Topologically robust defect states in two-fold $\mathcal{PT}$-symmetric systems

The emergence of topologically robust states has been understood in terms of the bulk-edge correspondence; they appear at the boundary between two insulating systems whose topology cannot be continuously deformed into another. However, is its converse true? Here, we find that at the boundary where two topologically trivial insulating system meets, a robust localized state can appear. We provide a new topological identification where two-fold $\mathcal{PT}$-symmetry of the system protects the emergence of such a state against continuous deformations. Moreover, we show the localization length of the topological defect state is insensitive to a wide range of parameters, and the defect states in the known systems fall into our topological identification.