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

Abstract

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-freeZ2topological 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.

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Plain Language Summary

Symmetry dictates how quantum-mechanical wave functions of elementary excitations behave, reflecting on all the physical observables. It is generally accepted that the symmetries of 3D solids are mathematically described by algebraic structures called crystallographic group theory. The theory successfully applies to a wide range of magnetic materials in relativistic quantum mechanics—known as magnetic groups—however, the symmetry description of magnetic materials in the nonrelativistic limit—known as spin groups—is seldom explored. Here, we build a bridge between this powerful but largely overlooked symmetry group and modern condensed-matter physics.

We pedagogically elucidate that more discrete or continuous spin rotations under a spatial rotation are permitted in spin groups compared with magnetic groups, where the spin and spatial rotations are completely locked to each other. We then, in the language of group theory, 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. These results directly give rise to our further discovery of emergent topological phases (states that are robust to perturbations) protected by new symmetries.

For future applications, the emergent topological classification is merely the tip of the iceberg, leaving fruitful physical properties induced by such an enhanced symmetry group to be further explored. Moreover, we provide a significantly expanded material pool containing candidates with both light elements and nontrivial band topology for material scientists to explore the novel effects of materials as well as potential applications in spintronics.