使用一个明确的1维模型，我们提供了直接的证据表明，一维拓扑相位从AIII和BDI对称类遵循$\mathbb Z$分类，即使在强无序制度时，费米能级嵌入在一个密集的本地化的频谱。我们分析的主要工具是缠绕数$ν$，在I.作者：J. L. Hughes和E.普罗丹，arXiv：1311.5233。对于这两个类，通过改变模型的参数和/或无序强度，级联的尖锐的拓扑转变$ν=0 \rightarrow ν=1 \rightarrow ν=2$生成，在该制度中，绝缘间隙完全填充的本地化频谱。我们证明了每个拓扑转变都伴随着一个安德森局域化-离域转变。此外，为了明确排除$\mathbb Z_2$分类，生成$ν=0$和$ν=2$之间的拓扑转换。这两个阶段也被发现是分开的安德森局域化-离域过渡，从而证明他们的独特身份。
英文摘要：
Using an explicit 1-dimensional model, we provide direct evidence that the one-dimensional topological phases from the AIII and BDI symmetry classes follow a $\mathbb Z$-classification, even in the strong disorder regime when the Fermi level is embedded in a dense localized spectrum. The main tool for our analysis is the winding number $ν$, in the non-commutative formulation introduced in I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan, arXiv:1311.5233. For both classes, by varying the parameters of the model and/or the disorder strength, a cascade of sharp topological transitions $ν=0 \rightarrow ν=1 \rightarrow ν=2$ is generated, in the regime where the insulating gap is completely filled with the localized spectrum. We demonstrate that each topological transition is accompanied by an Anderson localization-delocalization transition. Furthermore, to explicitly rule out a $\mathbb Z_2$ classification, a topological transition between $ν=0$ and $ν=2$ is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.
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No measurement induced phase transition in the entanglement dynamics of monitored non-interacting one-dimensional fermions in a disordered or quasiperiodic potential
Shanghai Center for Complex Physics, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院复杂物理中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院量子材料与技术研究所）；Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院凝聚态理论研究所）
Theory of Berry Singularity Markers: Diagnosing Topological Phase Transitions via Lock-In Tomography
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所理论物理重点实验室）
