使用基于散射理论的降维方案，我们表明，拓扑绝缘体和反射对称的超导体的分类表可以组织在两个周期2和四个周期8周期，类似于没有空间对称性的拓扑绝缘体和超导体的Bott周期性。借助于降维方案，可以从一维分类得到任意维分类$d \ge 1$.为此，我们提出了一种基于相对同伦群和正合序列的推导方法，对具有反射对称性的一维绝缘体和超导体进行分类.由此产生的分类与最近由Shiozaki和Sato [Phys.\修订版本\ B { \bf 90}，165114（2014）]。使用散射矩阵启发的方法，使我们能够解决第二个后裔$\bZ_2$阶段，其中拓扑非平凡的阶段是以前报道的易受扰动，打破平移对称性。
英文摘要：
Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d \ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant $\bZ_2$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.
Orbitals of Artificial Atoms in a Gapped Two-Dimensional Vacuum
Peter Grünberg Institut (PGI-3), Forschungszentrum Jülich（于利希研究中心）；Jülich Aachen Research Alliance (JARA)（亚琛于利希研究联盟）；RWTH Aachen University（亚琛工业大学）；National University of Singapore（新加坡国立大学）
Holographic Representations of Topological Quantum Criticality: Emergent Symmetry Approach around the Bott Clock
Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics, Southeast University（东南大学物理学院教育部量子材料与器件重点实验室）；Department of Physics and Astronomy, University of British Columbia（不列颠哥伦比亚大学物理与天文系）
Commuting-projector Hamiltonians for 2D topological insulators: edge physics and many-body invariants
受最近构造的二维（2D）时间反演不变拓扑超导体的投影哈密顿量的启发[Wang等人，Phys. Rev. B 98，094502（2018）]，我们引入了一个描述相互作用但精确可解的2D拓扑绝缘体的投影模型。我们明确地表明，我们的模型的带隙和无隙的边界是一致的带理论，弱相互作用的拓扑绝缘体。有趣的是，在某些晶格上，我们的时间反演对称模型也享有$\mathcal{CP}$对称性，从而直观地解释了$\mathcal{CP}$对称的拓扑绝缘体的体积不变量放在克莱因瓶系统。我们还简要地讨论了如何这些多体不变量可能能够表征模型与时间反演对称性。
