Subsymmetry protected topology in topological insulators and superconductors

Abstract

Exploration of topology protected by a certain symmetry is central in condensed matter physics. A recent idea of subsymmetry protected (SSP) topology—remains of a broken symmetry can still protect specific topological boundary states—has been developed and demonstrated in a photonic system [Nat. Phys. 19, 992 (2023)1745-247310.1038/s41567-023-02011-9], but not in topological superconductors. Here, we extend this idea further by applying subsymmetry protecting perturbation (SSPP) to one-dimensional topological insulating and superconducting systems using the Su-Schrieffer-Hegger (SSH) and Kitaev models. Using the tight-binding and low-energy effective theory, we show that the SSP boundary states retain topological properties while the SSPP results in the asymmetry of boundary states. For the SSH model, an SSP zero-energy edge state localized on one edge possesses quantized polarization. In contrast, the other edge state is perturbed to have nonzero energy, and its polarization is not quantized. For topological superconductors, SSP zero-energy Majorana boundary states for spinful Kitaev models emerge on only one edge, contrary to the conventional belief that Majorana fermions emerge at opposite edges. Our findings can be used as a platform to expand our understanding of topological materials as they broaden our understanding of the symmetry in a topological system and a method to engineer Majorana fermions.