Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a “bridge” between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, $E=0$. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number $\nu =1$ as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrumwide quantum critical delocalization of all states versus full localization except at the $E=0$ critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.

中文翻译：

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非维格纳-戴森拓扑绝缘体表面态的脆弱性

拓扑绝缘体和超导体支持扩展的表面态，防止静电无序的局部效应。具体来说，在属于对称性 A、AI 和 AII 类的维格纳-戴森绝缘体中，扩展表面态带连续连接到同样扩展的体态组，通过谱机制在不同表面之间形成“桥梁”流动。在这项工作中，我们表明这种机制在大多数非维格纳-戴森拓扑超导体和手性拓扑绝缘体中不存在。在这些系统中，精确地有一个点具有授予的扩展状态，即能带的中心，$乙=0$。远离它，状态是空间局部化的，或者可以通过添加空间局部势来实现。考虑AIII级三维绝缘子和匝数$\nu =1$作为一个典型案例研究，我们讨论了这种现象背后的物理原理及其方法和应用后果。特别是，我们表明，表面态描述中的低能狄拉克近似可能是危险的，因为它们往往会掩盖可局域性现象。我们还确定了根据贝里曲率定义的标记作为晶格模型中状态局部化程度的度量，并通过广泛的数值模拟支持我们的分析预测。这项工作的一个主要结论是，非维格纳-戴森拓扑绝缘体的表面现象学比其维格纳-戴森兄弟的表面现象学丰富得多，极端限制是所有状态的全谱量子临界离域与除了在$乙=0$临界点。作为我们研究的一部分，我们确定了区分传输或隧道光谱中这些不同替代方案的可能的实验特征。