The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting $C_4$-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas’s theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.

中文翻译：

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高阶对称保护拓扑相中的 Thouless 泵和体边界对应

体边界对应关系将量化边缘状态与物质拓扑相中的体拓扑不变量联系起来。在一维对称保护拓扑系统中，量化拓扑 Thouless 泵直接揭示了这一原理并提供了良好的数学基础。受对称保护的高阶物质拓扑相（HOSPT）也具有体边界对应，但它与量子化电荷传输的联系仍然难以捉摸。在这里，我们展示了量化的 Thouless 泵连接$C_4$- 对称 HOSPT 可以通过四个陈数的元组来描述，这些元组以与方向相关的方式测量量化的体电荷传输。此外，这个陈数元组允许预测 HOSPT 中分数角电荷的符号和值。我们表明，拓扑非平凡相可以通过四极和偶极配置来表征，从而为当前关于 HOSPT 体的多极性质的争论提供了新的思路。通过采用角周期边界条件，我们将 Restas 的理论推广到 HOSPT。我们的方法为理解一般 HOSPT 的拓扑不变量提供了一个简单的框架，并为深入描述未来的动力学实验铺平了道路。