaces as shown in Fig. 4(d).
Upon the introduction of SOC, the [100] easy axis breaks the four-fold lattice
rotation symmetry, yielding an OSSG subgroup that is isomorphic to the MSG 𝑃𝑚𝑚 0 𝑚 0 . As a result, the system is further classified as an SOM. Notably, a nonzero
net magnetization emerges parallel to the Néel vector, which is distinct from the
conventional ferromagnetic canting induced by Dzyaloshinskii-Moriya interactions,
where the net magnetization is perpendicular to the Néel vector. This effect arises from
SOC-induced Wyckoff splitting, in which the symmetry equivalence between two
opposite-spin sublattices is lifted. The two V atoms at spin Wyckoff position 2f are split
into two different magnetic Wyckoff positions, 1c and 1e.
The symmetry breaking also leads to an expansion of the reduced Brillouin zone
where the high-symmetry Γ-M path reduces to a set of general k-points (Fig. 4(b)). This
reduction lifts spin degeneracy throughout the whole Brillouin zone, allowing the 14 emergence of spin polarization S x . Although the {2 %** ‖𝑚 %** } symmetry ensures that the spin polarization at all high-symmetry points remains restricted to the S x component,
the further reduction of symmetry enables the emergence of an additional S y component at general k-points, while S z remains strictly forbidden by the {𝑚 **% ‖2 **% } symmetry. While MSG permits a uniform S x distribution across the Brillouin zone, the d-wave
character remains dominant because of the negligible SOC (Fig. 4(d)). The small S y
projection observed in the DFT calculations is also consistent with the symmetry
analysis. This evolution explains how the OSSG captures the symmetry-breaking
pathway and the resulting modulation of momentum-dependent spin textures, providing
a microscopic foundation for the design of tunneling magnetoresistance,
piezomagnetism, and spin-splitting torque.
The OSSG framework implemented in FINDSPINGROUP also facilitates the identification of coexisting ferroic orders and the exploration of magnetic switch
15 mechanisms. As stated before, this approach effectively distinguishes between spininduced and SOC-induced ferroelectricity, enabling the electric control of unconventional magnets [23-26]. For example, the non-magnetic state of MnSe 2 [54] crystallizes in the centrosymmetric space group Pa-3, which forbids spontaneous electric polarization. However, the emergence of a supercell antiferromagnetic order leads to a symmetry reduction as illustrated in Figs. 5(a) and 5(b). Symmetry analysis indicates that the spatial part of the OSSG is Pca2 1 , a polar group that permits a spontaneous electric polarization along z direction. This mechanism differs from traditional multiferroics governed by exchange striction models, such as MnS 2 and Cu 2 MnSnS 4 , where spatial inversion is broken while the Tτ symmetry is preserved, resulting in hidden altermagnets [55, 56]. In contrast, the magnetic ordering in MnSe 2 simultaneously breaks both, inducing a macroscopic ferroelectric state and enabling momentum-dependent spin splitting.
Beyond the coexistence of ferroic orders, FINDSPINGROUP enables the systematic identification of switching trajectories between symmetry-related domain states. By determining the coset representatives of the space group relative to the OSSG, 36 potential switching paths are identified. Subsequent calculations can be employed to identify the trajectories with the lowest energy barriers. For instance, considering the
represents a path where the spin polarization remains invariant during ferroelectric switching, whereas the latter corresponds to the simultaneous switch of electric polarization and spin splitting. Such symmetry analysis facilitates the design of spininduced ferroelectricity and the prediction of whether the ferroelectric state can control coexisting FM-like responses. By distinguishing between this exchange-driven mechanism and the more common SOC-induced ferroelectricity, the OSSG framework provides a microscopic roadmap for designing multiferroic materials where electric and magnetic degrees of freedom are coupled.
16
The all-in-all-out AFM CoNb₃S₆ (Fig. 6(a)) represents a typical case where OSSG
captures the coexistence of spin degenerate bands and the magnetic geometry induced
distinct feature of this OSSG is the coexistence of a hexagonal space group P622 with
a tetrahedral spin-space point group -43m, whose point-group components are
incompatible (Fig. 6(b)). Such a configuration goes beyond the description of 941 spin
point groups and can only be characterized by the SSG or OSSG. This example
demonstrates that spin point groups alone cannot exhaustively characterize the physical
properties of magnetic geometry, unlike classification using conventional point groups
and magnetic point groups.
Despite the absence of PT symmetry, the spin little co-group at the general k-point 17 is &&& 1 , which strictly forbids the spin polarization throughout the whole Brillouin zone
[14, 16, 57, 58]. However, while the electronic band structure of CoNb₃S₆ remains spin
degenerate in the absence of SOC, it supports a nonzero SOC-free AHE since its OSSG
corresponds to the ferromagnetic effective magnetic point group 62′2′ (Fig. 6(c) and
6(d)). This AHE originates from the non-coplanar magnetic geometry, while such a
response is forbidden in collinear and coplanar AFMs due to the presence of effective
time reversal symmetry from the spin-only group. The coexistence of spin degeneracy
and the AHE indicates that the physical origins and symmetry requirements of these
two effects are distinct [3].
When SOC is included, the OSSG corresponds to the chiral MSG 𝑃32 0 , which
allows the emergence of a net magnetization along the [001] direction and reclassifies the magnetic phase as SOM. Furthermore, SOC breaks the spin-translation group &&& 1 ,
leading to the emergence of SOC-induced spin polarization as shown in Fig. 6(e). Specifically, the presence of the 3 % % operation dictates that, for the Γ-A and H-K paths, only the S z component is permitted. In contrast, on the k y = 0 plane, the presence of the 𝑇2 % symmetry allows for the S y and S z components, while at general k-points,
can be output by FINDSPINGROUP.
The FINDSPINGROUP platform provides a systematic computational framework
for symmetry analysis and the exploration of physical properties in unconventional
magnets. By implementing the OSSG framework, it treats magnetic geometry and SOC
effects within a single workflow and combines standardized structural conventions,
spin site-symmetry groups, spin Brillouin zones, and spin little co-groups. Based on the
OSSG framework, the platform enables automated tracking of symmetry-breaking
pathways from the non-relativistic to the relativistic limit, facilitating the symmetry
classification of magnetic phases and the disentanglement of physical origins for
emergent phenomena such as spin splitting and the AHE. This scheme provides a clear
physical picture for understanding the interplay between magnetic geometry and SOC.
Furthermore, the introduction of the SCIF format standardizes spin-group symmetry
analysis in a manner analogous to conventional crystal structure analysis, enhancing
the efficiency of high-throughput screening and design of unconventional magnets.
Overall, the FINDSPINGROUP platform offers physical navigation for the high-
throughput discovery of unconventional magnets, facilitating the future development
of spintronic devices.
From the input magnetic structure, we determine a magnetic primitive cell and the
crystallographic operations preserving it using spglib [59]. The site moments are then
used to classify the magnetic configuration as collinear, coplanar, or non-coplanar and
to determine the admissible spin-space operations. The identified symmetry is the set
of all operations {𝑔 ! 6|𝑔 " |τ7 that leave the magnetic structure invariant, with action (1).
Classification is performed on the equivalence class of this operation set under 19 orientation-preserving affine changes of real-space coordinates, together with proper rotations of spin space. Writing an affine transformation as (𝐴, 𝑎) ∈ Aff + (3) and a spin-space rotation as 𝑄 ∈ SO(3) , two SSG descriptions are equivalent if their
operations are related by conjugation,
This removes basis, origin, and spin-frame choices while preserving the magnetic
symmetry.
The nomenclature is assigned only after reduction to the canonical database
representative of this equivalence class at database.findspingroup.com. Thus, the
reported label identifies an equivalence class rather than a raw input-dependent
realization. For certain coplanar families, especially those associated with nontrivial
spin-space point group 2 or 222, the coarse suffix P does not uniquely specify the full group 𝐺 '' = 𝐺 (' × 𝐺 ') . Although the nontrivial part 𝐺 (' is fixed, extending it by the coplanar spin-only part 𝐺 ') can yield, under the equivalence defined above, up to three inequivalent full-group representatives. These refined coplanar representatives
are distinguished in the database by the suffixes P1, P2, and P3.
Spin Wyckoff positions are obtained as site orbits under the OSSG. For a
representative site, the associated spin site-symmetry group is the subgroup of OSSG
operations that leaves the site invariant, and it determines the symmetry-allowed
moment components at that site. This fixes the splitting chain from the parent space
group to the OSSG and then to the corresponding MSG in one convention.
The spin Brillouin zone is constructed from the effective reciprocal-space action of the OSSG. For an OSSG operation {𝑔 ! ||𝑔 " |𝜏}, the induced action on a wave vector is
modulo reciprocal lattice vectors. The resulting effective k-space operations are
reduced to an arithmetic crystal class, which fixes the spin Brillouin-zone type, the
standardized primitive transformation and the tabulated high-symmetry k points. For a 20 given wave vector k, the spin little group is obtained from the subset of OSSG
operations that preserves k under the same action, modulo reciprocal lattice translations.
The corresponding spin little co-group is defined by removing the translational part of
this little group. In this way, the spin Brillouin zone, its labeled high-symmetry points,
and the associated spin little co-groups and magnetic little co-groups are determined in
the same oriented convention as the real-space symmetry.
To record information on magnetic materials and their OSSG symmetries, we
introduce the standardized .scif (spin crystallographic information file) format for
depositing and disseminating SSG-related data. Similar to .cif and .mcif, the .scif file
records the spin space group assignment together with the underlying crystallographic
structure of a material, while allowing symmetry information to be stored and
transformed across different conventions and settings. It further provides symmetry-
resolved information required for subsequent spin-group-based property calculations,
which can be directly accessed by tools such as STensor [33] for tensor-symmetry
analysis. In addition, .scif enables direct visualization of SSG symmetry with
compatible tools such as Jmol [41], facilitating intuitive inspection and verification.
The default tolerance parameters of the current implementation are used throughout: space_tol = 0.02 Å, mtol = 0.02 μ B , meigtol = 2 × 10 -5 , matrix_tol = 0.01,
and parser_atol = 0.02. Here, space_tol is the real-space distance tolerance used for
crystallographic symmetry matching, space-group determination, and standard-setting
construction; mtol is the magnetic-moment tolerance used for moment matching and
for classifying the magnetic configuration as collinear, coplanar, or non-coplanar;
meigtol is the eigenvalue tolerance used in point-group candidate construction;
matrix_tol is the tolerance used to compare operation matrices during standardization
and equivalence reduction; and parser_atol is the absolute tolerance used in SCIF
expansion to check whether symmetry-generated moments assigned to the same
crystallographic site remain numerically consistent.
21
All DFT calculations were performed using the projector augmented-wave method
as implemented in the Vienna ab initio simulation package [60, 61]. The generalized
gradient approximation of Perdew-Burke-Ernzerhof-type exchange-correlation
potential [62] was adopted. A plane-wave cutoff energy of 520 eV and an electronic
convergence criterion of 10 −7 eV were used. The self-consistent calculations were
performed with a Γ-centered 15×15×1 k-mesh, and the spin texture was calculated with
FINDSPINGROUP is an open-source platform, with the source code available at
https://github.com/LiuQH-lab/FindSpinGroup.
This work was supported by National Natural Science Foundation of China under Grant
Nos. 12525410, 12274194, 12574275 and 12534003, Guangdong Provincial Quantum
Science Strategic Initiative under Grant No. GDZX2401002, Guangdong Provincial
Key Laboratory for Computational Science and Material Design under Grant No.
2019B030301001, Shenzhen Science and Technology Program (Grant No.
RCJC20221008092722009 and No. 20231117091158001), the Innovative Team of
General Higher Educational Institutes in Guangdong Province (Grant No.
2020KCXTD001), the Open Fund of the State Key Laboratory of Spintronics Devices
and Technologies (Grant No. SPL-2407) and Center for Computational Science and
Engineering of Southern University of Science and Technology.
Q.L. conceived the project. Y.Y. and X.C. developed the theoretical framework and core
codebase and built the database. Y.Z. developed the spin-space-group-labeling code.
22 Y.H.L. calculated and curated the spin-Wyckoff-position data. R.X. calculated and
curated the spin-space-group operations data. J.L. provided theoretical guidance on
higher-order nonlinear tensors. Y.T.L. provided guidance on magnetic classification.
Y.Y., X.C., and Q.L. wrote the manuscript. All authors discussed the results and
approved the manuscript.
Conflict of interest statement. None declared.
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