Pairing Instabilities at the Edge and in the Bulk of a Topological Insulator*

The gapless edge modes of a topological insulator are hallmarks of its bulk topological nature. The fate of these edge modes in presence of interactions driving symmetry breaking is not well understood. Motivated by this, we study pairing instability of Kane-Mele model on a strip geometry with short-ranged attractive interaction within the Bogoliubov-de Gennes framework. We explore how the instabilities at the edge and the bulk are linked. In particular, we find that when the superconducting transition at the edge precedes the bulk transition, the edge modes may be gapped even when the bulk is topologically non-trivial. Additionally, we find that the pairing instability is strongly affected by the type of the edge (zigzag or armchair). Notably, the edge mode on an armchair edge is surprisingly robust against pairing. We also discuss the difference between the paired states from on-site and nearest neighbor interactions on both the edge and the bulk.