Quantum mechanics

Theoretical physics

Spin (aerodynamics)

Metallurgy

Materials science

Coupling (piping)

Symmetry group

Point group

Thermodynamics

Physics

Symmetry operation

Mathematics

Homogeneous space

Condensed matter physics

Geometry

Rotational symmetry

Symmetry (geometry)

Mechanics

Spin–orbit interaction

Group (periodic table)

Physical Review X

Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed-matter systems. Here, by combining symmetry analysis and model calculations, we elucidate that the crystallographic symmetry groups of a vast number of magnetic materials with light elements, in which the neglect of relativistic spin-orbit coupling (SOC) is an appropriate approximation, are considerably larger than the conventional magnetic groups. Thus, a symmetry description that involves partially decoupled spin and spatial rotations, dubbed spin group, is required. We derive the classifications of spin point groups describing coplanar and collinear magnetic structures, and the irreducible corepresentations of spin space groups illustrating more energy degeneracies that are disallowed by magnetic groups. One consequence of the spin group is the new antiunitary symmetries that protect SOC-free Z2 topological phases with unprecedented surface-node structures. Our work not only manifests the physical reality of materials with weak SOC, but also sheds light on the understanding of all solids with and without SOC by a unified group theory.Received 21 May 2021Revised 11 January 2022Accepted 18 February 2022DOI:https://doi.org/10.1103/PhysRevX.12.021016Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological phases of matterPhysical SystemsTopological materialsTechniquesDensity functional calculationsGroup theoryCondensed Matter, Materials & Applied Physics

Spin-Group Symmetry in Magnetic Materials with Negligible Spin-Orbit Coupling