The mechanism enabling antiferromagnets, including altermagnets, to exhibit a prominent anomalous Hall effect despite a vanishingly small net magnetization has long remained elusive. Here, by employing oriented spin group theory and spin-orbit-coupling tensor expansion, we systematically disentangle the perturbative behaviors of orbital and spin magnetizations with respect to spin-orbit coupling. Remarkably, we find that only if the opposite-spin sublattices are connected through a fourfold rotation, the orbital and spin magnetizations exhibit distinct perturbative orders. In these altermagnets, we further discover a coaxial Hall effect characterized by the induced spin and orbital magnetizations aligning parallel to the Néel vector, which we further demonstrate by first-principles calculations in the altermagnet KV_2Se_2O. This effect holds great promise for achieving deterministic switching of the Néel order under weak external fields. Our work provides a systematic symmetry approach to identify potential altermagnetic candidates combining a large anomalous Hall effect with minimal net magnetization, paving the way for high-performance, stray-field-free spintronic applications.

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.

Magnetism has seen substantial progress in recent decades, driven largely by its potential for next-generation storage devices. However, the classification of magnetic orders, even for fundamental concepts such as ferromagnetism (FM) and antiferromagnetism (AFM), remains a topic of active evolution, particularly with the discovery of unconventional magnetic materials and advances in antiferromagnetic spintronics1-4. Here we present a classification of magnetic order using the state-of-the-art spin space group (SSG) theory5-11. On the basis of whether the net spin magnetization is constrained to zero by the SSG framework, we systematically categorize magnetic orders into FM (including ferrimagnetism) and AFM. We further introduce an 'oriented spin space group' (OSSG) description, that is, a SSG with a fixed magnetic orientation, thereby unifying the SSG and magnetic space group (MSG)12-14 frameworks. This approach clearly reveals the symmetry-breaking pathway induced by spin-orbit coupling (SOC). On the basis of the proposed group framework, we identify a distinct magnetic phase, termed spin-orbit magnetism (SOM), in which the net spin magnetization is induced by SOC. Our work provides a comprehensive symmetry-based perspective for classifying magnetic order, offering fresh insights into unconventional magnets and broad applicability in spintronics and quantum materials design.

The interplay of symmetry and topology in crystal solids has given rise to various elementary excitations as quasiparticles. Among these, those with significant Berry-phase-related transport responses are of particular interest. In this study, we predict a type of quasiparticle called a topological charge quadrupole (TCQ), which is analogous to a charge quadrupole but consists of two closely packed pairs of Weyl points in momentum space, specifically in the half-Heusler antiferromagnet NdBiPt. Interestingly, the TCQ is protected by the spin-orbit U(1) quasi-symmetry rather than any exact crystallographic symmetries. This quasi-symmetry restricts the energy splitting induced by symmetry-lowering perturbations to a second-order effect. Furthermore, the closely located Berry curvature sources and sinks in the TCQ lead to a large Berry curvature dipole, resulting in a significant nonlinear Hall effect. Our work opens an avenue for designing unconventional quasiparticles utilizing quasi-symmetries and developing materials with enhanced nonlinear responses.

Magnonic systems provide a fertile playground for bosonic topology1, for example, Dirac2–6 and Weyl7,8 magnons, leading to a variety of exotic phenomena such as charge-free topologically protected boundary modes6,7, the magnon thermal Hall effect9 and the magnon spin Nernst effect10. However, their understanding has been hindered by the absence of fundamental symmetry descriptions of magnetic geometries and spin Hamiltonians primarily governed by isotropic Heisenberg interactions. The ensuing magnon dispersions enable gapless magnon band nodes that go beyond the scenario of representation theory of the magnetic space groups11,12, thus referred to as unconventional magnons. Here we developed spin space group13–17 theory to elucidate collinear magnetic configurations, classifying the 1,421 collinear spin space groups into 4 types, constructing their band representations and providing a comprehensive tabulation of unconventional magnons, such as duodecuple points, octuple nodal lines and charge-4 octuple points. On the basis of the MAGNDATA database18, we identified 498 collinear magnets with unconventional magnons, among which more than 200 magnon band structures were obtained by using first-principles calculations and linear spin wave theory. In addition, we evaluated the influence of the spin–orbit-coupling-induced exchange interaction in these magnets and found that more than 80 per cent are predominantly governed by the Heisenberg interactions, indicating that the spin space group serves as an ideal framework for describing magnon band nodes in most 3d, 4d and half-filled 4f collinear magnets. Spin space group theory is applied to identify more than 200 collinear magnets with unconventional magnons; high-throughput calculations with spin–orbit coupling find that most of these unconventional magnets are dictated by the Heisenberg exchange interaction.

We propose a novel ferroelectric switchable altermagnetism effect: the reversal of ferroelectric polarization is coupled to the switching of altermagnetic spin splitting. We demonstrate the design principles for the ferroelectric altermagnets and the additional symmetry constraints necessary for switching the spin splitting through flipping the electric polarization based on the state-of-the-art spin-group symmetry techniques. We find 22 ferroelectric altermagnets by screening through the 2001 experimental reported magnetic structures in the MAGNDATA database and identify two of them as ferroelectric switchable altermagnets. Using the hybrid improper ferroelectric material [C(NH_{2})_{3}] Cr(HCOO)_{3} as an example, we show how the altermagnetic spin splitting is tightly coupled to the ferroelectric polarization, providing an ideal platform for designing electric-field-controllable multiferroic devices. Finally, we find that such manipulation of altermagnetism can be detected by monitoring the physical quantities that are related to the nonvanishing Berry curvature dipole, such as the linearly polarized photogalvanic spin current.

Altermagnets, with spin splitting and vanishing magnetization, have been attributed to many fascinating phenomena and potential applications. In particular, integrating ferroelectricity with altermagnetism to enable magnetoelectric coupling and electric control of spin has drawn significant attention. However, its experimental realization and precise spin manipulation remain elusive. Here, by focusing on molecular ferroelectrics, the first discovered ferroelectrics renowned for their highly controllable molecular polarizations and structural flexibility, we reveal that these obstacles can be removed by an emergent multiferroic altermagnet with tunable spin polarization in a large class of fabricated organic materials. Using a symmetry-based design and a tight-binding model, we uncover the underlying mechanism of such molecular ferroelectric altermagnets and demonstrate how noncollinear molecular polarization can switch the spin polarization on and off and even reverse its sign, as detectable by the magneto-optical Kerr effect. From the first-principles calculations, we verify the feasibility of these materials in a series of well-established hybrid organic-inorganic perovskites and metal-organic frameworks. Our findings bridge molecular ferroelectrics and altermagnetic spintronics, highlighting an unexplored potential of multifunctional organic multiferroics.

Spintronic devices typically employ heterostructures with ferromagnets which break time-reversal symmetry and have non-vanishing magnetization. With the growing class of materials that support spin-polarized carriers, current, and excitations, it is possible to envision emerging spintronic applications that are not limited to magnetoresistance. Here, chiral perovskites with no net magnetization are considered, where the space-inversion and mirror symmetries are broken to induce chiral structure. The known importance of these perovskites is further expanded by the demonstration of the chiral-induced spin selectivity (CISS). However, the generation of the spin-polarized carriers across the interface with these chiral perovskites remains to be fully understood. These first-principles studies for 2D PbBr4-based chiral perovskites provide their electronic structure and an orbital-based symmetry analysis, which allows to establish an effective Hamiltonian to elucidate the underlying origin of their chirality. The same analysis is used for the Edelstein effect, responsible for electrical generation of the nonequilibrium spin polarization in many materials, which in chiral perovskites can be a mechanism contributing to CISS. Furthermore, by examining optical properties of chiral perovskites and the opportunity to use them to realize tunable altermagnets, another class of zero-magnetization spintronic materials, a versatile materials platform is put forth for unconventional spintronics.

The control of unconventional magnetism, which displays ferromagnetismlike properties with compensated magnetization, has drawn intense attention for advancing antiferromagnetic spintronics. Here, through symmetry analysis, we propose a general stacking rule, characterized by a connection operator linking two stacked bilayers, for controlling unconventional magnetism via sliding ferroelectricity. Such a rule enables the simultaneous switching of both electric polarization and nonrelativistic spin splitting or anomalous Hall effect in altermagnets, a class of collinear unconventional magnets. By comprehensively surveying the 80 layer groups, we identify all the stacking orders that allow for such two types of simultaneous switching. Furthermore, we extend the stacking rule to collinear compensated ferrimagnets, where the opposite-spin sublattices are not connected by any symmetry operator, yet the net magnetization remains zero. Combined with first-principles calculations, we demonstrate the sliding ferroelectric control of spin polarization and anomalous Hall effect in the altermagnetic AgF_{2} and Fe_{2}MoSe_{4} bilayers. Our Letter provides a symmetry strategy for achieving ferroelectric control of unconventional magnetism in bilayer systems and opens avenues for exploring new types of magnetoelectric coupling.

Excitons, bound electron-holes states, often dominate the optical response of two-dimensional (2D) materials and reflect their inherent properties, including spin-orbit coupling, magnetic ordering, or band topology. By focusing on a growing class of collinear antiferromagnets with a nonrelativistic spin splitting, referred to also as altermagnets (AM), we propose a theoretical framework based on the spin space group (SSG) to elucidate their resulting excitons. Our approach is illustrated on 2D AM with spin-polarized valleys, where we classify the combination of conduction and valence bands by the SSG representations into two cases that hosts bright s-like and p-like excitons, respectively. This analysis is further supported by effective Hamiltonians and the Bethe-Salpeter equation. We identify the excitonic optical selection rules from the calculated absorption spectra and the symmetry of bright excitons from their momentum space envelope functions. Our framework provides optical fingerprints for various cases of AM, while their tunability, such as the strain-induced valley splitting, is also transferred to excitons allowing, additionally, valley-polarized photocurrent generation.

Spin textures, i.e., the distribution of spin polarization vectors in reciprocal space, exhibit diverse patterns determined by symmetry constraints, resulting in a variety of spintronic phenomena. Here, we propose a universal theory to comprehensively describe the nature of spin textures by incorporating three symmetry flavors of reciprocal wavevector, atomic orbital and atomic site. Such approach enables us to establish a complete classification of spin textures constrained by the little co-group and predict unprecedentedly reported spin texture types, such as Zeeman-type spin splitting in antiferromagnets and quadratic spin texture. To illustrate the influence of atomic orbitals and sites on spin textures, we predict orbital-dependent spin texture and anisotropic spin-momentum-site locking effects, and corresponding material candidates validated through first-principles calculations. The comprehensive classification and the predicted new spin textures in realistic materials are expected to trigger future spin-based functionalities in electronics.

Spatial, momentum and energy separation of electronic spins in condensed matter systems guides the development of novel devices where spin-polarized current is generated and manipulated. Recent attention on a set of previously overlooked symmetry operations in magnetic materials leads to the emergence of a new type of spin splitting, enabling giant and momentum-dependent spin polarization of energy bands on selected antiferromagnets. Despite the ever-growing theoretical predictions, the direct spectroscopic proof of such spin splitting is still lacking. Here, we provide solid spectroscopic and computational evidence for the existence of such materials. In the noncoplanar antiferromagnet MnTe$_2$, the in-plane components of spin are found to be antisymmetric about the high-symmetry planes of the Brillouin zone, comprising a plaid-like spin texture in the antiferromagnetic (AFM) ground state. Such an unconventional spin pattern, further found to diminish at the high-temperature paramagnetic state, stems from the intrinsic AFM order instead of spin-orbit coupling (SOC). Our finding demonstrates a new type of quadratic spin texture induced by time-reversal breaking, placing AFM spintronics on a firm basis and paving the way for studying exotic quantum phenomena in related materials.

In solid state systems, group representation theory is powerful in characterizing the behavior of quasiparticles, notably the energy degeneracy. While conventional group theory is effective in answering yes-or-no questions related to symmetry breaking, its application to determining the magnitude of energy splitting resulting from symmetry lowering is limited. Here, we propose a theory on quasisymmetry and near degeneracy, thereby expanding the applicability of group theory to address questions regarding large-or-small energy splitting. Defined within the degenerate subspace of an unperturbed Hamiltonian, quasisymmetries form an enlarged symmetry group eliminating the first-order splitting. This framework ensures that the magnitude of splitting arises as a second-order effect of symmetry-lowering perturbations, such as external fields and spin-orbit coupling. We systematically tabulate the quasisymmetry groups within 32 crystallographic point groups and find all the possible unitary quasisymmetry group structures regarding double degeneracy. Applying our theory to the realistic material AgLa, we predict a "quasi-Dirac semimetal" phase characterized by two tiny-gap band anticrossings.

Quantum spin Hall insulators hosting edge spin currents hold great potential for low-power spintronic devices. In this paper, we present a universal approach to achieve a high and near-quantized spin Hall conductance plateau within a sizable bulk gap. Using a nonmagnetic four-band model Hamiltonian, we demonstrate that an even-spin Chern (ESC) insulator can be accessed by tuning the sign of spin-orbit coupling (SOC) within a crystal symmetry-enforced orbital doublet. With the assistance of a high-spin Chern number of CS = -2 and spin U (1) quasisymmetry, this orbital doublet driven ESC phase is endowed with the near double quantized spin Hall conductance. We identify 12 crystallographic point groups supporting such a sign-tunable SOC. Furthermore, we apply our theory to realistic examples, and show the phase transition from a trivial insulator governed by positive SOC in the RuI3 monolayer to an ESC insulator dominated by negative SOC in the RuBr3 monolayer. This orbital doublet driven ESC insulator, RuBr3, showcases nontrivial characteristics including helical edge states, near double quantized spin Hall conductance, and robust corner states. Our work provides different pathways in the pursuit of the long-sought quantum spin Hall insulators.

Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet RuO2, spiral spin polarization in the coplanar antiferromagnet CeAuAl3, and geometric Hall effect in the noncoplanar antiferromagnet CoNb3S6. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials.

Quantum spin Hall (QSH) effect, where electrons with opposite spin channels are deflected to opposite sides of a two-dimensional system with a quantized conductance, was believed to be characterized by a nontrivial topological index $Z_{2}$. However, spin mixing effects in realistic materials often lead to deviation of the spin Hall conductance from exact quantization. In this Letter, we present a universal symmetry indicator for diagnosing QSH effect in realistic materials, termed spin $U$(1) quasi-symmetry. Such a symmetry eliminates the first-order spin-mixing perturbation and thus protects the near-quantization of SHC, applicable to time-reversal-preserved cases with either $Z_{2}=1$ or $Z_{2}=0$, as well as time-reversal-broken scenarios. We propose that spin $U$(1) quasi-symmetry is hidden in the subspace spanned by the doublets with unquenched orbital momentum and emerges when SOC is present, which can be realized in 19 crystallographic point groups. Particularly, we identify a previous overlooked even spin Chern phase with a trivial $Z_{2}$ index as an ideal platform for achieving a near-double-quantized SHC, as exemplified by twisted bilayer transition metal dichalcogenides and monolayer RuBr$_{3}$. Our work offers a new perspective for understanding QSH effect and significantly expands the material pool for the screening of exemplary material candidates.

The ubiquitous Rashba spin splitting occurring around time-reversal-invariant momenta (TRIM) reflects the fundamental interplay between spin-orbit coupling (SOC) and crystalline symmetry. Unexpectedly, recent measurements reveal novel Rashba spin splitting around the momenta lacking time-reversal symmetry (non-TRIM), whose mechanism remains elusive. Here, we theoretically elucidate the microscopic origin of Rashba spin splitting around non-TRIM and identify the design principles for its appearance. Conventionally, a Zeeman-like spin splitting dominates around non-TRIM due to the broken time-reversal symmetry and diminishes the Rashba-like features. Our results show that the Bloch wave function with specific orbital representation at certain non-TRIM could eliminate the first-order SOC effect. As a result, the corresponding little group hosts a type of quasisymmetry that commutes with the dominating SOC Hamiltonian, leading to a tiny SOC gap and helical spin texture. Furthermore, such a quasi-symmetry-induced tiny gap could manifest large Berry curvature, which is desirable for various topological applications. Our work expands the concept of Rashba physics beyond the conventional TRIM scenario, paving an avenue for designing novel spin-splitting materials through the cooperation of symmetry and orbital engineering.

Topological magnons have received substantial interest for their potential in both fundamental research and device applications due to their exotic uncharged yet topologically protected boundary modes. However, their understanding has been impeded by the lack of fundamental symmetry descriptions of magnetic materials, of which the spin Hamiltonians are essentially determined by the isotropic Heisenberg interaction. The corresponding magnon band structures allows for more symmetry operations with separated spin and spatial operations, forming spin space groups (SSGs), than the conventional magnetic space groups. Here we developed spin space group (SSG) theory to describe collinear magnetic configurations, identifying all the 1421 collinear SSGs and categorizing them into four types, constructing band representations for these SSGs, and providing a full tabulation of SSGs with exotic nodal topology. Our representation theory perfectly explains the band degeneracies of previous experiments and identifies new magnons beyond magnetic space groups with topological charges, including duodecuple point, octuple nodal line and charge-4 octuple point. With an efficient algorithm that diagnoses topological magnons in collinear magnets, our work offers new pathways to exploring exotic phenomena of magnonic systems, with the potential to advance the next-generation spintronic devices.

The century-long development of surface sciences has witnessed the discoveries of a variety of quantum states. In the recently proposed “obstructed atomic insulators”, symmetric charges are pinned at virtual sites where no real atoms reside. The cleavage through these sites could lead to a set of obstructed surface states with partial electronic occupation. Here, utilizing scanning tunneling microscopy, angle-resolved photoemission spectroscopy and first-principles calculations, we observe spectroscopic signature of obstructed surface states in SrIn 2 P 2 . We find that a pair of surface states that are originated from the pristine obstructed surface states split in energy by a unique surface reconstruction. The upper branch is marked with a striking differential conductance peak followed by negative differential conductance, signaling its localized nature, while the lower branch is found to be highly dispersive. This pair of surface states is in consistency with our calculational results. Our finding not only demonstrates a surface quantum state induced by a new type of bulk-boundary correspondence, but also provides a platform for exploring efficient catalysts and related surface engineering.

Spin textures, i.e., the distribution of spin polarization vectors in reciprocal space, exhibit diverse patterns determined by symmetry constraints, resulting in a variety of spintronic phenomena. Here, we propose a universal theory to comprehensively describe the nature of spin textures by incorporating three symmetry flavors of reciprocal wavevector, atomic orbital and atomic site. Such approach enables us to establish a complete classification of spin textures constrained by the little co-group and predict unprecedentedly reported spin texture types, such as Zeeman-type spin splitting in antiferromagnets and quadratic spin texture. To examine the impact of atomic orbitals and sites, we predict orbital-dependent spin texture and anisotropic spin-momentum-site locking effects and corresponding material candidates validated through first-principles calculations. Our theory not only covers all possible spin textures in crystal solids described by magnetic space groups, but also introduces new possibilities for designing innovative spin textures by the coupling of multiple degrees of freedom.