Real-space screening of quantum spin-Hall insulators

It was shown by Prodan in 2009 that both magnetic and nonmagnetic two-dimensional insulators can support quantized spin-Chern number in the absence of spin-rotation symmetry as a bulk topological invariant, which is robust against impurity effects. Recent studies on higher-order and fragile topological insulators further demonstrate that the spin-Chern number can exist without gapless edge states. Therefore, the presence of generalized quantum spin-Hall states in real materials can be difficult to identify using only symmetry-based indicators and Wannier obstruction. Such phases require more sophisticated probes of bulk topology. Magnetic flux tube has emerged as one such singular real-space probe that leads to spin-charge separation and allows for a precise diagnosis of the spin-Chern number. In this paper, we develop an automated workflow to scan the database of experimentally relevant, large band-gap, two-dimensional insulators using magnetic flux tubes. The results reveal many material candidates of higher-order topological insulators possessing double spin-Chern number, including the 1H-${MX}_{2}$ family of transition metal dichalcogenides. Our paper has broad implications for current efforts to employ these materials for new platforms of moir\'e systems.