具有P和T对称性的完整拓扑分类：揭示K理论不可见的拓扑不变量
Complete Topological Classification with P and T Symmetries: Revealing a Topological Invariant Invisible to K-Theory
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- Department of Physics and HK Institute of Quantum Science & Technology, The University of Hong Kong（香港大学物理系及量子科学及技术研究所）
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中文总结 AI 辅助
本文针对同时具有P和T对称性的无自旋系统，通过第一性原理分析完整拓扑分类，揭示了K理论不可见的新拓扑不变量，证明存在K理论分类缺失的晶体拓扑相。
中文摘要 AI 辅助
K理论框架为十重对称性类提供了完整的拓扑分类，并已推广至包含晶体对称性的情形。本文表明，该分类在同时具有P（宇称）和T（时间反演）对称性的无自旋系统这一基础情形下仍不完备。我们通过对P和T对称能带的拓扑类进行第一性原理分析，即对相应的对称 clutching 数据进行分类，得到了完整分类。我们识别出一种拓扑不变量，当每个反演不变动量处的占据态具有一致的正宇称或一致的负宇称时，该不变量在K理论视角下不可见。在特殊情形下，例如所有反演不变动量都具有一致的正宇称时，该不变量可被解释为第二斯蒂费尔-惠特尼类，或定义在反演基本域（即布里渊区的一半）上的欧拉数。本工作不仅揭示了一种无法由P和T对称性下反演不变动量处的宇称谱确定的新拓扑不变量，还证明了存在K理论分类中缺失的晶体拓扑相。
英文摘要
The K-theoretic framework provides a complete topological classification of the tenfold symmetry classes and has been generalized to incorporate crystalline symmetries. Here, we show that this classification is incomplete even in the elementary case of spinless systems possessing both P and T symmetries. We obtain the complete classification through a first-principles analysis of the topological classes of P- and T-symmetric bands, namely, by classifying the corresponding symmetric clutching data. We identify a topological invariant that is invisible from the K-theoretic perspective when the occupied states at each inversion-invariant momentum have uniformly positive or uniformly negative parity. In special cases, such as when all inversion-invariant momenta have uniformly positive parity, this invariant can be interpreted as the second Stiefel--Whitney class or the Euler number defined over an inversion fundamental domain, namely, half of the Brillouin zone. Our work not only reveals a new topological invariant that cannot be determined from the parity spectra at inversion-invariant momenta under P and T symmetries, but also demonstrates the existence of crystalline topological phases that are absent from the K-theoretic classification.
