Spin-charge separation and spin-Chern number of disordered, higher-order topological insulators

According to the ten-fold classification scheme of topological insulators, only the time-reversal-symmetric, spin-orbit-coupled systems (class AII) can support stable, quantum spin Hall states at two spatial dimensions. In this work, we employ magnetic flux tube as a real-space probe to show that all three Wigner Dyson symmetry classes (A, AI, and AII) can display stable, quantum spin (or pseudo-spin) Hall insulators in the presence of disorder. This goal is accomplished by demonstrating spin-charge separation in the presence of flux tube. Our analysis is a manifestly gauge invariant method for addressing the magnitude of spin Chern number of first order and higher order topological insulators.