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.
Categories: cond-mat.mes-hall cond-mat.dis-nn cond-mat.mtrl-sci cond-mat.str-el