Boundary obstructed topological insulators are an unusual class of higher-order topological insulators with topological characteristics determined by the so-called Wannier bands. Boundary obstructed phases can harbor hinge/corner modes, but these modes can often be destabilized by a phase transition on the boundary instead of the bulk. While there has been much work on the stability of topological insulators in the presence disorder, the topology of a disordered Wannier band, and disorder-induced Wannier transitions have not been extensively studied. In this work, we focus on the simplest example of a Wannier topological insulator: a mirror-symmetric $\pi$-flux ladder in 1D. We find that the Wannier topology is robust to disorder, and derive a real-space renormalization group procedure to understand a new type of strong disorder-induced transition between non-trivial and trivial Wannier topological phases. We also establish a connection between the Wannier topology of the ladder and the energy band topology of a related system with a physical boundary cut, something which has generally been conjectured for clean models, but has not been studied in the presence of disorder.

中文翻译：

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π磁通梯中的Wannier带跃迁

边界受阻拓扑绝缘体是一类特殊的高阶拓扑绝缘体，其拓扑特征由所谓的Wannier谱带确定。边界受阻相可以包含铰链/角模式，但是这些模式通常可以通过边界而不是整体上的相变来稳定。尽管在存在障碍中拓扑绝缘体的稳定性方面有很多工作，但对无序Wannier谱带的拓扑结构以及由无序诱发的Wannier跃迁尚未进行深入研究。在这项工作中，我们将重点放在Wannier拓扑绝缘子的最简单示例上：镜像对称$\pi$-一维通量梯子。我们发现，Wannier拓扑对无序具有鲁棒性，并推导了实空间重归一化组过程，以了解新型的由强无序引起的非平凡和平凡的Wannier拓扑阶段之间的过渡。我们还建立了梯子的Wannier拓扑与具有物理边界切线的相关系统的能带拓扑之间的连接，通常已将其推测为干净的模型，但尚未进行无序研究。