Topological hierarchy matters–topological matters with superlattices of defects

Topological insulators/superconductors are new states of quantum matter with metallic edge/surface states. In this paper, we review the defects effect in these topological states and study new types of topological matters–topological hierarchy matters. We find that both topological defects (quantized vortices) and non topological defects (vacancies) can induce topological mid-gap states in the topological hierarchy matters after considering the superlattice of defects. These topological mid-gap states have nontrivial topological properties, including the nonzero Chern number and the gapless edge states. Effective tight-binding models are obtained to describe the topological mid-gap states in the topological hierarchy matters.