Enumeration and Representation Theory of Spin Space Groups,Physical Review X

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Enumeration and Representation Theory of Spin Space Groups
Physical Review X ( IF 11.6 ) Pub Date : 2024-08-28 , DOI: 10.1103/physrevx.14.031038
Xiaobing Chen ₁ , Jun Ren ₁ , Yanzhou Zhu ₁ , Yutong Yu ₁ , Ao Zhang ₁ , Pengfei Liu ₁ , Jiayu Li ₁ , Yuntian Liu ₁ , Caiheng Li ₁ , Qihang Liu _{1,

1,

1}

Affiliation

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

R u O 2

, spiral spin polarization in the coplanar antiferromagnet

C e A u A l 3

, and geometric Hall effect in the noncoplanar antiferromagnet

C o N b 3 S 6

. 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.

中文翻译：

自旋空间群的枚举和表示理论

所有磁性有序材料的基本物理性质，如相变、电子结构和自旋激发，最终被认为都依赖于磁性空间群的对称理论。最近，人们发现，需要一个更全面的组，称为自旋空间群（SSG），它结合了单独的自旋和空间操作，以充分表征磁性有序材料的几何形状和基本特性。然而，SSG 的基本理论很少得到发展。在这项工作中，我们提出了对 SSG 的列举和表示理论的系统研究。从 230 个晶体空间群和有限平移群（最多为 8 个）开始，我们在四索引命名法和国际符号下建立了超过 100 000 个 SSG 的广泛集合。然后，我们确定了特别适用于共线、共面和非共面磁性配置的不等价 SSG。为了便于识别 SSG，我们开发了一个在线程序，可以确定任何磁性有序晶体的 SSG 对称性。此外，我们在 SSG 框架内推导出了小群在动量空间中的不可约共表示。最后，我们通过几个具有代表性的材料示例来说明磁性空间群框架之外的 SSG 对称性和物理效应，包括候选交替磁体