Hierarchy of Surface Modes in Disordered Symmetry Protected Topological Insulators

Symmetry protected topological (SPT) phases of matter are defined by their resiliency towards local perturbations such as unwanted noise, which make them desirable candidates for encoding quantum memories. We investigate the response of protection against disorder on bulk and surface modes in the chiral symmetric category of topological insulator models, and find robust signatures of topological witnesses in strong disorder. We find that a hierarchy of surface modes emerges within extended chiral symmetric models with disorder, and we analytically derive their multi-hopping winding number. We demonstrate phase transitions from the SPT regime to the disordered Dyson regime, where we find that the localized zero-energy states grow logarithmically. Our result can be used to theoretically and technologically inform the design of quantum many-body systems in disordered environments.