National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093, Nanjing, China.
Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore, 487372, Singapore.
Nat Commun. 2023 Feb 10;14(1):743. doi: 10.1038/s41467-023-36447-7.
A peculiar feature of quantum states is that they may embody so-called projective representations of symmetries rather than ordinary representations. Projective representations of space groups-the defining symmetry of crystals-remain largely unexplored. Despite recent advances in artificial crystals, whose intrinsic gauge structures necessarily require a projective description, a unified theory is yet to be established. Here, we establish such a unified theory by exhaustively classifying and representing all 458 projective symmetry algebras of time-reversal-invariant crystals from 17 wallpaper groups in two dimensions-189 of which are algebraically non-equivalent. We discover three physical signatures resulting from projective symmetry algebras, including the shift of high-symmetry momenta, an enforced nontrivial Zak phase, and a spinless eight-fold nodal point. Our work offers a theoretical foundation for the field of artificial crystals and opens the door to a wealth of topological states and phenomena beyond the existing paradigms.
量子态的一个奇特特征是，它们可能体现出所谓的对称的射影表示，而不是普通的表示。空间群的射影表示——晶体的定义对称性——在很大程度上仍未被探索。尽管最近在人工晶体方面取得了进展，其内在的规范结构必然需要射影描述，但尚未建立统一的理论。在这里，我们通过详尽地分类和表示二维 17 个壁群中 17 个时间反演不变晶体的所有 458 个射影对称代数，建立了这样一个统一的理论——其中 189 个在代数上是不等价的。我们发现了三个源于射影对称代数的物理特征，包括高对称动量的移动、强制的非平凡 Zak 相位和无自旋的八重节点。我们的工作为人工晶体领域提供了理论基础，并为超越现有范例的丰富拓扑态和现象开辟了道路。
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