自旋空间群的射影表示理论

中文摘要 AI 辅助

自旋空间群为具有弱自旋轨道耦合的磁性晶体提供了自然的对称性框架，但提取其物理后果需要一般性的不可约表示理论。核心困难在于自旋空间群是射影表示的：其因子系统可使晶格平移非对易，在动量空间中诱导非对称形态作用，并修改小余群的射影结构。这些效应超出了常规空间群和双群表示理论的范畴。在此，我们利用Mackey群扩张理论，为所有共线、共面和非共面自旋空间群（包括反幺正对称性和一般因子系统）发展了一个统一的构造性框架。一个核心技术结果是将相关因子系统$\nu$规范分解为平移因子$\sigma$、将平移与点群操作耦合的混合因子$\gamma$以及点群因子$\alpha$。该分解清晰展示了普通表示理论如何被修改：$\sigma$决定射影平移代数和适当的布里渊区，$\gamma$控制动量空间群作用并可能使其非对称形态，$\alpha$对小余群的因子系统有贡献。在此基础上，我们通过动量空间轨道上的诱导构造了所有射影不可约余表示。该框架识别了自旋空间群可实现哪些非对称形态动量空间对称性，并揭示了为紧致平坦流形而非环面的布里渊空间、对称性强制的Zak相位、重构的高对称动量和能带简并，以及新型准粒子。我们的结果为弱自旋轨道耦合磁性材料的系统研究奠定了表示论基础。

英文摘要

Spin space groups provide the natural symmetry framework for magnetic crystals with weak spin-orbit coupling, but extracting their physical consequences requires a general theory of irreducible representations. The central difficulty is that spin space groups are represented projectively: their factor systems can render lattice translations noncommuting, induce nonsymmorphic actions in momentum space, and modify the projective structure of little cogroups. These effects lie beyond conventional space-group and double-group representation theory. Here, using Mackey's theory of group extensions, we develop a unified constructive framework for all collinear, coplanar, and noncoplanar spin space groups, including antiunitary symmetries and general factor systems. A central technical result is a canonical decomposition of the relevant factor system $ν$ into a translational factor $σ$, a mixed factor $γ$ coupling translations to point-group operations, and a point-group factor $α$. This decomposition makes transparent how ordinary representation theory is modified: $σ$ determines the projective translation algebra and the appropriate Brillouin zone, $γ$ controls the momentum-space group action and can make it nonsymmorphic, and $α$ contributes to the factor systems of little cogroups. On this basis, we construct all projective irreducible corepresentations by induction over momentum-space orbits. The framework identifies which nonsymmorphic momentum-space symmetries can be realized by spin space groups and reveals Brillouin spaces that are compact flat manifolds rather than tori, symmetry-enforced Zak phases, reconstructed high-symmetry momenta and band degeneracies, and new types of quasiparticles. Our results establish the representation-theoretic foundation for systematic studies of weak-spin-orbit-coupled magnetic materials.

发表机构

• Lanzhou University of Technology（兰州理工大学）
• The University of Hong Kong（香港大学）

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29 pages