Topological hierarchy matters — topological matters with superlattices of defects**Project supported by the National Basic Research Program of China (Grant Nos. 2011CB921803 and 2012CB921704), the National Natural Science Foundation of China (Grant Nos. 11174035, 11474025, 11404090, and 11674026), the Natural Science Foundation of Hebei Province, China (Grant No. A2015205189), the Hebei Education Department Natural Science Foundation, China (Grant No. QN2014022), and the Specialized Research Fund for the Doctoral Program of Higher Education, China.

Abstract

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.