We show that the $\pi$ flux and the dislocation represent topological observables that probe two-dimensional topological order through binding of the zero-energy modes. We analytically demonstrate that $\pi$ flux hosts a Kramers pair of zero modes in the topological $\Gamma$ (Berry phase Skyrmion at the zero momentum) and $M$ (Berry phase Skyrmion at a finite momentum) phases of the $M\mathrm{\text{−}}B$ model introduced for the HgTe quantum spin Hall insulator. Furthermore, we analytically show that the dislocation acts as a $\pi$ flux, but only so in the $M$ phase. Our numerical analysis confirms this through a Kramers pair of zero modes bound to a dislocation appearing in the $M$ phase only, and further demonstrates the robustness of the modes to disorder and the Rashba coupling. Finally, we conjecture that by studying the zero modes bound to dislocations all translationally distinguishable two-dimensional topological band insulators can be classified.

• Received 19 August 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.106403