Sub-symmetry Protected Topology in Topological Insulators and Superconductors

TL;DR

This paper investigates how sub-symmetry protection can preserve and modify boundary states in topological insulators and superconductors, revealing asymmetric boundary states and new ways to engineer Majorana fermions.

Contribution

It extends the concept of sub-symmetry-protected topology to 1D topological systems using SSH and Kitaev models, demonstrating asymmetric boundary states and novel Majorana fermion configurations.

Findings

SSP boundary states retain topological properties

SSP induces asymmetry in boundary states

Majorana states appear on only one edge in SSP conditions

Abstract

Exploration of topology protected by a certain symmetry is central in condensed matter physics. A recent idea of sub-symmetry-protected (SSP) topology--remains of a broken symmetry can still protect specific topological boundary states--has been developed and demonstrated in an optical system [Nat. Phys. 19, 992-998 (2023)]. Here, we extend this idea further by applying sub-symmetry-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 non-zero energy, and its polarization is not quantized. For topological superconductors, zero-energy SSP 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.