Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders,Physical Review X

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Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders
Physical Review X ( IF 11.6 ) Pub Date : 2024-08-28 , DOI: 10.1103/physrevx.14.031039
Yi Jiang _{1,

2,

3,

4} , Ziyin Song _{1,

2,

3} , Tiannian Zhu _{1,

2,

3} , Zhong Fang _{1,

2} , Hongming Weng _{1,

2,

5} , Zheng-Xin Liu ₆ , Jian Yang _{1,

2} , Chen Fang _{1,

2,

2,

5,

7}

Affiliation

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 spatial and spin rotations. SSGs give 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. , a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has nonrelativistic spin plitting, is well described by SSGs. Because of 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 consider only SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157 289 noncoplanar, 24 788 coplanar-noncollinear, and 1421 collinear SSGs. The enumerated SSGs are stored in an online database with a user-friendly interface. We develop an algorithm to identify SSGs for realistic materials and find SSGs for 1626 magnetic materials. We also discuss several potential applications of SSGs, including the representation theory, topological states protected by SSGs, structures of spin textures, and refinement of magnetic neutron diffraction patterns using SSGs. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials.

Published by the American Physical Society

2024

中文翻译：

自旋空间群的枚举：磁阶对称性的完整描述

三维周期标量场的对称性由 230 个空间群（SG）描述。然而，三维周期（伪）矢量场的对称性是由自旋空间群（SSG）描述的，SSG 最初用于描述磁序的对称性。在 SSG 中，实空间自由度和自旋自由度是解锁的，因为操作可以具有不同的空间和自旋旋转。SSG 给出了磁性结构的完整对称性描述，并在具有弱自旋轨道耦合的磁有序系统中行程电子的能带理论中具有自然应用。，一个最近提出的概念，属于对称补偿共线磁序，但具有非相对论性自旋波纹，SSG 对此进行了很好的描述。由于 SSG 数量众多且集团结构复杂，因此尚未系统地列举 SSG。在这项工作中，我们基于 SG 的不变子群穷尽了 SSG，自旋操作由不变子群的商群的三维（3D）真实表示构建。对于共线和共面磁阶次，自旋运算可以简化为低维实数表示。由于 SSG 的数量是无限的，因此我们只考虑描述磁性晶胞的 SSG，最高可达晶体晶胞的 12 倍。我们获得了 157 289 个非共面、24 788 个共面非共线和 1421 个共线 SSG。列举的 SSG 存储在具有用户友好界面的在线数据库中。我们开发了一种算法来识别真实材料的 SSG，并找到 1626 磁性材料的 SSG。我们还讨论了 SSG 的几种潜在应用，包括表示理论、SSG 保护的拓扑状态、自旋织构结构以及使用 SSG 对磁中子衍射图样的细化。我们的结果为进一步研究磁有序材料的对称性和拓扑结构提供了坚实的起点。美国物理学会 2024 年出版

更新日期：2024-08-28

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