Time-Reversal Soliton Pair in Two-Dimensional Topological Insulating Systems

Solitons on the edges of two-dimensional systems with non-trivial topology, formed through the one-dimensional mass-kink mechanism, play an important role in driving the emergence of higher-order topological phases. In this connection, the existing work has focused on gaping a single edge Dirac cone by time-reversal symmetry breaking perturbations, which are not suitable for the edge solitons in time-reversal symmetric systems with multiple edge Dirac cones. Here, we discuss the mass-kink mechanism in systems where time-reversal symmetric perturbations open the gaps of time-reversal related Dirac cones. We thus explain the appearance of pairwise corner modes and predict the value of the corner charges. Furthermore, we have developed an efficient-numerical method based on real-space renormalization group using Green's functions to calculate the phase difference of the mass terms between the adjacent edges without using nano-disks and k·p modeling. Using this technique, we demonstrate that the in-gap corner modes and the corner charges in monolayer α-Sb are generated by the mass-kink mechanism that originates from gapping two pairs of edge Dirac cones with Sz-mixing spin-orbit coupling.