Field theory arguments suggest the possibility of $\mathbb{Z}$-classification of quantum spin Hall effect with magnetic flux tubes, that cause separation of spin and charge degrees of freedom, and pumping of spin or Kramers pair. However, the \emph{proof of principle} demonstration of spin-charge separation is yet to be accomplished for realistic, \emph{ab initio} band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on $\beta$-bismuthene to demonstrate spin-charge separation, and quantized pumping of spin for three insulating states that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of topologically non-trivial bands, supporting even integer winding numbers, which cannot be inferred from symmetry-based indicators. Our work sets a new standard for prediction of two-dimensional, quantum spin-Hall materials, based on precise bulk invariant and universal topological response.

中文翻译：

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$β$-铋的自旋电荷分离和量子自旋霍尔效应

场论论证表明，使用磁通量管对量子自旋霍尔效应进行分类的可能性，这会导致自旋和电荷自由度的分离，以及自旋或克莱默对的泵浦。然而，对于缺乏自旋守恒定律的自旋轨道耦合材料的现实的 \emph {ab initio} 能带结构，自旋电荷分离的 \emph {原理证明} 演示尚未完成。在这项工作中，我们用磁通管在 $\beta$-铋上进行了思想实验，以证明自旋电荷分离，以及可以通过调整填充分数来访问的三种绝缘状态的自旋量子泵浦。通过对动量空间拓扑和实空间响应的综合分析，我们确定了拓扑非平凡带的重要作用，支持甚至整数绕组数，这不能从基于对称的指标中推断出来。我们的工作基于精确的体不变量和通用拓扑响应，为二维量子自旋霍尔材料的预测设立了新标准。