Enumeration of spin-space groups: Towards a complete description of symmetries of magnetic orders

Abstract

Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo-) vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spacial and spin rotations. SSGs gives a complete symmetry description of magnetic structures, and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling. Altermagnetism, a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has non-relativistic spin splitting, is well described by SSGs. Due to the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower dimensional real representations. As the number of SSGs is infinite, we only consider SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157,289 non-coplanar, 24,788 coplanar-non-collinear, and 1,421 collinear SSGs. The enumerated SSGs are stored in an online database at \url{https://cmpdc.iphy.ac.cn/ssg} with a user-friendly interface. We also develop an algorithm to identify SSG for realistic materials and find SSGs for 1,626 magnetic materials. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials.

arXiv commentThe manuscript is revised according to referees' reports

References in corpus (70)

• Topological Insulators in Three Dimensions
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Insulators with Inversion Symmetry
• Dirac semimetal in three dimensions
• Topological quantum chemistry
• Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals
• The (High Quality) Topological Materials In The World
• Altermagnetism: spin-momentum locked phase protected by non-relativistic symmetries
• Topological nodal line semimetals with and without spin-orbital coupling
• Catalogue of Topological Electronic Materials
• Crystal Hall effect in Collinear Antiferromagnets
• Anomalous Hall effect arising from noncollinear antiferromagnetism
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Antiferromagnetic topological insulators
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Observation of the Anomalous Hall Effect in a Collinear Antiferromagnet
• Efficient Electrical Spin-Splitter Based on Non-Relativistic Collinear Antiferromagnetism
• Momentum-Dependent Spin Splitting by Collinear Antiferromagnetic Ordering
• Progress and prospects in magnetic topological materials
• Observation of spin splitting torque in a collinear antiferromagnet RuO2
• High-throughput calculations of magnetic topological materials
• Tilted spin current generated by the collinear antiferromagnet RuO2
• Spontaneous anomalous Hall effect arising from an unconventional compensated magnetic phase in a semiconductor
• Observation of spin-splitter torque in collinear antiferromagnetic RuO${}_2$
• Cluster multipole theory for anomalous Hall effect in antiferromagnets
• Bottom-up design of spin-split and reshaped electronic band structures in spin-orbit-coupling free antiferromagnets: Procedure on the basis of augmented multipoles
• Strong, anisotropic anomalous Hall effect and spin Hall effect in chiral antiferromagnetic compounds Mn${}_{3}X$ ($X$ = Ge, Sn, Ga, Ir, Rh and Pt)
• Orbital ferromagnetism and anomalous Hall effect in antiferromagnets on distorted fcc lattice
• Quantitative mappings between symmetry and topology in solids
• Dirac Fermions in Antiferromagnetic Semimetal
• Magnetic Topological Quantum Chemistry
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling
• Observation of plaid-like spin splitting in a noncoplanar antiferromagnet
• Spin-group symmetry in magnetic materials with negligible spin-orbit coupling
• Enumeration and representation theory of spin space groups
• X-ray Magnetic Circular Dichroism in Altermagnetic $\alpha$-MnTe
• Andreev reflection at the altermagnet-superconductor interface
• Spin Space Groups: Full Classification and Applications
• Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups
• Dirac semimetal in type IV magnetic space group
• Band Representations and Topological Quantum Chemistry
• Quantum Anomalous Hall Effect in Antiferromagnetism
• Topological Insulators with Commensurate Antiferromagnetism
• Diagnosis for topological semimetals in the absence of spin-orbital coupling
• Topological correspondence between magnetic space group representations
• Magnetic superspace groups and symmetry constraints in incommensurate magnetic phases
• Multifold nodal points in magnetic materials
• Systematic investigation of emergent particles in type-III magnetic space groups
• Semimetal with both Rarita-Schwinger-Weyl and Weyl excitations
• Rarita-Schwinger-Weyl semimetal in Jeff=3/2 electron systems
• Encyclopedia of emergent particles in type-IV magnetic space groups
• Exhaustive constructions of effective models in 1651 magnetic space groups
• Topological classification and diagnosis in magnetically ordered electronic materials
• Real-space recipes for general topological crystalline states
• Complete classification of band nodal structures
• Gradual pressure-induced change in the magnetic structure of the non-collinear antiferromagnet Mn${}_3$Ge
• Dynamic paramagnon-polarons in altermagnets
• Chiral Dirac-like fermion in spin-orbit-free antiferromagnetic semimetals
• Symmetry-protected Nodal Points and Nodal Lines in Magnetic Materials
• Chiral Dirac fermion in a collinear antiferromagnet
• MSGCorep: A package for corepresentations of magnetic space groups
• Triple nodal points characterized by their nodal-line structure in all magnetic space groups
• Irreducible Projective Representations and Their Physical Applications
• Hund nodal line semimetals: The case of twisted magnetic phase in the double-exchange model
• A $k\cdot p$ effective Hamiltonian generator
• A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the $k\cdot p$ Perturbation for Anti-unitary Symmetry Groups
• All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups

Cited by in corpus (71)

• Antiferroelectric Altermagnets: Antiferroelectricity Alters Magnets
• Enumeration and representation theory of spin space groups
• Ferroelectric switchable altermagnetism
• Three-dimensional mapping of the altermagnetic spin splitting in CrSb
• Spin Space Groups: Full Classification and Applications
• Catalog of Unconventional Magnons in Collinear Magnets
• Two-Dimensional Ferroelectric Altermagnets: From Model to Material Realization
• AI-accelerated Discovery of Altermagnetic Materials
• Spontaneous Anomalous Hall Effect in Two-Dimensional Altermagnets
• Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling
• Elasto-Hall conductivity and the anomalous Hall effect in altermagnets
• Odd-parity Magnetism Driven by Antiferromagnetic Exchange
• The Crystallographic Spin Point Groups and their Representations
• G-type Antiferromagnetic BiFeO${}_3$ is a Multiferroic $g$-wave Altermagnet
• Nonlinear Superconducting Magnetoelectric Effect
• Topological responses from gapped Weyl points in 2D altermagnets
• Design of altermagnetic models from spin clusters
• Magnetic geometry induced quantum geometry and nonlinear transports
• Fragile Unconventional Magnetism in RuO${}_2$ by Proximity to Landau-Pomeranchuk Instability
• Two-dimensional Dual-Switchable Ferroelectric Altermagnets: Altering Electrons and Magnons
• X-type stacking in cross-chain antiferromagnets
• Tuning of altermagnetism by strain
• Quantum transport theory for unconventional magnets: Interplay of altermagnetism and p-wave magnetism with superconductivity
• Multiferroic collinear antiferromagnet with hidden altermagnetic split
• FeGe as a building block for the kagome 1:1, 1:6:6, and 1:3:5 families: hidden d-orbital decoupling of flat band sectors, effective models and interaction Hamiltonians
• Anisotropic spin-polarized conductivity in collinear altermagnets
• Gapped nodal planes drive a large topological Nernst effect in a chiral lattice antiferromagnet
• Multiferroic nematic d-wave altermagnetism driven by orbital-order on the honeycomb lattice
• Pseudo-Ising superconductivity induced by $p$-wave magnetism
• Field-sensitive dislocation bound states in two-dimensional $d$-wave altermagnets
• The electronic structures, magnetic transition and Fermi surface instability of room-temperature altermagnet KV${}_2$Se${}_2$O
• Constructions and Applications of Irreducible Representations of Spin-Space Groups
• Light-induced Odd-parity Magnetism in Conventional Collinear Antiferromagnets
• Light-induced odd-parity altermagnets on dimerized lattices
• Direct ab initio calculation of magnons in altermagnets: method, spin-space symmetry aspects, and application to MnTe
• Spin-Selective Second-Order Topological Insulators Enabling Cornertronics in 2D Altermagnets
• Symmetry Classification for Alternating Excitons in Two-Dimensional Altermagnets
• Superconducting Order Parameters in Spin Space Groups: Methodology and Application
• Anisotropic Piezomagnetism in Noncollinear Antiferromagnets
• Tunable altermagnetism via inter-chain engineering in parallelassembled atomic chains
• Real Chern insulators in two-dimensional altermagnetic Fe${}_2$S${}_2$O and Fe${}_2$Se${}_2$O
• Altermagnetic Weyl node-network semimetals protected by spin symmetry
• Valley-dependent electronic properties in two-dimensional altermagnetic iron-based transition metal chalcogenides
• Elastic Quantum Criticality in Nematics and Altermagnets via the Elasto-Caloric Effect
• Magnetic-field-tuned randomness in inhomogeneous altermagnets
• Anisotropic spin dynamics in antiferromagnets with a nonrelativistic spin splitting
• Unlocking Doping Effects on Altermagnetism in MnTe: Emergence of Quasi-altermagnetism
• EuAuSb: An odd-parity helical variation on altermagnetism
• Two-Terminal Electrical Detection of the Néel Vector via Longitudinal Antiferromagnetic Nonreciprocal Transport
• $d$-Wave Polarization-Spin Locking in Two-Dimensional Altermagnets
• Dirac fermions in the altermagnet Ce4Sb3
• Odd-Parity Magnetism in Fe-Based Superconductors
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Topological Anderson insulators by latent symmetry
• Field-driven band asymmetry and non-reciprocal transport in a helimagnet
• Finite-momentum mixed singlet-triplet pairing in chiral antiferromagnets induced by even-parity spin texture
• Quantum spin models of commensurate $p$-wave magnets
• Enhancement of Josephson Supercurrent in a $\pi$-Junction state by Chiral Antiferromagnetism
• Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation
• Orbital altermagnetism on the kagome lattice and possible application to $A$V${}_3$Sb${}_5$
• Non-Hermitian higher-order topological insulators enabled by altermagnet engineering
• Nonrelativistic Functional Properties in Collinear Antiferromagnets Based on Multipole Representation Theory
• Effect of symmetry breaking on altermagnetism in CrSb and Formation of fragmented nodal curves
• Spontaneous emergence of altermagnetism in the single-orbital extended Hubbard model
• Altermagnetism in Folded Iron Chain Systems
• Nonlinear opto-magnetic signature of d-wave altermagnets
• Two-Dimensional Magnetic Materials from Ferromagnetism to Altermagnetism: Fabrication, Symmetry, Evidence, and Device Readiness
• Non-self-averaging topological Anderson insulator
• Competitive Orders in Altermagnetic Chiral Magnons
• Revealing Spin and Spatial Symmetry Decoupling: New Insights into Magnetic Systems with Dzyaloshinskii-Moriya Interaction
• Magnon Thermal Hall Effect Induced By Symmetric Exchange Interaction